├── .github
    └── workflows
    │   └── lean.yml
├── .gitignore
├── .vscode
    └── extensions.json
├── README.md
├── coq
    ├── LICENSE
    ├── README.md
    ├── setup.sh
    └── src
    │   ├── putnam_1962_a2.v
    │   ├── putnam_1962_a4.v
    │   ├── putnam_1962_a5.v
    │   ├── putnam_1962_a6.v
    │   ├── putnam_1962_b2.v
    │   ├── putnam_1962_b5.v
    │   ├── putnam_1962_b6.v
    │   ├── putnam_1963_a2.v
    │   ├── putnam_1963_a4.v
    │   ├── putnam_1963_b1.v
    │   ├── putnam_1963_b2.v
    │   ├── putnam_1963_b3.v
    │   ├── putnam_1963_b5.v
    │   ├── putnam_1963_b6.v
    │   ├── putnam_1964_a1.v
    │   ├── putnam_1964_a2.v
    │   ├── putnam_1964_a4.v
    │   ├── putnam_1964_a5.v
    │   ├── putnam_1964_a6.v
    │   ├── putnam_1964_b1.v
    │   ├── putnam_1964_b2.v
    │   ├── putnam_1964_b3.v
    │   ├── putnam_1964_b5.v
    │   ├── putnam_1964_b6.v
    │   ├── putnam_1965_a2.v
    │   ├── putnam_1965_a3.v
    │   ├── putnam_1965_a4.v
    │   ├── putnam_1965_a5.v
    │   ├── putnam_1965_a6.v
    │   ├── putnam_1965_b2.v
    │   ├── putnam_1965_b3.v
    │   ├── putnam_1965_b4.v
    │   ├── putnam_1965_b5.v
    │   ├── putnam_1965_b6.v
    │   ├── putnam_1966_a1.v
    │   ├── putnam_1966_a3.v
    │   ├── putnam_1966_a4.v
    │   ├── putnam_1966_a5.v
    │   ├── putnam_1966_a6.v
    │   ├── putnam_1966_b2.v
    │   ├── putnam_1966_b3.v
    │   ├── putnam_1966_b4.v
    │   ├── putnam_1966_b6.v
    │   ├── putnam_1967_a1.v
    │   ├── putnam_1967_a2.v
    │   ├── putnam_1967_a3.v
    │   ├── putnam_1967_a4.v
    │   ├── putnam_1967_a6.v
    │   ├── putnam_1967_b2.v
    │   ├── putnam_1967_b3.v
    │   ├── putnam_1967_b4.v
    │   ├── putnam_1967_b5.v
    │   ├── putnam_1967_b6.v
    │   ├── putnam_1968_a1.v
    │   ├── putnam_1968_a2.v
    │   ├── putnam_1968_a3.v
    │   ├── putnam_1968_a4.v
    │   ├── putnam_1968_a5.v
    │   ├── putnam_1968_b2.v
    │   ├── putnam_1968_b4.v
    │   ├── putnam_1969_a4.v
    │   ├── putnam_1969_a5.v
    │   ├── putnam_1969_a6.v
    │   ├── putnam_1969_b3.v
    │   ├── putnam_1969_b5.v
    │   ├── putnam_1970_a1.v
    │   ├── putnam_1970_a2.v
    │   ├── putnam_1970_a4.v
    │   ├── putnam_1970_b1.v
    │   ├── putnam_1970_b2.v
    │   ├── putnam_1970_b3.v
    │   ├── putnam_1970_b4.v
    │   ├── putnam_1970_b5.v
    │   ├── putnam_1971_a1.v
    │   ├── putnam_1971_a2.v
    │   ├── putnam_1971_a3.v
    │   ├── putnam_1971_a4.v
    │   ├── putnam_1971_a6.v
    │   ├── putnam_1971_b1.v
    │   ├── putnam_1971_b2.v
    │   ├── putnam_1971_b6.v
    │   ├── putnam_1972_a1.v
    │   ├── putnam_1972_a2.v
    │   ├── putnam_1972_a3.v
    │   ├── putnam_1972_a5.v
    │   ├── putnam_1972_b1.v
    │   ├── putnam_1972_b2.v
    │   ├── putnam_1972_b3.v
    │   ├── putnam_1972_b5.v
    │   ├── putnam_1972_b6.v
    │   ├── putnam_1973_a3.v
    │   ├── putnam_1973_a4.v
    │   ├── putnam_1973_b1.v
    │   ├── putnam_1973_b2.v
    │   ├── putnam_1973_b3.v
    │   ├── putnam_1973_b4.v
    │   ├── putnam_1974_a1.v
    │   ├── putnam_1974_a3.v
    │   ├── putnam_1974_a4.v
    │   ├── putnam_1974_a6.v
    │   ├── putnam_1974_b3.v
    │   ├── putnam_1974_b5.v
    │   ├── putnam_1974_b6.v
    │   ├── putnam_1975_a1.v
    │   ├── putnam_1975_a2.v
    │   ├── putnam_1975_a3.v
    │   ├── putnam_1975_a5.v
    │   ├── putnam_1975_b5.v
    │   ├── putnam_1975_b6.v
    │   ├── putnam_1976_a3.v
    │   ├── putnam_1976_a6.v
    │   ├── putnam_1976_b5.v
    │   ├── putnam_1976_b6.v
    │   ├── putnam_1977_a1.v
    │   ├── putnam_1977_a2.v
    │   ├── putnam_1977_a3.v
    │   ├── putnam_1977_a4.v
    │   ├── putnam_1977_a5.v
    │   ├── putnam_1977_b1.v
    │   ├── putnam_1977_b5.v
    │   ├── putnam_1978_a1.v
    │   ├── putnam_1978_a3.v
    │   ├── putnam_1978_a5.v
    │   ├── putnam_1978_a6.v
    │   ├── putnam_1978_b2.v
    │   ├── putnam_1978_b3.v
    │   ├── putnam_1978_b4.v
    │   ├── putnam_1978_b5.v
    │   ├── putnam_1978_b6.v
    │   ├── putnam_1979_a1.v
    │   ├── putnam_1979_a2.v
    │   ├── putnam_1979_a3.v
    │   ├── putnam_1979_a5.v
    │   ├── putnam_1979_a6.v
    │   ├── putnam_1979_b2.v
    │   ├── putnam_1979_b6.v
    │   ├── putnam_1980_a2.v
    │   ├── putnam_1980_a3.v
    │   ├── putnam_1980_a4.v
    │   ├── putnam_1980_a5.v
    │   ├── putnam_1980_b1.v
    │   ├── putnam_1980_b3.v
    │   ├── putnam_1980_b6.v
    │   ├── putnam_1981_a1.v
    │   ├── putnam_1981_a3.v
    │   ├── putnam_1981_b1.v
    │   ├── putnam_1981_b2.v
    │   ├── putnam_1981_b3.v
    │   ├── putnam_1981_b5.v
    │   ├── putnam_1982_a2.v
    │   ├── putnam_1982_a3.v
    │   ├── putnam_1982_a5.v
    │   ├── putnam_1982_a6.v
    │   ├── putnam_1982_b5.v
    │   ├── putnam_1983_a1.v
    │   ├── putnam_1983_a3.v
    │   ├── putnam_1983_a4.v
    │   ├── putnam_1983_a5.v
    │   ├── putnam_1983_a6.v
    │   ├── putnam_1983_b4.v
    │   ├── putnam_1983_b5.v
    │   ├── putnam_1984_a2.v
    │   ├── putnam_1984_a5.v
    │   ├── putnam_1984_b1.v
    │   ├── putnam_1984_b2.v
    │   ├── putnam_1984_b5.v
    │   ├── putnam_1985_a1.v
    │   ├── putnam_1985_a3.v
    │   ├── putnam_1985_a4.v
    │   ├── putnam_1985_a5.v
    │   ├── putnam_1985_a6.v
    │   ├── putnam_1985_b2.v
    │   ├── putnam_1985_b3.v
    │   ├── putnam_1985_b5.v
    │   ├── putnam_1986_a1.v
    │   ├── putnam_1986_a2.v
    │   ├── putnam_1986_a3.v
    │   ├── putnam_1986_a6.v
    │   ├── putnam_1986_b2.v
    │   ├── putnam_1986_b4.v
    │   ├── putnam_1987_a3.v
    │   ├── putnam_1987_b1.v
    │   ├── putnam_1987_b2.v
    │   ├── putnam_1987_b4.v
    │   ├── putnam_1988_a2.v
    │   ├── putnam_1988_a3.v
    │   ├── putnam_1988_a4.v
    │   ├── putnam_1988_a5.v
    │   ├── putnam_1988_b1.v
    │   ├── putnam_1988_b2.v
    │   ├── putnam_1988_b3.v
    │   ├── putnam_1988_b4.v
    │   ├── putnam_1988_b6.v
    │   ├── putnam_1989_a2.v
    │   ├── putnam_1989_a3.v
    │   ├── putnam_1990_a1.v
    │   ├── putnam_1990_a2.v
    │   ├── putnam_1990_a5.v
    │   ├── putnam_1990_b1.v
    │   ├── putnam_1990_b2.v
    │   ├── putnam_1990_b3.v
    │   ├── putnam_1990_b5.v
    │   ├── putnam_1991_a3.v
    │   ├── putnam_1991_a5.v
    │   ├── putnam_1991_b1.v
    │   ├── putnam_1991_b2.v
    │   ├── putnam_1991_b4.v
    │   ├── putnam_1991_b5.v
    │   ├── putnam_1991_b6.v
    │   ├── putnam_1992_a1.v
    │   ├── putnam_1992_a2.v
    │   ├── putnam_1992_a3.v
    │   ├── putnam_1992_a4.v
    │   ├── putnam_1992_a5.v
    │   ├── putnam_1992_b1.v
    │   ├── putnam_1993_a2.v
    │   ├── putnam_1993_a4.v
    │   ├── putnam_1993_a5.v
    │   ├── putnam_1993_b1.v
    │   ├── putnam_1993_b4.v
    │   ├── putnam_1993_b5.v
    │   ├── putnam_1994_a1.v
    │   ├── putnam_1994_b1.v
    │   ├── putnam_1994_b2.v
    │   ├── putnam_1994_b3.v
    │   ├── putnam_1994_b4.v
    │   ├── putnam_1994_b5.v
    │   ├── putnam_1994_b6.v
    │   ├── putnam_1995_a1.v
    │   ├── putnam_1995_a2.v
    │   ├── putnam_1995_a4.v
    │   ├── putnam_1995_a5.v
    │   ├── putnam_1995_b4.v
    │   ├── putnam_1995_b6.v
    │   ├── putnam_1996_a1.v
    │   ├── putnam_1996_a3.v
    │   ├── putnam_1996_a4.v
    │   ├── putnam_1996_a5.v
    │   ├── putnam_1996_a6.v
    │   ├── putnam_1996_b2.v
    │   ├── putnam_1996_b3.v
    │   ├── putnam_1996_b4.v
    │   ├── putnam_1997_a3.v
    │   ├── putnam_1997_a4.v
    │   ├── putnam_1997_a5.v
    │   ├── putnam_1997_a6.v
    │   ├── putnam_1997_b1.v
    │   ├── putnam_1997_b2.v
    │   ├── putnam_1997_b3.v
    │   ├── putnam_1997_b5.v
    │   ├── putnam_1998_a3.v
    │   ├── putnam_1998_a4.v
    │   ├── putnam_1998_b1.v
    │   ├── putnam_1998_b4.v
    │   ├── putnam_1998_b5.v
    │   ├── putnam_1998_b6.v
    │   ├── putnam_1999_a2.v
    │   ├── putnam_1999_a3.v
    │   ├── putnam_1999_a4.v
    │   ├── putnam_1999_a5.v
    │   ├── putnam_1999_a6.v
    │   ├── putnam_1999_b2.v
    │   ├── putnam_1999_b4.v
    │   ├── putnam_1999_b6.v
    │   ├── putnam_2000_a1.v
    │   ├── putnam_2000_a2.v
    │   ├── putnam_2000_a4.v
    │   ├── putnam_2000_a6.v
    │   ├── putnam_2000_b1.v
    │   ├── putnam_2000_b2.v
    │   ├── putnam_2000_b4.v
    │   ├── putnam_2000_b5.v
    │   ├── putnam_2001_a1.v
    │   ├── putnam_2001_a3.v
    │   ├── putnam_2001_a5.v
    │   ├── putnam_2001_b2.v
    │   ├── putnam_2001_b3.v
    │   ├── putnam_2001_b4.v
    │   ├── putnam_2001_b5.v
    │   ├── putnam_2001_b6.v
    │   ├── putnam_2002_a1.v
    │   ├── putnam_2002_b3.v
    │   ├── putnam_2003_a1.v
    │   ├── putnam_2003_a2.v
    │   ├── putnam_2003_a3.v
    │   ├── putnam_2003_a4.v
    │   ├── putnam_2003_b1.v
    │   ├── putnam_2003_b3.v
    │   ├── putnam_2003_b4.v
    │   ├── putnam_2003_b6.v
    │   ├── putnam_2004_a3.v
    │   ├── putnam_2004_a4.v
    │   ├── putnam_2004_a6.v
    │   ├── putnam_2004_b1.v
    │   ├── putnam_2004_b2.v
    │   ├── putnam_2004_b5.v
    │   ├── putnam_2005_a1.v
    │   ├── putnam_2005_a3.v
    │   ├── putnam_2005_a5.v
    │   ├── putnam_2005_b1.v
    │   ├── putnam_2005_b2.v
    │   ├── putnam_2005_b3.v
    │   ├── putnam_2005_b4.v
    │   ├── putnam_2006_a5.v
    │   ├── putnam_2006_b2.v
    │   ├── putnam_2006_b5.v
    │   ├── putnam_2006_b6.v
    │   ├── putnam_2007_a4.v
    │   ├── putnam_2007_b1.v
    │   ├── putnam_2007_b2.v
    │   ├── putnam_2007_b3.v
    │   ├── putnam_2008_a1.v
    │   ├── putnam_2008_a4.v
    │   ├── putnam_2008_b2.v
    │   ├── putnam_2008_b4.v
    │   ├── putnam_2008_b5.v
    │   ├── putnam_2008_b6.v
    │   ├── putnam_2009_a1.v
    │   ├── putnam_2009_a2.v
    │   ├── putnam_2009_a4.v
    │   ├── putnam_2009_b1.v
    │   ├── putnam_2009_b5.v
    │   ├── putnam_2009_b6.v
    │   ├── putnam_2010_a2.v
    │   ├── putnam_2010_a4.v
    │   ├── putnam_2010_a6.v
    │   ├── putnam_2010_b1.v
    │   ├── putnam_2010_b2.v
    │   ├── putnam_2010_b4.v
    │   ├── putnam_2010_b5.v
    │   ├── putnam_2011_a2.v
    │   ├── putnam_2011_a3.v
    │   ├── putnam_2011_b1.v
    │   ├── putnam_2011_b3.v
    │   ├── putnam_2011_b5.v
    │   ├── putnam_2011_b6.v
    │   ├── putnam_2012_a2.v
    │   ├── putnam_2012_a3.v
    │   ├── putnam_2012_a6.v
    │   ├── putnam_2012_b1.v
    │   ├── putnam_2012_b4.v
    │   ├── putnam_2013_a3.v
    │   ├── putnam_2013_a6.v
    │   ├── putnam_2013_b1.v
    │   ├── putnam_2013_b2.v
    │   ├── putnam_2013_b4.v
    │   ├── putnam_2013_b5.v
    │   ├── putnam_2014_a1.v
    │   ├── putnam_2014_a3.v
    │   ├── putnam_2014_a4.v
    │   ├── putnam_2014_a5.v
    │   ├── putnam_2014_b2.v
    │   ├── putnam_2014_b4.v
    │   ├── putnam_2015_a2.v
    │   ├── putnam_2015_a3.v
    │   ├── putnam_2015_a4.v
    │   ├── putnam_2015_a5.v
    │   ├── putnam_2015_b1.v
    │   ├── putnam_2015_b4.v
    │   ├── putnam_2015_b5.v
    │   ├── putnam_2016_a2.v
    │   ├── putnam_2016_a3.v
    │   ├── putnam_2016_a6.v
    │   ├── putnam_2016_b1.v
    │   ├── putnam_2016_b2.v
    │   ├── putnam_2016_b5.v
    │   ├── putnam_2016_b6.v
    │   ├── putnam_2017_a1.v
    │   ├── putnam_2017_a2.v
    │   ├── putnam_2017_a3.v
    │   ├── putnam_2017_a4.v
    │   ├── putnam_2017_b2.v
    │   ├── putnam_2017_b3.v
    │   ├── putnam_2017_b4.v
    │   ├── putnam_2017_b6.v
    │   ├── putnam_2018_a1.v
    │   ├── putnam_2018_a3.v
    │   ├── putnam_2018_a5.v
    │   ├── putnam_2018_b1.v
    │   ├── putnam_2018_b2.v
    │   ├── putnam_2018_b3.v
    │   ├── putnam_2018_b4.v
    │   ├── putnam_2018_b6.v
    │   ├── putnam_2019_a1.v
    │   ├── putnam_2019_a6.v
    │   ├── putnam_2019_b2.v
    │   ├── putnam_2019_b5.v
    │   ├── putnam_2020_a1.v
    │   ├── putnam_2020_a2.v
    │   ├── putnam_2020_a3.v
    │   ├── putnam_2020_a6.v
    │   ├── putnam_2020_b1.v
    │   ├── putnam_2020_b5.v
    │   ├── putnam_2020_b6.v
    │   ├── putnam_2021_a2.v
    │   ├── putnam_2021_a4.v
    │   ├── putnam_2021_a5.v
    │   ├── putnam_2021_a6.v
    │   ├── putnam_2021_b2.v
    │   ├── putnam_2021_b4.v
    │   ├── putnam_2022_a1.v
    │   ├── putnam_2022_a2.v
    │   ├── putnam_2022_a6.v
    │   ├── putnam_2022_b1.v
    │   ├── putnam_2022_b2.v
    │   ├── putnam_2022_b4.v
    │   ├── putnam_2022_b6.v
    │   ├── putnam_2023_a1.v
    │   ├── putnam_2023_a2.v
    │   ├── putnam_2023_a3.v
    │   ├── putnam_2023_b2.v
    │   └── putnam_2023_b5.v
├── docs
    ├── .gitignore
    ├── README.md
    ├── index.html
    ├── leaderboard.html
    ├── results.json
    └── static
    │   ├── css
    │       ├── bulma-carousel.min.css
    │       ├── bulma-slider.min.css
    │       ├── bulma.css.map.txt
    │       ├── bulma.min.css
    │       ├── fontawesome.all.min.css
    │       └── index.css
    │   ├── images
    │       └── icon.svg
    │   └── js
    │       ├── bulma-carousel.js
    │       ├── bulma-carousel.min.js
    │       ├── bulma-slider.js
    │       ├── bulma-slider.min.js
    │       ├── fontawesome.all.min.js
    │       └── index.js
├── informal
    ├── README.md
    └── putnam.json
├── isabelle
    ├── LICENSE
    ├── README.md
    ├── putnam_1962_a1.thy
    ├── putnam_1962_a2.thy
    ├── putnam_1962_a3.thy
    ├── putnam_1962_a4.thy
    ├── putnam_1962_a5.thy
    ├── putnam_1962_a6.thy
    ├── putnam_1962_b1.thy
    ├── putnam_1962_b2.thy
    ├── putnam_1962_b3.thy
    ├── putnam_1962_b5.thy
    ├── putnam_1962_b6.thy
    ├── putnam_1963_a2.thy
    ├── putnam_1963_a3.thy
    ├── putnam_1963_a4.thy
    ├── putnam_1963_a6.thy
    ├── putnam_1963_b1.thy
    ├── putnam_1963_b2.thy
    ├── putnam_1963_b3.thy
    ├── putnam_1963_b5.thy
    ├── putnam_1963_b6.thy
    ├── putnam_1964_a1.thy
    ├── putnam_1964_a2.thy
    ├── putnam_1964_a3.thy
    ├── putnam_1964_a4.thy
    ├── putnam_1964_a5.thy
    ├── putnam_1964_a6.thy
    ├── putnam_1964_b1.thy
    ├── putnam_1964_b2.thy
    ├── putnam_1964_b3.thy
    ├── putnam_1964_b5.thy
    ├── putnam_1964_b6.thy
    ├── putnam_1965_a1.thy
    ├── putnam_1965_a2.thy
    ├── putnam_1965_a3.thy
    ├── putnam_1965_a4.thy
    ├── putnam_1965_a5.thy
    ├── putnam_1965_a6.thy
    ├── putnam_1965_b1.thy
    ├── putnam_1965_b2.thy
    ├── putnam_1965_b3.thy
    ├── putnam_1965_b4.thy
    ├── putnam_1965_b5.thy
    ├── putnam_1965_b6.thy
    ├── putnam_1966_a1.thy
    ├── putnam_1966_a2.thy
    ├── putnam_1966_a3.thy
    ├── putnam_1966_a4.thy
    ├── putnam_1966_a5.thy
    ├── putnam_1966_a6.thy
    ├── putnam_1966_b1.thy
    ├── putnam_1966_b2.thy
    ├── putnam_1966_b3.thy
    ├── putnam_1966_b4.thy
    ├── putnam_1966_b5.thy
    ├── putnam_1966_b6.thy
    ├── putnam_1967_a1.thy
    ├── putnam_1967_a2.thy
    ├── putnam_1967_a3.thy
    ├── putnam_1967_a4.thy
    ├── putnam_1967_a5.thy
    ├── putnam_1967_a6.thy
    ├── putnam_1967_b1.thy
    ├── putnam_1967_b2.thy
    ├── putnam_1967_b3.thy
    ├── putnam_1967_b4.thy
    ├── putnam_1967_b5.thy
    ├── putnam_1967_b6.thy
    ├── putnam_1968_a1.thy
    ├── putnam_1968_a2.thy
    ├── putnam_1968_a3.thy
    ├── putnam_1968_a4.thy
    ├── putnam_1968_a5.thy
    ├── putnam_1968_a6.thy
    ├── putnam_1968_b2.thy
    ├── putnam_1968_b4.thy
    ├── putnam_1968_b5.thy
    ├── putnam_1968_b6.thy
    ├── putnam_1969_a1.thy
    ├── putnam_1969_a2.thy
    ├── putnam_1969_a4.thy
    ├── putnam_1969_a5.thy
    ├── putnam_1969_a6.thy
    ├── putnam_1969_b1.thy
    ├── putnam_1969_b2.thy
    ├── putnam_1969_b3.thy
    ├── putnam_1969_b5.thy
    ├── putnam_1969_b6.thy
    ├── putnam_1970_a1.thy
    ├── putnam_1970_a2.thy
    ├── putnam_1970_a3.thy
    ├── putnam_1970_a4.thy
    ├── putnam_1970_b1.thy
    ├── putnam_1970_b2.thy
    ├── putnam_1970_b3.thy
    ├── putnam_1970_b4.thy
    ├── putnam_1970_b5.thy
    ├── putnam_1970_b6.thy
    ├── putnam_1971_a1.thy
    ├── putnam_1971_a2.thy
    ├── putnam_1971_a3.thy
    ├── putnam_1971_a4.thy
    ├── putnam_1971_a5.thy
    ├── putnam_1971_a6.thy
    ├── putnam_1971_b1.thy
    ├── putnam_1971_b2.thy
    ├── putnam_1971_b3.thy
    ├── putnam_1971_b6.thy
    ├── putnam_1972_a1.thy
    ├── putnam_1972_a2.thy
    ├── putnam_1972_a3.thy
    ├── putnam_1972_a5.thy
    ├── putnam_1972_a6.thy
    ├── putnam_1972_b1.thy
    ├── putnam_1972_b2.thy
    ├── putnam_1972_b3.thy
    ├── putnam_1972_b4.thy
    ├── putnam_1972_b5.thy
    ├── putnam_1972_b6.thy
    ├── putnam_1973_a1.thy
    ├── putnam_1973_a2.thy
    ├── putnam_1973_a3.thy
    ├── putnam_1973_a4.thy
    ├── putnam_1973_a6.thy
    ├── putnam_1973_b1.thy
    ├── putnam_1973_b2.thy
    ├── putnam_1973_b3.thy
    ├── putnam_1973_b4.thy
    ├── putnam_1974_a1.thy
    ├── putnam_1974_a3.thy
    ├── putnam_1974_a4.thy
    ├── putnam_1974_a6.thy
    ├── putnam_1974_b1.thy
    ├── putnam_1974_b2.thy
    ├── putnam_1974_b3.thy
    ├── putnam_1974_b4.thy
    ├── putnam_1974_b5.thy
    ├── putnam_1974_b6.thy
    ├── putnam_1975_a1.thy
    ├── putnam_1975_a2.thy
    ├── putnam_1975_a3.thy
    ├── putnam_1975_a4.thy
    ├── putnam_1975_a5.thy
    ├── putnam_1975_b1.thy
    ├── putnam_1975_b2.thy
    ├── putnam_1975_b3.thy
    ├── putnam_1975_b4.thy
    ├── putnam_1975_b5.thy
    ├── putnam_1975_b6.thy
    ├── putnam_1976_a2.thy
    ├── putnam_1976_a3.thy
    ├── putnam_1976_a4.thy
    ├── putnam_1976_a6.thy
    ├── putnam_1976_b1.thy
    ├── putnam_1976_b2.thy
    ├── putnam_1976_b3.thy
    ├── putnam_1976_b5.thy
    ├── putnam_1976_b6.thy
    ├── putnam_1977_a1.thy
    ├── putnam_1977_a2.thy
    ├── putnam_1977_a3.thy
    ├── putnam_1977_a4.thy
    ├── putnam_1977_a5.thy
    ├── putnam_1977_a6.thy
    ├── putnam_1977_b1.thy
    ├── putnam_1977_b3.thy
    ├── putnam_1977_b5.thy
    ├── putnam_1977_b6.thy
    ├── putnam_1978_a1.thy
    ├── putnam_1978_a2.thy
    ├── putnam_1978_a3.thy
    ├── putnam_1978_a4.thy
    ├── putnam_1978_a5.thy
    ├── putnam_1978_a6.thy
    ├── putnam_1978_b2.thy
    ├── putnam_1978_b3.thy
    ├── putnam_1978_b4.thy
    ├── putnam_1978_b5.thy
    ├── putnam_1978_b6.thy
    ├── putnam_1979_a1.thy
    ├── putnam_1979_a2.thy
    ├── putnam_1979_a3.thy
    ├── putnam_1979_a4.thy
    ├── putnam_1979_a5.thy
    ├── putnam_1979_a6.thy
    ├── putnam_1979_b2.thy
    ├── putnam_1979_b3.thy
    ├── putnam_1979_b5.thy
    ├── putnam_1979_b6.thy
    ├── putnam_1980_a2.thy
    ├── putnam_1980_a3.thy
    ├── putnam_1980_a4.thy
    ├── putnam_1980_a5.thy
    ├── putnam_1980_a6.thy
    ├── putnam_1980_b1.thy
    ├── putnam_1980_b3.thy
    ├── putnam_1980_b4.thy
    ├── putnam_1980_b5.thy
    ├── putnam_1980_b6.thy
    ├── putnam_1981_a1.thy
    ├── putnam_1981_a3.thy
    ├── putnam_1981_a5.thy
    ├── putnam_1981_b1.thy
    ├── putnam_1981_b2.thy
    ├── putnam_1981_b3.thy
    ├── putnam_1981_b4.thy
    ├── putnam_1981_b5.thy
    ├── putnam_1982_a2.thy
    ├── putnam_1982_a3.thy
    ├── putnam_1982_a4.thy
    ├── putnam_1982_a5.thy
    ├── putnam_1982_a6.thy
    ├── putnam_1982_b2.thy
    ├── putnam_1982_b3.thy
    ├── putnam_1982_b4.thy
    ├── putnam_1982_b5.thy
    ├── putnam_1983_a1.thy
    ├── putnam_1983_a3.thy
    ├── putnam_1983_a4.thy
    ├── putnam_1983_a5.thy
    ├── putnam_1983_a6.thy
    ├── putnam_1983_b2.thy
    ├── putnam_1983_b4.thy
    ├── putnam_1983_b5.thy
    ├── putnam_1983_b6.thy
    ├── putnam_1984_a2.thy
    ├── putnam_1984_a3.thy
    ├── putnam_1984_a5.thy
    ├── putnam_1984_a6.thy
    ├── putnam_1984_b1.thy
    ├── putnam_1984_b2.thy
    ├── putnam_1984_b3.thy
    ├── putnam_1984_b5.thy
    ├── putnam_1985_a1.thy
    ├── putnam_1985_a3.thy
    ├── putnam_1985_a4.thy
    ├── putnam_1985_a5.thy
    ├── putnam_1985_a6.thy
    ├── putnam_1985_b1.thy
    ├── putnam_1985_b2.thy
    ├── putnam_1985_b3.thy
    ├── putnam_1985_b5.thy
    ├── putnam_1985_b6.thy
    ├── putnam_1986_a1.thy
    ├── putnam_1986_a2.thy
    ├── putnam_1986_a3.thy
    ├── putnam_1986_a4.thy
    ├── putnam_1986_a5.thy
    ├── putnam_1986_a6.thy
    ├── putnam_1986_b1.thy
    ├── putnam_1986_b2.thy
    ├── putnam_1986_b3.thy
    ├── putnam_1986_b4.thy
    ├── putnam_1986_b5.thy
    ├── putnam_1986_b6.thy
    ├── putnam_1987_a1.thy
    ├── putnam_1987_a2.thy
    ├── putnam_1987_a4.thy
    ├── putnam_1987_a5.thy
    ├── putnam_1987_a6.thy
    ├── putnam_1987_b1.thy
    ├── putnam_1987_b2.thy
    ├── putnam_1987_b3.thy
    ├── putnam_1987_b4.thy
    ├── putnam_1987_b5.thy
    ├── putnam_1987_b6.thy
    ├── putnam_1988_a1.thy
    ├── putnam_1988_a2.thy
    ├── putnam_1988_a3.thy
    ├── putnam_1988_a4.thy
    ├── putnam_1988_a5.thy
    ├── putnam_1988_a6.thy
    ├── putnam_1988_b1.thy
    ├── putnam_1988_b2.thy
    ├── putnam_1988_b3.thy
    ├── putnam_1988_b4.thy
    ├── putnam_1988_b5.thy
    ├── putnam_1988_b6.thy
    ├── putnam_1989_a1.thy
    ├── putnam_1989_a2.thy
    ├── putnam_1989_a3.thy
    ├── putnam_1989_a6.thy
    ├── putnam_1989_b1.thy
    ├── putnam_1989_b2.thy
    ├── putnam_1989_b3.thy
    ├── putnam_1989_b4.thy
    ├── putnam_1989_b6.thy
    ├── putnam_1990_a1.thy
    ├── putnam_1990_a2.thy
    ├── putnam_1990_a4.thy
    ├── putnam_1990_a5.thy
    ├── putnam_1990_a6.thy
    ├── putnam_1990_b1.thy
    ├── putnam_1990_b2.thy
    ├── putnam_1990_b3.thy
    ├── putnam_1990_b4.thy
    ├── putnam_1990_b5.thy
    ├── putnam_1991_a2.thy
    ├── putnam_1991_a3.thy
    ├── putnam_1991_a4.thy
    ├── putnam_1991_a5.thy
    ├── putnam_1991_a6.thy
    ├── putnam_1991_b1.thy
    ├── putnam_1991_b2.thy
    ├── putnam_1991_b4.thy
    ├── putnam_1991_b5.thy
    ├── putnam_1991_b6.thy
    ├── putnam_1992_a1.thy
    ├── putnam_1992_a2.thy
    ├── putnam_1992_a3.thy
    ├── putnam_1992_a4.thy
    ├── putnam_1992_a5.thy
    ├── putnam_1992_b1.thy
    ├── putnam_1992_b2.thy
    ├── putnam_1992_b3.thy
    ├── putnam_1992_b4.thy
    ├── putnam_1992_b5.thy
    ├── putnam_1992_b6.thy
    ├── putnam_1993_a1.thy
    ├── putnam_1993_a2.thy
    ├── putnam_1993_a3.thy
    ├── putnam_1993_a4.thy
    ├── putnam_1993_a5.thy
    ├── putnam_1993_a6.thy
    ├── putnam_1993_b1.thy
    ├── putnam_1993_b3.thy
    ├── putnam_1993_b4.thy
    ├── putnam_1993_b5.thy
    ├── putnam_1993_b6.thy
    ├── putnam_1994_a1.thy
    ├── putnam_1994_a3.thy
    ├── putnam_1994_a4.thy
    ├── putnam_1994_a5.thy
    ├── putnam_1994_a6.thy
    ├── putnam_1994_b1.thy
    ├── putnam_1994_b2.thy
    ├── putnam_1994_b3.thy
    ├── putnam_1994_b4.thy
    ├── putnam_1994_b5.thy
    ├── putnam_1994_b6.thy
    ├── putnam_1995_a1.thy
    ├── putnam_1995_a2.thy
    ├── putnam_1995_a3.thy
    ├── putnam_1995_a4.thy
    ├── putnam_1995_a5.thy
    ├── putnam_1995_a6.thy
    ├── putnam_1995_b1.thy
    ├── putnam_1995_b3.thy
    ├── putnam_1995_b4.thy
    ├── putnam_1995_b6.thy
    ├── putnam_1996_a2.thy
    ├── putnam_1996_a3.thy
    ├── putnam_1996_a4.thy
    ├── putnam_1996_a5.thy
    ├── putnam_1996_a6.thy
    ├── putnam_1996_b1.thy
    ├── putnam_1996_b2.thy
    ├── putnam_1996_b3.thy
    ├── putnam_1996_b4.thy
    ├── putnam_1996_b5.thy
    ├── putnam_1997_a3.thy
    ├── putnam_1997_a4.thy
    ├── putnam_1997_a5.thy
    ├── putnam_1997_a6.thy
    ├── putnam_1997_b1.thy
    ├── putnam_1997_b2.thy
    ├── putnam_1997_b3.thy
    ├── putnam_1997_b4.thy
    ├── putnam_1997_b5.thy
    ├── putnam_1998_a2.thy
    ├── putnam_1998_a3.thy
    ├── putnam_1998_a4.thy
    ├── putnam_1998_a5.thy
    ├── putnam_1998_a6.thy
    ├── putnam_1998_b1.thy
    ├── putnam_1998_b2.thy
    ├── putnam_1998_b4.thy
    ├── putnam_1998_b5.thy
    ├── putnam_1998_b6.thy
    ├── putnam_1999_a1.thy
    ├── putnam_1999_a2.thy
    ├── putnam_1999_a3.thy
    ├── putnam_1999_a4.thy
    ├── putnam_1999_a5.thy
    ├── putnam_1999_a6.thy
    ├── putnam_1999_b2.thy
    ├── putnam_1999_b3.thy
    ├── putnam_1999_b4.thy
    ├── putnam_1999_b5.thy
    ├── putnam_1999_b6.thy
    ├── putnam_2000_a1.thy
    ├── putnam_2000_a2.thy
    ├── putnam_2000_a4.thy
    ├── putnam_2000_a5.thy
    ├── putnam_2000_a6.thy
    ├── putnam_2000_b1.thy
    ├── putnam_2000_b2.thy
    ├── putnam_2000_b3.thy
    ├── putnam_2000_b4.thy
    ├── putnam_2000_b5.thy
    ├── putnam_2001_a1.thy
    ├── putnam_2001_a3.thy
    ├── putnam_2001_a5.thy
    ├── putnam_2001_b1.thy
    ├── putnam_2001_b2.thy
    ├── putnam_2001_b3.thy
    ├── putnam_2001_b4.thy
    ├── putnam_2001_b5.thy
    ├── putnam_2001_b6.thy
    ├── putnam_2002_a1.thy
    ├── putnam_2002_a2.thy
    ├── putnam_2002_a3.thy
    ├── putnam_2002_a5.thy
    ├── putnam_2002_a6.thy
    ├── putnam_2002_b3.thy
    ├── putnam_2002_b5.thy
    ├── putnam_2002_b6.thy
    ├── putnam_2003_a1.thy
    ├── putnam_2003_a2.thy
    ├── putnam_2003_a3.thy
    ├── putnam_2003_a4.thy
    ├── putnam_2003_a5.thy
    ├── putnam_2003_a6.thy
    ├── putnam_2003_b1.thy
    ├── putnam_2003_b2.thy
    ├── putnam_2003_b3.thy
    ├── putnam_2003_b4.thy
    ├── putnam_2003_b5.thy
    ├── putnam_2003_b6.thy
    ├── putnam_2004_a1.thy
    ├── putnam_2004_a3.thy
    ├── putnam_2004_a4.thy
    ├── putnam_2004_a5.thy
    ├── putnam_2004_a6.thy
    ├── putnam_2004_b1.thy
    ├── putnam_2004_b2.thy
    ├── putnam_2004_b4.thy
    ├── putnam_2004_b5.thy
    ├── putnam_2004_b6.thy
    ├── putnam_2005_a1.thy
    ├── putnam_2005_a2.thy
    ├── putnam_2005_a3.thy
    ├── putnam_2005_a4.thy
    ├── putnam_2005_a5.thy
    ├── putnam_2005_b1.thy
    ├── putnam_2005_b2.thy
    ├── putnam_2005_b3.thy
    ├── putnam_2005_b4.thy
    ├── putnam_2005_b5.thy
    ├── putnam_2005_b6.thy
    ├── putnam_2006_a1.thy
    ├── putnam_2006_a3.thy
    ├── putnam_2006_a4.thy
    ├── putnam_2006_a5.thy
    ├── putnam_2006_b1.thy
    ├── putnam_2006_b2.thy
    ├── putnam_2006_b3.thy
    ├── putnam_2006_b4.thy
    ├── putnam_2006_b5.thy
    ├── putnam_2006_b6.thy
    ├── putnam_2007_a1.thy
    ├── putnam_2007_a2.thy
    ├── putnam_2007_a3.thy
    ├── putnam_2007_a4.thy
    ├── putnam_2007_a5.thy
    ├── putnam_2007_b1.thy
    ├── putnam_2007_b2.thy
    ├── putnam_2007_b3.thy
    ├── putnam_2007_b4.thy
    ├── putnam_2007_b5.thy
    ├── putnam_2007_b6.thy
    ├── putnam_2008_a1.thy
    ├── putnam_2008_a3.thy
    ├── putnam_2008_a4.thy
    ├── putnam_2008_a5.thy
    ├── putnam_2008_a6.thy
    ├── putnam_2008_b1.thy
    ├── putnam_2008_b2.thy
    ├── putnam_2008_b3.thy
    ├── putnam_2008_b4.thy
    ├── putnam_2008_b5.thy
    ├── putnam_2008_b6.thy
    ├── putnam_2009_a1.thy
    ├── putnam_2009_a2.thy
    ├── putnam_2009_a3.thy
    ├── putnam_2009_a4.thy
    ├── putnam_2009_a5.thy
    ├── putnam_2009_b1.thy
    ├── putnam_2009_b2.thy
    ├── putnam_2009_b3.thy
    ├── putnam_2009_b4.thy
    ├── putnam_2009_b5.thy
    ├── putnam_2009_b6.thy
    ├── putnam_2010_a1.thy
    ├── putnam_2010_a2.thy
    ├── putnam_2010_a4.thy
    ├── putnam_2010_a5.thy
    ├── putnam_2010_a6.thy
    ├── putnam_2010_b1.thy
    ├── putnam_2010_b2.thy
    ├── putnam_2010_b3.thy
    ├── putnam_2010_b4.thy
    ├── putnam_2010_b5.thy
    ├── putnam_2010_b6.thy
    ├── putnam_2011_a1.thy
    ├── putnam_2011_a2.thy
    ├── putnam_2011_a3.thy
    ├── putnam_2011_a4.thy
    ├── putnam_2011_a5.thy
    ├── putnam_2011_a6.thy
    ├── putnam_2011_b1.thy
    ├── putnam_2011_b2.thy
    ├── putnam_2011_b3.thy
    ├── putnam_2011_b4.thy
    ├── putnam_2011_b5.thy
    ├── putnam_2011_b6.thy
    ├── putnam_2012_a1.thy
    ├── putnam_2012_a2.thy
    ├── putnam_2012_a3.thy
    ├── putnam_2012_a4.thy
    ├── putnam_2012_a5.thy
    ├── putnam_2012_a6.thy
    ├── putnam_2012_b1.thy
    ├── putnam_2012_b3.thy
    ├── putnam_2012_b4.thy
    ├── putnam_2012_b5.thy
    ├── putnam_2012_b6.thy
    ├── putnam_2013_a2.thy
    ├── putnam_2013_a3.thy
    ├── putnam_2013_a4.thy
    ├── putnam_2013_a5.thy
    ├── putnam_2013_a6.thy
    ├── putnam_2013_b1.thy
    ├── putnam_2013_b2.thy
    ├── putnam_2013_b3.thy
    ├── putnam_2013_b4.thy
    ├── putnam_2013_b5.thy
    ├── putnam_2014_a1.thy
    ├── putnam_2014_a2.thy
    ├── putnam_2014_a3.thy
    ├── putnam_2014_a4.thy
    ├── putnam_2014_a5.thy
    ├── putnam_2014_a6.thy
    ├── putnam_2014_b1.thy
    ├── putnam_2014_b2.thy
    ├── putnam_2014_b3.thy
    ├── putnam_2014_b4.thy
    ├── putnam_2014_b6.thy
    ├── putnam_2015_a1.thy
    ├── putnam_2015_a2.thy
    ├── putnam_2015_a3.thy
    ├── putnam_2015_a4.thy
    ├── putnam_2015_a5.thy
    ├── putnam_2015_a6.thy
    ├── putnam_2015_b1.thy
    ├── putnam_2015_b2.thy
    ├── putnam_2015_b3.thy
    ├── putnam_2015_b4.thy
    ├── putnam_2015_b5.thy
    ├── putnam_2015_b6.thy
    ├── putnam_2016_a1.thy
    ├── putnam_2016_a2.thy
    ├── putnam_2016_a3.thy
    ├── putnam_2016_a5.thy
    ├── putnam_2016_a6.thy
    ├── putnam_2016_b1.thy
    ├── putnam_2016_b2.thy
    ├── putnam_2016_b3.thy
    ├── putnam_2016_b4.thy
    ├── putnam_2016_b5.thy
    ├── putnam_2016_b6.thy
    ├── putnam_2017_a1.thy
    ├── putnam_2017_a2.thy
    ├── putnam_2017_a3.thy
    ├── putnam_2017_a4.thy
    ├── putnam_2017_b1.thy
    ├── putnam_2017_b2.thy
    ├── putnam_2017_b3.thy
    ├── putnam_2017_b4.thy
    ├── putnam_2017_b6.thy
    ├── putnam_2018_a1.thy
    ├── putnam_2018_a2.thy
    ├── putnam_2018_a3.thy
    ├── putnam_2018_a4.thy
    ├── putnam_2018_a5.thy
    ├── putnam_2018_a6.thy
    ├── putnam_2018_b1.thy
    ├── putnam_2018_b2.thy
    ├── putnam_2018_b3.thy
    ├── putnam_2018_b4.thy
    ├── putnam_2018_b5.thy
    ├── putnam_2018_b6.thy
    ├── putnam_2019_a1.thy
    ├── putnam_2019_a3.thy
    ├── putnam_2019_a5.thy
    ├── putnam_2019_a6.thy
    ├── putnam_2019_b1.thy
    ├── putnam_2019_b2.thy
    ├── putnam_2019_b3.thy
    ├── putnam_2019_b4.thy
    ├── putnam_2019_b5.thy
    ├── putnam_2019_b6.thy
    ├── putnam_2020_a1.thy
    ├── putnam_2020_a2.thy
    ├── putnam_2020_a3.thy
    ├── putnam_2020_a5.thy
    ├── putnam_2020_a6.thy
    ├── putnam_2020_b1.thy
    ├── putnam_2020_b4.thy
    ├── putnam_2020_b5.thy
    ├── putnam_2020_b6.thy
    ├── putnam_2021_a1.thy
    ├── putnam_2021_a2.thy
    ├── putnam_2021_a3.thy
    ├── putnam_2021_a4.thy
    ├── putnam_2021_a5.thy
    ├── putnam_2021_a6.thy
    ├── putnam_2021_b2.thy
    ├── putnam_2021_b3.thy
    ├── putnam_2021_b4.thy
    ├── putnam_2021_b5.thy
    ├── putnam_2022_a1.thy
    ├── putnam_2022_a2.thy
    ├── putnam_2022_a3.thy
    ├── putnam_2022_a6.thy
    ├── putnam_2022_b1.thy
    ├── putnam_2022_b2.thy
    ├── putnam_2022_b3.thy
    ├── putnam_2022_b4.thy
    ├── putnam_2022_b5.thy
    ├── putnam_2022_b6.thy
    ├── putnam_2023_a1.thy
    ├── putnam_2023_a2.thy
    ├── putnam_2023_a3.thy
    ├── putnam_2023_a5.thy
    ├── putnam_2023_a6.thy
    ├── putnam_2023_b1.thy
    ├── putnam_2023_b2.thy
    ├── putnam_2023_b4.thy
    ├── putnam_2023_b5.thy
    ├── putnam_2023_b6.thy
    └── scripts
    │   ├── convert_to_json.py
    │   └── rewrite_solutions.py
└── lean4
    ├── LICENSE
    ├── README.md
    ├── check_docstrings.lean
    ├── lake-manifest.json
    ├── lakefile.lean
    ├── lean-toolchain
    ├── scripts
        ├── extract_to_json.py
        ├── generate_files_by_year.py
        └── rewrite_solutions.py
    └── src
        ├── putnam_1962_a1.lean
        ├── putnam_1962_a2.lean
        ├── putnam_1962_a3.lean
        ├── putnam_1962_a4.lean
        ├── putnam_1962_a5.lean
        ├── putnam_1962_a6.lean
        ├── putnam_1962_b1.lean
        ├── putnam_1962_b2.lean
        ├── putnam_1962_b3.lean
        ├── putnam_1962_b5.lean
        ├── putnam_1962_b6.lean
        ├── putnam_1963_a2.lean
        ├── putnam_1963_a3.lean
        ├── putnam_1963_a4.lean
        ├── putnam_1963_a6.lean
        ├── putnam_1963_b1.lean
        ├── putnam_1963_b2.lean
        ├── putnam_1963_b3.lean
        ├── putnam_1963_b5.lean
        ├── putnam_1963_b6.lean
        ├── putnam_1964_a1.lean
        ├── putnam_1964_a2.lean
        ├── putnam_1964_a3.lean
        ├── putnam_1964_a4.lean
        ├── putnam_1964_a5.lean
        ├── putnam_1964_a6.lean
        ├── putnam_1964_b1.lean
        ├── putnam_1964_b2.lean
        ├── putnam_1964_b3.lean
        ├── putnam_1964_b4.lean
        ├── putnam_1964_b5.lean
        ├── putnam_1964_b6.lean
        ├── putnam_1965_a1.lean
        ├── putnam_1965_a2.lean
        ├── putnam_1965_a3.lean
        ├── putnam_1965_a4.lean
        ├── putnam_1965_a5.lean
        ├── putnam_1965_a6.lean
        ├── putnam_1965_b1.lean
        ├── putnam_1965_b2.lean
        ├── putnam_1965_b3.lean
        ├── putnam_1965_b4.lean
        ├── putnam_1965_b5.lean
        ├── putnam_1965_b6.lean
        ├── putnam_1966_a1.lean
        ├── putnam_1966_a2.lean
        ├── putnam_1966_a3.lean
        ├── putnam_1966_a4.lean
        ├── putnam_1966_a5.lean
        ├── putnam_1966_a6.lean
        ├── putnam_1966_b1.lean
        ├── putnam_1966_b2.lean
        ├── putnam_1966_b3.lean
        ├── putnam_1966_b4.lean
        ├── putnam_1966_b5.lean
        ├── putnam_1966_b6.lean
        ├── putnam_1967_a1.lean
        ├── putnam_1967_a2.lean
        ├── putnam_1967_a3.lean
        ├── putnam_1967_a4.lean
        ├── putnam_1967_a5.lean
        ├── putnam_1967_a6.lean
        ├── putnam_1967_b1.lean
        ├── putnam_1967_b2.lean
        ├── putnam_1967_b3.lean
        ├── putnam_1967_b4.lean
        ├── putnam_1967_b5.lean
        ├── putnam_1967_b6.lean
        ├── putnam_1968_a1.lean
        ├── putnam_1968_a2.lean
        ├── putnam_1968_a3.lean
        ├── putnam_1968_a4.lean
        ├── putnam_1968_a5.lean
        ├── putnam_1968_a6.lean
        ├── putnam_1968_b1.lean
        ├── putnam_1968_b2.lean
        ├── putnam_1968_b4.lean
        ├── putnam_1968_b5.lean
        ├── putnam_1968_b6.lean
        ├── putnam_1969_a1.lean
        ├── putnam_1969_a2.lean
        ├── putnam_1969_a4.lean
        ├── putnam_1969_a5.lean
        ├── putnam_1969_a6.lean
        ├── putnam_1969_b1.lean
        ├── putnam_1969_b2.lean
        ├── putnam_1969_b3.lean
        ├── putnam_1969_b4.lean
        ├── putnam_1969_b5.lean
        ├── putnam_1969_b6.lean
        ├── putnam_1970_a1.lean
        ├── putnam_1970_a2.lean
        ├── putnam_1970_a3.lean
        ├── putnam_1970_a4.lean
        ├── putnam_1970_b1.lean
        ├── putnam_1970_b2.lean
        ├── putnam_1970_b3.lean
        ├── putnam_1970_b4.lean
        ├── putnam_1970_b5.lean
        ├── putnam_1970_b6.lean
        ├── putnam_1971_a1.lean
        ├── putnam_1971_a2.lean
        ├── putnam_1971_a3.lean
        ├── putnam_1971_a4.lean
        ├── putnam_1971_a5.lean
        ├── putnam_1971_a6.lean
        ├── putnam_1971_b1.lean
        ├── putnam_1971_b2.lean
        ├── putnam_1971_b3.lean
        ├── putnam_1971_b6.lean
        ├── putnam_1972_a1.lean
        ├── putnam_1972_a2.lean
        ├── putnam_1972_a3.lean
        ├── putnam_1972_a5.lean
        ├── putnam_1972_a6.lean
        ├── putnam_1972_b1.lean
        ├── putnam_1972_b2.lean
        ├── putnam_1972_b3.lean
        ├── putnam_1972_b4.lean
        ├── putnam_1972_b5.lean
        ├── putnam_1972_b6.lean
        ├── putnam_1973_a1.lean
        ├── putnam_1973_a2.lean
        ├── putnam_1973_a3.lean
        ├── putnam_1973_a4.lean
        ├── putnam_1973_a6.lean
        ├── putnam_1973_b1.lean
        ├── putnam_1973_b2.lean
        ├── putnam_1973_b3.lean
        ├── putnam_1973_b4.lean
        ├── putnam_1974_a1.lean
        ├── putnam_1974_a3.lean
        ├── putnam_1974_a4.lean
        ├── putnam_1974_a6.lean
        ├── putnam_1974_b1.lean
        ├── putnam_1974_b2.lean
        ├── putnam_1974_b3.lean
        ├── putnam_1974_b4.lean
        ├── putnam_1974_b5.lean
        ├── putnam_1974_b6.lean
        ├── putnam_1975_a1.lean
        ├── putnam_1975_a2.lean
        ├── putnam_1975_a3.lean
        ├── putnam_1975_a4.lean
        ├── putnam_1975_a5.lean
        ├── putnam_1975_b1.lean
        ├── putnam_1975_b2.lean
        ├── putnam_1975_b3.lean
        ├── putnam_1975_b4.lean
        ├── putnam_1975_b5.lean
        ├── putnam_1975_b6.lean
        ├── putnam_1976_a2.lean
        ├── putnam_1976_a3.lean
        ├── putnam_1976_a4.lean
        ├── putnam_1976_a6.lean
        ├── putnam_1976_b1.lean
        ├── putnam_1976_b2.lean
        ├── putnam_1976_b3.lean
        ├── putnam_1976_b5.lean
        ├── putnam_1976_b6.lean
        ├── putnam_1977_a1.lean
        ├── putnam_1977_a2.lean
        ├── putnam_1977_a3.lean
        ├── putnam_1977_a4.lean
        ├── putnam_1977_a5.lean
        ├── putnam_1977_a6.lean
        ├── putnam_1977_b1.lean
        ├── putnam_1977_b3.lean
        ├── putnam_1977_b5.lean
        ├── putnam_1977_b6.lean
        ├── putnam_1978_a1.lean
        ├── putnam_1978_a2.lean
        ├── putnam_1978_a3.lean
        ├── putnam_1978_a4.lean
        ├── putnam_1978_a5.lean
        ├── putnam_1978_a6.lean
        ├── putnam_1978_b2.lean
        ├── putnam_1978_b3.lean
        ├── putnam_1978_b4.lean
        ├── putnam_1978_b5.lean
        ├── putnam_1978_b6.lean
        ├── putnam_1979_a1.lean
        ├── putnam_1979_a2.lean
        ├── putnam_1979_a3.lean
        ├── putnam_1979_a4.lean
        ├── putnam_1979_a5.lean
        ├── putnam_1979_a6.lean
        ├── putnam_1979_b2.lean
        ├── putnam_1979_b3.lean
        ├── putnam_1979_b5.lean
        ├── putnam_1979_b6.lean
        ├── putnam_1980_a2.lean
        ├── putnam_1980_a3.lean
        ├── putnam_1980_a4.lean
        ├── putnam_1980_a5.lean
        ├── putnam_1980_a6.lean
        ├── putnam_1980_b1.lean
        ├── putnam_1980_b3.lean
        ├── putnam_1980_b4.lean
        ├── putnam_1980_b5.lean
        ├── putnam_1980_b6.lean
        ├── putnam_1981_a1.lean
        ├── putnam_1981_a3.lean
        ├── putnam_1981_a5.lean
        ├── putnam_1981_b1.lean
        ├── putnam_1981_b2.lean
        ├── putnam_1981_b3.lean
        ├── putnam_1981_b4.lean
        ├── putnam_1981_b5.lean
        ├── putnam_1982_a2.lean
        ├── putnam_1982_a3.lean
        ├── putnam_1982_a4.lean
        ├── putnam_1982_a5.lean
        ├── putnam_1982_a6.lean
        ├── putnam_1982_b2.lean
        ├── putnam_1982_b3.lean
        ├── putnam_1982_b4.lean
        ├── putnam_1982_b5.lean
        ├── putnam_1983_a1.lean
        ├── putnam_1983_a3.lean
        ├── putnam_1983_a4.lean
        ├── putnam_1983_a5.lean
        ├── putnam_1983_a6.lean
        ├── putnam_1983_b2.lean
        ├── putnam_1983_b4.lean
        ├── putnam_1983_b5.lean
        ├── putnam_1983_b6.lean
        ├── putnam_1984_a2.lean
        ├── putnam_1984_a3.lean
        ├── putnam_1984_a5.lean
        ├── putnam_1984_a6.lean
        ├── putnam_1984_b1.lean
        ├── putnam_1984_b2.lean
        ├── putnam_1984_b3.lean
        ├── putnam_1984_b5.lean
        ├── putnam_1985_a1.lean
        ├── putnam_1985_a3.lean
        ├── putnam_1985_a4.lean
        ├── putnam_1985_a5.lean
        ├── putnam_1985_a6.lean
        ├── putnam_1985_b1.lean
        ├── putnam_1985_b2.lean
        ├── putnam_1985_b3.lean
        ├── putnam_1985_b5.lean
        ├── putnam_1985_b6.lean
        ├── putnam_1986_a1.lean
        ├── putnam_1986_a2.lean
        ├── putnam_1986_a3.lean
        ├── putnam_1986_a4.lean
        ├── putnam_1986_a5.lean
        ├── putnam_1986_a6.lean
        ├── putnam_1986_b1.lean
        ├── putnam_1986_b2.lean
        ├── putnam_1986_b3.lean
        ├── putnam_1986_b4.lean
        ├── putnam_1986_b5.lean
        ├── putnam_1986_b6.lean
        ├── putnam_1987_a1.lean
        ├── putnam_1987_a2.lean
        ├── putnam_1987_a4.lean
        ├── putnam_1987_a5.lean
        ├── putnam_1987_a6.lean
        ├── putnam_1987_b1.lean
        ├── putnam_1987_b2.lean
        ├── putnam_1987_b3.lean
        ├── putnam_1987_b4.lean
        ├── putnam_1987_b5.lean
        ├── putnam_1987_b6.lean
        ├── putnam_1988_a1.lean
        ├── putnam_1988_a2.lean
        ├── putnam_1988_a3.lean
        ├── putnam_1988_a4.lean
        ├── putnam_1988_a5.lean
        ├── putnam_1988_a6.lean
        ├── putnam_1988_b1.lean
        ├── putnam_1988_b2.lean
        ├── putnam_1988_b3.lean
        ├── putnam_1988_b4.lean
        ├── putnam_1988_b5.lean
        ├── putnam_1988_b6.lean
        ├── putnam_1989_a1.lean
        ├── putnam_1989_a2.lean
        ├── putnam_1989_a3.lean
        ├── putnam_1989_a6.lean
        ├── putnam_1989_b1.lean
        ├── putnam_1989_b2.lean
        ├── putnam_1989_b3.lean
        ├── putnam_1989_b4.lean
        ├── putnam_1989_b6.lean
        ├── putnam_1990_a1.lean
        ├── putnam_1990_a2.lean
        ├── putnam_1990_a4.lean
        ├── putnam_1990_a5.lean
        ├── putnam_1990_a6.lean
        ├── putnam_1990_b1.lean
        ├── putnam_1990_b2.lean
        ├── putnam_1990_b3.lean
        ├── putnam_1990_b4.lean
        ├── putnam_1990_b5.lean
        ├── putnam_1991_a2.lean
        ├── putnam_1991_a3.lean
        ├── putnam_1991_a4.lean
        ├── putnam_1991_a5.lean
        ├── putnam_1991_a6.lean
        ├── putnam_1991_b1.lean
        ├── putnam_1991_b2.lean
        ├── putnam_1991_b4.lean
        ├── putnam_1991_b5.lean
        ├── putnam_1991_b6.lean
        ├── putnam_1992_a1.lean
        ├── putnam_1992_a2.lean
        ├── putnam_1992_a3.lean
        ├── putnam_1992_a4.lean
        ├── putnam_1992_a5.lean
        ├── putnam_1992_b1.lean
        ├── putnam_1992_b2.lean
        ├── putnam_1992_b3.lean
        ├── putnam_1992_b4.lean
        ├── putnam_1992_b5.lean
        ├── putnam_1992_b6.lean
        ├── putnam_1993_a1.lean
        ├── putnam_1993_a2.lean
        ├── putnam_1993_a3.lean
        ├── putnam_1993_a4.lean
        ├── putnam_1993_a5.lean
        ├── putnam_1993_a6.lean
        ├── putnam_1993_b1.lean
        ├── putnam_1993_b3.lean
        ├── putnam_1993_b4.lean
        ├── putnam_1993_b5.lean
        ├── putnam_1993_b6.lean
        ├── putnam_1994_a1.lean
        ├── putnam_1994_a3.lean
        ├── putnam_1994_a4.lean
        ├── putnam_1994_a5.lean
        ├── putnam_1994_a6.lean
        ├── putnam_1994_b1.lean
        ├── putnam_1994_b2.lean
        ├── putnam_1994_b3.lean
        ├── putnam_1994_b4.lean
        ├── putnam_1994_b5.lean
        ├── putnam_1994_b6.lean
        ├── putnam_1995_a1.lean
        ├── putnam_1995_a2.lean
        ├── putnam_1995_a3.lean
        ├── putnam_1995_a4.lean
        ├── putnam_1995_a5.lean
        ├── putnam_1995_a6.lean
        ├── putnam_1995_b1.lean
        ├── putnam_1995_b3.lean
        ├── putnam_1995_b4.lean
        ├── putnam_1995_b6.lean
        ├── putnam_1996_a2.lean
        ├── putnam_1996_a3.lean
        ├── putnam_1996_a4.lean
        ├── putnam_1996_a5.lean
        ├── putnam_1996_a6.lean
        ├── putnam_1996_b1.lean
        ├── putnam_1996_b2.lean
        ├── putnam_1996_b3.lean
        ├── putnam_1996_b4.lean
        ├── putnam_1996_b5.lean
        ├── putnam_1997_a3.lean
        ├── putnam_1997_a4.lean
        ├── putnam_1997_a5.lean
        ├── putnam_1997_a6.lean
        ├── putnam_1997_b1.lean
        ├── putnam_1997_b2.lean
        ├── putnam_1997_b3.lean
        ├── putnam_1997_b4.lean
        ├── putnam_1997_b5.lean
        ├── putnam_1998_a2.lean
        ├── putnam_1998_a3.lean
        ├── putnam_1998_a4.lean
        ├── putnam_1998_a5.lean
        ├── putnam_1998_a6.lean
        ├── putnam_1998_b1.lean
        ├── putnam_1998_b2.lean
        ├── putnam_1998_b4.lean
        ├── putnam_1998_b5.lean
        ├── putnam_1998_b6.lean
        ├── putnam_1999_a1.lean
        ├── putnam_1999_a2.lean
        ├── putnam_1999_a3.lean
        ├── putnam_1999_a4.lean
        ├── putnam_1999_a5.lean
        ├── putnam_1999_a6.lean
        ├── putnam_1999_b2.lean
        ├── putnam_1999_b3.lean
        ├── putnam_1999_b4.lean
        ├── putnam_1999_b5.lean
        ├── putnam_1999_b6.lean
        ├── putnam_2000_a1.lean
        ├── putnam_2000_a2.lean
        ├── putnam_2000_a4.lean
        ├── putnam_2000_a5.lean
        ├── putnam_2000_a6.lean
        ├── putnam_2000_b1.lean
        ├── putnam_2000_b2.lean
        ├── putnam_2000_b3.lean
        ├── putnam_2000_b4.lean
        ├── putnam_2000_b5.lean
        ├── putnam_2001_a1.lean
        ├── putnam_2001_a3.lean
        ├── putnam_2001_a5.lean
        ├── putnam_2001_b1.lean
        ├── putnam_2001_b2.lean
        ├── putnam_2001_b3.lean
        ├── putnam_2001_b4.lean
        ├── putnam_2001_b5.lean
        ├── putnam_2001_b6.lean
        ├── putnam_2002_a1.lean
        ├── putnam_2002_a2.lean
        ├── putnam_2002_a3.lean
        ├── putnam_2002_a5.lean
        ├── putnam_2002_a6.lean
        ├── putnam_2002_b3.lean
        ├── putnam_2002_b5.lean
        ├── putnam_2002_b6.lean
        ├── putnam_2003_a1.lean
        ├── putnam_2003_a2.lean
        ├── putnam_2003_a3.lean
        ├── putnam_2003_a4.lean
        ├── putnam_2003_a5.lean
        ├── putnam_2003_a6.lean
        ├── putnam_2003_b1.lean
        ├── putnam_2003_b2.lean
        ├── putnam_2003_b3.lean
        ├── putnam_2003_b4.lean
        ├── putnam_2003_b5.lean
        ├── putnam_2003_b6.lean
        ├── putnam_2004_a1.lean
        ├── putnam_2004_a3.lean
        ├── putnam_2004_a4.lean
        ├── putnam_2004_a5.lean
        ├── putnam_2004_a6.lean
        ├── putnam_2004_b1.lean
        ├── putnam_2004_b2.lean
        ├── putnam_2004_b4.lean
        ├── putnam_2004_b5.lean
        ├── putnam_2004_b6.lean
        ├── putnam_2005_a1.lean
        ├── putnam_2005_a2.lean
        ├── putnam_2005_a3.lean
        ├── putnam_2005_a4.lean
        ├── putnam_2005_a5.lean
        ├── putnam_2005_b1.lean
        ├── putnam_2005_b2.lean
        ├── putnam_2005_b3.lean
        ├── putnam_2005_b4.lean
        ├── putnam_2005_b5.lean
        ├── putnam_2005_b6.lean
        ├── putnam_2006_a1.lean
        ├── putnam_2006_a3.lean
        ├── putnam_2006_a4.lean
        ├── putnam_2006_a5.lean
        ├── putnam_2006_b1.lean
        ├── putnam_2006_b2.lean
        ├── putnam_2006_b3.lean
        ├── putnam_2006_b4.lean
        ├── putnam_2006_b5.lean
        ├── putnam_2006_b6.lean
        ├── putnam_2007_a1.lean
        ├── putnam_2007_a2.lean
        ├── putnam_2007_a3.lean
        ├── putnam_2007_a4.lean
        ├── putnam_2007_a5.lean
        ├── putnam_2007_b1.lean
        ├── putnam_2007_b2.lean
        ├── putnam_2007_b3.lean
        ├── putnam_2007_b4.lean
        ├── putnam_2007_b5.lean
        ├── putnam_2007_b6.lean
        ├── putnam_2008_a1.lean
        ├── putnam_2008_a3.lean
        ├── putnam_2008_a4.lean
        ├── putnam_2008_a5.lean
        ├── putnam_2008_a6.lean
        ├── putnam_2008_b1.lean
        ├── putnam_2008_b2.lean
        ├── putnam_2008_b3.lean
        ├── putnam_2008_b4.lean
        ├── putnam_2008_b5.lean
        ├── putnam_2008_b6.lean
        ├── putnam_2009_a1.lean
        ├── putnam_2009_a2.lean
        ├── putnam_2009_a3.lean
        ├── putnam_2009_a4.lean
        ├── putnam_2009_a5.lean
        ├── putnam_2009_b1.lean
        ├── putnam_2009_b2.lean
        ├── putnam_2009_b3.lean
        ├── putnam_2009_b4.lean
        ├── putnam_2009_b5.lean
        ├── putnam_2009_b6.lean
        ├── putnam_2010_a1.lean
        ├── putnam_2010_a2.lean
        ├── putnam_2010_a3.lean
        ├── putnam_2010_a4.lean
        ├── putnam_2010_a5.lean
        ├── putnam_2010_a6.lean
        ├── putnam_2010_b1.lean
        ├── putnam_2010_b2.lean
        ├── putnam_2010_b3.lean
        ├── putnam_2010_b4.lean
        ├── putnam_2010_b5.lean
        ├── putnam_2010_b6.lean
        ├── putnam_2011_a1.lean
        ├── putnam_2011_a2.lean
        ├── putnam_2011_a3.lean
        ├── putnam_2011_a4.lean
        ├── putnam_2011_a5.lean
        ├── putnam_2011_a6.lean
        ├── putnam_2011_b1.lean
        ├── putnam_2011_b2.lean
        ├── putnam_2011_b3.lean
        ├── putnam_2011_b4.lean
        ├── putnam_2011_b5.lean
        ├── putnam_2011_b6.lean
        ├── putnam_2012_a1.lean
        ├── putnam_2012_a2.lean
        ├── putnam_2012_a3.lean
        ├── putnam_2012_a4.lean
        ├── putnam_2012_a5.lean
        ├── putnam_2012_a6.lean
        ├── putnam_2012_b1.lean
        ├── putnam_2012_b3.lean
        ├── putnam_2012_b4.lean
        ├── putnam_2012_b5.lean
        ├── putnam_2012_b6.lean
        ├── putnam_2013_a2.lean
        ├── putnam_2013_a3.lean
        ├── putnam_2013_a4.lean
        ├── putnam_2013_a5.lean
        ├── putnam_2013_a6.lean
        ├── putnam_2013_b1.lean
        ├── putnam_2013_b2.lean
        ├── putnam_2013_b3.lean
        ├── putnam_2013_b4.lean
        ├── putnam_2013_b5.lean
        ├── putnam_2014_a1.lean
        ├── putnam_2014_a2.lean
        ├── putnam_2014_a3.lean
        ├── putnam_2014_a4.lean
        ├── putnam_2014_a5.lean
        ├── putnam_2014_a6.lean
        ├── putnam_2014_b1.lean
        ├── putnam_2014_b2.lean
        ├── putnam_2014_b3.lean
        ├── putnam_2014_b4.lean
        ├── putnam_2014_b6.lean
        ├── putnam_2015_a1.lean
        ├── putnam_2015_a2.lean
        ├── putnam_2015_a3.lean
        ├── putnam_2015_a4.lean
        ├── putnam_2015_a5.lean
        ├── putnam_2015_a6.lean
        ├── putnam_2015_b1.lean
        ├── putnam_2015_b2.lean
        ├── putnam_2015_b3.lean
        ├── putnam_2015_b4.lean
        ├── putnam_2015_b5.lean
        ├── putnam_2015_b6.lean
        ├── putnam_2016_a1.lean
        ├── putnam_2016_a2.lean
        ├── putnam_2016_a3.lean
        ├── putnam_2016_a5.lean
        ├── putnam_2016_a6.lean
        ├── putnam_2016_b1.lean
        ├── putnam_2016_b2.lean
        ├── putnam_2016_b3.lean
        ├── putnam_2016_b4.lean
        ├── putnam_2016_b5.lean
        ├── putnam_2016_b6.lean
        ├── putnam_2017_a1.lean
        ├── putnam_2017_a2.lean
        ├── putnam_2017_a3.lean
        ├── putnam_2017_a4.lean
        ├── putnam_2017_b1.lean
        ├── putnam_2017_b2.lean
        ├── putnam_2017_b3.lean
        ├── putnam_2017_b4.lean
        ├── putnam_2017_b6.lean
        ├── putnam_2018_a1.lean
        ├── putnam_2018_a2.lean
        ├── putnam_2018_a3.lean
        ├── putnam_2018_a4.lean
        ├── putnam_2018_a5.lean
        ├── putnam_2018_a6.lean
        ├── putnam_2018_b1.lean
        ├── putnam_2018_b2.lean
        ├── putnam_2018_b3.lean
        ├── putnam_2018_b4.lean
        ├── putnam_2018_b5.lean
        ├── putnam_2018_b6.lean
        ├── putnam_2019_a1.lean
        ├── putnam_2019_a3.lean
        ├── putnam_2019_a4.lean
        ├── putnam_2019_a5.lean
        ├── putnam_2019_a6.lean
        ├── putnam_2019_b1.lean
        ├── putnam_2019_b2.lean
        ├── putnam_2019_b3.lean
        ├── putnam_2019_b4.lean
        ├── putnam_2019_b5.lean
        ├── putnam_2019_b6.lean
        ├── putnam_2020_a1.lean
        ├── putnam_2020_a2.lean
        ├── putnam_2020_a3.lean
        ├── putnam_2020_a5.lean
        ├── putnam_2020_a6.lean
        ├── putnam_2020_b1.lean
        ├── putnam_2020_b4.lean
        ├── putnam_2020_b5.lean
        ├── putnam_2020_b6.lean
        ├── putnam_2021_a1.lean
        ├── putnam_2021_a2.lean
        ├── putnam_2021_a3.lean
        ├── putnam_2021_a4.lean
        ├── putnam_2021_a5.lean
        ├── putnam_2021_a6.lean
        ├── putnam_2021_b2.lean
        ├── putnam_2021_b3.lean
        ├── putnam_2021_b4.lean
        ├── putnam_2021_b5.lean
        ├── putnam_2022_a1.lean
        ├── putnam_2022_a2.lean
        ├── putnam_2022_a3.lean
        ├── putnam_2022_a4.lean
        ├── putnam_2022_a5.lean
        ├── putnam_2022_a6.lean
        ├── putnam_2022_b1.lean
        ├── putnam_2022_b2.lean
        ├── putnam_2022_b3.lean
        ├── putnam_2022_b4.lean
        ├── putnam_2022_b5.lean
        ├── putnam_2022_b6.lean
        ├── putnam_2023_a1.lean
        ├── putnam_2023_a2.lean
        ├── putnam_2023_a3.lean
        ├── putnam_2023_a4.lean
        ├── putnam_2023_a5.lean
        ├── putnam_2023_a6.lean
        ├── putnam_2023_b1.lean
        ├── putnam_2023_b2.lean
        ├── putnam_2023_b3.lean
        ├── putnam_2023_b4.lean
        ├── putnam_2023_b5.lean
        ├── putnam_2023_b6.lean
        ├── putnam_2024_a1.lean
        ├── putnam_2024_a2.lean
        ├── putnam_2024_a3.lean
        ├── putnam_2024_a4.lean
        ├── putnam_2024_a5.lean
        ├── putnam_2024_a6.lean
        ├── putnam_2024_b1.lean
        ├── putnam_2024_b3.lean
        ├── putnam_2024_b4.lean
        ├── putnam_2024_b5.lean
        └── putnam_2024_b6.lean


/.gitignore:
--------------------------------------------------------------------------------
 1 | # Ignore the lake file
 2 | .lake/
 3 | *.olean
 4 | *.olean.*
 5 | 
 6 | lean4/putnam_all*
 7 | lean4/solutions_replaced/*
 8 | lean4/solutions_replaced_new/
 9 | 
10 | informal/join_all_jsons*
11 | 
12 | isabelle_archive/
13 | lean4_archive/
14 | 
15 | split_into_files.py


--------------------------------------------------------------------------------
/.vscode/extensions.json:
--------------------------------------------------------------------------------
1 | {
2 |     "recommendations": [
3 |         "leanprover.lean4"
4 |     ]
5 | }
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1962_a5.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Open Scope nat_scope.
 8 | 
 9 | Definition putnam_1962_a5_solution : nat -> nat := fun n : nat => n * (n + 1) * 2 ^ (n - 2).
10 | Theorem putnam_1962_a5
11 |     : forall n : nat, n >= 2 ->
12 |         putnam_1962_a5_solution n = \sum_(1 <= k < n.+1) (binomial n k * k ^ 2).
13 | Proof. Admitted.
14 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1962_a6.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles QArith.
2 | Theorem putnam_1962_a6
3 |     (A : Ensemble Q)
4 |     (hSSadd : forall a b : Q, (A a /\ A b) -> A (a + b))
5 |     (hSSprod : forall a b : Q, (A a /\ A b) -> A (a * b))
6 |     (hSScond : forall r : Q, (A r \/ A (-r) \/ r = 0) /\ ~(A r \/ A (-r)) /\ ~(A r /\ r = 0) /\ ~(A (-r) /\ r = 0))
7 |     : A = (fun r : Q => r > 0).
8 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1962_b2.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | From mathcomp Require Import reals.
 3 | From mathcomp Require Import classical_sets.
 4 | 
 5 | Set Implicit Arguments.
 6 | Unset Strict Implicit.
 7 | Unset Printing Implicit Defensive.
 8 | 
 9 | Open Scope ring_scope.
10 | Open Scope classical_set_scope.
11 | 
12 | Variable R : realType.
13 | Theorem putnam_1962_b2
14 |     : exists f : R -> set nat, forall a b : R, a < b -> f a `<` f b.
15 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1965_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Coquelicot.Hierarchy Reals.
2 | 
3 | Local Coercion INR : nat >-> R.
4 | 
5 | Theorem putnam_1965_a2
6 |     : forall n : nat, gt n 0 -> sum_n (fun r : nat => ((n - 2 * r) * (C n r) / n) ^ 2) ((n - 1) / 2) = (C (2 * n - 2) (n - 1)) / n.
7 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1965_a4.v:
--------------------------------------------------------------------------------
1 | From mathcomp Require Import ssrbool ssreflect ssrfun fintype.
2 | Variables G B : finType.
3 | Theorem putnam_1965_a4
4 |     (dances : G -> B -> Prop)
5 |     (h : ~(exists b : B, forall g : G, dances g b) /\ (forall g : G, exists b : B, dances g b))
6 |     (nonempty : inhabited G /\ inhabited B)
7 |     : exists (g h : G) (b c : B), dances g b /\ dances h c /\ ~dances h b /\ ~dances g c.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1965_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Finite_sets. From mathcomp Require Import fintype perm.
2 | Definition putnam_1965_a5_solution : nat -> nat := (fun n : nat => 2 ^ (n - 1)).
3 | Theorem putnam_1965_a5
4 |     : forall n : nat, n > 0 -> cardinal {perm 'I_n} (fun p : {perm 'I_n} => forall m : 'I_n, m > 0 -> exists k : 'I_n, k < m /\ (nat_of_ord (p m) = (p k) + 1 \/ nat_of_ord (p m) = (p k) - 1)) (putnam_1965_a5_solution n).
5 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1965_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Finite_sets Reals Coquelicot.Coquelicot.
2 | Theorem putnam_1965_b3
3 |     : cardinal (R * R * R) (fun abc : R * R * R => let '(a, b, c) := abc in (IZR (floor a) = a /\ IZR (floor b) = b /\ IZR (floor c) = c /\ a > 0 /\ a <= b /\ c > 0 /\ a ^ 2 + b ^ 2 = c ^ 2 /\ a * b / 2 = 2 * (a + b + c))) 3.
4 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1966_a1.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_ssreflect ssrnum ssralg ssrint.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Local Open Scope ring_scope.
 8 | 
 9 | Theorem putnam_1966_a1
10 |     (f : nat -> int := fun n => \sum_(0 <= m < n + 1) (if (~~odd m) then (m%:Z)/2 else (m%:Z-1)/2))
11 |     : forall x y : nat, (gt x 0) -> (gt y 0) -> gt x y -> (x * y)%:Z = f (Nat.add x y) - f (Nat.sub x y).
12 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1967_b5.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_ssreflect all_algebra.
 2 | From mathcomp Require Import reals.
 3 | 
 4 | Set Implicit Arguments.
 5 | Unset Strict Implicit.
 6 | Unset Printing Implicit Defensive.
 7 | 
 8 | Local Open Scope ring_scope.
 9 | 
10 | Variable R : realType.
11 | Theorem putnam_1967_b5
12 |     (f : nat -> R := fun n => \sum_(0 <= i < n) ((binomial (n + i - 1) i)%:R / (2 ^+ (n + i))))
13 |     : forall n, gt n 0 -> 1%R/2 = f n.
14 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1968_a2.v:
--------------------------------------------------------------------------------
 1 | Require Import ZArith Reals QArith Qabs.
 2 | 
 3 | Local Coercion inject_Z : Z >-> Q.
 4 | Local Coercion Q2R : Q >-> R.
 5 | 
 6 | Theorem putnam_1968_a2
 7 |     (a b c d e f : Z)
 8 |     (eps : R)
 9 |     (hne : Z.mul a d <> Z.mul b c)
10 |     (heps : Rgt eps 0)
11 |     : exists r s : Q, 0 < Qabs (r * a + s * b - e) /\ Rlt (Qabs (r * a + s * b - e)) eps /\ 0 < Qabs (r * c + s * d - f) /\ Rlt (Qabs (r * c + s * d - f)) eps.
12 | Proof. Admitted.
13 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1971_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles RelationClasses.
2 | Theorem putnam_1971_b1
3 |     (S : Type)
4 |     (op : S -> S -> S)
5 |     (hself : forall x : S, op x x = x)
6 |     (h2 : forall x y z : S, op (op x y) z = op (op y z) x)
7 |     : (forall x y z : S, op (op x y) z = op x (op y z)) /\ (forall x y : S, op x y = op y x).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1972_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles Reals FinFun.
2 | Theorem putnam_1972_a2
3 |     : (forall (S : Type) (Smul : S -> S -> S), (forall x y : S, (Smul x (Smul x y) = y /\ Smul (Smul y x) x = y)) -> (forall x y : S, Smul x y = Smul y x)) /\
4 |         (exists (S : Type) (Smul : S -> S -> S), (forall x y : S, (Smul x (Smul x y) = y /\ Smul (Smul y x) x = y)) /\ ~(forall x y z : S, Smul x (Smul y z) = Smul (Smul x y) z)).
5 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1972_a5.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Local Open Scope ring_scope.
 8 | 
 9 | Theorem putnam_1972_a5
10 |     (n : nat)
11 |     (hn : gt n 1)
12 |     : ~~(n%:Z %| 2 ^+ n - 1)%Z.
13 | Proof. Admitted.
14 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1972_b3.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import fingroup.
 2 | Open Scope group_scope.
 3 | (* Note: This formalization is only for finite groups (due to mathcomp), but this is sufficiently general since the group generated by A and B is finite. *)
 4 | Variable T : finGroupType.
 5 | Theorem putnam_1972_b3
 6 |     (G : {group T})
 7 |     (A B : T)
 8 |     (hab : A * B * A = B * A ^+ 2 * B /\ A ^+ 3 = 1 /\ exists n : nat, n > 0 /\ B ^+ (2*n - 1) = 1)
 9 |     : (B = 1).
10 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1972_b5.v:
--------------------------------------------------------------------------------
1 | From GeoCoq Require Import Main.Tarski_dev.Ch16_coordinates_with_functions.
2 | Context `{T3D:Tarski_3D}.
3 | Theorem putnam_1972_b5
4 |     (A B C D : Tpoint)
5 |     (hnonplanar : ~Coplanar A B C D)
6 |     (hangles : Ang A B C = Ang C D A /\ Ang B C D = Ang D A B)
7 |     : (Cong A B C D /\ Cong B C D A).
8 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1973_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals QArith Coquelicot.Complex. From Coqtail Require Import Cpow.
2 | Theorem putnam_1973_b2
3 |     (z : C)
4 |     (hzrat : exists q1 q2 : Q, Re z = Q2R q1 /\ Im z = Q2R q2)
5 |     (hznorm : Cmod z = 1%R)
6 |     : forall n : nat, exists q1 q2 : Q, Cmod (Cpow z (2 * n) - 1) = Q2R q1 /\ Cmod (1 / (Cpow z (2 * n) - 1)) = Q2R q2.
7 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1976_b6.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Local Open Scope nat_scope.
 8 | 
 9 | Theorem putnam_1976_b6
10 | (sigma : nat -> int := fun N => \sum_(d <- divisors N) d)
11 | : forall N : nat, sigma N = (N.*2 + 1) -> (exists m : nat, odd m = true /\ N = m ^ 2).
12 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1977_a5.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Local Open Scope nat_scope.
 8 | 
 9 | Theorem putnam_1977_a5
10 |     (p n m : nat)
11 |     (hp : prime p)
12 |     (hmgen : ge m n)
13 |     : 'C(mult p m, mult p n) = 'C(m, n) %[mod p].
14 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1979_b6.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | From mathcomp Require Import reals complex.
 3 | 
 4 | Set Implicit Arguments.
 5 | Unset Strict Implicit.
 6 | Unset Printing Implicit Defensive.
 7 | 
 8 | Local Open Scope ring_scope.
 9 | 
10 | Variable R : realType.
11 | Theorem putnam_1979_b6
12 |     (l : seq R[i])
13 |     : `|Re (sqrtc (\sum_(z <- l) (z ^+ 2)))| <= \sum_(z <- l) `|Re z|.
14 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1981_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_1981_a3_solution := False.
3 | Theorem putnam_1981_a3
4 |     : (exists r : R, Lim (fun k => exp (-1*k) * (RInt (fun x => (RInt (fun y => (exp x - exp y) / (x - y)) 0 k)) 0 k)) p_infty = r) <-> putnam_1981_a3_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1981_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_1981_b1_solution := -1.
3 | Theorem putnam_1981_b1
4 |     : Lim_seq (fun n => 1/(pow (INR n) 5) * (sum_n (fun r => (sum_n (fun s => 5 * pow (INR r) 4 - 18  * pow (INR r) 2 * pow (INR s) 2 + 5 * pow (INR s) 4) n)) n)) = putnam_1981_b1_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1981_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals.
2 | Open Scope R.
3 | Definition putnam_1981_b2_solution := 12 - 8 * sqrt 2.
4 | Theorem putnam_1981_b2
5 |     (f : R -> R -> R -> R := fun a b c => pow (a-1) 2 + pow (b / a - 1) 2 + pow (c / b - 1) 2 + pow (4 / c - 1) 2)
6 |     : (forall (a b c: R), 1 <= a /\ a <= b /\ b <= c /\ c <= 4 -> putnam_1981_b2_solution <= f a b c) /\
7 |     (exists (a b c: R), 1 <= a /\ a <= b /\ b <= c /\ c <= 4 -> putnam_1981_b2_solution  = f a b c).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1981_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Nat ZArith Znumtheory. From mathcomp Require Import div.
2 | Open Scope nat_scope.
3 | Theorem putnam_1981_b3
4 |     : ~ exists (N: nat),
5 |     forall (n: nat), 
6 |     (forall (p: nat), prime (Z.of_nat p) /\ p %| pow n 2 + 3 = true -> exists (m: nat), p %| pow m 2 + 3 = true /\ pow m 2 < n) ->
7 |     n < N.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1982_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1982_a2_solution := True.
4 | Theorem putnam_1982_a2
5 |     (B : nat -> R -> R := fun n x => sum_n (fun m => Rpower (INR m) x) n)
6 |     (f : nat -> R := fun n => B n (ln 2 / ln (INR n)) / ((INR n) * Rpower (ln 2 / ln (INR n)) 2))
7 |     : ex_series (fun n => if (lt_dec n 2) then 0 else f n) <-> putnam_1982_a2_solution.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1982_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1982_a3_solution := PI / 2 * ln PI.
4 | Theorem putnam_1982_a3
5 |     (f : R -> R := fun x => (atan (PI * x) - atan x) / x)
6 |     : Lim (fun n => RInt f 0 n) p_infty = putnam_1982_a3_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1982_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals.
2 | Open Scope R.
3 | Theorem putnam_1982_a5 
4 |     (a b c d: nat)
5 |     (hpos : Nat.lt 0 a /\ Nat.lt 0 b /\ Nat.lt 0 c /\ Nat.lt 0 d)
6 |     (habcd : Nat.le (Nat.add a c) 1982 /\ INR a / INR b + INR c / INR d < 1)
7 |     : 1 - INR a / INR b - INR c / INR d > 1/pow 1983 3.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1982_b5.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Theorem putnam_1982_b5
 4 |     (F := fix f (n: nat) (x: R) :=
 5 |         match n with 
 6 |         | O => exp 1
 7 |         | S n' => ln x / ln (f n' x)
 8 |     end)
 9 |     : forall (x: R), x > Rpower (exp 1) (exp 1) -> 
10 |     ex_finite_lim_seq (fun n => F n x) /\ 
11 |     let g (x: R) := Lim_seq (fun n => F n x) in
12 |     continuity_pt g x.
13 | Proof. Admitted.
14 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1983_a1.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Ensembles Finite_sets. From mathcomp Require Import div.
2 | Definition putnam_1983_a1_solution := 2301.
3 | Theorem putnam_1983_a1
4 |     (E: Ensemble nat := fun n => n %| 10^(40) = true \/ n %| 20^(30) = true)
5 |     : cardinal nat E putnam_1983_a1_solution.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1983_a4.v:
--------------------------------------------------------------------------------
 1 | Require Import Binomial Reals Coquelicot.Coquelicot.
 2 | Open Scope nat_scope.
 3 | Theorem putnam_1983_a4
 4 |     (k : nat)
 5 |     (kpos : k > 0)
 6 |     (m : nat := 6 * k - 1)
 7 |     (s : R := sum_n_m (fun j => Rmult ((-1) ^ (j + 1)) (Binomial.C m (3 * j - 1))) 1 (2*k - 1))
 8 |     : s <> R0.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1983_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Nat ZArith Reals Coquelicot.Coquelicot.
2 | Definition putnam_1983_a5_solution := True.
3 | Theorem putnam_1983_a5
4 |     : (exists a : R, a > 0 /\ forall n : nat, gt n 0 -> Z.Even (floor (a^n) - Z.of_nat n)) <-> putnam_1983_a5_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1983_a6.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_1983_a6_solution := 2 / 9.
3 | Theorem putnam_1983_a6
4 |     (F : R -> R := fun a => (a ^ 4 / exp (a ^ 3)) * RInt (fun x => RInt (fun y => exp (x ^ 3 + y ^ 3)) 0 (a - x)) 0 a)
5 |     : filterlim F (Rbar_locally p_infty) (locally putnam_1983_a6_solution).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1983_b4.v:
--------------------------------------------------------------------------------
 1 | Require Import ZArith Reals Coquelicot.Coquelicot.
 2 | Open Scope Z_scope.
 3 | Theorem putnam_1983_b4
 4 |     (m : Z)
 5 |     (hm : m > 0)
 6 |     (f : Z -> Z := fun n => n + floor (sqrt (IZR n)))
 7 |     (A := fix a (n: nat) :=
 8 |         match n with
 9 |         | O => m
10 |         | S n' => f (a n')
11 |     end)
12 |     : exists (i : nat) (s : Z), A i = s^2.
13 | Proof. Admitted.
14 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1983_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1983_b5_solution := ln (4/PI).
4 | Theorem putnam_1983_b5
5 |     (mindist : R -> R := fun x => Rmin (x - IZR (floor x)) (IZR (floor (x+1)) - x))
6 |     : Lim_seq (fun n => (1/INR n) * (RInt (fun x => mindist (INR n/x)) 1 (INR n))) = putnam_1983_b5_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1984_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope nat_scope.
3 | (* Uses division by 0 in the first term of the series. *)
4 | Definition putnam_1984_a2_solution := 2%R.
5 | Theorem putnam_1984_a2
6 |     (f : nat -> R := fun n => Rdiv (pow 6 n) ((pow 3 (n+1) - pow 2 (n+1)) * (pow 3 n - pow 2 n)))
7 |     : Series f = putnam_1984_a2_solution.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1984_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Factorial Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1984_a5_solution := INR (fact 9 * fact 8 * fact 4) / INR (fact 25).
4 | Theorem putnam_1984_a5
5 |     : RInt (fun z => RInt (fun y => RInt (fun x => x * pow y 9 * pow z 8 * pow (1 - x - y - z) 4) 0 (1 - y - z)) 0 (1 - z)) 0 1 = putnam_1984_a5_solution.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1984_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1984_b2_solution := 8.
4 | Theorem putnam_1984_b2
5 |     (f : R -> R -> R := fun x y => pow (x - y) 2 + pow (sqrt (2 - pow x 2) - 9 / y) 2)
6 |     : (forall x y : R, 0 < x < sqrt 2 /\ y > 0 -> f x y >= putnam_1984_b2_solution) /\
7 |     (exists x y : R, 0 < x < sqrt 2 /\ y > 0 /\ f x y = putnam_1984_b2_solution).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1985_a3.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1985_a3_solution (x: R) := exp x - 1.
 4 | Theorem putnam_1985_a3
 5 |     (x: R) 
 6 |     (A := fix a (i j: nat) :=
 7 |         match (i,j) with
 8 |         | (i, 0) => x/pow 2 i
 9 |         | (i, S j') => pow (a i j') 2 + 2 * a i j'
10 |     end) 
11 |     : Lim_seq (fun n => A n n) = putnam_1985_a3_solution x.
12 | Proof. Admitted.
13 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1985_a4.v:
--------------------------------------------------------------------------------
 1 | Require Import Ensembles Nat Coquelicot.Coquelicot.
 2 | Definition putnam_1985_a4_solution := fun k => k = 87.
 3 | Theorem putnam_1985_a4
 4 |     (a : nat -> nat := fix a (n: nat) :=
 5 |         match n with
 6 |         | O => 3
 7 |         | S n' => 3 ^ (a n')
 8 |     end)
 9 |     : let E: Ensemble nat := fun k => k < 100 /\ forall (n: nat), exists (i: nat), ge i n /\ (a i) mod 100 = k in
10 |     E = putnam_1985_a4_solution.
11 | Proof. Admitted.
12 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1985_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Reals List Coquelicot.Coquelicot.
2 | Open Scope nat_scope.
3 | Definition putnam_1985_a5_solution (n : nat) := n = 3 \/ n = 4 \/ n = 7 \/ n = 8.
4 | Theorem putnam_1985_a5
5 |     (F : nat -> R -> R := fun n x => fold_right Rmult 1%R (map (fun i : nat => cos (INR i * x)) (seq 1 n)))
6 |     : forall (n: nat), 1 <= n <= 10 -> (RInt (F n) 0 (2*PI) <> 0%R <-> putnam_1985_a5_solution n).
7 | Proof. Admitted. 


--------------------------------------------------------------------------------
/coq/src/putnam_1985_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Ensembles Finite_sets.
2 | Theorem putnam_1985_b3
3 |     (a : nat -> nat -> nat)
4 |     (apos : forall m n : nat, a m n > 0)
5 |     (ha : forall k : nat, k > 0 -> cardinal (nat * nat) (fun t => let (m, n) := t in m > 0 /\ n > 0 /\ a m n = k) 8)
6 |     : exists m n : nat, m > 0 /\ n > 0 /\ a m n > m * n.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1985_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1985_b5_solution := sqrt (PI / 1985) * exp (-3970).
4 | Theorem putnam_1985_b5
5 |     : filterlim (fun n : R => RInt (fun x => Rpower x (-1/2) * exp (-1985 * (x + 1/x))) 0 n) (Rbar_locally p_infty) (locally putnam_1985_b5_solution).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1986_a1.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Ensembles Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1986_a1_solution := 18.
 4 | Theorem putnam_1986_a1
 5 |     (f : R -> R := fun x => pow x 3 - 3 * x)
 6 |     (T : Ensemble R := fun x => pow x 4 + 36 <= 13 * pow x 2)
 7 |     : (forall x : R, In R T x -> putnam_1986_a1_solution >= f x) /\
 8 |     (exists x : R, In R T x /\ putnam_1986_a1_solution = f x).
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1986_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Nat.
2 | Definition putnam_1986_a2_solution := 3.
3 | Theorem putnam_1986_a2
4 |     : (10 ^ (20000) / (10 ^ (100) + 3)) mod 10 = putnam_1986_a2_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1986_a3.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1986_a3_solution := PI / 2.
 4 | Theorem putnam_1986_a3
 5 |     (cot : R -> R)
 6 |     (fcot : cot = fun t => cos t / sin t)
 7 |     (arccot : R -> R)
 8 |     (harccot : forall t : R, t >= 0 -> 0 < arccot t <= PI / 2 /\ cot (arccot t) = t)
 9 |     : Series (fun n => arccot (pow (INR n) 2 + INR n + 1)) = putnam_1986_a3_solution.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1986_b4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Ranalysis Coquelicot.Coquelicot.
2 | Definition putnam_1986_b4_solution := True.
3 | Theorem putnam_1986_b4
4 |     (G : R -> R)
5 |     (hGeq : forall (r: R), exists (m n: Z), G r = Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
6 |     (hGlb : forall (r: R), forall (m n: Z), G r <= Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
7 |     : filterlim G (Rbar_locally p_infty) (locally 0) <-> putnam_1986_b4_solution.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1987_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_1987_b1_solution := 1.
3 | Theorem putnam_1987_b1
4 |     : RInt (fun x => ln (9 - x) ^ (1/2) / ( ln (9 - x) ^ (1/2)  + ln (x + 3) ^ (1/2))) 2 4 = putnam_1987_b1_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1987_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Binomial Reals Coquelicot.Coquelicot.
2 | Theorem putnam_1987_b2
3 |     : forall (n r s: nat), ge n (r + s) -> 
4 |     sum_n (fun i => Binomial.C s i / Binomial.C n (r + i)) s = (INR n + 1)/((INR n + 1 - INR s) * Binomial.C (n - s) r).
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1988_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1988_a3_solution (a: R) := a > 1/2.
4 | Theorem putnam_1988_a3
5 |    : forall (a: R), ex_finite_lim_seq (fun m => sum_n_m (fun n => Rpower (1/INR n * (1 / sin (1/INR n)) - 1) a) 1 m) <-> putnam_1988_a3_solution a.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1988_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot. From mathcomp Require Import fintype.
2 | Definition putnam_1988_a4_solution : Prop * Prop := (True, False).
3 | Theorem putnam_1988_a4
4 |     (p : nat -> Prop := fun n : nat => (forall color : (R * R) -> 'I_n, exists p q : R * R, color p = color q /\ norm (fst p - fst q, snd p - snd q) = 1))
5 |     : let (a, b) := putnam_1988_a4_solution in ((p 3%nat) <-> a) /\ ((p 9%nat) <-> b).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1988_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Basics Reals.
2 | Open Scope R.
3 | Theorem putnam_1988_a5
4 |     : exists! (f: R -> R), 
5 |     (forall (x: R), x <= 0 -> f x = 0) /\
6 |     forall (x: R), x > 0 -> 
7 |     (f x > 0 /\ (compose f f) x = 6 * x - f x).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1988_b1.v:
--------------------------------------------------------------------------------
1 | Require Import ZArith Znumtheory.
2 | Open Scope Z.
3 | Theorem putnam_1988_b1
4 |     : forall (n: Z), n > 3 /\ ~ prime n -> exists (a b c: Z), a > 0 /\ b > 0 /\ c > 0 /\ n = a * b + b * c + c * a + 1.    
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1988_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1988_b2_solution := True.
4 | Theorem putnam_1988_b2
5 |     : (forall (a: R), a >= 0 -> forall (x: R), pow (x + 1) 2 >= a * (a + 1) ->
6 |     pow x 2 >= a * (a - 1)) <-> putnam_1988_b2_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1988_b6.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles Finite_sets ZArith.
2 | Open Scope Z.
3 | Theorem putnam_1988_b6
4 |     (triangular : Z -> Prop := fun n => exists (m: Z), m >= 0 /\ n = (m * (m + 1)) / 2)
5 |     (E: Ensemble (Z*Z) := fun '(a, b) => forall (m: Z), m > 0 -> (triangular m <-> triangular (Z.mul a m + b)))
6 |     : ~ exists (n: nat), cardinal (Z*Z) E n.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1989_a2.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1989_a2_solution (a b: R) := (exp (pow (a*b) 2) - 1)/(a * b).
 4 | Theorem putnam_1989_a2
 5 |     (a b: R)
 6 |     (abpos : a > 0 /\ b > 0)
 7 |     (f : R -> R -> R := fun x y => Rmax (pow (b*x) 2) (pow (a*y) 2))
 8 |     : RInt (fun x => (RInt (fun y => exp (f x y)) 0 b)) 0 a = putnam_1989_a2_solution a b.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1989_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot. From Coqtail Require Import Cpow.
2 | Open Scope C.
3 | Theorem putnam_1989_a3
4 |     (f : C -> C := fun z => 11 * Cpow z 10 + 10 * Ci * Cpow z 9 + 10 * Ci * z - 11)
5 |     : forall (x: C), f x = 0 -> Cmod x = R1.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1990_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1990_a2_solution := True.
4 | Theorem putnam_1990_a2
5 |     (numform : R -> Prop := fun x => exists (n m: nat), x = Rpower (INR n) (1/3) - Rpower (INR m) (1/3))
6 |     : (exists (s: nat -> R), (forall (i: nat), numform (s i)) /\ is_lim_seq s (sqrt 2)) <-> putnam_1990_a2_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1990_a5.v:
--------------------------------------------------------------------------------
1 | From mathcomp Require Import matrix ssralg.
2 | Open Scope ring_scope.
3 | Definition putnam_1990_a5_solution := False.
4 | Theorem putnam_1990_a5 
5 |     : (forall (R : ringType) (n: nat) (A B : 'M[R]_n), n >= 1 -> mulmx (mulmx (mulmx A B) A) B = 0 -> mulmx (mulmx (mulmx B A) B) A = 0) <-> putnam_1990_a5_solution.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1990_b1.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1990_b1_solution (f: R -> R) := f = (fun x => sqrt 1990 * exp x) \/ f = (fun x => -sqrt 1990 * exp x).
 4 | Theorem putnam_1990_b1
 5 |     (f : R -> R)
 6 |     (fderiv : (forall x : R, ex_derive f x) /\ continuity (Derive f))
 7 |     : (forall x, pow (f x) 2 = RInt (fun t => pow (f t) 2 + pow ((Derive f) t) 2) 0 x + 1990) <->
 8 |     putnam_1990_b1_solution f.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1991_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1991_a5_solution := 1/3.
4 | Theorem putnam_1991_a5
5 |     (f : R -> R := fun y : R => RInt (fun x => sqrt (pow x 4 + pow (y - pow y 2) 2)) 0 y)
6 |     : (exists (y : R), 0 <= y <= 1 /\ putnam_1991_a5_solution = f y) /\ (forall (y : R), 0 <= y <= 1 -> putnam_1991_a5_solution >= f y).
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1991_b4.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Reals ZArith Znumtheory Binomial Coquelicot.Coquelicot.
2 | Theorem putnam_1991_b4
3 |     (p: nat)
4 |     (hp : odd p = true /\ prime (Z.of_nat p)) 
5 |     (expr : R := sum_n (fun j => Binomial.C p j * Binomial.C (p + j) j) p)
6 |     : (floor expr) mod (Z.pow (Z.of_nat p) 2) = Z.add (Z.pow 2 (Z.of_nat p)) 1.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1992_a1.v:
--------------------------------------------------------------------------------
1 | Require Import Basics ZArith.
2 | Open Scope Z_scope.
3 | Theorem putnam_1992_a1 
4 |     (f : Z -> Z) 
5 |     : f = (fun n => 1 - n) <-> ((forall n : Z, f (f n) = n) /\ (forall n : Z, f (f (n + 2) + 2) = n) /\ f 0 = 1).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1992_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Binomial Factorial Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1992_a2_solution := 1992.
4 | Theorem putnam_1992_a2
5 |     (C : R -> R := fun a => (Derive_n (fun x => Rpower (1 + x) a) 1992) 0 / INR (fact 1992))
6 |     : RInt (fun y => C (-y - 1) * sum_n_m (fun k => 1 / (y + INR k)) 1 1992) 0 1 = putnam_1992_a2_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1993_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Theorem putnam_1993_a2 
4 |     (x : nat -> R)
5 |     (xnonzero : forall n : nat, x n <> 0)
6 |     (hx : forall n : nat, ge n 1 -> pow (x n) 2 - x (pred n) * x (S n) = 1)
7 |     : exists a : R, forall n : nat, ge n 1 -> x (S n) = a * x n - x (pred n).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1993_a5.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Theorem putnam_1993_a5 
 4 |     (f : R -> R := fun x => (pow x 2 - x) / (pow x 3 - 3 * x + 1))
 5 |     : exists p q : Z,
 6 |     RInt (fun x => pow (f x) 2)  (-100) (-10) +
 7 |     RInt (fun x => pow (f x) 2) (1/101) (1/11) +
 8 |     RInt (fun x => pow (f x) 2) (101/100) (11/10) =
 9 |     IZR p /IZR q.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1993_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals.
2 | Open Scope R.
3 | Definition putnam_1993_b1_solution : nat := 3987.
4 | Theorem putnam_1993_b1
5 |     (cond : nat -> Prop := fun n => gt n 0 /\ forall (m: nat), and (lt 0 m) (lt m 1993) -> exists (k: nat), INR m / 1993 < INR k / INR n < INR (m + 1) / 1994)
6 |     : cond putnam_1993_b1_solution /\ forall (n: nat), cond n -> ge n putnam_1993_b1_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1993_b5.v:
--------------------------------------------------------------------------------
1 | Require Import ZArith Reals Coquelicot.Coquelicot. From mathcomp Require Import fintype.
2 | Theorem putnam_1993_b5
3 |     (pdists : ('I_4 -> (R * R)) -> Prop)
4 |     (hpdists : forall p : 'I_4 -> (R * R), pdists p = (forall i j : 'I_4, i <> j -> (exists k : Z, IZR k = norm (fst (p i) - fst (p j), (snd (p i) - snd (p j))) /\ Z.odd k = true)))
5 |     : ~ (exists p : 'I_4 -> (R * R), pdists p).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1994_a1.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Theorem putnam_1994_a1
4 |     (a : nat -> R)
5 |     (ha : forall n : nat, gt n 0 -> 0 < a n <= a (mul 2 n) + a (add (mul 2 n) 1))
6 |     : ~ ex_finite_lim_seq (fun n => sum_n (fun m => a m) n).
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1994_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles Finite_sets ZArith.
2 | Open Scope Z.
3 | Definition putnam_1994_b1_solution (n: Z) := 315 <= n <= 325 \/ 332 <= n <= 350.
4 | Theorem putnam_1994_b1
5 |     (n : Z)
6 |     (nwithin : Prop := cardinal nat (fun m => Z.abs (n - (Z.of_nat m) ^ 2) <= 250) 15)
7 |     : (n > 0 /\ nwithin) <-> putnam_1994_b1_solution n.
8 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1994_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles Reals Finite_sets Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1994_b2_solution (c: R) := c < 243 / 8.
4 | Theorem putnam_1994_b2
5 |     (hintersect : R -> Prop := fun c => exists(m b: R), cardinal R (fun x => m * x + b = pow x 4 + 9 * pow x 3 + c * pow x 2 + 9 * x + 4) 4)
6 |     : forall (c: R), hintersect c <-> putnam_1994_b2_solution c.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1994_b6.v:
--------------------------------------------------------------------------------
1 | Require Import ZArith.
2 | Open Scope Z_scope.
3 | Theorem putnam_1994_b6
4 |     (n : Z -> Z := fun a => 101 * a - 100 * 2^a)
5 |     : forall a b c d : Z, 0 <= a <= 99 /\ 0 <= b <= 99 /\ 0 <= c <= 99 /\ 0 <= d <= 99 /\ 
6 |     (n a + n b) mod 10100 = (n c + n d) mod 10100 -> (a = c /\ b = d) \/ (a = d /\ b = c).
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1995_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1995_a2_solution : R -> R -> Prop := fun (a b: R) => a = b.
4 | Theorem putnam_1995_a2
5 |     : forall (a b: R), a > 0 /\ b > 0 -> ((exists limit : R, filterlim (fun n : R => RInt (fun x => sqrt (sqrt (x + a) - sqrt x) - sqrt (sqrt x - (x - b))) b n) (Rbar_locally p_infty) (locally limit)) <-> putnam_1995_a2_solution a b).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1995_b4.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals ZArith Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1995_b4_solution : Z * Z * Z * Z := (3%Z,1%Z,5%Z,2%Z).
 4 | Theorem putnam_1995_b4
 5 |     (contfrac : R)
 6 |     (hcontfrac : contfrac = 2207 - 1/contfrac)
 7 |     (hcontfrac' : 1 < contfrac)
 8 |     : let (abc, d) := putnam_1995_b4_solution in let (ab, c) := abc in let (a, b) := ab in 
 9 |     pow contfrac (1 / 8) = (IZR a + IZR b * sqrt (IZR c))/IZR d.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1996_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Binomial Reals Znumtheory Coquelicot.Coquelicot. From mathcomp Require Import div.
2 | Open Scope R.
3 | Theorem putnam_1996_a5 
4 |     (p : nat)
5 |     (hp : prime (Z.of_nat p) /\ gt p 3)
6 |     (k : nat := Z.to_nat (floor (2 * INR p / 3)))
7 |     : exists (m : nat), sum_n_m (fun i => Binomial.C p i) 1 k = INR m * pow (INR p) 2.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1996_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot. From mathcomp Require Import bigop.
2 | Open Scope R.
3 | Theorem putnam_1996_b2
4 |     (n : nat)
5 |     (prododd : R := INR (\prod_(1 <= i < (n + 1)) (2 * i - 1)))
6 |     (npos : gt n 0)
7 |     : Rpower ((2 * INR n - 1) / exp 1) ((2 * INR n - 1) / 2) < prododd < Rpower ((2 * INR n + 1) / exp 1) ((2 * INR n + 1) / 2).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1997_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Ensembles Finite_sets List Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1997_a5_solution := True.
4 | Theorem putnam_1997_a5
5 |     (E: Ensemble (list nat) := fun l => length l = 10%nat /\ (forall i : nat, lt i 10 -> gt (nth i l 0%nat) 0) /\ sum_n (fun i => 1/ INR (nth i l 0%nat)) 9 = 1)
6 |     (m: nat)
7 |     : cardinal (list nat) E m -> (odd m = true <-> putnam_1997_a5_solution).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1997_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1997_b1_solution : nat -> R := (fun n : nat => INR n).
4 | Theorem putnam_1997_b1
5 |     (rnd : R -> R := fun x => Rmin (Rabs (IZR (floor x) - x)) (Rabs (IZR (floor (x + 1)) - x)))
6 |     : forall (n: nat), gt n 0 -> sum_n_m (fun m => Rmin (rnd (INR m / (6 * INR n))) (rnd (INR m / (3 * INR n)))) 1 (6 * n - 1) = putnam_1997_b1_solution n.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1997_b2.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Theorem putnam_1997_b2 
 4 |     (f g : R -> R)
 5 |     (hg : forall x : R, g x >= 0)
 6 |     (hfdiff : forall x : R, ex_derive f x /\ ex_derive_n f 2 x)
 7 |     (hfg : forall x : R, f x + Derive_n f 2 x = -x * g x * Derive f x)
 8 |     : exists M : R, (forall x : R, -M <= abs (f x) <= M).
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1997_b5.v:
--------------------------------------------------------------------------------
 1 | Require Import Nat.
 2 | Theorem putnam_1997_b5
 3 |     (tetration := fix tetration' (b m: nat) : nat :=
 4 |       match m with
 5 |       | O => 1
 6 |       | S m' => b ^ (tetration' b m')
 7 |     end)
 8 |     (n : nat)
 9 |     (hn : n >= 2)
10 |     : (tetration 2 n) mod n = (tetration 2 (n-1)) mod n.
11 | Proof. Admitted.
12 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1998_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Theorem putnam_1998_a3
4 |     (f : R -> R)
5 |     (hf : (forall x : R, ex_derive f x /\ ex_derive_n f 2 x /\ ex_derive_n f 3 x) /\ continuity (Derive_n f 3))
6 |     : exists (a: R), (Derive_n f 0) a * (Derive_n f 1) a * (Derive_n f 2) a * (Derive_n f 3) a >= 0.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1998_b1.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Open Scope R.
 3 | Definition putnam_1998_b1_solution : R := 6.
 4 | Theorem putnam_1998_b1
 5 |     (f : R -> R := fun x => ((x + 1 / x) ^ 6 - (x ^ 6 + 1 / (x ^ 6)) - 2) / ((x + 1 / x) ^ 3 + (x ^ 3 + 1 / (x ^ 3))))
 6 |     (m : R)
 7 |     (hm : exists (x: R), x > 0 /\ f x = m) 
 8 |     (hmub : forall (x: R), x > 0 -> f x >= m) 
 9 |     : m = putnam_1998_b1_solution.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1998_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Definition putnam_1998_b5_solution : nat := 1.
4 | Theorem putnam_1998_b5
5 |     (N : R := sum_n (fun i => 10^i) 1997)
6 |     : Z.to_nat (floor (10^1000 * sqrt N)) mod 10 = putnam_1998_b5_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1998_b6.v:
--------------------------------------------------------------------------------
1 | Require Import Reals.
2 | Open Scope Z.
3 | Theorem putnam_1998_b6
4 |     (a b c : Z)
5 |     : exists (n: Z), n > 0 /\ (forall (m : Z), n * n * n + a * n * n + b * n + c <> m * m).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1999_a2.v:
--------------------------------------------------------------------------------
1 | From mathcomp.analysis Require Import reals. From mathcomp Require Import ssrnat ssrnum ssralg fintype poly.
2 | Open Scope ring_scope.
3 | Theorem putnam_1999_a2
4 |     (R : realType)
5 |     (p : {poly R})
6 |     (hpos : forall x, (p.[x] >= 0) = true)
7 |     : exists (k : nat) (f : nat -> {poly R}), p = \sum_(i < k) ((f i)*(f i)).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1999_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_1999_a3
3 |     (f : R -> R := fun x => 1/(1 - 2 * x - x^2))
4 |     (a : nat -> R)
5 |     (hf : exists epsilon : R, epsilon > 0 /\ (forall x : R, 0 <= Rabs (x) < epsilon -> filterlim (fun n : nat => sum_n (fun i => a i * x^i) n) eventually (locally (f x))))
6 |     : forall n : nat, exists m : nat, (a n)^2 + (a (S n))^2 = a m.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1999_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_1999_a4_solution : R := 9/32.
3 | Theorem putnam_1999_a4: 
4 |     Series (fun m => Series (fun n => (INR (m + 1) ^ 2 * INR (n + 1)) / (3 ^ (m + 1) * (INR (n + 1) * 3 ^ (m + 1) + INR (m + 1) * 3 ^ (n + 1))))) = putnam_1999_a4_solution.
5 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_1999_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_1999_a5
3 |     (p : (nat -> R) -> R -> R := fun a x => sum_n (fun i => a i * x ^ i) 1999)
4 |     : exists (c: R), forall (a: nat -> R), Rabs (p a 0) <= c * RInt (fun x => Rabs (p a x)) (-1) 1.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1999_a6.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Theorem putnam_1999_a6
 3 |     (A : nat -> R := fix a (n: nat) :=
 4 |         match n with 
 5 |         | O => 1
 6 |         | S O => 2
 7 |         | S (S O) => 24
 8 |         | S (S ((S n'') as n') as n) => (6 * (a n) ^ 2 * a n'' - 8 * a n * (a n') ^ 2) / (a n' * a n'')
 9 |     end)
10 |     : forall (n: nat), exists (k: Z), A n = INR (n + 1) * IZR k.
11 | Proof. Admitted.
12 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1999_b4.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Derive.
 2 | Open Scope R_scope.
 3 | Theorem putnam_1999_b4
 4 |     (f: R -> R)
 5 |     (f_cont : (forall n : nat, (le 1 n /\ le n 3) -> (forall x : R, ex_derive_n f n x)) /\ continuity (Derive_n f 3))
 6 |     (f_pos : forall (x: R), f x > 0 /\ (Derive_n f 1) x > 0 /\ (Derive_n f 2) x > 0 /\ (Derive_n f 3) x > 0)
 7 |     (hf : forall (x: R), (Derive_n f 3) x <= f x)
 8 |     : forall (x: R), (Derive_n f 1) x < 2 * f x.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_1999_b6.v:
--------------------------------------------------------------------------------
1 | Require Import Reals List Znumtheory.
2 | Theorem putnam_1999_b6
3 |     (A : list Z)
4 |     (Age1 : forall (x: Z), In x A -> x > 1)
5 |     (hgcd : forall (n: Z), exists (s: Z), In s A /\ (Zis_gcd s n 1 \/ Zis_gcd s n s))
6 |     : exists (s: Z) (t: Z) (p: Z), In s A /\ In t A /\ Zis_gcd s t p /\ prime p.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2000_a1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2000_a1_solution : R -> (R -> Prop) := (fun A : R => (fun x : R => 0 < x < A ^ 2)).
3 | Theorem putnam_2000_a1
4 |     (A: R)
5 |     (hA : A > 0)
6 |     : forall SS : R, ((exists x : nat -> R, (forall j : nat, x j > 0) /\ Series x = A /\ Series (fun j => (x j) ^ 2) = SS) <-> (putnam_2000_a1_solution A) SS).
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2000_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals.
2 | Open Scope Z.
3 | Theorem putnam_2000_a2
4 |     : forall (m: Z), exists (n: Z), n >= m /\ 
5 |     exists (a1 a2 b1 b2 c1 c2: Z), n = a1*a1 + a2*a2 /\ n+1 = b1*b1 + b2*b2 /\ n+2 = c1*c1 + c2*c2. 
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2000_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2000_a4
3 |     : ex_finite_lim_seq (fun B : nat => RInt (fun x : R => sin x * sin (x ^ 2)) 0 (INR B)).
4 | Proof. Admitted.
5 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2000_a6.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Local Open Scope ring_scope.
 8 | 
 9 | Theorem putnam_2000_a6
10 |     (f : {poly int})
11 |     (a : nat -> int)
12 |     (ha0 : a 0%nat = 0)
13 |     (ha : forall n : nat, a (n.+1) = f.[a n])
14 |     : (exists m : nat, gt m 0 /\ a m = 0) -> (a 1%nat = 0 \/ a 2%nat = 0).
15 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_2000_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Reals.
2 | Open Scope R.
3 | Theorem putnam_2000_b2 
4 |     : forall (n m: nat), and (ge n m) (ge m 1) -> exists (c: Z), (INR (gcd m n) / INR n) * Binomial.C n m = IZR c.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2000_b4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2000_b4
3 |     (f : R -> R)
4 |     (hf : forall x : R, f (2*x*x-1) = 2*x*(f x))
5 |     (f_cont : continuity f)
6 |     : forall x : R, -1 <= x <= 1 -> f x = 0. 
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2001_a1.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles RelationClasses.
2 | Theorem putnam_2001_a1
3 |     (A : Type)
4 |     (op : A->A->A)
5 |     (hop : forall (a b: A), op (op a b) a = b)
6 |     : forall (a b: A), op a (op b a) = b.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2001_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Nat.
2 | Theorem putnam_2001_a5 
3 |     : exists! (a n: nat), a > 0 /\ n > 0 /\ a ^ (n + 1) - (a + 1) ^ n = 2001.
4 | Proof. Admitted.
5 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2001_b2.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Definition putnam_2001_b2_solution : R -> R -> Prop := (fun x y : R => x = (3 ^ (1 / 5) + 1) / 2 /\ y = (3 ^ (1 / 5) - 1) / 2).
 3 | Theorem putnam_2001_b2
 4 |     (x y : R)
 5 |     (hx : x <> 0)
 6 |     (hy : y <> 0)
 7 |     : (1 / x + 1 / (2 * y) = (x ^ 2 + 3 * y ^ 2) * (3 * x ^ 2 + y ^ 2) /\
 8 |     1 / x - 1 / (2 * y) = 2 * (y ^ 4 - x ^ 4)) <-> putnam_2001_b2_solution x y.
 9 | Proof. Admitted.
10 | 
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2001_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2001_b3_solution : R := 3.
3 | Theorem putnam_2001_b3
4 |     (closest : nat -> R := (fun n : nat => IZR (floor (sqrt (INR n) + 0.5))))
5 |     : Series (fun n : nat => sum_n_m (fun n' : nat => (Rpower 2 (closest n') + Rpower 2 (-closest n')) / (2 ^ n')) 1 n) = putnam_2001_b3_solution.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2001_b5.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Theorem putnam_2001_b5
 3 |     (a b: R) 
 4 |     (g: R -> R)
 5 |     (hab : 0 < a < 1/2 /\ 0 < b < 1/2)
 6 |     (gcont : continuity g)
 7 |     (hg : forall (x: R), g (g x) = a * g x + b * x)
 8 |     : exists c : R, forall x : R, g x = c * x.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2002_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope R.
3 | Theorem putnam_2002_b3 
4 |     : forall (n: nat), gt n 1 -> let n := INR n in 1 / (2 * n * exp 1) < 1 / (exp 1) - Rpower (1 - 1 / n) n < 1 / (n * (exp 1)).
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2003_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2003_a3_solution := 2 * sqrt 2 - 1.
3 | Theorem putnam_2003_a3
4 |     (f : R -> R := fun x => Rabs (sin x + cos x + tan x + 1 / tan x + 1 / cos x  + 1 / sin x))
5 |     : (exists x : R, f x = putnam_2003_a3_solution) /\ (forall x : R, f x >= putnam_2003_a3_solution).
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2003_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2003_a4
3 |     (a b c A B C: R)
4 |     (ha : a <> 0)
5 |     (hA : A <> 0)
6 |     (hp : forall (x: R), Rabs (a * x ^ 2 + b * x + c) <= Rabs (A * x ^ 2 + B * x + C))
7 |     : Rabs (b ^ 2 - 4 * a * c) <= Rabs (B ^ 2 - 4 * A * C).
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2003_b3.v:
--------------------------------------------------------------------------------
 1 | From mathcomp Require Import all_algebra all_ssreflect.
 2 | 
 3 | Set Implicit Arguments.
 4 | Unset Strict Implicit.
 5 | Unset Printing Implicit Defensive.
 6 | 
 7 | Local Open Scope nat_scope.
 8 | 
 9 | Theorem putnam_2003_b3
10 |     (n : nat)
11 |     : n `! = \prod_(1 <= i < n.+1) (foldl (fun x y => lcmn x y) 1%nat (iota 1 (n%/i))).
12 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_2003_b6.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2003_b6
3 |     (f : R -> R)
4 |     (hf : continuity f)
5 |     : RInt (fun x => RInt (fun y => Rabs (f x + f y)) 0 1) 0 1 >= RInt (fun x => Rabs (f x)) 0 1.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2004_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2004_a3
3 |     (u : nat -> R)
4 |     (hubase : u O = 1 /\ u (S O) = 1 /\ u (S (S O)) = 1)
5 |     (hudet : forall n : nat, u n * u (Nat.add n 3) - u (Nat.add n 1) * u (Nat.add n 2) = INR (fact n))
6 |     : forall n : nat, exists m : Z, u n = IZR m.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2004_a6.v:
--------------------------------------------------------------------------------
1 | Require Import Basics Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2004_a6
3 |     (f: R -> R -> R) 
4 |     (hf : forall x y : R, (0 <= x <= 1 /\ 0 <= y <= 1) -> continuity_2d_pt f x y)
5 |     : RInt (fun y => (RInt (fun x => f x y) 0 1)^2) 0 1 + RInt (fun x => (RInt (fun y => f x y) 0 1)^2) 0 1 <= 
6 |     (RInt (fun x => RInt (fun y => f x y) 0 1) 0 1) ^ 2 + RInt (fun x => RInt (fun y => (f x y) ^ 2) 0 1) 0 1.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2004_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Reals QArith Coquelicot.Coquelicot.
2 | Theorem putnam_2004_b1
3 |     (c : nat -> Z)
4 |     (n : nat)
5 |     (r : Q)
6 |     (Preq0 : sum_n (fun i => IZR (c i) * (Q2R r) ^ i) n = 0)
7 |     : forall i : nat, lt i n -> exists m : Z, IZR m = sum_n (fun j => IZR (c (sub n j)) * (Q2R r) ^ (i + 1 - j)) i. 
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2004_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2004_b2
3 |     (m n : nat)
4 |     (mnpos : gt m 0 /\ gt n 0)
5 |     : INR (fact (m + n)) / INR (m + n) ^ (m + n) < INR (fact m) / INR m ^ m * INR (fact n) / INR n ^ n.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2005_a5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2005_a5_solution := PI * ln 2 / 8.
3 | Theorem putnam_2005_a5
4 |     : RInt (fun x => ln (x + 1) / (x ^ 2 + 1)) 0 1 = putnam_2005_a5_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2005_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2005_b3_solution (f : R -> R) := exists c d : R, c > 0 /\ d > 0 /\ (d = 1 -> c = 1) /\ forall x : R, x > 0 -> f x = c * Rpower x d.
3 | Theorem putnam_2005_b3
4 |     (f : R -> R)
5 |     (hf : forall x : R, x > 0 -> f x > 0)
6 |     (hf' : forall x : R, x > 0 -> ex_derive f x)
7 |     : (exists a : R, a > 0 /\ forall x : R, x > 0 -> Derive f (a / x) = x / f x) <-> putnam_2005_b3_solution f.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2007_b1.v:
--------------------------------------------------------------------------------
 1 | Require Import Nat Reals Coquelicot.Coquelicot. From mathcomp Require Import div.
 2 | Theorem putnam_2007_b1
 3 |     (c: nat -> nat)
 4 |     (n: nat)
 5 |     (hn : gt n 0)
 6 |     (k: nat)
 7 |     (f : nat -> R := fun x => sum_n (fun i => INR (mul (c i) (x ^ i))) (k + 1))
 8 |     : forall (x: nat), gt (c x) 0 ->
 9 |     Z.to_nat (floor (f n)) %| Z.to_nat (floor (f (Z.to_nat (floor (f n + 1))))) = true <-> n = 1%nat.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2007_b3.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Definition putnam_2007_b3_solution := let a := (1 + sqrt 5) / 2 in (2 ^ 2006 / sqrt 5) * (a ^ 3997 - Rpower a (-3997)).
 3 | Theorem putnam_2007_b3
 4 |     (X := fix x (n: nat) :=
 5 |         match n with
 6 |         | O => 1
 7 |         | S n' => 3 * x n' + IZR (floor (x n' * sqrt 5))
 8 |     end)
 9 |     : X 2007%nat = putnam_2007_b3_solution.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2008_a1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2008_a1
3 |     (f : R -> R -> R)
4 |     (hf : forall (x y z: R), f x y + f y z + f z x = 0)
5 |     : exists (g: R -> R), forall (x y: R), f x y = g x - g y.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2008_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2008_a4_solution := False.
3 | Theorem putnam_2008_a4
4 |     (f : R -> R)
5 |     (hf : forall (x : R), f x = (if (Rle_dec x (exp 1)) then x else x * f (ln x)))
6 |     : ex_lim_seq (fun nInc => sum_n_m (fun n => 1 / f (INR n)) 1 nInc) <-> putnam_2008_a4_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2008_b2.v:
--------------------------------------------------------------------------------
 1 | Require Import Factorial Reals Coquelicot.Coquelicot.
 2 | Definition putnam_2008_b2_solution := -1.
 3 | Theorem putnam_2008_b2
 4 |     (f := fix F (n: nat) (x: R) :=
 5 |         match n with
 6 |         | O => ln x
 7 |         | S n' => RInt (fun t => F n' t) 0 x
 8 |     end)
 9 |     : Lim_seq (fun n => INR (fact n) * f n 1 / ln (INR n)) = putnam_2008_b2_solution. 
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2009_a1.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals GeoCoq.Main.Tarski_dev.Ch16_coordinates_with_functions.
 2 | Context `{T2D:Tarski_2D} `{TE:@Tarski_euclidean Tn TnEQD}.
 3 | Open Scope R.
 4 | Definition putnam_2009_a1_solution := True.
 5 | Theorem putnam_2009_a1
 6 |     : (forall f: Tpoint -> R, 
 7 |     (forall (A B C D: Tpoint), Square A B C D -> 
 8 |     f A + f B + f C + f D = 0) ->
 9 |     forall (P : Tpoint), f P = 0) <->
10 |     putnam_2009_a1_solution.
11 | Proof. Admitted.
12 | 


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/coq/src/putnam_2009_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Ensembles QArith.
2 | Definition putnam_2009_a4_solution := False.
3 | Theorem putnam_2009_a4
4 |     : (forall (E: Ensemble Q), (E 0 /\ forall (q: Q), (E q -> E (q + 1) /\ E (q - 1)) /\ (E q /\ q <> 0 /\ q <> 1 -> E (1 / (q * (q - 1))))) -> forall (q: Q), E q) <-> putnam_2009_a4_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2009_b1.v:
--------------------------------------------------------------------------------
1 | From mathcomp Require Import ssrbool seq ssrnat prime rat ssralg ssrnum ssrint.
2 | 
3 | Local Open Scope ring_scope.
4 | 
5 | Theorem putnam_2009_b1 :
6 |   let fact_prod (ls : seq nat) : rat := \prod_(i <- ls) (i`!)%:Q in
7 |   forall q : rat, q > 0 -> exists n d : seq nat,
8 |   all prime (n ++ d) /\ fact_prod n / fact_prod d = q.
9 | Proof. Admitted.


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/coq/src/putnam_2009_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2009_b5
3 |     (f: R -> R) 
4 |     (hf : forall (x: R), 1 < x -> (ex_derive f x /\ Derive f x = (x ^ 2 - (f x) ^ 2) / (x ^ 2 * ((f x) ^ 2 + 1))))
5 |     : is_lim f p_infty p_infty.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2010_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2010_a2_solution (f: R -> R) := exists (c d: R), f = (fun x => c * x + d).
3 | Theorem putnam_2010_a2
4 |     (f: R -> R)
5 |     : (forall (x: R) (n: nat), gt n 0 -> ex_derive f x /\ Derive f x = (f (x + (INR n)) - f x) / (INR n)) <-> putnam_2010_a2_solution f.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2010_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Znumtheory.
2 | Open Scope Z.
3 | Theorem putnam_2010_a4
4 |     : forall (n: Z), n > 0 -> ~ prime (10^(10^(10^n)) + 10^(10^n) + 10^n - 1). 
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2010_a6.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2010_a6
3 |     (f: R -> R)
4 |     (hf : (forall (x: R), x >= 0 -> (forall (y: R), x < y -> f x > f y) /\ continuity_pt f x) /\ is_lim f p_infty 0)
5 |     : ~ ex_finite_lim (fun nInc => RInt (fun x => (f x - f (x + 1)) / f x) 0 nInc) p_infty.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2010_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2010_b1_solution := False.
3 | Theorem putnam_2010_b1
4 |     : (exists (a: nat -> R), forall (m: nat), gt m 0 -> Series (fun i => (a i) ^ m) = (INR m)) <-> putnam_2010_b1_solution. 
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2010_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2010_b5_solution := False.
3 | Theorem putnam_2010_b5
4 |     : (exists (f: R -> R), forall (x y: R), (x < y -> f x < f y) /\ ex_derive f x /\ Derive f x = f (f x)) <-> putnam_2010_b5_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2011_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | (* Note: There may be multiple possible correct answers. *)
3 | Definition putnam_2011_a3_solution := (-1, 2 / PI).
4 | Theorem putnam_2011_a3
5 |     : let (c, L) := putnam_2011_a3_solution in L > 0 /\ is_lim_seq (fun r => (Rpower (INR r) c * RInt (fun x => x ^ r * sin x) 0 (PI / 2)) / (RInt (fun x => x ^ r * cos x) 0 (PI / 2))) L.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2011_b1.v:
--------------------------------------------------------------------------------
1 | Require Import Reals ZArith Coquelicot.Coquelicot.
2 | Theorem putnam_2011_b1
3 |     : forall (h k: Z), Z.gt h 0 /\ Z.gt k 0 -> forall (ep: R), ep > 0 -> exists (m n: Z), Z.gt m 0 /\ Z.gt n 0 /\ ep < Rabs (IZR h * sqrt (IZR m) - IZR k * sqrt (IZR n)) < 2 * ep.
4 | Proof. Admitted.
5 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2011_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2011_b3_solution := True.
3 | Theorem putnam_2011_b3
4 |     : (forall (f g : R -> R), g 0 <> 0 /\ continuity_pt g 0 /\ ex_derive (fun x : R => f x * g x) 0 /\ ex_derive (fun x : R => f x / g x) 0 -> ex_derive f 0) <-> putnam_2011_b3_solution.
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2011_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Theorem putnam_2011_b5
3 |     (a: nat -> R)
4 |     (ha : exists (A: R), forall (n: nat), exists (L : R), is_lim_seq (fun nInc => (RInt (fun x => (sum_n_m (fun i => 1 / (1 + (x - a i) ^ 2)) 1 n) ^ 2) (-1 * INR nInc) (INR nInc))) L /\ L <= A * INR n)
5 |     : exists (B: R), B > 0 /\ forall (n: nat), sum_n_m (fun i => (sum_n_m (fun j => 1 + (a i - a j) ^ 2)) 1 n) 1 n >= B * INR n ^ 3.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2011_b6.v:
--------------------------------------------------------------------------------
1 | Require Import Nat List Factorial Ensembles Finite_sets Reals Znumtheory ZArith Coquelicot.Coquelicot.
2 | Open Scope nat_scope.
3 | Theorem putnam_2011_b6 
4 |     (p: nat)
5 |     (hp : prime (Z.of_nat p) /\ odd p = true)
6 |     : exists (E: Ensemble nat), (forall (n: nat), E n -> lt n p) /\ cardinal nat E ((p + 1) / 2) /\
7 |     forall (n: nat), E n -> Z.to_nat (floor (sum_n (fun k => INR (fact k * n ^ k)) (p - 1))) mod p <> 0.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2012_b4.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Lim_seq.
 2 | Open Scope R.
 3 | Definition putnam_2012_b4_solution := True.
 4 | Theorem putnam_2012_b4
 5 |     (A := fix a (n: nat) : R :=
 6 |         match n with
 7 |         | O => 1
 8 |         | S n' => a n' + Rpower (exp 1) ((-1) * a n')
 9 |     end)
10 |     (B : nat -> R := fun n => A n - ln (INR n))
11 |     : ex_finite_lim_seq B <-> putnam_2012_b4_solution.
12 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_2013_a3.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Theorem putnam_2013_a3
 3 |     (a: nat -> R)
 4 |     (x: R)
 5 |     (n: nat)
 6 |     (hx : 0 < x < 1)
 7 |     (ha : sum_n (fun i => a i / (1 - x ^ (i + 1))) n = 0)
 8 |     : exists (y: R), 0 < y < 1 /\ sum_n (fun i => a i * y ^ i) n = 0.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2014_a1.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Factorial Znumtheory Coquelicot.Derive.
 2 | Theorem putnam_2014_a1
 3 |     (f : R -> R := fun x => (1 - x + x^2) * (exp x))
 4 |     (hf : forall x : R, forall n : nat, ex_derive_n f n x)
 5 |     (coeff : nat -> R := fun i => Derive_n f i 0 / INR (fact i))
 6 |     (n : nat)
 7 |     (hcoeff : coeff n <> 0)
 8 |     : exists a b: Z, b <> 0%Z /\ coeff n = IZR a / IZR b /\ Zis_gcd a b 1 /\ (a = 1%Z \/ prime a).
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2014_b4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Open Scope C.
3 | Theorem putnam_2014_b4
4 |     (n : nat)
5 |     (npos : ge n 0)
6 |     (P : C -> C := fun r => sum_n (fun k => (2 * ((INR k) * (INR n - INR k)) * r ^ k)) n)
7 |     : forall r : C, P r = 0 -> (exists (t : R), r = RtoC t). 
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2016_a3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.RInt Coquelicot.Hierarchy.
2 | Definition putnam_2016_a3_solution :=  3 * PI / 8.
3 | Theorem putnam_2016_a3
4 |     (f: R -> R)
5 |     (hf : forall x : R, x <> 0 -> f x + f (1 - 1/x) = atan x)
6 |     : RInt f 0 1 = putnam_2016_a3_solution.
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2016_b1.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Definition putnam_2016_b1_solution := 1.
 3 | Theorem putnam_2016_b1
 4 |     (x : nat -> R := fix x (n: nat) :=
 5 |         match n with
 6 |         | O => 1
 7 |         | S n' => ln (exp (x n') - x n')
 8 |     end)
 9 |     : Series x = putnam_2016_b1_solution.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2016_b5.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Rpower.
2 | Open Scope R.
3 | Definition putnam_2016_b5_solution : (R -> R) -> Prop := fun f => exists (c: R), c > 0 /\ forall (x: R), x > 1 -> f x = Rpower x c .
4 | Theorem putnam_2016_b5
5 |     (f : R -> R)
6 |     (fle : Prop := (forall x : R, x > 1 -> f x > 1) /\ (forall x y : R, (x > 1 /\ y > 1 /\ x*x <= y <= x*x*x) -> ((f x) * (f x) <= f y <= (f x) * (f x) * (f x))))
7 |     : fle <-> putnam_2016_b5_solution f.
8 | Proof. Admitted.


--------------------------------------------------------------------------------
/coq/src/putnam_2016_b6.v:
--------------------------------------------------------------------------------
1 | Require Import List Reals Coquelicot.Hierarchy Coquelicot.Series. 
2 | Definition putnam_2016_b6_solution := 1.
3 | Theorem putnam_2016_b6: 
4 |     Series (fun k => (-1)^k/(INR k+1) * Series (fun n => 1/(INR k*(2^n)+1))) = putnam_2016_b6_solution.
5 | Proof. Admitted. 
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2017_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Series.
2 | Open Scope R_scope.
3 | Theorem putnam_2017_b3
4 |     (c : nat -> R)
5 |     (hc : forall n, c n = 0 \/ c n = 1)
6 |     : Series (fun n => c n * (2/3)^n) = 3/2 -> (~ exists a b: Z, b <> 0%Z /\ (Series (fun n => c n * (1/2)^n)) = IZR a / IZR b).
7 | Proof. Admitted.
8 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2017_b4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2017_b4_solution := (ln 2 / ln 10) ^ 2.
3 | Theorem putnam_2017_b4
4 |     : Lim_seq (fun n => sum_n (fun k => let k := INR k in (3 * ln (4 * k + 2) / (4 * k + 2) - ln (4 * k + 3) / (4 * k + 3) - ln (4 * k + 4) / (4 * k + 4) - ln (4 * k + 5) / (4 * k + 5))) n) = putnam_2017_b4_solution .
5 | Proof. Admitted.
6 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2018_a5.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Derive.
 2 | Open Scope R_scope.
 3 | Theorem putnam_2018_a5
 4 |     (f: R -> R)
 5 |     (h0 : f 0 = 0)
 6 |     (h1 : f 1 = 1)
 7 |     (hpos : forall x : R, f x >= 0)
 8 |     (hf : forall x : R, forall n : nat, ex_derive_n f n x)
 9 |     : exists (n : nat) (x : R), gt n 0 /\ Derive_n f n x < 0.
10 | Proof. Admitted.
11 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2018_b2.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot. From Coqtail Require Import Cpow.
 2 | Open Scope C_scope.
 3 | Theorem putnam_2018_b2
 4 |     (n : nat)
 5 |     (hn : gt n 0)
 6 |     (f : nat -> C -> C)
 7 |     (hf : forall z : C, f n z = sum_n_m (fun i => (((RtoC (INR n)) - (RtoC (INR i))) * z ^ i)) 0 (n-1))
 8 |     : forall (z : C), Cnorm z <= 1 -> f n z <> 0.
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2018_b3.v:
--------------------------------------------------------------------------------
1 | Require Import Nat Ensembles. From mathcomp Require Import div seq ssrnat ssrbool.
2 | Definition putnam_2018_b3_solution := fun n => n = 2^2 \/ n = 2^4 \/ n = 2^16 \/ n = 2^(256).
3 | Theorem putnam_2018_b3
4 |     (E : Ensemble nat := fun n => n > 0 /\ (n < 10^(100)) /\ (n %| 2^n) /\ ((n-1) %| (2^n-1)) /\ ((n-2) %| (2^n-2)))
5 |     : E = putnam_2018_b3_solution.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2018_b6.v:
--------------------------------------------------------------------------------
 1 | Require Import Nat List Ensembles Finite_sets Reals.
 2 | Theorem putnam_2018_b6
 3 |     (E: Ensemble (list nat) := fun l => length l = 2018 /\ 
 4 |         forall (n: nat), (List.In n l) -> (n = 1 \/ n = 2 \/ n = 3 \/ n = 4 \/ n = 5 \/ n = 6 \/ n = 10) /\
 5 |         fold_left Nat.add l 0 = 3860
 6 |     )
 7 |     (n : nat)
 8 |     : cardinal (list nat) E n -> Rle (INR n) (Rmult (Rpower 2 3860) (Rpower (Rdiv (INR 2018) (INR 2048)) 2018)).
 9 | Proof. Admitted.
10 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2019_a1.v:
--------------------------------------------------------------------------------
1 | Require Import ZArith Ensembles Coquelicot.Coquelicot.
2 | Open Scope Z.
3 | Definition putnam_2019_a1_solution : Ensemble Z := fun n => Z.ge n 0 /\ n mod 9 <> 3 /\ n mod 9 <> 6.
4 | Theorem putnam_2019_a1 
5 |     (A : Ensemble Z)
6 |     (hA : forall (n: Z), A n <-> exists (A B C: Z), A >= 0 /\ B >= 0 /\ C >= 0 /\ A ^ 3 + B ^ 3 + C ^ 3 - 3 * A * B * C = n)
7 |     : A = putnam_2019_a1_solution.
8 | Proof. Admitted.
9 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2019_b2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2019_b2_solution := 8 / PI ^ 3.
3 | Theorem putnam_2019_b2 
4 |     (a : nat -> R := fun n => sum_n_m (fun k => let k := INR k in let n := INR n in (sin (2 * (k + 1) * PI / (2 * n))) / ((cos ((k - 1) * PI / (2 * n))) ^ 2 * (cos ((k * PI) / (2 * n))) ^ 2)) 1 (n - 1))
5 |     : Lim_seq (fun n => a n / INR n ^ 3) = putnam_2019_b2_solution.
6 | Proof. Admitted.
7 | 


--------------------------------------------------------------------------------
/coq/src/putnam_2020_a2.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2020_a2_solution := fun k => 4 ^ k.
3 | Theorem putnam_2020_a2  
4 |     : (fun k => sum_n (fun j => 2 ^ (k - j) * Binomial.C (k + j) j) k) = putnam_2020_a2_solution.
5 | Proof. Admitted.
6 | 


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/coq/src/putnam_2020_a3.v:
--------------------------------------------------------------------------------
 1 | Require Import Reals Coquelicot.Coquelicot.
 2 | Definition putnam_2020_a3_solution := False.
 3 | Theorem putnam_2020_a3 
 4 |     (a : nat -> R := fix a (n: nat) := 
 5 |         match n with
 6 |         | O => PI / 2
 7 |         | S n' => sin (a n')
 8 |     end)
 9 |     : ex_lim_seq (fun n => (a n) ^ 2) <-> putnam_2020_a3_solution.
10 | Proof. Admitted.
11 | 


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/coq/src/putnam_2020_b5.v:
--------------------------------------------------------------------------------
1 | From mathcomp Require Import fintype seq ssrbool. Require Import Reals Coquelicot.Complex.
2 | Open Scope C.
3 | Theorem putnam_2020_b5
4 |     (z1 z2 z3 z4 : C)
5 |     (hznorm : Cmod z1 = 1%R /\ Cmod z2 = 1%R /\ Cmod z3 = 1%R /\ Cmod z4 = 1%R)
6 |     (hzne : z1 <> RtoC 1 /\ z2 <> RtoC 1 /\ z3 <> RtoC 1 /\ z4 <> RtoC 1)
7 |     : RtoC 3 - z1 - z2 - z3 - z4 + z1 * z2 * z3 * z4 <> RtoC 0.
8 | Proof. Admitted.
9 | 


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/coq/src/putnam_2020_b6.v:
--------------------------------------------------------------------------------
1 | Require Import Reals. From Coquelicot Require Import Coquelicot Hierarchy Rcomplements. 
2 | Open Scope R.
3 | Theorem putnam_2020_b6 
4 |     (A : nat -> R := fun k => (-1)^(Z.to_nat (floor (INR k * (sqrt 2 - 1)))))
5 |     (B : nat -> R := fun n => sum_n_m A 1 n)
6 |     : forall (n: nat), B n >= 0.
7 | Proof. Admitted. 


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/coq/src/putnam_2021_a4.v:
--------------------------------------------------------------------------------
1 | Require Import Reals Coquelicot.Coquelicot.
2 | Definition putnam_2021_a4_solution := (sqrt 2 / 2) * PI * ln 2 / ln 10.
3 | Theorem putnam_2021_a4 
4 |     (I : nat -> R := fun r => RInt (fun x => RInt (fun y => (1 + 2 * x ^ 2) / (1 + x ^ 4 + 6 * x ^ 2 * y ^ 2 + y ^ 4) - (1 + y ^ 2) / (2  + x ^ 4 + y ^ 4)) 0 (sqrt (INR r ^ 2 - x ^ 2))) 0 r)
5 |     : Lim_seq I = putnam_2021_a4_solution.
6 | Proof. Admitted.
7 | 


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/coq/src/putnam_2023_b5.v:
--------------------------------------------------------------------------------
1 | Require Import PeanoNat. From mathcomp Require Import div fintype perm ssrbool.
2 | Definition putnam_2023_b5_solution (n : nat) := n = 1 \/ n mod 4 = 2.
3 | Theorem putnam_2023_b5
4 |     (p : nat -> Prop := fun n => forall m : nat, coprime m n -> exists (π: {perm 'I_n}), forall (k: 'I_n), nat_of_ord (π (π k)) = (m*(nat_of_ord k))%%n)
5 |     : forall n : nat, (n > 0 /\ p n) <-> putnam_2023_b5_solution n.
6 | Proof. Admitted. 


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/docs/.gitignore:
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1 | .DS_store
2 | .idea
3 | 


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/isabelle/README.md:
--------------------------------------------------------------------------------
1 | The Isabelle formalizations largely rely on the HOL Library, and are present in a problem-per-file format by default.
2 | 
3 | We also note that while the dependencies listed in each formalization are sufficient to *state* the problem, one may need further mathematical theory developed in Isabelle to write a proof. We do not believe this will be the case for many problems, but please do let us know if you have particular feedback on this or with any other concerns.
4 | 


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/isabelle/putnam_1962_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1962_a1 imports Complex_Main
 2 | "HOL-Analysis.Finite_Cartesian_Product"
 3 | "HOL-Analysis.Convex"
 4 | begin
 5 | 
 6 | theorem putnam_1962_a1:
 7 |   fixes S :: "(real^2) set"
 8 |   assumes hS : "card S = 5"
 9 |     and hnoncol : "\<forall> s \<subseteq> S. card s = 3 \<longrightarrow> \<not>collinear s"
10 |   shows "\<exists> T \<subseteq> S. card T = 4 \<and> \<not>(\<exists> t \<in> T. t \<in> convex hull (T - {t}))"
11 |   sorry
12 | 
13 | end


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/isabelle/putnam_1962_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1962_a5 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1962_a5_solution :: "nat \<Rightarrow> nat" where "putnam_1962_a5_solution \<equiv> undefined"
 5 | (* \<lambda> n :: nat. n * (n + 1) * 2^(n - 2) *)
 6 | theorem putnam_1962_a5:
 7 |   shows "\<forall> n \<ge> 2. putnam_1962_a5_solution n = (\<Sum> k \<in> {1..n}. (n choose k) * k^2)"
 8 |   sorry
 9 | 
10 | end


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/isabelle/putnam_1962_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1962_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1962_b1:
 5 |   fixes p :: "nat \<Rightarrow> real \<Rightarrow> real"
 6 |     and x y :: real
 7 |     and n :: nat
 8 |   assumes h0 : "p 0 = (\<lambda> x :: real. 1)"
 9 |     and hp : "\<forall> n > 0. p n = (\<lambda> x :: real. \<Prod> i \<in> {0..n-1}. (x - i))"
10 |   shows "p n (x + y) = (\<Sum> k \<in> {0..n}. (n choose k) * (p k x) * (p (n - k) y))"
11 |   sorry
12 | 
13 | end


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/isabelle/putnam_1962_b2.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1962_b2 imports Complex_Main
2 | begin
3 | 
4 | (* Note: Uses nat set instead of set of positive integers, the problem is the same *)
5 | theorem putnam_1962_b2:
6 |   shows "\<exists> f :: real \<Rightarrow> nat set. \<forall> a b :: real. (a < b \<longrightarrow> (f a \<subset> f b))" 
7 |   sorry
8 | 
9 | end


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/isabelle/putnam_1962_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1962_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1962_b5:
 5 |   fixes n :: "int"
 6 |   assumes ng1 : "n > 1"
 7 |   shows "(3 * (real_of_int n) + 1) / (2 * n + 2) < (\<Sum> i \<in> {1..n}. (((real_of_int i) / n) powr (real_of_int n))) \<and> 
 8 |   (\<Sum> i \<in> {1..n}. (((real_of_int i) / n) powr (real_of_int n))) < 2"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1963_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1963_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1963_a2:
 5 |   fixes f::"nat\<Rightarrow>nat"
 6 |   assumes hfpos : "\<forall>n. f n > 0"
 7 |     and hfinc : "strict_mono_on {1..} f"
 8 |     and hf2 : "f 2 = 2"
 9 |     and hfmn : "\<forall>m n. m > 0 \<longrightarrow> n > 0 \<longrightarrow> coprime m n \<longrightarrow> f (m * n) = f m * f n"
10 |   shows "\<forall>n > 0. f n = n"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1963_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1963_b1 imports Complex_Main "HOL-Computational_Algebra.Polynomial"
 2 | begin
 3 | 
 4 | definition putnam_1963_b1_solution::int where "putnam_1963_b1_solution \<equiv> undefined"
 5 | (* 2 *)
 6 | theorem putnam_1963_b1:
 7 |   shows "\<forall>a::int. ([: (real_of_int a), -1, 1 :] dvd ([: 90, 1 :] + monom 1 13)) \<longleftrightarrow> a = putnam_1963_b1_solution"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1963_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1963_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1963_b5:
 5 |   fixes a::"int\<Rightarrow>real"
 6 |   assumes haineq : "\<forall>n \<ge> 1. \<forall>k::int. (n \<le> k \<and> k \<le> 2 * n) \<longrightarrow> (0 \<le> a k \<and> a k \<le> 100 * a n)"
 7 |     and haseries : "summable a"
 8 |   shows "(\<lambda>n. n * a n) \<longlonglongrightarrow> 0"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1964_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1964_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1964_b3:
 5 |   fixes f :: "real \<Rightarrow> real"
 6 |   assumes hf: "continuous_on UNIV f \<and> (\<forall>\<alpha>::real>0. filterlim (\<lambda>n::nat. f (n*\<alpha>)) (nhds 0) at_top)"
 7 |   shows "filterlim f (nhds 0) at_top"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1964_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1964_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1964_b5:
 5 |   fixes a :: "nat \<Rightarrow> nat"
 6 |   and b :: "nat \<Rightarrow> nat"
 7 |   assumes ha: "strict_mono a \<and> (\<forall>n::nat. a n > 0)"
 8 |   and hb: "b 0 = a 0 \<and> (\<forall>n::nat. b (n+1) = lcm (b n) (a (n+1)))"
 9 |   shows "summable (\<lambda>n::nat. 1 / b n)"
10 |   sorry
11 | 
12 | end
13 | 


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/isabelle/putnam_1965_a2.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1965_a2 imports
2 | Complex_Main
3 | begin
4 | 
5 | theorem putnam_1965_a2:
6 |   shows "\<forall> n > 0. (\<Sum> r = 0 .. (n - 1) div 2. ((n - 2 * r) * (n choose r) / n) ^ 2) = ((2 * n - 2) choose (n - 1)) / n"
7 |   sorry
8 | 
9 | end


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/isabelle/putnam_1965_b3.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1965_b3 imports
2 | Complex_Main
3 | begin
4 | 
5 | theorem putnam_1965_b3:
6 |   shows "card {(a :: int, b :: int, c :: int). a > 0 \<and> a \<le> b \<and> c > 0 \<and> a ^ 2 + b ^ 2 = c ^ 2 \<and> a * b / 2 = 2 * (a + b + c)} = 3"
7 |   sorry
8 | 
9 | end


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/isabelle/putnam_1966_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_a1 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1966_a1:
 6 |   fixes f :: "int \<Rightarrow> int"
 7 |   defines "f \<equiv> \<lambda> n :: int. \<Sum> m = 0..n. (if even m then m div 2 else (m - 1) div 2)"
 8 |   shows "\<forall> x y :: int. (x > 0 \<and> y > 0 \<and> x > y) \<longrightarrow> x * y = f (x + y) - f (x - y)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1966_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_a3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1966_a3:
 6 |   fixes x :: "nat \<Rightarrow> real"
 7 |   assumes hx1: "0 < x 1 \<and> x 1 < 1"
 8 |   and hxi: "\<forall> n \<ge> 1. x (n + 1) = x n * (1 - x n)"
 9 |   shows "(\<lambda> n. n * x n) \<longlonglongrightarrow> 1"
10 |   sorry
11 | 
12 | end


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/isabelle/putnam_1966_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_a4 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1966_a4:
 6 |   fixes a :: "nat \<Rightarrow> int"
 7 |   assumes ha1: "a 1 = 2"
 8 |   and hai: "\<forall> n \<ge> 1. a (n + 1) = (if (\<exists> m :: int. m ^ 2 = a n + 1) then a n + 2 else a n + 1)"
 9 |   shows "\<forall> n \<ge> 1. a n = n + round (sqrt n)"
10 |   sorry
11 | 
12 | end


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/isabelle/putnam_1966_a6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_a6 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1966_a6:
 6 |   fixes a :: "nat \<Rightarrow> nat \<Rightarrow> real"
 7 |   assumes ha: "\<forall> n \<ge> 1. a n n = n \<and> (\<forall> m \<ge> 1. m < n \<longrightarrow> a n m = m * sqrt (1 + a n (m + 1)))"
 8 |   shows "(\<lambda> n. a n 1) \<longlonglongrightarrow> 3"
 9 |   sorry
10 | 
11 | end


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/isabelle/putnam_1966_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_b2 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1966_b2:
 6 |   fixes S :: "int \<Rightarrow> int set"
 7 |   defines "S \<equiv> \<lambda> n :: int. {n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9}"
 8 |   shows "\<forall> n :: int. n > 0 \<longrightarrow> (\<exists> k \<in> S n. \<forall> m \<in> S n. k \<noteq> m \<longrightarrow> coprime m k)"
 9 |   sorry
10 | 
11 | end


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/isabelle/putnam_1966_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_b3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1966_b3:
 6 |   fixes p :: "nat \<Rightarrow> real"
 7 |   assumes hpos: "\<forall> n :: nat. p n > 0"
 8 |   and hconv: "convergent (\<lambda> m. \<Sum> n = 1..m. 1 / p n)"
 9 |   shows "convergent (\<lambda> m. \<Sum> n = 1..m. (p n) * n ^ 2 / (\<Sum> i = 1..n. p i) ^ 2)"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1966_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1966_b6 imports
 2 | Complex_Main
 3 | "HOL-Analysis.Derivative"
 4 | begin
 5 | 
 6 | theorem putnam_1966_b6:
 7 |   fixes y :: "real \<Rightarrow> real"
 8 |   assumes hy: "y differentiable_on UNIV \<and> (deriv y) differentiable_on UNIV"
 9 |   and diffeq: "\<forall> x :: real. (deriv (deriv y)) x + (exp x) * (y x) = 0"
10 |   shows "\<exists> r s N :: real. \<forall> x > N. r \<le> y x \<and> y x \<le> s"
11 |   sorry
12 | 
13 | end


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/isabelle/putnam_1967_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1967_a4 imports Complex_Main
 2 | "HOL-Analysis.Interval_Integral"
 3 | begin
 4 | 
 5 | theorem putnam_1967_a4:
 6 |   fixes lambda :: real
 7 |   assumes hlambda: "lambda > 1/2"
 8 |   shows "\<not>(\<exists>u::real\<Rightarrow>real. \<forall>x::real\<in>{0..1}. u x = 1 + lambda * (interval_lebesgue_integral lebesgue x 1 (\<lambda>y::real. u y * u (y - x))))"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1967_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1967_a5 imports Complex_Main
 2 | "HOL-Analysis.Lebesgue_Measure"
 3 | begin
 4 | 
 5 | theorem putnam_1967_a5:
 6 |   fixes R :: "(real^2) set"
 7 |   assumes hR : "convex R \<and> emeasure lebesgue R > pi / 4"
 8 |   shows "\<exists> P \<in> R. \<exists> Q \<in> R. dist P Q = 1"
 9 |   sorry
10 | 
11 | end


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/isabelle/putnam_1967_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1967_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1967_b5:
 5 |   fixes n :: "nat"
 6 |   assumes hn : "n > 0"
 7 |   shows "1/(2 :: real) = (\<Sum> i \<in> {0::nat..<n}. ((n + i - 1) choose i) * ((2 :: real) powr (-(int n) - i)))"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1968_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1968_a1 imports
 2 | Complex_Main
 3 | "HOL-Analysis.Interval_Integral"
 4 | begin
 5 | 
 6 | theorem putnam_1968_a1:
 7 |   shows "22 / 7 - pi = interval_lebesgue_integral lebesgue 0 1 (\<lambda> x. x ^ 4 * (1 - x) ^ 4 / (1 + x ^ 2))"
 8 |   sorry
 9 | 
10 | end


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/isabelle/putnam_1968_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1968_a3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1968_a3:
 6 |   shows "\<exists> l :: ('a::finite) set list. l!0 = {} \<and> (\<forall> S :: 'a set. \<exists>! i \<in> {0 ..< length l}. l!i = S) \<and>
 7 |           (\<forall> i \<in> {0 ..< length l - 1}. \<exists> a :: 'a. (a \<notin> l!i \<and> l!(i+1) = insert a (l!i)) \<or> (a \<notin> l!(i+1) \<and> l!i = insert a (l!(i+1))))"
 8 |   sorry
 9 | 
10 | end


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/isabelle/putnam_1969_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1969_a4 imports Complex_Main
 2 | "HOL-Analysis.Interval_Integral"
 3 | begin
 4 | 
 5 | theorem putnam_1969_a4:
 6 |   shows "filterlim (\<lambda>n::int. (\<Sum>i::int=1..n. (-1) powi (i+1) * i powi (-i))) (nhds (interval_lebesgue_integral lebesgue 0 1 (\<lambda>x::real. x powr x))) at_top"
 7 |   sorry
 8 | 
 9 | end
10 | 


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/isabelle/putnam_1969_a6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1969_a6 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1969_a6:
 5 |   fixes x :: "nat \<Rightarrow> real"
 6 |   and y :: "nat \<Rightarrow> real"
 7 |   assumes hy1: "\<forall>n::nat\<ge>2. y n = x (n-1) + 2*(x n)"
 8 |   and hy2: "\<exists>c::real. filterlim y (nhds c) at_top"
 9 |   shows "\<exists>C::real. filterlim x (nhds C) at_top"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1969_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1969_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1969_b1:
 5 |   fixes n :: nat
 6 |   assumes hnpos: "n > 0"
 7 |   and hn: "24 dvd (n + 1)"
 8 |   shows "24 dvd (\<Sum> {d::nat. d dvd n})"
 9 |   sorry
10 | 
11 | end
12 | 


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/isabelle/putnam_1969_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1969_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1969_b3:
 5 |   fixes T :: "nat \<Rightarrow> real"
 6 |   assumes hT1: "\<forall>n::nat. (n \<ge> 1 \<longrightarrow> (T n) * (T (n+1)) = n)"
 7 |   and hT2: "filterlim (\<lambda>n::nat. (T n) / (T (n+1))) (nhds 1) at_top"
 8 |   shows "pi*(T 1)^2 = 2"
 9 |   sorry
10 | 
11 | end
12 | 


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/isabelle/putnam_1970_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1970_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1970_a2:
 5 |   fixes A B C D E F G :: "real"
 6 |   assumes hle : "B^2 - 4*A*C < 0"
 7 |   shows "\<exists> \<delta> > 0. \<not> (\<exists> x y :: real. x^2 + y^2 \<noteq> 0 \<and> x^2 + y^2 < \<delta>^2 \<and>
 8 |     A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y+ F*x*y^2 + G*y^3 = 0)"
 9 |   sorry
10 | 
11 | end


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/isabelle/putnam_1970_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1970_a4 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1970_a4:
 5 |   fixes x :: "nat \<Rightarrow> real"
 6 |   assumes hxlim: "filterlim (\<lambda>n::nat. x n - x (n-2)) (nhds 0) at_top"
 7 |   shows "filterlim (\<lambda>n::nat. (x n - x (n-1)) / n) (nhds 0) at_top"
 8 |   sorry
 9 | 
10 | end
11 | 


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/isabelle/putnam_1970_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1970_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1970_b1_solution :: real where "putnam_1970_b1_solution \<equiv> undefined"
 5 | (* exp (2*ln 5 - 4 + 2*arctan 2) *)
 6 | theorem putnam_1970_b1:
 7 |   shows "filterlim (\<lambda>n::nat. 1/(n^4) * (\<Prod>i::nat=1..(2*n). (n^2 + i^2) powr (1/n))) (nhds putnam_1970_b1_solution) at_top"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1970_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1970_b2 imports Complex_Main
 2 | "HOL-Analysis.Derivative"
 3 | "HOL-Computational_Algebra.Polynomial"
 4 | begin
 5 | 
 6 | theorem putnam_1970_b2:
 7 |   fixes T :: "real"
 8 |     and H :: "real poly"
 9 |   assumes hT : "T > 0"
10 |     and hH : "degree H \<le> 3"
11 |   shows "(poly H (-T/sqrt 3) + poly H (T/sqrt 3))/2 = 1/(2*T) * (interval_lebesgue_integral lebesgue (-T) T (poly H))"
12 |   sorry
13 | 
14 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1970_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1970_b3 imports Complex_Main
 2 | "HOL-Analysis.Finite_Cartesian_Product"
 3 | begin
 4 | 
 5 | theorem putnam_1970_b3:
 6 |   fixes S :: "(real^2) set"
 7 |   and a b :: real
 8 |   assumes hab: "a < b"
 9 |   and hS: "\<forall>s\<in>S. s$1 \<in> {a<..<b}"
10 |   and hSclosed: "closed S"
11 |   shows "closed {y::real. (\<exists>s\<in>S. y = s$2)}"
12 |   sorry
13 | 
14 | end
15 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1971_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1971_a2 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | definition putnam_1971_a2_solution :: "(real poly) set" where "putnam_1971_a2_solution \<equiv> undefined"
 6 | (* {monom 1 1} *)
 7 | theorem putnam_1971_a2:
 8 |   shows "\<forall>P::real poly. ((poly P 0 = 0 \<and> (\<forall>x::real. poly P (x^2 + 1) = (poly P x)^2 + 1)) \<longleftrightarrow> P \<in> putnam_1971_a2_solution)"
 9 |   sorry
10 | 
11 | end
12 | 


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/isabelle/putnam_1971_a6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1971_a6 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1971_a6:
 5 |   fixes c :: real
 6 |   assumes hc: "\<forall>n::nat. (n > 0 \<longrightarrow> (\<exists>m::int. n powr c = m))"
 7 |   shows "\<exists>m::int. c = m"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1971_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1971_b3 imports Complex_Main
 2 | "HOL-Analysis.Lebesgue_Measure"
 3 | begin
 4 | 
 5 | theorem putnam_1971_b3:
 6 |   fixes T :: "real"
 7 |   assumes hT : "T > 0"
 8 |   shows "emeasure lebesgue {t :: real. t \<ge> T \<and> \<lfloor>t\<rfloor> = 2 * \<lfloor>t - T\<rfloor>} = 1"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1971_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1971_b6 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1971_b6:
 5 |   fixes \<delta> :: "nat \<Rightarrow> nat"
 6 |   defines "\<delta> \<equiv> (\<lambda>n::nat. (GREATEST t::nat. odd t \<and> t dvd n))"
 7 |   shows "\<forall>x::nat. (x > 0 \<longrightarrow> \<bar>(\<Sum>i::nat=1..x. (\<delta> i)/(i :: real)) - 2*x/3\<bar> < 1)"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1972_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1972_a5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1972_a5:
 5 |   fixes n :: nat
 6 |   assumes hn: "n > 1"
 7 |   shows "\<not>(n dvd (2^n - 1))"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1972_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1972_b3 imports Complex_Main
 2 | "HOL-Algebra.Group"
 3 | begin
 4 | 
 5 | theorem putnam_1972_b3:
 6 |   fixes G (structure)
 7 |   and A B :: 'a
 8 |   assumes Ggroup: "group G"
 9 |   and abinG: "A \<in> carrier G \<and> B \<in> carrier G"
10 |   and hab: "A\<otimes>B\<otimes>A = B\<otimes>A[^]2\<otimes>B \<and> A[^]3 = \<one> \<and> (\<exists>n::nat. n > 0 \<and> B[^](2*n-1) = \<one>)"
11 |   shows "B = \<one>"
12 |   sorry
13 | 
14 | end
15 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1972_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1972_b4 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | theorem putnam_1972_b4:
 6 |   fixes n :: nat
 7 |   assumes hn: "n > 1"
 8 |   shows "\<exists>P::int poly poly poly. \<forall>x::int. x = poly (poly (poly P (monom (monom (x + x^(n+2)) 0) 0)) (monom (x^(n+1)) 0)) (x^n)"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1973_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1973_a3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1973_a3:
 5 |   fixes b :: "nat \<Rightarrow> real"
 6 |   defines "b \<equiv> (\<lambda>n::nat. (LEAST knk::real. (\<exists>k::int>0. knk = k + n/k)))"
 7 |   shows "\<forall>n::nat. (n > 0 \<longrightarrow> \<lfloor>b n\<rfloor> = \<lfloor>sqrt (4*n + 1)\<rfloor>)"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1973_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1973_a4 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1973_a4_solution :: nat where "putnam_1973_a4_solution \<equiv> undefined"
 5 | (* 3 *)
 6 | theorem putnam_1973_a4:
 7 |   fixes f :: "real \<Rightarrow> real"
 8 |   defines "f \<equiv> (\<lambda>x::real. 2 powr x - 1 - x^2)"
 9 |   shows "putnam_1973_a4_solution = card {x::real. (f x = 0)}"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1973_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1973_b2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1973_b2:
 5 |   fixes z :: complex
 6 |   assumes hzrat: "\<exists>q1 q2::rat. Re z = q1 \<and> Im z = q2"
 7 |   and hznorm: "norm z = 1"
 8 |   shows "\<forall>n::int. (\<exists>q::rat. norm (z powi (2*n) - 1) = q)"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1973_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1973_b3 imports Complex_Main
 2 | "HOL-Computational_Algebra.Primes"
 3 | begin
 4 | 
 5 | theorem putnam_1973_b3:
 6 |   fixes p :: nat
 7 |   assumes pgt1: "p > 1"
 8 |     and hprime: "\<forall>x::int\<in>{0..<p}. prime (x^2 - x + p)"
 9 |   shows "\<exists>!triple::int\<times>int\<times>int. let (a,b,c) = triple in (b^2 - 4*a*c = 1 - 4*p \<and> 0 < a \<and> a \<le> c \<and> -a \<le> b \<and> b < a)"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1974_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1974_b2 imports
 2 | Complex_Main
 3 | "HOL-Analysis.Derivative"
 4 | begin
 5 | 
 6 | theorem putnam_1974_b2:
 7 |   fixes y :: "real \<Rightarrow> real"
 8 |   assumes ycontdiff: "y C1_differentiable_on UNIV"
 9 |   and limy: "((\<lambda> x :: real. (deriv y x) ^ 2 + (y x) ^ 3) \<longlongrightarrow> 0) at_top"
10 |   shows "(y \<longlongrightarrow> 0) at_top \<and> ((deriv y) \<longlongrightarrow> 0) at_top"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1974_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1974_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1974_b3:
 5 |   fixes \<alpha> :: real
 6 |   assumes ha: "cos (pi*\<alpha>) = 1/3"
 7 |   shows "\<alpha> \<notin> \<rat>"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1974_b5.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1974_b5 imports Complex_Main
2 | begin
3 | 
4 | theorem putnam_1974_b5:
5 |   shows "\<forall>n::nat\<ge>0. (\<Sum>i::nat=0..n. (n^i)/(fact i)) > (exp n)/2"
6 |   sorry
7 | 
8 | end
9 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1975_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1975_b5 imports Complex_Main "HOL-Analysis.Derivative"
 2 | begin
 3 | 
 4 | theorem putnam_1975_b5:
 5 |   fixes f::"nat\<Rightarrow>real\<Rightarrow>real"
 6 |   assumes h0 : "\<forall>x::real. f 0 x = exp x"
 7 |     and hf : "\<forall>n::nat. \<forall>x::real. f (n+1) x = x * (deriv (f n) x)"
 8 |   shows "(\<Sum>n::nat. (f n 1) / (fact n)) = exp (exp 1)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1975_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1975_b6 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1975_b6:
 6 |   fixes s :: "nat \<Rightarrow> real"
 7 |   defines "s \<equiv> \<lambda> n. \<Sum> i = 1 .. n. 1 / i"
 8 |   shows "(\<forall> n :: nat. n > 1 \<longrightarrow> n * (n + 1) powr (1 / n) < n + s n) \<and> (\<forall> n :: nat. n > 2 \<longrightarrow> (n - 1) * n powr (-1 / n - 1) < n - s n)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1976_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1976_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1976_b5_solution::"nat\<Rightarrow>(real \<Rightarrow> real)" where "putnam_1976_b5_solution \<equiv> undefined"
 5 | (* \<lambda>n. \<lambda>x. fact n *)
 6 | theorem putnam_1976_b5:
 7 |   shows "\<forall>n::nat. (\<lambda>x. (\<Sum>k=0..n. ((-1)^k * (n choose k)) * (x - k)^n)) = putnam_1976_b5_solution n"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1977_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1977_a5 imports Complex_Main "HOL-Number_Theory.Cong"
 2 | begin
 3 | 
 4 | theorem putnam_1977_a5:
 5 |   fixes p m n::nat
 6 |   assumes hp : "prime p"
 7 |     and hmgen : "m \<ge> n"
 8 |   shows "[(p*m) choose (p * n) = m choose n] (mod p)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1977_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1977_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1977_b1_solution::real where "putnam_1977_b1_solution \<equiv> undefined"
 5 | (* 2/3 *)
 6 | theorem putnam_1977_b1:
 7 |   shows "(\<lambda>N. \<Prod>n=2..N. (n^3 - 1) / (n^3 + 1)) \<longlonglongrightarrow> putnam_1977_b1_solution"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1977_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1977_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1977_b5:
 5 |   fixes n :: "nat"
 6 |     and a :: "nat \<Rightarrow> real"
 7 |     and A :: "real"
 8 |   assumes hn : "n > 1"
 9 |     and hA : "A + (\<Sum> i \<in> {1..n}. (a i)^2) < 1/(real n - 1) * (\<Sum> i \<in> {1..n}. a i)^2"
10 |   shows "\<forall> i \<in> {1 :: nat..n}. \<forall> j \<in> {1 :: nat..n}. i < j \<longrightarrow> A < 2 * (a i) * (a j)"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1978_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1978_a1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1978_a1:
 5 |   fixes S T :: "nat set"
 6 |   defines "S \<equiv> {k :: nat. \<exists> j :: nat \<le> 33. k = 3 * j + 1}"
 7 |   assumes hT : "T \<subset> S \<and> card T = 20"
 8 |   shows "\<exists> m \<in> T. \<exists> n \<in> T. m \<noteq> n \<and> m + n = 104"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1978_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1978_a5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1978_a5:
 5 |   fixes n :: "nat"
 6 |     and \<mu> :: "real"
 7 |     and a :: "nat \<Rightarrow> real"
 8 |   defines "\<mu> \<equiv> (\<Sum> i \<in> {1..n}. a i)/(real n)"
 9 |   assumes npos : "n > 0"
10 |     and ha : "\<forall> i :: nat. a i \<in> {0 :: real<..<pi}"
11 |   shows "(\<Prod> i \<in> {1..n}. sin (a i) / (a i)) \<le> (sin \<mu> / \<mu>)^n"
12 |   sorry
13 | 
14 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1978_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1978_b2 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1978_b2_solution :: "rat" where
 5 | "putnam_1978_b2_solution \<equiv> undefined"
 6 | (* (rat_of_nat 7)/4 *)
 7 | theorem putnam_1978_b2:
 8 |   shows "(\<Sum> i \<in> {1 :: nat..}. (\<Sum> j \<in> {1 :: nat..}. (rat_of_nat 1)/(rat_of_nat (i^2 * j + 2 * i * j + i * j^2)))) = putnam_1978_b2_solution"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1978_b4.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1978_b4 imports Complex_Main
2 | begin
3 | 
4 | theorem putnam_1978_b4:
5 |   shows "\<forall>N::real. \<exists> a b c d::int. a > N \<and> b > N \<and> c > N \<and> d > N \<and> a^2 + b^2 + c^2 + d^2 = a*b*c + a*b*d + a*c*d + b*c*d"
6 |   sorry
7 | 
8 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1978_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1978_b6 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1978_b6:
 5 |   fixes a::"nat\<Rightarrow>nat\<Rightarrow>real" and m n::nat
 6 |   assumes ha : "\<forall>i j. a i j \<in> {0..1::real}"
 7 |     and mnpos : "m > 0 \<and> n > 0"
 8 |   shows "(\<Sum>i=1..n. (\<Sum>j=1..m*i. a i j / i))^2 \<le> 2 * m * (\<Sum>i=1..n. (\<Sum>j=1..m*i. a i j))"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1979_a6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1979_a6 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1979_a6:
 6 |   fixes n :: nat
 7 |   and p :: "nat \<Rightarrow> real"
 8 |   assumes hp: "\<forall> i \<in> {1..n}. p i \<in> {0..1}"
 9 |   shows "\<exists> x \<in> {0..1}. (\<forall> i \<in> {1..n}. x \<noteq> p i) \<and> (\<Sum> i = 1..n. 1 / \<bar>x - p i\<bar>) \<le> 8 * n * (\<Sum> i = 1..n. 1 / (2 * i - 1))"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1979_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1979_b6 imports Complex_Main
 2 | begin
 3 | 
 4 | (* Note: Problem dimension boosted from Fin n *)
 5 | theorem putnam_1979_b6:
 6 |   fixes n :: "nat"
 7 |     and z :: "nat \<Rightarrow> complex"
 8 |   shows "\<bar>Re (csqrt (\<Sum> i \<in> {1::nat..n}. (z i)^2))\<bar> \<le> (\<Sum> i \<in> {1..n}. \<bar>Re (z i)\<bar>)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1980_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1980_a3 imports
 2 | Complex_Main
 3 | "HOL-Analysis.Interval_Integral"
 4 | begin
 5 | 
 6 | definition putnam_1980_a3_solution :: real where "putnam_1980_a3_solution \<equiv> undefined"
 7 | (* pi / 4 *)
 8 | theorem putnam_1980_a3:
 9 |   shows "interval_lebesgue_integral lebesgue 0 (pi / 2) (\<lambda> x :: real. 1 / (1 + (tan x) powr (sqrt 2))) = putnam_1980_a3_solution"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1980_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1980_b1 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_1980_b1_solution :: "real set" where "putnam_1980_b1_solution \<equiv> undefined"
 6 | (* {c :: real. c \<ge> 1 / 2} *)
 7 | theorem putnam_1980_b1:
 8 |   fixes c :: real
 9 |   shows "(\<forall> x :: real. (exp x + exp (-x)) / 2 \<le> exp (c * x ^ 2)) \<longleftrightarrow> c \<in> putnam_1980_b1_solution"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1981_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1981_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1981_b1_solution::real where "putnam_1981_b1_solution \<equiv> undefined"
 5 | (* -1 *)
 6 | theorem putnam_1981_b1:
 7 |   fixes f::"nat\<Rightarrow>real"
 8 |   defines "f \<equiv> \<lambda>n. 1/n^5 * (\<Sum>h=1..n. (\<Sum>k=1..n. (5*h^4 -18*h^2*k^2 + 5*k^4)))"
 9 |   shows "f \<longlonglongrightarrow> putnam_1981_b1_solution"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1981_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1981_b3 imports Complex_Main "HOL-Computational_Algebra.Primes"
 2 | begin
 3 | 
 4 | theorem putnam_1981_b3:
 5 |   fixes P::"nat\<Rightarrow>bool"
 6 |   defines "P \<equiv> \<lambda>n. \<forall>p::nat. (prime p \<and> p dvd (n^2 + 3)) \<longrightarrow> (\<exists>k::nat. k^2 < n \<and> p dvd (k^2 + 3))"
 7 |   shows "\<forall>n::nat. \<exists>m::nat. m > n \<and> P m"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1982_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1982_a3 imports Complex_Main
 2 | "HOL-Analysis.Interval_Integral"
 3 | 
 4 | begin
 5 | 
 6 | definition putnam_1982_a3_solution :: "real" where
 7 | "putnam_1982_a3_solution \<equiv> undefined"
 8 | (* (pi / 2) * ln pi *)
 9 | theorem putnam_1982_a3:
10 |   shows "filterlim (\<lambda> t :: real. (interval_lebesgue_integral lebesgue 0 t (\<lambda> x. (arctan (pi * x) - arctan x)/x))) (nhds putnam_1982_a3_solution) at_top"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1982_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1982_a5 imports Complex_Main
 2 | 
 3 | begin
 4 | 
 5 | theorem putnam_1982_a5:
 6 |   fixes a b c d :: "nat"
 7 |   assumes hpos : "a > 0 \<and> b > 0 \<and> c > 0 \<and> d > 0"
 8 |     and hac : "a + c \<le> 1982"
 9 |     and hfrac : "(real a) / b + (real c) / d < 1"
10 |   shows "(1 - (real a)/b - (real c)/d) > 1 / 1983^3"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1983_a1.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1983_a1 imports Complex_Main
2 | begin
3 | 
4 | definition putnam_1983_a1_solution :: "nat" where "putnam_1983_a1_solution \<equiv> undefined"
5 | (* 2301 *)
6 | theorem putnam_1983_a1:
7 |   shows "card {n :: nat. n > 0 \<and> (n dvd 10^40 \<or> n dvd 20^30)} = putnam_1983_a1_solution"
8 |   sorry
9 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1983_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1983_a3 imports Complex_Main
 2 | "HOL-Number_Theory.Cong"
 3 | begin
 4 | 
 5 | theorem putnam_1983_a3:
 6 |   fixes p :: "nat"
 7 |     and F :: "nat \<Rightarrow> nat"
 8 |   defines "F \<equiv> \<lambda> n::nat. (\<Sum> i=0..(p-2). ((i + 1) * n^i))"
 9 |   assumes hp : "odd p \<and> prime p"
10 |   shows "\<forall> a b :: nat. ((a < p \<and> b < p \<and> a \<noteq> b) \<longrightarrow> \<not>([F a = F b] (mod p)))"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1983_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1983_a4 imports Complex_Main
 2 | 
 3 | begin
 4 | 
 5 | theorem putnam_1983_a4:
 6 |   fixes k m :: "nat"
 7 |     and S :: "int"
 8 |   defines "m \<equiv> 6*k-1"
 9 |     and "S \<equiv> \<Sum> j::nat \<in> {1..2*k-1}. (-1)^(j+1) * (m choose (3 * j -1))"
10 |   assumes kpos : "k > 0"
11 |   shows "S \<noteq> 0"
12 |   sorry
13 | 
14 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1983_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1983_a5 imports Complex_Main
 2 | 
 3 | begin
 4 | 
 5 | definition putnam_1983_a5_solution :: "bool" where
 6 | "putnam_1983_a5_solution \<equiv> undefined"
 7 | (* True *)
 8 | theorem putnam_1983_a5:
 9 |   shows "putnam_1983_a5_solution \<longleftrightarrow> (\<exists> \<alpha> :: real. \<alpha> > 0 \<and> (\<forall> n :: nat. n > 0 \<longrightarrow> even (\<lfloor>\<alpha>^n\<rfloor> - n)))" 
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1983_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1983_b4 imports Complex_Main
 2 | 
 3 | begin
 4 | 
 5 | theorem putnam_1983_b4:
 6 |   fixes a f :: "nat \<Rightarrow> nat"
 7 |   defines "f \<equiv> \<lambda> n :: nat. n + (nat \<lfloor>sqrt (real n)\<rfloor>)"
 8 |   assumes ha0 : "a 0 > 0"
 9 |     and han : "\<forall> n :: nat. a (n + 1) = f (a n)"
10 |   shows "\<exists> i :: nat. \<exists> s :: nat. a i = s^2"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1984_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1984_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1984_a2_solution :: "rat" where "putnam_1984_a2_solution \<equiv> undefined"
 5 | (* 2 *)
 6 | theorem putnam_1984_a2:
 7 |   shows "(\<Sum> (k :: nat) \<in> {1..}. 6^k/((3^(k+1) - 2^(k+1))*(3^k - 2^k))) = putnam_1984_a2_solution"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1985_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1985_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1985_b3:
 5 |   fixes a::"nat\<Rightarrow>nat\<Rightarrow>nat"
 6 |   assumes apos : "\<forall>m n::nat. a m n > 0"
 7 |     and ha : "\<forall>k::nat. k > 0 \<longrightarrow> card {(m::nat, n::nat). m > 0 \<and> n > 0 \<and> a m n = k} = 8"
 8 |   shows "\<exists>m n::nat. m > 0 \<and> n > 0 \<and> a m n > m * n"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1986_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1986_a1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1986_a1_solution::real where "putnam_1986_a1_solution \<equiv> undefined"
 5 | (* 18 *)
 6 | theorem putnam_1986_a1:
 7 |   fixes f::"real\<Rightarrow>real"
 8 |   defines "f \<equiv> \<lambda>x. x^3 - 3 * x"
 9 |   shows "putnam_1986_a1_solution = (GREATEST y. (\<exists>x. y = f x \<and> x^4 + 36 \<le> 13 * x^2))"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1986_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1986_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1986_a2_solution::nat where "putnam_1986_a2_solution \<equiv> undefined"
 5 | (* 3 *)
 6 | theorem putnam_1986_a2:
 7 |   shows "putnam_1986_a2_solution = \<lfloor>(10^20000) / (10^100 + 3)\<rfloor> mod 10"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1986_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1986_a3 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1986_a3_solution::real where "putnam_1986_a3_solution \<equiv> undefined"
 5 | (* pi / 2 *)
 6 | theorem putnam_1986_a3:
 7 |   fixes arccot::"real\<Rightarrow>real"
 8 |   defines "arccot \<equiv> \<lambda>y. (THE x. 0 < x \<and> x \<le> pi/2 \<and> cot x = y)"
 9 |   shows "(\<Sum>n::nat. arccot (n^2 + n + 1)) = putnam_1986_a3_solution"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1986_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1986_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1986_b1_solution :: "real" where "putnam_1986_b1_solution \<equiv> undefined"
 5 | (* 2 / 5 *)
 6 | theorem putnam_1986_b1:
 7 |   fixes b h :: "real"
 8 |   assumes hbh : "b > 0 \<and> h > 0 \<and> b^2 + h^2 = 2^2"
 9 |     and areaeq : "b*h = 0.5 * b * (1 - h/2)"
10 |   shows "h = putnam_1986_b1_solution"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1987_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1987_b1 imports Complex_Main "HOL-Analysis.Interval_Integral"
 2 | begin
 3 | 
 4 | definition putnam_1987_b1_solution::real where "putnam_1987_b1_solution \<equiv> undefined"
 5 | (* 1 *)
 6 | theorem putnam_1987_b1:
 7 |   shows "putnam_1987_b1_solution =
 8 |      interval_lebesgue_integral lebesgue 2 4 (\<lambda>x. sqrt (ln (9-x)) / (sqrt (ln (9-x)) + sqrt (ln (x+3))))"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1987_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1987_b2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1987_b2:
 5 |   fixes r s t::nat
 6 |   assumes hsum : "r + s \<le> t"
 7 |   shows "(\<Sum>i=0..s. (s choose i) / (t choose (r+i))) = (t+1) / ((t+1-s) * ((t-s) choose r))"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1987_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1987_b3 imports Complex_Main
 2 | "HOL-Algebra.Ring"
 3 | begin
 4 | 
 5 | theorem putnam_1987_b3:
 6 |   fixes F :: "'f :: field"
 7 |   assumes hF : "(1 :: 'f) + 1 \<noteq> 0"
 8 |   shows "{(x :: 'f, y :: 'f). x^2 + y^2 = 1} = {(1, 0)} \<union> {s :: 'f \<times> 'f. \<exists> r \<in> {r' :: 'f. r' ^ 2 \<noteq> -1}. s = ((r ^ 2 - 1) / (r ^ 2 + 1), (2 * r) / (r ^ 2 + 1))}"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1988_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1988_a3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_1988_a3_solution :: "real set" where "putnam_1988_a3_solution \<equiv> undefined"
 6 | (* {x :: real. x > 1 / 2} *)
 7 | theorem putnam_1988_a3:
 8 |   shows "{x :: real. convergent (\<lambda> t :: nat. (\<Sum> n = 1..t. (1 / n) / sin (1 / n) - 1) powr x)} = putnam_1988_a3_solution"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1988_a5.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1988_a5 imports
2 | Complex_Main
3 | begin
4 | 
5 | theorem putnam_1988_a5:
6 |   shows "\<exists> f :: real \<Rightarrow> real. (\<forall> x > 0. f (f x) = 6 * x - f x \<and> f x > 0) \<and> (\<forall> g :: real \<Rightarrow> real. (\<forall> x > 0. g (g x) = 6 * x - g x \<and> g x > 0) \<longrightarrow> (\<forall> x > 0. f x = g x))"
7 |   sorry
8 | 
9 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1988_b1.thy:
--------------------------------------------------------------------------------
1 | theory putnam_1988_b1 imports
2 | Complex_Main
3 | begin
4 | 
5 | theorem putnam_1988_b1:
6 |   shows "\<forall> (a :: int) \<ge> 2. \<forall> (b :: int) \<ge> 2. \<exists> x y z:: int. x > 0 \<and> y > 0 \<and> z > 0 \<and>  a * b = x * y + x * z + y * z + 1"
7 |   sorry
8 | 
9 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1988_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1988_b2 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_1988_b2_solution :: bool where "putnam_1988_b2_solution \<equiv> undefined"
 6 | (* True *)
 7 | theorem putnam_1988_b2:
 8 |   shows "(\<forall> x y :: real. (y \<ge> 0 \<and> y * (y + 1) \<le> (x + 1) ^ 2) \<longrightarrow> y * (y - 1) \<le> x ^ 2) \<longleftrightarrow> putnam_1988_b2_solution"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1988_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1988_b6 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1988_b6:
 6 |   fixes trinums :: "int set"
 7 |   defines "trinums \<equiv> {t :: int. \<exists> n :: nat. t = (n * (n + 1)) / 2}"
 8 |   shows "infinite {(a :: int, b :: int). \<forall> t > 0. (a * t + b) \<in> trinums \<longleftrightarrow> t \<in> trinums}"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1989_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1989_a3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1989_a3:
 6 |   fixes z :: complex
 7 |   assumes hz: "11 * z ^ 10 + 10 * \<i> * z ^ 9 + 10 * \<i> * z - 11 = 0"
 8 |   shows "norm z = 1"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1990_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1990_b2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1990_b2:
 5 |   fixes x z::real and P::"nat\<Rightarrow>real"
 6 |   defines "P \<equiv> \<lambda>j. (\<Prod>i=0..<j. (1 - z * x^i)) / (\<Prod>i=1..j. (z - x^i))"
 7 |   assumes xlt1 : "abs(x) < 1"
 8 |     and zgt1 : "abs(z) > 1"
 9 |   shows "1 + (\<Sum>j::nat. (1 + x^(j+1)) * P (j+1)) = 0"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1990_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1990_b3 imports Complex_Main "HOL-Analysis.Finite_Cartesian_Product"
 2 | begin
 3 | 
 4 | theorem putnam_1990_b3:
 5 |   fixes S::"(nat^2^2) set"       
 6 |   assumes hS : "\<forall>A \<in> S. (\<forall>i j :: 2. ((\<exists>x::nat. A$i$j = x^2) \<and> A$i$j \<le> 200))"
 7 |   shows "card S > 50387 \<longrightarrow> (\<exists>A \<in> S. \<exists>B \<in> S. A \<noteq> B \<and> A ** B = B ** A)"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1991_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1991_b4 imports
 2 | Complex_Main
 3 | "HOL-Computational_Algebra.Primes"
 4 | "HOL-Number_Theory.Cong"
 5 | begin
 6 | 
 7 | theorem putnam_1991_b4:
 8 |   fixes p :: nat
 9 |   assumes podd: "odd p"
10 |   and pprime: "prime p"
11 |   shows "[(\<Sum> j = 0 .. p. (p choose j) * ((p + j) choose j)) = 2 ^ p + 1] (mod p ^ 2)"
12 |   sorry
13 | 
14 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1992_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1992_a1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1992_a1:
 5 |   fixes f :: "int \<Rightarrow> int"
 6 |   shows "(f = (\<lambda>n::int. 1 - n)) \<longleftrightarrow> ((\<forall>n::int. f (f n) = n) \<and> (\<forall>n::int. f (f (n+2) + 2) = n) \<and> f 0 = 1)"
 7 |   sorry
 8 | 
 9 | end
10 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1992_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1992_b2 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | theorem putnam_1992_b2:
 6 |   fixes Q :: "nat \<Rightarrow> nat \<Rightarrow> nat"
 7 |   defines "Q \<equiv> (\<lambda>n k::nat. coeff ((monom 1 0 + monom 1 1 + monom 1 2 + monom 1 3)^n) k)"
 8 |   shows "\<forall>n k::nat. Q n k = (\<Sum>j::nat=0..k. (n choose j) * (if 2 * j \<le> k then n choose (k - 2*j) else 0))"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1993_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1993_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1993_a2:
 5 |   fixes x::"nat\<Rightarrow>real"
 6 |   assumes xn0 : "\<forall>n::nat. x n \<noteq> 0"
 7 |     and hx : "\<forall>n \<ge> 1. (x n)^2 - x (n-1) * x (n+1) = 1"
 8 |   shows "\<exists>a::real. \<forall>n \<ge> 1. x (n+1) = a * x n - x (n-1)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1994_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1994_a1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1994_a1:
 5 |   fixes a :: "nat \<Rightarrow> real"
 6 |   assumes ha: "\<forall>n::nat\<ge>1. 0 < a n \<and> a n \<le> a (2*n) + a (2*n+1)"
 7 |   shows "filterlim (\<lambda>N::nat. (\<Sum>n::nat=1..N. a n)) at_top at_top"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1994_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1994_a4 imports Complex_Main
 2 | "HOL-Analysis.Finite_Cartesian_Product"
 3 | begin
 4 | 
 5 | theorem putnam_1994_a4:
 6 |   fixes A B :: "int^2^2"
 7 |   assumes ABinv: "invertible A \<and> invertible (A + B) \<and> invertible (A + 2*B) \<and> invertible (A + 3*B) \<and> invertible (A + 4*B)"
 8 |   shows "invertible (A + 5*B)"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1994_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1994_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1994_b5:
 5 |   fixes f :: "real \<Rightarrow> int \<Rightarrow> int"
 6 |   and n :: nat
 7 |   assumes hf: "\<forall>(\<alpha>::real)(x::int). f \<alpha> x = \<lfloor>\<alpha>*x\<rfloor>"
 8 |   and npos: "n > 0"
 9 |   shows "\<exists>\<alpha>::real. \<forall>k::nat\<in>{1..n}. (((f \<alpha>)^^k) (n^2) = n^2 - k) \<and> (f (\<alpha>^k) (n^2) = n^2 - k)"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1994_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1994_b6 imports Complex_Main
 2 | "HOL-Number_Theory.Cong"
 3 | begin
 4 | 
 5 | theorem putnam_1994_b6:
 6 |   fixes n :: "nat \<Rightarrow> int"
 7 |   assumes hn: "\<forall>a::nat. n a = 101*a - 100 * 2^a"
 8 |   shows "\<forall>a::nat\<in>{0..99}. \<forall>b::nat\<in>{0..99}. \<forall>c::nat\<in>{0..99}. \<forall>d::nat\<in>{0..99}. (([n a + n b = n c + n d] (mod 10100)) \<longrightarrow> {a, b} = {c, d})"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1996_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1996_a5 imports Complex_Main
 2 | "HOL-Computational_Algebra.Primes"
 3 | begin
 4 | 
 5 | theorem putnam_1996_a5:
 6 |   fixes p k :: "nat"
 7 |   defines "k \<equiv> nat \<lfloor>2 * p / 3\<rfloor>"
 8 |   assumes hpprime : "prime p"
 9 |     and hpge3 : "p > 3"
10 |   shows "(p^2) dvd (\<Sum> i \<in> {1 :: nat..k}. p choose i)"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1996_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1996_b2 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_1996_b2:
 6 |   fixes n :: nat
 7 |   and prododd :: real
 8 |   defines "prododd \<equiv> \<Prod> i \<in> {1 .. 2 * n - 1}. if odd i then i else 1"
 9 |   assumes npos: "n > 0"          
10 |   shows "((2 * n - 1) / exp 1) powr ((2 * n - 1) / 2) < prododd \<and> prododd < ((2 * n + 1) / exp 1) powr ((2 * n + 1) / 2)"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1998_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1998_a3 imports Complex_Main
 2 | "HOL-Analysis.Derivative"
 3 | begin
 4 | 
 5 | theorem putnam_1998_a3:
 6 |   fixes f :: "real \<Rightarrow> real"
 7 |   assumes hf : "\<forall>n::nat\<in>{0..2}. ((deriv^^n) f) C1_differentiable_on UNIV"
 8 |   shows "\<exists> a :: real. f a * deriv f a * (deriv^^2) f a * (deriv^^3) f a \<ge> 0"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1998_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1998_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1998_b1_solution::real where "putnam_1998_b1_solution \<equiv> undefined"
 5 | (* 6 *)
 6 | theorem putnam_1998_b1:
 7 |   shows "putnam_1998_b1_solution = (LEAST y. (\<exists>x::real > 0. y = ((x+1/x)^6 - (x^6 + 1/x^6) - 2) / ((x+1/x)^3 + (x^3 + 1/x^3))))"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1998_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1998_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_1998_b5_solution::nat where "putnam_1998_b5_solution \<equiv> undefined"
 5 | (* 1 *)
 6 | theorem putnam_1998_b5:
 7 |   fixes N::nat
 8 |   defines "N \<equiv> (\<Sum>i=0..<1998. 10^i)"
 9 |   shows "putnam_1998_b5_solution = (\<lfloor>10^1000 * sqrt N\<rfloor> mod 10)"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1998_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1998_b6 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_1998_b6:
 5 |   fixes a b c :: int
 6 |   shows "\<exists> n:: int > 0 . sqrt (n^3 + a * n^2 + b * n + c) \<notin> \<int>"
 7 |   sorry
 8 | 
 9 | end
10 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1999_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1999_a2 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | theorem putnam_1999_a2:
 6 |   fixes p :: "real poly"
 7 |   assumes hpos : "\<forall>x. poly p x \<ge> 0"
 8 |   shows "\<exists>S :: real poly set . finite S \<and> (\<forall>x. poly p x = (\<Sum>s \<in> S. (poly s x)^2))"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_1999_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1999_a4 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_1999_a4_solution :: real where "putnam_1999_a4_solution \<equiv> undefined"
 6 | (* 9 / 32 *)
 7 | theorem putnam_1999_a4:
 8 |   shows "(\<Sum> m :: nat. \<Sum> n :: nat. ((m + 1) ^ 2 * (n + 1)) / (3 ^ (m + 1) * ((n + 1) * 3 ^ (m + 1) + (m + 1) * 3 ^ (n + 1)))) = putnam_1999_a4_solution"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1999_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1999_a5 imports
 2 | Complex_Main
 3 | "HOL-Computational_Algebra.Polynomial"
 4 | "HOL-Analysis.Interval_Integral"
 5 | begin
 6 | 
 7 | theorem putnam_1999_a5:
 8 |   shows "\<exists> C :: real. \<forall> p :: real poly. degree p = 1999 \<longrightarrow> \<bar>poly p 0\<bar> \<le> C * (interval_lebesgue_integral lebesgue (-1) 1 (\<lambda> x. \<bar>poly p x\<bar>))"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_1999_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_1999_b6 imports Complex_Main
 2 | "HOL-Computational_Algebra.Primes"
 3 | begin
 4 | 
 5 | theorem putnam_1999_b6:
 6 |   fixes S::"int set"
 7 |   assumes S_fin: "finite S"
 8 |     and Sge1: "\<forall>s \<in> S. s > 1"
 9 |     and hgcd: "\<forall>n::int. \<exists>s \<in> S. (gcd s n) = 1 \<or> (gcd s n) = s"
10 |   shows "\<exists> s \<in> S. \<exists> t \<in> S. prime (gcd s t)"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2000_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2000_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | definition sum_of_squares:: "int \<Rightarrow> bool" where
 5 |   "sum_of_squares n \<equiv> \<exists>a b::int. n = a^2 + b^2"
 6 | 
 7 | theorem putnam_2000_a2:
 8 |   shows "\<forall>n :: int. \<exists>m::int. m > n \<and> 
 9 |     sum_of_squares m \<and> sum_of_squares (m+1) \<and> sum_of_squares (m+2)"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2000_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2000_a4 imports
 2 | Complex_Main
 3 | "HOL-Analysis.Interval_Integral"
 4 | begin
 5 | 
 6 | theorem putnam_2000_a4:
 7 |   shows "\<exists> y :: real. ((\<lambda> B. interval_lebesgue_integral lebesgue 0 B (\<lambda> x. sin x * sin (x ^ 2))) \<longlongrightarrow> y) at_top"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2000_a6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2000_a6 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | theorem putnam_2000_a6:
 6 |   fixes f :: "int poly"
 7 |   and a :: "nat \<Rightarrow> int"
 8 |   assumes ha0: "a 0 = 0"
 9 |   and ha: "\<forall>n::nat. a (n + 1) = poly f (a n)"
10 |   shows "(\<exists>m::nat>0. a m = 0) \<longrightarrow> (a 1 = 0 \<or> a 2 = 0)"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2000_b2.thy:
--------------------------------------------------------------------------------
1 | theory putnam_2000_b2 imports Complex_Main
2 | begin
3 | 
4 | theorem putnam_2000_b2:
5 |   shows "\<forall>m n::nat. (m \<ge> 1 \<longrightarrow> n \<ge> m \<longrightarrow> n dvd ((gcd m n) * (n choose m)))"
6 |   sorry
7 | 
8 | end
9 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2000_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2000_b4 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2000_b4:
 5 |   fixes f :: "real \<Rightarrow> real"
 6 |   assumes hf : "\<forall>x::real. f (2 * x^2 - 1) = 2 * x * f x"
 7 |     and f_cont : "continuous_on UNIV f"
 8 |   shows "\<forall>x::real. x \<ge> -1 \<and> x \<le> 1 \<longrightarrow> f x = 0"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2001_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2001_a1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2001_a1:
 5 |   fixes op :: "'a => 'a => 'a"
 6 |   assumes hop : "\<forall>a b :: 'a. op (op a b) a = b"
 7 |   shows "\<forall>a b :: 'a. op a (op b a) = b"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2001_a5.thy:
--------------------------------------------------------------------------------
1 | theory putnam_2001_a5 imports Complex_Main
2 | begin
3 | 
4 | theorem putnam_2001_a5:
5 |   shows "\<exists>! (a :: int). \<exists>! (n :: nat). a > 0 \<and> n > 0 \<and> a^(n+1) - (a+1)^n = 2001"
6 |   sorry
7 | 
8 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2001_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2001_b3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_2001_b3_solution :: real where "putnam_2001_b3_solution \<equiv> undefined"
 6 | (* 3 *)
 7 | theorem putnam_2001_b3:
 8 |   shows "(\<Sum> n :: nat. (2 powr (round (sqrt (n + 1))) + 2 powr (- round (sqrt (n + 1)))) / 2 powr (n + 1)) = putnam_2001_b3_solution"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2001_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2001_b5 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_2001_b5:
 6 |   fixes a b :: real
 7 |   and g :: "real \<Rightarrow> real"
 8 |   assumes abint: "a \<in> {0 <..< 1 / 2} \<and> b \<in> {0 <..< 1 / 2}"
 9 |   and gcont: "continuous_on UNIV g"
10 |   and hg : "\<forall> x :: real. g (g x) = a * g x + b * x"
11 |   shows "\<exists> c :: real. \<forall> x :: real. g x = c * x"
12 |   sorry
13 | 
14 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2002_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2002_a3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2002_a3:
 5 |   fixes n Tn :: "int"
 6 |   defines "Tn \<equiv> card {S :: int set. S \<subseteq> {1..n} \<and> S \<noteq> {} \<and> (\<exists> k :: int. k = (real 1)/(card S) * (\<Sum> s \<in> S. s))}"
 7 |   assumes hn : "n \<ge> 2"
 8 |   shows "even (Tn - n)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2002_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2002_a5 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_2002_a5:
 6 |   fixes a :: "nat \<Rightarrow> rat"
 7 |   assumes ha: "a 0 = 1 \<and> (\<forall> n :: nat. a (2 * n + 1) = a n \<and> a (2 * n + 2) = a n + a (n + 1))"
 8 |   shows "\<forall> q :: rat. q > 0 \<longrightarrow> q \<in> {a (n - 1) / a n | n :: nat. n \<ge> 1}"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2002_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2002_b3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_2002_b3:
 6 |   fixes e :: real
 7 |   and f :: "int \<Rightarrow> real"
 8 |   defines "e \<equiv> exp 1"
 9 |   and "f \<equiv> \<lambda> n :: int. 1 / e - (1 - 1 / n) powi n" 
10 |   shows "\<forall> n :: int. n > 1 \<longrightarrow> 1 / (2 * n * e) < f n \<and> f n < 1 / (n * e)"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2002_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2002_b5 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | fun digits :: "nat \<Rightarrow> nat \<Rightarrow> nat list" where
 6 |   "digits b n = (if b < 2 then (replicate n 1) else (if n < b then [n] else [n mod b] @ digits b (n div b)))"
 7 | theorem putnam_2002_b5:
 8 |   shows "\<exists> n :: nat. card {b :: nat. b \<ge> 1 \<and> length (digits b n) = 3 \<and> digits b n = rev (digits b n)} \<ge> 2002"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2003_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2003_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | (* Note: Boosted domain to infinite set *)
 5 | theorem putnam_2003_a2:
 6 |   fixes n::nat and a b::"nat\<Rightarrow>real"
 7 |   assumes npos: "n \<ge> 1"
 8 |   and abnneg: "\<forall>i \<in> {0..<n}. a i \<ge> 0 \<and> b i \<ge> 0"
 9 |   shows "((\<Prod>i=0..<n. a i) powr (1/n)) + ((\<Prod>i=0..<n. b i) powr (1/n)) \<le> ((\<Prod>i=0..<n. (a i + b i)) powr (1/n))"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2003_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2003_a3 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_2003_a3_solution::real where "putnam_2003_a3_solution \<equiv> undefined"
 5 | (* 2 * sqrt 2 - 1 *)
 6 | theorem putnam_2003_a3:
 7 |   fixes f::"real\<Rightarrow>real"
 8 |   defines "f \<equiv> \<lambda>x::real. abs(sin x + cos x + tan x + 1 / tan x + 1 / cos x + 1 / sin x)"
 9 |   shows "putnam_2003_a3_solution = (LEAST y. \<exists>x::real. f x = y)"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2003_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2003_a4 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2003_a4:
 5 |   fixes a b c A B C::real
 6 |   assumes aAne0 : "a \<noteq> 0 \<and> A \<noteq> 0"
 7 |     and hle : "\<forall>x::real. abs(a * x^2 + b * x + c) \<le> abs(A * x^2 + B * x + C)"
 8 |   shows "abs(b^2 - 4 * a * c) \<le> abs(B^2 - 4 * A * C)"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2003_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2003_b1 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | definition putnam_2003_b1_solution :: bool where "putnam_2003_b1_solution \<equiv> undefined"
 6 | (* False *)
 7 | theorem putnam_2003_b1:
 8 |   shows "(\<exists>a b c d::real poly. (\<forall>x y::real. 1 + x*y + x^2*y^2 = poly a x * poly c y + poly b x * poly d y)) \<longleftrightarrow> putnam_2003_b1_solution"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2003_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2003_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2003_b3:
 5 |   fixes n::nat
 6 |   assumes npos : "n > 0"
 7 |   shows "fact n = (\<Prod>i=1..n. Lcm {1..\<lfloor>n / i\<rfloor>})"
 8 |   sorry
 9 | 
10 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2004_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2004_a3 imports Complex_Main
 2 | "HOL-Analysis.Determinants"
 3 | begin
 4 | 
 5 | theorem putnam_2004_a3:
 6 |   fixes u::"nat\<Rightarrow>real"
 7 |   assumes hubase : "u 0 = 1 \<and> u 1 = 1 \<and> u 2 = 1"
 8 |     and hudet : "\<forall>n::nat. det (vector[vector[u n, u (n+1)], vector[u (n+2), u (n+3)]]) = fact n"
 9 |   shows "\<forall>n::nat. u n \<in> \<int>"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2004_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2004_b2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2004_b2:
 5 |   fixes m n::nat
 6 |   assumes mnpos : "m > 0 \<and> n > 0"
 7 |   shows "fact (m+n) / (m+n)^(m+n) < (fact m / m^m) * (fact n / n^n)"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2005_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2005_a5 imports Complex_Main
 2 | "HOL-Analysis.Lebesgue_Measure"
 3 | "HOL-Analysis.Interval_Integral"
 4 | begin
 5 | 
 6 | definition putnam_2005_a5_solution :: "real" where "putnam_2005_a5_solution \<equiv> undefined"
 7 | (* pi * (ln 2) / 8 *)
 8 | theorem putnam_2005_a5:
 9 |   shows "interval_lebesgue_integral lebesgue 0 1 (\<lambda> x :: real. (ln (x + 1))/(x^2 + 1)) = putnam_2005_a5_solution" 
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2005_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2005_b4 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2005_b4:
 5 |   fixes m n::nat 
 6 |     and f::"nat\<Rightarrow>nat\<Rightarrow>nat"
 7 |   assumes mnpos : "m > 0 \<and> n > 0"
 8 |     and hf : "\<forall>m' > 0. \<forall>n' > 0. f m' n' = card {x::int list. size x = n' \<and> (\<Sum>i=0..<n'. abs(x!i)) \<le> m'}"
 9 |   shows "f m n = f n m"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2006_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2006_a3 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_2006_a3:
 6 |   fixes x :: "nat \<Rightarrow> nat"
 7 |   assumes hxlo: "\<forall> k \<in> {1 .. 2006}. x k = k"
 8 |   and hxhi: "\<forall> k \<ge> 2006. x (k + 1) = x k + x (k - 2005)"
 9 |   shows "\<exists> i :: nat. i > 0 \<and> (\<forall> j \<in> {0 ..< 2005}. 2006 dvd x (i + j))"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2006_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2006_b2 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | theorem putnam_2006_b2:
 6 |   fixes n :: nat
 7 |   and X :: "real set"
 8 |   assumes npos: "n > 0"
 9 |   and hXcard: "finite X \<and> card X = n"
10 |   shows "\<exists> S \<subseteq> X. (S \<noteq> {}) \<and> (\<exists> m :: int. \<bar>m + (\<Sum> s \<in> S. s)\<bar> \<le> 1 / (n + 1))"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2007_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2007_a5 imports Complex_Main
 2 | "HOL-Algebra.Multiplicative_Group"
 3 | "HOL-Computational_Algebra.Primes"
 4 | begin
 5 | 
 6 | theorem putnam_2007_a5:
 7 |   fixes G (structure)
 8 |   and p :: nat
 9 |   and n :: nat
10 |   assumes hG: "group G \<and> finite (carrier G)"
11 |   and hp: "prime p"
12 |   and hn: "n \<equiv> card {g::'a. g \<in> carrier G \<and> (group.ord G) g = p}"
13 |   shows "n = 0 \<or> p dvd (n+1)"
14 |   sorry
15 | 
16 | end
17 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2007_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2007_b5 imports
 2 | Complex_Main
 3 | "HOL-Computational_Algebra.Polynomial"
 4 | begin
 5 | 
 6 | theorem putnam_2007_b5:
 7 |   fixes k :: nat
 8 |   assumes kpos: "k > 0"
 9 |   shows "\<exists> P :: real poly list. length P = k \<and> (\<forall> n :: int. \<lfloor>n / k\<rfloor> ^ k = (\<Sum> i = 0..(k - 1). poly (P!i) n * \<lfloor>n / k\<rfloor> ^ i))"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2008_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2008_a1 imports Complex_Main
 2 | 
 3 | begin
 4 | 
 5 | theorem putnam_2008_a1:
 6 |   fixes f :: "real \<Rightarrow> real \<Rightarrow> real"
 7 |   assumes hf : "\<forall> x y z :: real. f x y + f y z + f z x = 0"
 8 |   shows "\<exists> g :: real \<Rightarrow> real. \<forall> x y :: real. f x y = g x - g y"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2009_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2009_b5 imports Complex_Main
 2 | "HOL-Analysis.Derivative"
 3 | begin
 4 | 
 5 | (* uses (real \<Rightarrow> real) instead of ({1<..} \<Rightarrow> real) *)
 6 | theorem putnam_2009_b5:
 7 |   fixes f :: "real \<Rightarrow> real"
 8 |   assumes hfdiff: "f differentiable_on {1<..}"
 9 |   and f: "\<forall>x::real>1. deriv f x = (x^2 - (f x)^2) / (x^2 * ((f x)^2 + 1))"
10 |   shows "filterlim f at_top at_top"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2010_a4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2010_a4 imports Complex_Main
 2 | "HOL-Computational_Algebra.Primes"
 3 | begin
 4 | 
 5 | theorem putnam_2010_a4:
 6 |   fixes n :: nat
 7 |   assumes n_pos : "n > 0"
 8 |   shows "\<not> prime (10^(10^(10^n)) + 10^(10^n) + 10^n - 1)"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2010_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2010_a5 imports Complex_Main
 2 | "HOL-Analysis.Cross3"
 3 | "HOL-Algebra.Group"
 4 | begin
 5 | 
 6 | theorem putnam_2010_a5:
 7 |   fixes S :: "(real^3) monoid"
 8 |   assumes hgroup : "Group.group S"
 9 |     and hcross: "\<forall>a \<in> carrier S. \<forall>b \<in> carrier S. cross3 a b = a \<otimes>\<index> b \<or> cross3 a b = 0"
10 |   shows "\<forall>a \<in> carrier S. \<forall>b \<in> carrier S. cross3 a b = 0"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2010_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2010_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_2010_b1_solution :: bool where "putnam_2010_b1_solution \<equiv> undefined"
 5 | (* False *)
 6 | theorem putnam_2010_b1:
 7 |   shows "(\<exists>a::nat\<Rightarrow>real. \<forall>m::nat>0. (\<Sum>i::nat. (a i)^m) = m) \<longleftrightarrow> putnam_2010_b1_solution"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2010_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2010_b5 imports Complex_Main
 2 | "HOL-Analysis.Derivative"
 3 | begin
 4 | 
 5 | definition putnam_2010_b5_solution :: bool where "putnam_2010_b5_solution \<equiv> undefined"
 6 | (* False *)
 7 | theorem putnam_2010_b5:
 8 |   shows "(\<exists>f::real\<Rightarrow>real. strict_mono_on UNIV f \<and> f differentiable_on UNIV \<and> (\<forall>x::real. deriv f x = f (f x))) \<longleftrightarrow> putnam_2010_b5_solution"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2011_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2011_b1 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2011_b1:
 5 |   fixes h k :: nat
 6 |   assumes hkpos: "h > 0 \<and> k > 0"
 7 |   shows "\<forall>\<epsilon>>0. \<exists>m n::nat. m > 0 \<and> n > 0 \<and> \<epsilon> < \<bar>h * sqrt m - k * sqrt n\<bar> \<and> \<bar>h * sqrt m - k * sqrt n\<bar> < 2*\<epsilon>"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2011_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2011_b6 imports Complex_Main
 2 | "HOL-Computational_Algebra.Primes"
 3 | begin
 4 | 
 5 | theorem putnam_2011_b6:
 6 |   fixes p :: nat
 7 |   assumes hp: "odd p \<and> prime p"
 8 |   shows "card {n::nat\<in>{0..(p-1)}. \<not>(p dvd (\<Sum>k::nat=0..(p-1). fact k * n^k))} \<ge> (p + 1)/2"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2012_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2012_a2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2012_a2:
 5 |   fixes Smul :: "'S \<Rightarrow> 'S \<Rightarrow> 'S" (infixl "\<^bold>*" 70)
 6 |   and a b c :: 'S
 7 |   assumes Smulasg: "abel_semigroup Smul"
 8 |   and hS: "\<forall>x y::'S. \<exists>z::'S. x \<^bold>* z = y"
 9 |   and habc: "a \<^bold>* c = b \<^bold>* c"
10 |   shows "a = b"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2012_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2012_b4 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_2012_b4_solution::bool where "putnam_2012_b4_solution \<equiv> undefined"
 5 | (* True *)
 6 | theorem putnam_2012_b4:
 7 |   fixes a::"nat\<Rightarrow>real"
 8 |   assumes ha0 : "a 0 = 1"
 9 |     and han : "\<forall>n::nat. a (n+1) = a n + exp (-a n)"
10 |   shows "(convergent (\<lambda>n. a n - ln n)) \<longleftrightarrow> putnam_2012_b4_solution"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2014_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2014_b4 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | theorem putnam_2014_b4:
 6 |   fixes n :: nat
 7 |   and P :: "complex poly"
 8 |   assumes npos: "n > 0"
 9 |   and Px: "degree P = n \<and> (\<forall>k::nat\<in>{0..n}. coeff P k = 2 ^ (k*(n-k)))"
10 |   shows "\<forall>r::complex. (poly P r = 0 \<longrightarrow> Im r = 0)"
11 |   sorry
12 | 
13 | end
14 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2015_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2015_a3 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_2015_a3_solution :: complex where "putnam_2015_a3_solution \<equiv> undefined"
 5 | (* 13725 *)
 6 | theorem putnam_2015_a3:
 7 |   shows "log 2 (Re (\<Prod>a::nat=1..2015. \<Prod>b::nat=1..2015. (1 + exp (2*pi*\<i>*a*b/2015)))) = putnam_2015_a3_solution"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2016_a1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2016_a1 imports
 2 | Complex_Main
 3 | "HOL-Computational_Algebra.Polynomial"
 4 | begin
 5 | 
 6 | definition putnam_2016_a1_solution :: nat where "putnam_2016_a1_solution \<equiv> undefined"
 7 | (* 8 *)
 8 | theorem putnam_2016_a1:
 9 |   shows "putnam_2016_a1_solution = (LEAST j :: nat. j > 0 \<and> (\<forall> P :: int poly. \<forall> k :: int. 2016 dvd poly ((pderiv ^^ j) P) k))"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2016_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2016_b1 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_2016_b1_solution :: real where "putnam_2016_b1_solution \<equiv> undefined"
 6 | (* exp 1 - 1 *)
 7 | theorem putnam_2016_b1:
 8 |   fixes x :: "nat \<Rightarrow> real"
 9 |   assumes hx0: "x 0 = 1"
10 |   and hxn: "\<forall> n :: nat. x (n + 1) = ln (exp (x n) - x n)"
11 |   shows "(\<Sum> n :: nat. x n) = putnam_2016_b1_solution"
12 |   sorry
13 | 
14 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2016_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2016_b6 imports Complex_Main
 2 | "HOL-Analysis.Infinite_Sum"
 3 | begin
 4 | 
 5 | definition putnam_2016_b6_solution :: real where "putnam_2016_b6_solution \<equiv> undefined"
 6 | (* 1 *)
 7 | theorem putnam_2016_b6:
 8 |   shows "(\<Sum>\<^sub>\<infinity>k::nat\<in>{1..}. ((-1)^(k-1) / k) * (\<Sum>n::nat. 1 / (k*2^n + 1))) = putnam_2016_b6_solution"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2017_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2017_a2 imports Complex_Main
 2 | "HOL-Computational_Algebra.Polynomial"
 3 | begin
 4 | 
 5 | theorem putnam_2017_a2:
 6 |   fixes Q :: "nat \<Rightarrow> real poly"
 7 |   assumes hQbase: "\<forall>x::real. Q 0 = monom 1 0 \<and> Q 1 = monom 1 1"
 8 |   and hQn: "\<forall>n::nat. Q (n + 2) * Q n = (Q (n-1))^2 - 1"
 9 |   shows "\<forall>n::nat>0. \<exists>P::int poly. Q n = map_poly real_of_int P"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2017_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2017_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2017_b3:
 5 |   fixes f :: "real \<Rightarrow> real" and c :: "nat \<Rightarrow> real"
 6 |   defines hf: "f \<equiv> \<lambda>x. \<Sum>i::nat.  c (i) * x^i"
 7 |   assumes hc: "\<forall>n. c n = 0 \<or> c n = 1"
 8 |     and "f (2/3) = 3/2"
 9 |   shows "f (1/2) \<notin> \<rat>"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2017_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2017_b4 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_2017_b4_solution :: real where "putnam_2017_b4_solution \<equiv> undefined"
 6 | (* (ln 2) ^ 2 *)
 7 | theorem putnam_2017_b4:
 8 |   shows "(\<Sum> k :: nat. 3 * ln (4 * k + 2) / (4 * k + 2) - ln (4 * k + 3) / (4 * k + 3) - ln (4 * k + 4) / (4 * k - 4) - ln (4 * k + 5) / (4 * k + 5)) = putnam_2017_b4_solution"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2018_a5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2018_a5 imports Complex_Main
 2 | "HOL-Analysis.Derivative"
 3 | begin
 4 | 
 5 | theorem putnam_2018_a5:
 6 |   fixes f :: "real\<Rightarrow>real"
 7 |   assumes f_diff : "\<forall>n. (deriv^^n) f differentiable_on UNIV"
 8 |     and f0 : "f 0 = 0"
 9 |     and f1 : "f 1 = 1"
10 |     and fpos : "\<forall>x. f x \<ge> 0"
11 |   shows "\<exists>n > 0. \<exists>x::real. (deriv^^n) f x < 0"
12 |   sorry
13 | 
14 | end
15 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2018_b2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2018_b2 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2018_b2:
 5 |   fixes n :: nat and f_n :: "complex \<Rightarrow> complex" and z :: complex
 6 |   defines "f_n \<equiv> \<lambda>x::complex. \<Sum>i=0..(n-1). (n - i) * x^i"
 7 |   assumes npos: "n > 0" and zunit: "norm(z) \<le> 1"
 8 |   shows "f_n(z) \<noteq> 0"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2018_b3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2018_b3 imports Complex_Main
 2 | begin
 3 | 
 4 | definition putnam_2018_b3_solution::"int set" where "putnam_2018_b3_solution \<equiv> undefined"
 5 | (* {2^2, 2^4, 2^16, 2^256} *)
 6 | theorem putnam_2018_b3:
 7 |   shows "putnam_2018_b3_solution \<equiv> {n::int. n > 3 \<and> n < 10^100 \<and> 
 8 |     n dvd 2^(nat n) \<and> (n-1) dvd (2^(nat n) - 1) \<and> (n-2) dvd (2^(nat n) - 2)}"
 9 |   sorry
10 | 
11 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2018_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2018_b4 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2018_b4:
 5 |   fixes a::real and f::"nat \<Rightarrow> real"
 6 |   assumes f0 : "f 0 = 1"
 7 |     and f1 : "f 1 = a"
 8 |     and f2 : "f 2 = a"
 9 |     and fn : "\<forall>n \<ge> 2. f (n+1) = 2  * f n * f (n-1) - f (n-2)"
10 |     and fex0 : "\<exists>n. f n = 0"
11 |   shows "\<exists>T>0. \<forall>n. f (n + T) = f n"
12 |   sorry
13 | 
14 | end
15 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2020_a2.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2020_a2 imports
 2 | Complex_Main
 3 | begin
 4 | 
 5 | definition putnam_2020_a2_solution :: "nat \<Rightarrow> nat" where "putnam_2020_a2_solution \<equiv> undefined"
 6 | (* \<lambda> k. 4 ^ k *)
 7 | theorem putnam_2020_a2:
 8 |   fixes k :: nat
 9 |   shows "(\<Sum> j = 0..k. 2 ^ (k - j) * (k + j choose j)) = putnam_2020_a2_solution k"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2020_b5.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2020_b5 imports Complex_Main
 2 | begin
 3 | 
 4 | (* Note: Boosted domain to infinite set *)
 5 | theorem putnam_2020_b5:
 6 |   fixes z :: "nat \<Rightarrow> complex"
 7 |   assumes hz: "\<forall>n \<in> {1..4}. norm (z n) = 1"
 8 |     and hzne1: "\<forall>n \<in> {1..4}. z n \<noteq> 1"
 9 |   shows "3 - z 1 - z 2 - z 3 - z 4 + z 1 * z 2 * z 3 * z 4 \<noteq> 0"
10 |   sorry
11 | 
12 | end
13 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2020_b6.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2020_b6 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2020_b6:
 5 |   fixes n :: nat
 6 |   assumes npos: "n > 0"
 7 |   shows "(\<Sum>k::nat=1..n. (-1) ^ (nat \<lfloor>k * (sqrt 2 - 1)\<rfloor>)) \<ge> 0"
 8 |   sorry
 9 | 
10 | end
11 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2021_b4.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2021_b4 imports Complex_Main
 2 | begin
 3 | 
 4 | theorem putnam_2021_b4:
 5 |   fixes f :: "nat\<Rightarrow>nat"
 6 |   assumes hf : "\<forall>x\<ge>2. f x = f (x-1) + f (x-2)"
 7 |     and f01 : "f 0 = 0 \<and> f 1 = 1"
 8 |   shows "\<forall>m > 2. \<exists>p. (\<Prod>k=1..(f m - 1). k^k) mod f m = f p"
 9 |   sorry
10 | 
11 | end
12 | 


--------------------------------------------------------------------------------
/isabelle/putnam_2022_a3.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2022_a3 imports Complex_Main
 2 | "HOL-Number_Theory.Cong"
 3 | begin
 4 | 
 5 | theorem putnam_2022_a3:
 6 |   fixes p f::int
 7 |   defines "f \<equiv> card {a :: nat \<Rightarrow> int. \<forall> n :: nat. a n \<in> {1..(p-1)} \<and> [a n * a (n + 2) = 1 + a (n + 1)] (mod p)}"
 8 |   assumes hp : "prime p \<and> p > 5"
 9 |   shows "[f = 0] (mod 5) \<or> [f = 2] (mod 5)"
10 |   sorry
11 | 
12 | end


--------------------------------------------------------------------------------
/isabelle/putnam_2022_b1.thy:
--------------------------------------------------------------------------------
 1 | theory putnam_2022_b1 imports Complex_Main "HOL-Computational_Algebra.Polynomial"
 2 | begin
 3 | 
 4 | theorem putnam_2022_b1:
 5 |   fixes P :: "int poly"
 6 |     and b :: "nat \<Rightarrow> real"
 7 |   assumes Pconst: "coeff P 0 = 0"
 8 |     and podd: "odd (coeff P 1)"
 9 |     and hb: "\<forall>x::real. exp (poly (map_poly real_of_int P) x) = (\<Sum> n :: nat. ((b n) * x^n))"
10 |   shows "\<forall>k::nat. b k \<noteq> 0"
11 |   sorry
12 | 
13 | end


--------------------------------------------------------------------------------
/lean4/lean-toolchain:
--------------------------------------------------------------------------------
1 | leanprover/lean4:v4.17.0


--------------------------------------------------------------------------------
/lean4/src/putnam_1962_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MeasureTheory
 4 | 
 5 | /--
 6 | Given five points in a plane, no three of which lie on a straight line, show that some four of these points form the vertices of a convex quadrilateral.
 7 | -/
 8 | theorem putnam_1962_a1
 9 | (S : Set (ℝ × ℝ))
10 | (hS : S.ncard = 5)
11 | (hnoncol : ∀ s ⊆ S, s.ncard = 3 → ¬Collinear ℝ s)
12 | : ∃ T ⊆ S, T.ncard = 4 ∧ ¬∃ t ∈ T, t ∈ convexHull ℝ (T \ {t}) :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1962_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_1962_a5_solution : ℕ → ℕ := sorry
 4 | -- fun n : ℕ => n * (n + 1) * 2^(n - 2)
 5 | /--
 6 | Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
 7 | -/
 8 | theorem putnam_1962_a5
 9 | : ∀ n ≥ 2, putnam_1962_a5_solution n = ∑ k ∈ Finset.Icc 1 n, Nat.choose n k * k^2 :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1962_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MeasureTheory
 4 | 
 5 | /--
 6 | Prove that for every integer $n$ greater than 1: \[ \frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^n + \left(\frac{2}{n} \right)^n + \cdots + \left(\frac{n}{n} \right)^n < 2. \]
 7 | -/
 8 | theorem putnam_1962_b5
 9 | (n : ℤ)
10 | (ng1 : n > 1)
11 | : (3 * (n : ℝ) + 1) / (2 * n + 2) < ∑ i : Finset.Icc 1 n, ((i : ℝ) / n) ^ (n : ℝ) ∧ ∑ i : Finset.Icc 1 n, ((i : ℝ) / n) ^ (n : ℝ) < 2 :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1963_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Polynomial
 4 | 
 5 | abbrev putnam_1963_b1_solution : ℤ := sorry
 6 | -- 2
 7 | /--
 8 | For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
 9 | -/
10 | theorem putnam_1963_b1
11 | : ∀ a : ℤ, (X^2 - X + (C a)) ∣ (X ^ 13 + X + (C 90)) ↔ a = putnam_1963_b1_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1964_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Function Filter Topology
 4 | 
 5 | /--
 6 | Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous and for every $\alpha > 0$, $\lim_{n \to \infty} f(n\alpha) = 0$. Prove that $\lim_{x \to \infty} f(x) = 0$.
 7 | -/
 8 | theorem putnam_1964_b3
 9 | (f : ℝ → ℝ)
10 | (hf : Continuous f ∧ ∀ α > 0, Tendsto (fun n : ℕ ↦ f (n * α)) atTop (𝓝 0))
11 | : (Tendsto f atTop (𝓝 0)) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1965_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open EuclideanGeometry
 4 | 
 5 | /--
 6 | Prove that $$\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \left(\frac{n - 2r}{n} {n \choose r}\right)^2 = \frac{1}{n} {{2n - 2} \choose {n - 1}}$$ for every positive integer $n$.
 7 | -/
 8 | theorem putnam_1965_a2
 9 | : ∀ n > 0, ∑ r ∈ Finset.Icc 0 ((n - 1)/2), ((n - 2*r) * Nat.choose n r / (n : ℚ))^2 = (Nat.choose (2*n - 2) (n - 1))/(n : ℚ) :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1965_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open EuclideanGeometry Topology Filter Complex
 4 | 
 5 | /--
 6 | Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter.
 7 | -/
 8 | theorem putnam_1965_b3
 9 | : {(a, b, c) : ℤ × ℤ × ℤ | a > 0 ∧ a ≤ b ∧ c > 0 ∧ a^2 + b^2 = c^2 ∧ a*b/(2 : ℚ) = 2*(a + b + c)}.ncard = 3 :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1965_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open EuclideanGeometry Topology Filter Complex SimpleGraph.Walk
 4 | 
 5 | /--
 6 | Prove that, if $4E \le V^2$, there exists a graph with $E$ edges and $V$ vertices with no triangles (cycles of length $3$).
 7 | -/
 8 | theorem putnam_1965_b5
 9 | {K : Type*}
10 | [Fintype K]
11 | (V E : ℕ)
12 | (hV : V = Nat.card K)
13 | (hE: 4*E ≤ V^2)
14 | : ∃ G : SimpleGraph K, G.edgeSet.ncard = E ∧ ∀ a : K, ∀ w : G.Walk a a, w.length ≠ 3 :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1966_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | If $0 < x_1 < 1$ and $x_{n+1} = x_n(1 - x_n)$ for all $n \ge 1$, prove that $\lim_{n \to \infty} nx_n = 1$.
 7 | -/
 8 | theorem putnam_1966_a3
 9 | (x : ℕ → ℝ)
10 | (hx1 : 0 < x 1 ∧ x 1 < 1)
11 | (hxi : ∀ n ≥ 1, x (n + 1) = (x n) * (1 - (x n)))
12 | : Tendsto (fun n : ℕ => n * (x n)) atTop (𝓝 1) :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1966_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Prove that the $n$th item in the ascending list of non-perfect-square positive integers equals $n + \{\sqrt{n}\}$, where $\{m\}$ denotes the closest integer to $m$.
 7 | -/
 8 | theorem putnam_1966_a4
 9 | (a : ℕ → ℤ)
10 | (ha1 : a 1 = 2)
11 | (hai : ∀ n ≥ 1, a (n + 1) = (if ∃ m : ℤ, m^2 = a n + 1 = True then a n + 2 else a n + 1))
12 | : ∀ n ≥ 1, a n = n + round (Real.sqrt n) :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1966_a6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$
 7 | -/
 8 | theorem putnam_1966_a6
 9 | (a : ℕ → (ℕ → ℝ))
10 | (ha : ∀ n ≥ 1, a n n = n ∧ ∀ m ≥ 1, m < n → a n m = m * Real.sqrt (1 + a n (m + 1)))
11 | : Tendsto (fun n => a n 1) atTop (𝓝 3) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1966_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Prove that, for any ten consecutive integers, at least one is relatively prime to all of the others.
 5 | -/
 6 | theorem putnam_1966_b2
 7 | (S : ℤ → Set ℤ)
 8 | (hS : S = fun n : ℤ => {n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9})
 9 | : ∀ n : ℤ, n > 0 → (∃ k ∈ S n, ∀ m ∈ S n, k ≠ m → IsCoprime m k) :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1966_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Prove that any solution $y(x)$ to the differential equation $y'' + e^{x}y = 0$ remains bounded as $x$ goes to $+\infty$.
 7 | -/
 8 | theorem putnam_1966_b6
 9 | (y : ℝ → ℝ)
10 | (hy : Differentiable ℝ y ∧ Differentiable ℝ (deriv y))
11 | (diffeq : deriv (deriv y) + Real.exp * y = 0)
12 | : ∃ r s N : ℝ, ∀ x > N, r ≤ y x ∧ y x ≤ s :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1967_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Topology Filter
 4 | 
 5 | /--
 6 | Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u$ such that for all $x$ in the closed interval $0 \leq x \leq 1$, $u(x)=1+\lambda\int_x^1 u(y)u(y-x)\,dy$.
 7 | -/
 8 | theorem putnam_1967_a4
 9 | (lambda : ℝ)
10 | (hlambda : lambda > 1 / 2)
11 | : ¬∃ u : ℝ → ℝ, ∀ x ∈ Set.Icc 0 1, u x = 1 + lambda * (∫ y in Set.Ioo x 1, u y * u (y - x)) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1967_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Topology Filter
 4 | 
 5 | /--
 6 | Prove that any convex region in the Euclidean plane with area greater than $\pi/4$ contains a pair of points exactly $1$ unit apart.
 7 | -/
 8 | theorem putnam_1967_a5
 9 | (R : Set (EuclideanSpace ℝ (Fin 2)))
10 | (hR : Convex ℝ R ∧ (MeasureTheory.volume R).toReal > Real.pi / 4)
11 | : ∃ P ∈ R, ∃ Q ∈ R, dist P Q = 1 :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1967_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Topology Filter
 4 | 
 5 | /--
 6 | For any positive integer $n$, prove that the sum of the first $n$ terms of the bimonial expansion of $(2 - 1)^{-n}$ (starting with the maximal exponent of $2$) is $\frac{1}{2}.$
 7 | -/
 8 | theorem putnam_1967_b5
 9 | (n : ℕ)
10 | (hn : n > 0)
11 | : (1 : ℚ)/2 = ∑ i ∈ Finset.range n, (Nat.choose (n + i - 1) i) * (2 : ℚ)^(-(n : ℤ) - i) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1968_a1.lean:
--------------------------------------------------------------------------------
1 | import Mathlib
2 | 
3 | /--
4 | Prove that $$\frac{22}{7} - \pi = \int_{0}^{1} \frac{x^4(1 - x)^4}{1 + x^2} dx$$.
5 | -/
6 | theorem putnam_1968_a1
7 | : 22/7 - Real.pi = ∫ x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2) :=
8 | sorry
9 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1968_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Finset
 4 | 
 5 | /--
 6 | Prove that the sum of the squares of the distances between any $n$ points on the unit sphere $\{(x, y, z) \mid x^2 + y^2 + z^2 = 1\}$ is at most $n^2$.
 7 | -/
 8 | theorem putnam_1968_a4
 9 | (n : ℕ)
10 | (S : Fin n → (EuclideanSpace ℝ (Fin 3)))
11 | (hS : ∀ i : Fin n, dist 0 (S i) = 1)
12 | : ∑ i : Fin n, ∑ j : Fin n, (if i < j then (dist (S i) (S j))^2 else (0 : ℝ)) ≤ n^2 :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1968_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Finset Polynomial Topology Filter Metric
 4 | 
 5 | /--
 6 | Prove that no sequence $\{K_n\}_{n=0}^{\infty}$ of compact (closed and bounded) sets of rational numbers has the property that every compact set of rational numbers is contained by at least one $K_n$.
 7 | -/
 8 | theorem putnam_1968_b6
 9 | : ¬∃ K : ℕ → Set ℚ, (∀ n : ℕ, IsCompact (K n)) ∧ (∀ S : Set ℚ, IsCompact S → ∃ n : ℕ, S ⊆ K n) :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1969_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Matrix Filter Topology Set Nat
 4 | 
 5 | /--
 6 | Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$.
 7 | -/
 8 | theorem putnam_1969_a4
 9 | : Tendsto (fun n => ∑ i ∈ Finset.Icc (1 : ℤ) n, (-1)^(i+1)*(i : ℝ)^(-i)) atTop (𝓝 (∫ x in Ioo (0 : ℝ) 1, x^x)) :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1969_a6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Matrix Filter Topology Set Nat
 4 | 
 5 | /--
 6 | Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges.
 7 | -/
 8 | theorem putnam_1969_a6
 9 | (x : ℕ → ℝ)
10 | (y : ℕ → ℝ)
11 | (hy1 : ∀ n ≥ 2, y n = x (n-1) + 2 * (x n))
12 | (hy2 : ∃ c : ℝ, Tendsto y atTop (𝓝 c))
13 | : ∃ C : ℝ, Tendsto x atTop (𝓝 C) :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1969_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Matrix Filter Topology Set Nat
 4 | 
 5 | /--
 6 | Let $n$ be a positive integer such that $n+1$ is divisible by $24$. Prove that the sum of all the divisors of $n$ is divisible by $24$.
 7 | -/
 8 | theorem putnam_1969_b1
 9 | (n : ℕ)
10 | (hnpos : n > 0)
11 | (hn : 24 ∣ n + 1)
12 | : 24 ∣ ∑ d ∈ divisors n, d :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1969_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Matrix Filter Topology Set Nat
 4 | 
 5 | /--
 6 | Suppose $T$ is a sequence which satisfies $T_n * T_{n+1} = n$ whenever $n \geq 1$, and also $\lim_{n \to \infty} \frac{T_n}{T_{n+1}} = 1. Show that $\pi * T_1^2 = 2$.
 7 | -/
 8 | theorem putnam_1969_b3
 9 | (T : ℕ → ℝ)
10 | (hT1 : ∀ n : ℕ, n ≥ 1 → (T n) * (T (n + 1)) = n)
11 | (hT2 : Tendsto (fun n => (T n)/(T (n + 1))) atTop (𝓝 1))
12 | : Real.pi * (T 1)^2 = 2 :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1970_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Metric Set EuclideanGeometry Filter Topology
 4 | 
 5 | /--
 6 | Suppose $(x_n)$ is a sequence such that $\lim_{n \to \infty} (x_n - x_{n-2} = 0$. Prove that $\lim_{n \to \infty} \frac{x_n - x_{n-1}}{n} = 0$.
 7 | -/
 8 | theorem putnam_1970_a4
 9 | (x : ℕ → ℝ)
10 | (hxlim : Tendsto (fun n => x (n+2) - x n) atTop (𝓝 0))
11 | : Tendsto (fun n => (x (n+1) - x (n))/(n+1)) atTop (𝓝 0) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1970_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Metric Set EuclideanGeometry Filter Topology
 4 | 
 5 | /--
 6 | A closed subset $S$ of $\mathbb{R}^2$ lies in $a < x < b$. Show that its projection on the y-axis is closed.
 7 | -/
 8 | theorem putnam_1970_b3
 9 | (S : Set (ℝ × ℝ))
10 | (a b : ℝ)
11 | (hab : a < b)
12 | (hS : ∀ s ∈ S, s.1 ∈ Ioo a b)
13 | (hSclosed : IsClosed S)
14 | : IsClosed {y | ∃ x : ℝ, ⟨x,y⟩ ∈ S} :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1971_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set
 4 | 
 5 | abbrev putnam_1971_a2_solution : Set (Polynomial ℝ) := sorry
 6 | -- {Polynomial.X}
 7 | /--
 8 | Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.
 9 | -/
10 | theorem putnam_1971_a2
11 |     (P : Polynomial ℝ) :
12 |     (P.eval 0 = 0 ∧ (∀ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) ↔ P ∈ putnam_1971_a2_solution :=
13 |   sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1971_a6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set MvPolynomial
 4 | 
 5 | /--
 6 | Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.
 7 | -/
 8 | theorem putnam_1971_a6
 9 | (c : ℝ)
10 | (hc : ∀ n : ℤ, n > 0 → ∃ m : ℤ, (n : ℝ)^c = m)
11 | : ∃ m : ℤ, m ≥ 0 ∧ c = m :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1971_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set MvPolynomial
 4 | 
 5 | /--
 6 | Let $\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\sum_{n = 1}^x \delta(n)/n - 2x/3| < 1$ for all positive integers $x$.
 7 | -/
 8 | theorem putnam_1971_b6
 9 | (δ : ℤ → ℤ)
10 | (hδ : δ = fun n => sSup {t | Odd t ∧ t ∣ n})
11 | : ∀ x : ℤ, x > 0 → |∑ i ∈ Finset.Icc 1 x, (δ i)/(i : ℚ) - 2*x/3| < 1 :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1972_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open EuclideanGeometry Filter Topology Set
 4 | 
 5 | /--
 6 | Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$.
 7 | -/
 8 | theorem putnam_1972_a5
 9 | (n : ℕ)
10 | (hn : n > 1)
11 | : ¬((n : ℤ) ∣ 2^n - 1) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1972_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open EuclideanGeometry Filter Topology Set MeasureTheory Metric
 4 | 
 5 | /--
 6 | Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$, and $B^{2n-1} = 1$ for some positive integer $n$. Prove that $B = 1$.
 7 | -/
 8 | theorem putnam_1972_b3
 9 | (G : Type*) [Group G]
10 | (A B : G)
11 | (hab : A * B * A = B * A^2 * B ∧ A^3 = 1 ∧ (∃ n : ℤ, n > 0 ∧ B^(2*n - 1) = 1))
12 | : B = 1 :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1972_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open EuclideanGeometry Set Metric
 4 | 
 5 | /--
 6 | Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$.
 7 | -/
 8 | theorem putnam_1972_b5
 9 | (A B C D : EuclideanSpace ℝ (Fin 3))
10 | (hnonplanar : ¬Coplanar ℝ {A, B, C, D})
11 | (hangles : ∠ A B C = ∠ C D A ∧ ∠ B C D = ∠ D A B)
12 | : dist A B = dist C D ∧ dist B C = dist D A :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1973_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Set MeasureTheory Topology Filter
 4 | 
 5 | abbrev putnam_1973_a4_solution : ℕ := sorry
 6 | -- 3
 7 | /--
 8 | How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?
 9 | -/
10 | theorem putnam_1973_a4
11 | (f : ℝ → ℝ)
12 | (hf : f = fun x => 2^x - 1 - x^2)
13 | : putnam_1973_a4_solution = {x : ℝ | f x = 0}.ncard :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1973_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Set MeasureTheory Topology Filter
 4 | 
 5 | /--
 6 | Let $z = x+iy$ be a complex number with $x$ and $y$ rational and with $\| z \| = 1$. Show thaat the number $\| z^{2n} - 1 \|$ is rational for every integer $n$.
 7 | -/
 8 | theorem putnam_1973_b2
 9 | (z : ℂ)
10 | (hzrat : ∃ q1 q2 : ℚ, z.re = q1 ∧ z.im = q2)
11 | (hznorm : ‖z‖ = 1)
12 | : ∀ n : ℤ, ∃ q : ℚ, ‖z^(2*n) - 1‖ = q :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1974_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Nat Polynomial Filter Topology
 4 | 
 5 | /--
 6 | Prove that if $\alpha$ is a real number such that $\cos (\pi \alpha) = 1/3$, the $\alpha$ is irrational.
 7 | -/
 8 | theorem putnam_1974_b3
 9 | (α : ℝ)
10 | (ha : Real.cos (Real.pi * α) = (1 : ℝ)/3)
11 | : Irrational α :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1974_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Nat Polynomial Filter Topology
 4 | 
 5 | /--
 6 | Show that $1 + (n/1!) + (n^2/2!) + \dots + (n^n/n!) > e^n/2$ for every integer $n \geq 0$.
 7 | -/
 8 | theorem putnam_1974_b5
 9 | : ∀ n ≥ 0, ∑ i ∈ Finset.Icc (0 : ℕ) n, (n^i : ℝ)/(Nat.factorial i) > (Real.exp n)/2 :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1976_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_1976_a3_solution : Set (ℕ × ℕ × ℕ × ℕ) := sorry
 4 | -- {(3, 2, 2, 3), (2, 3, 3, 2)}
 5 | /--
 6 | Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.
 7 | -/
 8 | theorem putnam_1976_a3
 9 | : {(p, r, q, s) : ℕ × ℕ × ℕ × ℕ | Nat.Prime p ∧ Nat.Prime q ∧ r > 1 ∧ s > 1 ∧ |(p^r : ℤ) - q^s| = 1} = putnam_1976_a3_solution :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1976_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Polynomial Filter Topology ProbabilityTheory MeasureTheory
 4 | 
 5 | noncomputable abbrev putnam_1976_b5_solution : ℕ → Polynomial ℤ := sorry
 6 | -- fun n => C (Nat.factorial n)
 7 | /--
 8 | Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$
 9 | -/
10 | theorem putnam_1976_b5
11 | : ∀ n : ℕ, ∑ k ∈ Finset.range (n + 1), C ((-(1 : ℤ))^k * Nat.choose n k) * (X - (C (k : ℤ)))^n = putnam_1976_b5_solution n :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1977_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open RingHom Set
 4 | 
 5 | noncomputable abbrev putnam_1977_a4_solution : RatFunc ℝ := sorry
 6 | -- RatFunc.X / (1 - RatFunc.X)
 7 | /--
 8 | Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
 9 | -/
10 | theorem putnam_1977_a4 :
11 |     ∀ x ∈ Ioo 0 1,
12 |       putnam_1977_a4_solution.eval (id ℝ) x = ∑' n : ℕ, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1)) :=
13 |   sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1977_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open RingHom Set Nat
 4 | 
 5 | /--
 6 | Let $p$ be a prime and $m \geq n$ be non-negative integers. Show that $\binom{pm}{pn} = \binom{m}{n} \pmod p$, where $\binom{m}{n}$ is the binomial coefficient.
 7 | -/
 8 | theorem putnam_1977_a5
 9 | (p m n : ℕ)
10 | (hp : Nat.Prime p)
11 | (hmgen : m ≥ n)
12 | : (choose (p * m) (p * n) ≡ choose m n [MOD p]) :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1977_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open RingHom Set Nat Filter Topology
 4 | 
 5 | noncomputable abbrev putnam_1977_b1_solution : ℝ := sorry
 6 | -- 2 / 3
 7 | /--
 8 | Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.
 9 | -/
10 | theorem putnam_1977_b1
11 | : Tendsto (fun N ↦ ∏ n ∈ Finset.Icc (2 : ℤ) N, ((n : ℝ) ^ 3 - 1) / (n ^ 3 + 1)) atTop (𝓝 putnam_1977_b1_solution) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1978_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $S = \{1, 4, 7, 10, 13, 16, \dots , 100\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$.
 5 | -/
 6 | theorem putnam_1978_a1
 7 | (S T : Set ℤ)
 8 | (hS : S = {k | ∃ j : ℤ, 0 ≤ j ∧ j ≤ 33 ∧ k = 3 * j + 1})
 9 | (hT : T ⊆ S ∧ T.ncard = 20)
10 | : (∃ m ∈ T, ∃ n ∈ T, m ≠ n ∧ m + n = 104) :=
11 | sorry
12 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1978_a6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Real
 4 | 
 5 | /--
 6 | Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart.
 7 | -/
 8 | theorem putnam_1978_a6
 9 | (S : Finset (EuclideanSpace ℝ (Fin 2)))
10 | (n : ℕ)
11 | (hn : n = S.card)
12 | (npos : n > 0)
13 | : ({pair : Set (EuclideanSpace ℝ (Fin 2)) | ∃ P ∈ S, ∃ Q ∈ S, pair = {P, Q} ∧ dist P Q = 1}.ncard < 2 * (n : ℝ) ^ ((3 : ℝ) / 2)) :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1978_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Real
 4 | 
 5 | abbrev putnam_1978_b2_solution : ℚ := sorry
 6 | -- 7 / 4
 7 | /--
 8 | Find
 9 | \[
10 | \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}.
11 | \]
12 | -/
13 | theorem putnam_1978_b2
14 | : (∑' i : ℕ+, ∑' j : ℕ+, (1 : ℚ) / (i ^ 2 * j + 2 * i * j + i * j ^ 2) = putnam_1978_b2_solution) :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1978_b4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Real Filter Topology Polynomial
 4 | 
 5 | /--
 6 | Show that we can find integers $a, b, c, d$ such that $a^2 + b^2 + c^2 + d^2 = abc + abd + acd + bcd$, and the smallest of $a, b, c, d$ is arbitrarily large.
 7 | -/
 8 | theorem putnam_1978_b4 :
 9 |   ∀ N : ℝ,
10 |     ∃ a b c d : ℤ,
11 |       a > N ∧ b > N ∧ c > N ∧ d > N ∧
12 |       a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = a * b * c + a * b * d + a * c * d + b * c * d :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1979_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_1979_a2_solution : ℝ → Prop := sorry
 4 | -- fun k : ℝ => k ≥ 0
 5 | /--
 6 | For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?
 7 | -/
 8 | theorem putnam_1979_a2
 9 | : ∀ k : ℝ, (∃ f : ℝ → ℝ, Continuous f ∧ ∀ x : ℝ, f (f x) = k*x^9) ↔ putnam_1979_a2_solution k :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1980_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | noncomputable abbrev putnam_1980_a3_solution : ℝ := sorry
 4 | -- Real.pi / 4
 5 | /--
 6 | Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
 7 | -/
 8 | theorem putnam_1980_a3
 9 | : ∫ x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)) = putnam_1980_a3_solution :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1980_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Real
 4 | 
 5 | abbrev putnam_1980_b1_solution : Set ℝ := sorry
 6 | -- {c : ℝ | c ≥ 1 / 2}
 7 | /--
 8 | For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
 9 | -/
10 | theorem putnam_1980_b1
11 | (c : ℝ)
12 | : (∀ x : ℝ, (exp x + exp (-x)) / 2 ≤ exp (c * x ^ 2)) ↔ c ∈ putnam_1980_b1_solution :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1982_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Function Filter Topology Polynomial Real
 4 | 
 5 | noncomputable abbrev putnam_1982_a3_solution : ℝ := sorry
 6 | -- (Real.pi / 2) * log Real.pi
 7 | /--
 8 | Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$.
 9 | -/
10 | theorem putnam_1982_a3 :
11 |   Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 putnam_1982_a3_solution) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1982_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$.
 5 | -/
 6 | theorem putnam_1982_a5
 7 | (a b c d : ℤ)
 8 | (hpos : a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0)
 9 | (hac : a + c ≤ 1982)
10 | (hfrac : (a : ℝ) / b + (c : ℝ) / d < 1)
11 | : (1 - (a : ℝ) / b - (c : ℝ) / d > 1 / 1983 ^ 3) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1983_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_1983_a1_solution : ℕ := sorry
 4 | -- 2301
 5 | /--
 6 | How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?
 7 | -/
 8 | theorem putnam_1983_a1
 9 | : {n : ℤ | n > 0 ∧ (n ∣ 10 ^ 40 ∨ n ∣ 20 ^ 30)}.encard = putnam_1983_a1_solution :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1983_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat
 4 | 
 5 | /--
 6 | Prove that for $m = 5 \pmod 6$,
 7 | \[
 8 | \binom{m}{2} - \binom{m}{5} + \binom{m}{8} - \binom{m}{11} + ... - \binom{m}{m-6} + \binom{m}{m-3} \neq 0.
 9 | \]
10 | -/
11 | theorem putnam_1983_a4
12 | (k m : ℕ)
13 | (S : ℤ)
14 | (kpos : k > 0)
15 | (hm : m = 6 * k - 1)
16 | (hS : S = ∑ j ∈ Finset.Icc 1 (2 * k - 1), (-1 : ℤ) ^ (j + 1) * choose m (3 * j - 1))
17 | : (S ≠ 0) :=
18 | sorry
19 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1983_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat
 4 | 
 5 | abbrev putnam_1983_a5_solution : Prop := sorry
 6 | -- True
 7 | /--
 8 | Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)
 9 | -/
10 | theorem putnam_1983_a5 :
11 |     (∃ α : ℝ, α > 0 ∧ ∀ n : ℕ, n > 0 → Even (⌊α ^ n⌋ - n)) ↔ putnam_1983_a5_solution :=
12 |   sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1984_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_1984_a2_solution : ℚ := sorry
 4 | -- 2
 5 | /--
 6 | Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.
 7 | -/
 8 | theorem putnam_1984_a2
 9 | : ∑' k : Set.Ici 1, (6 ^ (k : ℕ) / ((3 ^ ((k : ℕ) + 1) - 2 ^ ((k : ℕ) + 1)) * (3 ^ (k : ℕ) - 2 ^ (k : ℕ)))) = putnam_1984_a2_solution :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1986_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_1986_a2_solution : ℕ := sorry
 4 | -- 3
 5 | /--
 6 | What is the units (i.e., rightmost) digit of
 7 | \[
 8 | \left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
 9 | \]
10 | -/
11 | theorem putnam_1986_a2
12 | : (Nat.floor ((10 ^ 20000 : ℝ) / (10 ^ 100 + 3)) % 10 = putnam_1986_a2_solution) :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1987_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MvPolynomial Real Nat
 4 | 
 5 | abbrev putnam_1987_b1_solution : ℝ := sorry
 6 | -- 1
 7 | /--
 8 | Evaluate
 9 | \[
10 | \int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
11 | \]
12 | -/
13 | theorem putnam_1987_b1
14 | : (∫ x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))) = putnam_1987_b1_solution) :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1988_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MeasureTheory
 4 | 
 5 | abbrev putnam_1988_a1_solution : ℝ := sorry
 6 | -- 6
 7 | /--
 8 | Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$.
 9 | -/
10 | theorem putnam_1988_a1
11 |     (R : Set (Fin 2 → ℝ))
12 |     (hR : R = {p | |p 0| - |p 1| ≤ 1 ∧ |p 1| ≤ 1}) :
13 |     (volume R).toReal = putnam_1988_a1_solution :=
14 |   sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1988_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Filter Topology
 4 | 
 5 | /--
 6 | A \emph{composite} (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.
 7 | -/
 8 | theorem putnam_1988_b1
 9 | : ∀ a ≥ 2, ∀ b ≥ 2, ∃ x y z : ℤ, x > 0 ∧ y > 0 ∧ z > 0 ∧ a * b = x * y + x * z + y * z + 1 :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1988_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Filter Topology
 4 | 
 5 | abbrev putnam_1988_b2_solution : Prop := sorry
 6 | -- True
 7 | /--
 8 | Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
 9 | -/
10 | theorem putnam_1988_b2
11 | : (∀ x y : ℝ, (y ≥ 0 ∧ y * (y + 1) ≤ (x + 1) ^ 2) → (y * (y - 1) ≤ x ^ 2)) ↔ putnam_1988_b2_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1989_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | open Complex
 3 | 
 4 | /--
 5 | Prove that if
 6 | \[
 7 | 11z^{10}+10iz^9+10iz-11=0,
 8 | \]
 9 | then $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.)
10 | -/
11 | theorem putnam_1989_a3
12 | (z : ℂ)
13 | (hz : 11 * z ^ 10 + 10 * I * z ^ 9 + 10 * I * z - 11 = 0)
14 | : (‖z‖ = 1) :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1991_b4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology
 4 | 
 5 | /--
 6 | Suppose $p$ is an odd prime. Prove that $\sum_{j=0}^p \binom{p}{j}\binom{p+j}{j} \equiv 2^p+1 \pmod{p^2}$.
 7 | -/
 8 | theorem putnam_1991_b4
 9 | (p : ℕ)
10 | (podd : Odd p)
11 | (pprime : Prime p)
12 | : (∑ j : Fin (p + 1), (p.choose j) * ((p + j).choose j)) ≡ (2 ^ p + 1) [MOD (p ^ 2)] :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1993_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $(x_n)_{n \geq 0}$ be a sequence of nonzero real numbers such that $x_n^2-x_{n-1}x_{n+1}=1$ for $n=1,2,3,\dots$. Prove there exists a real number $a$ such that $x_{n+1}=ax_n-x_{n-1}$ for all $n \geq 1$.
 5 | -/
 6 | theorem putnam_1993_a2
 7 | (x : ℕ → ℝ)
 8 | (xnonzero : ∀ n : ℕ, x n ≠ 0)
 9 | (hx : ∀ n ≥ 1, (x n) ^ 2 - x (n - 1) * x (n + 1) = 1)
10 | : ∃ a : ℝ, ∀ n ≥ 1, x (n + 1) = a * x n - x (n - 1) :=
11 | sorry
12 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1993_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
 5 | -/
 6 | theorem putnam_1993_b5:
 7 |   ¬∃ p : Fin 4 → (EuclideanSpace ℝ (Fin 2)),
 8 |     ∀ i j, i ≠ j →
 9 |       (∃ n : ℤ, dist (p i) (p j) = n ∧ Odd n) :=
10 |   sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1994_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology
 4 | 
 5 | /--
 6 | Suppose that a sequence $a_1,a_2,a_3,\dots$ satisfies $0< a_n \leq a_{2n}+a_{2n+1}$ for all $n \geq 1$. Prove that the series $\sum_{n=1}^\infty a_n$ diverges.
 7 | -/
 8 | theorem putnam_1994_a1
 9 |     (a : ℕ → ℝ)
10 |     (ha : ∀ n ≥ 1, 0 < a n ∧ a n ≤ a (2 * n) + a (2 * n + 1)) :
11 |     Tendsto (fun N : ℕ => ∑ n : Set.Icc 1 N, a n) atTop atTop :=
12 |   sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1994_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology
 4 | 
 5 | abbrev putnam_1994_b1_solution : Set ℤ := sorry
 6 | -- {n : ℤ | (315 ≤ n ∧ n ≤ 325) ∨ (332 ≤ n ∧ n ≤ 350)}
 7 | /--
 8 | Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.
 9 | -/
10 | theorem putnam_1994_b1
11 |     (n : ℤ) :
12 |     n ∈ putnam_1994_b1_solution ↔
13 |     (0 < n ∧ {m : ℕ | |n - m ^ 2| ≤ 250}.encard = 15) :=
14 |   sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1994_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology
 4 | 
 5 | /--
 6 | For any integer $a$, set $n_a=101a-100 \cdot 2^a$. Show that for $0 \leq a,b,c,d \leq 99$, $n_a+n_b \equiv n_c+n_d \pmod{10100}$ implies $\{a,b\}=\{c,d\}$.
 7 | -/
 8 | theorem putnam_1994_b6
 9 | (n : ℕ → ℤ)
10 | (hn : ∀ a : ℕ, n a = 101 * a - 100 * 2 ^ a)
11 | : ∀ a b c d : Set.Icc 0 99, (n a + n b ≡ n c + n d [ZMOD 10100]) → (({a, b} : Set (Set.Icc 0 99)) = {c, d}) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1996_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Function
 4 | 
 5 | /--
 6 | If $p$ is a prime number greater than 3 and $k = \lfloor 2p/3 \rfloor$, prove that the sum \[\binom p1 + \binom p2 + \cdots + \binom pk \] of binomial coefficients is divisible by $p^2$.
 7 | -/
 8 | theorem putnam_1996_a5
 9 | (p : ℕ)
10 | (hpprime : Prime p)
11 | (hpge3 : p > 3)
12 | (k : ℕ)
13 | (hk : k = Nat.floor (2*p/(3 : ℚ)))
14 | : p^2 ∣ ∑ i ∈ Finset.Icc 1 k, Nat.choose p i :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1997_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | def tetration : ℕ → ℕ → ℕ
 4 |   | _, 0 => 1
 5 |   | b, (m + 1) => b^(tetration b m)
 6 | /--
 7 | Prove that for $n\geq 2$, \[\overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n$ terms}} \equiv \overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n-1$ terms}} \quad \pmod{n}.\]
 8 | -/
 9 | theorem putnam_1997_b5
10 | (n : ℕ)
11 | (hn : n ≥ 2)
12 | : tetration 2 n ≡ tetration 2 (n-1) [MOD n] :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1998_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $f$ be a real function on the real line with continuous third derivative.  Prove that there exists a point $a$ such that \[f(a)\cdot f'(a) \cdot f''(a) \cdot f'''(a)\geq 0 .\]
 5 | -/
 6 | theorem putnam_1998_a3
 7 | (f : ℝ → ℝ)
 8 | (hf : ContDiff ℝ 3 f)
 9 | : ∃ a : ℝ, (f a) * (deriv f a) * (iteratedDeriv 2 f a) * (iteratedDeriv 3 f a) ≥ 0 :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1998_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Function Metric
 4 | 
 5 | abbrev putnam_1998_b1_solution : ℝ := sorry
 6 | -- 6
 7 | /--
 8 | Find the minimum value of \[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.
 9 | -/
10 | theorem putnam_1998_b1
11 | : sInf {((x + 1/x)^6 - (x^6 + 1/x^6) - 2)/((x + 1/x)^3 + (x^3 + 1/x^3)) | x > (0 : ℝ)} = putnam_1998_b1_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1998_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Function Metric
 4 | 
 5 | /--
 6 | Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.
 7 | -/
 8 | theorem putnam_1998_b6
 9 | : ∀ a b c : ℤ, ∃ n : ℤ, n > 0 ∧ ¬(∃ k : ℤ, k = Real.sqrt (n^3 + a * n^2 + b * n + c)) :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1999_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $p(x)$ be a polynomial that is nonnegative for all real $x$.  Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that \[p(x) =  \sum_{j=1}^k (f_j(x))^2.\]
 5 | -/
 6 | theorem putnam_1999_a2
 7 | (p : Polynomial ℝ)
 8 | (hp : ∀ x, p.eval x ≥ 0)
 9 | : ∃ᵉ (k) (f : Fin k → Polynomial ℝ),
10 |     k > 0 ∧
11 |     ∀ x : ℝ, p.eval x = ∑ j : Fin k, ((f j).eval x) ^ 2 :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_1999_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology Metric
 4 | 
 5 | /--
 6 | Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree 1999, then \[|p(0)|\leq C \int_{-1}^1 |p(x)|\,dx.\]
 7 | -/
 8 | theorem putnam_1999_a5
 9 | : ∃ C : ℝ, ∀ p : Polynomial ℝ, p.degree = 1999 → ‖p.eval 0‖ ≤ C * ∫ x in (-1)..1, ‖p.eval x‖ :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2000_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.
 7 | -/
 8 | theorem putnam_2000_a2 :
 9 |   ∀ n : ℕ,
10 |     ∃ N : ℤ,
11 |       ∃ i : Fin 6 → ℕ,
12 |         N > n ∧
13 |         N = (i 0)^2 + (i 1)^2 ∧
14 |         N + 1 = (i 2)^2 + (i 3)^2 ∧
15 |         N + 2 = (i 4)^2 + (i 5)^2 :=
16 |   sorry
17 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2000_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Show that the improper integral $\lim_{B \to \infty} \int_0^B \sin(x)\sin(x^2)\,dx$ converges.
 7 | -/
 8 | theorem putnam_2000_a4
 9 | : ∃ y : ℝ, Tendsto (fun B : ℝ => ∫ x in Set.Ioo 0 B, Real.sin x * Real.sin (x ^ 2)) atTop (𝓝 y) :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2000_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Nat
 4 | 
 5 | /--
 6 | Prove that the expression
 7 | \[
 8 | \frac{gcd(m,n)}{n}\binom{n}{m}
 9 | \]
10 | is an integer for all pairs of integers $n\geq m\geq 1$.
11 | -/
12 | theorem putnam_2000_b2
13 | : (∀ m n : ℕ, m ≥ 1 → n ≥ m → n ∣ Nat.gcd m n * Nat.choose n m) :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2000_b4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Nat Set Function
 4 | 
 5 | /--
 6 | Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$.  Show that $f(x)=0$ for $-1\leq x\leq 1$.
 7 | -/
 8 | theorem putnam_2000_b4
 9 | (f : ℝ → ℝ)
10 | (hfcont : Continuous f)
11 | (hf : ∀ x, f (2 * x ^ 2 - 1) = 2 * x * f x)
12 | : ∀ x, x ∈ Icc (-1) 1 → f x = 0 :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2001_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$.  Assume $(a*b)*a=b$ for all $a,b\in S$.  Prove that $a*(b*a)=b$ for all $a,b\in S$.
 7 | -/
 8 | theorem putnam_2001_a1
 9 | (S : Type*)
10 | [Mul S]
11 | (hS : ∀ a b : S, (a * b) * a = b)
12 | : ∀ a b : S, a * (b * a) = b :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2001_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Polynomial Set
 4 | 
 5 | /--
 6 | Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
 7 | -/
 8 | theorem putnam_2001_a5
 9 | : ∃! an : ℤ × ℕ, let (a, n) := an; a > 0 ∧ n > 0 ∧ a^(n+1) - (a+1)^n = 2001 :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2002_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Set Topology Filter
 4 | 
 5 | /--
 6 | Show that, for all integers $n > 1$,
 7 | \[
 8 | \frac{1}{2ne} < \frac{1}{e} - \left( 1 - \frac{1}{n} \right)^n
 9 | < \frac{1}{ne}.
10 | \]
11 | -/
12 | theorem putnam_2002_b3
13 | (e : ℝ)
14 | (he : e = Real.exp 1)
15 | (f : ℤ → ℝ)
16 | (hf : f = fun n : ℤ => 1/e - (1 - 1/n)^n)
17 | : ∀ n : ℤ, n > 1 → 1/(2*n*e) < f n ∧ f n < 1/(n*e) :=
18 | sorry
19 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2003_a4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MvPolynomial
 4 | 
 5 | /--
 6 | Suppose that $a,b,c,A,B,C$ are real numbers, $a \ne 0$ and $A \ne 0$, such that $|ax^2+bx+c| \leq |Ax^2+Bx+C|$ for all real numbers $x$. Show that $|b^2-4ac| \leq |B^2-4AC|$.
 7 | -/
 8 | theorem putnam_2003_a4
 9 | (a b c A B C : ℝ)
10 | (aAne0 : a ≠ 0 ∧ A ≠ 0)
11 | (hle : ∀ x : ℝ, |a * x ^ 2 + b * x + c| ≤ |A * x ^ 2 + B * x + C|)
12 | : |b ^ 2 - 4 * a * c| ≤ |B ^ 2 - 4 * A * C| :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2003_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MvPolynomial Set
 4 | 
 5 | abbrev putnam_2003_b1_solution : Prop := sorry
 6 | -- False
 7 | /--
 8 | Do there exist polynomials $a(x), b(x), c(y), d(y)$ such that \[ 1 + xy + x^2y^2 = a(x)c(y) + b(x)d(y)\] holds identically?
 9 | -/
10 | theorem putnam_2003_b1
11 | : (∃ a b c d : Polynomial ℝ, (∀ x y : ℝ, 1 + x * y + x ^ 2 * y ^ 2 = a.eval x * c.eval y + b.eval x * d.eval y)) ↔ putnam_2003_b1_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2003_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MvPolynomial Set Nat
 4 | 
 5 | /--
 6 | Show that for each positive integer $n$, $n!=\prod_{i=1}^n \text{lcm}\{1,2,\dots,\lfloor n/i \rfloor\}$. (Here lcm denotes the least common multiple, and $\lfloor x \rfloor$ denotes the greatest integer $\leq x$.)
 7 | -/
 8 | theorem putnam_2003_b3 (n : ℕ) :
 9 |     n ! = ∏ i ∈ Finset.Icc 1 n, ((List.range ⌊n / i⌋₊).map succ).foldl Nat.lcm 1 :=
10 |   sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2003_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open MvPolynomial Set Nat
 4 | 
 5 | /--
 6 | Let  $f(x)$  be  a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \]
 7 | -/
 8 | theorem putnam_2003_b6
 9 | (f : ℝ → ℝ)
10 | (hf : Continuous f)
11 | : (∫ x in (0 : ℝ)..1, (∫ y in (0 : ℝ)..1, |f x + f y|)) ≥ (∫ x in (0 : ℝ)..1, |f x|) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2004_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Topology Filter
 4 | 
 5 | /--
 6 | Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$.
 7 | -/
 8 | theorem putnam_2004_b2
 9 | (m n : ℕ)
10 | (mnpos : m > 0 ∧ n > 0)
11 | : ((m + n)! / ((m + n) ^ (m + n) : ℚ)) < (((m)! / (m ^ m : ℚ)) * ((n)! / (n ^ n : ℚ))) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2005_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat
 4 | 
 5 | /--
 6 | Show that every positive integer is a sum of one or more numbers of the form $2^r 3^s$, where $r$ and $s$ are nonnegative integers and no summand divides another.
 7 | -/
 8 | theorem putnam_2005_a1
 9 | : ∀ n : ℤ, n > 0 → (∃ k : ℕ, ∃ a : Fin k → Fin 2 → ℕ, n = ∑ i : Fin k, 2^(a i 0)*3^(a i 1) ∧
10 | (∀ i j : Fin k, i ≠ j → ¬(2^(a i 0)*3^(a i 1) ∣ 2^(a j 0)*3^(a j 1)))) :=
11 | sorry
12 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2005_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat Set
 4 | 
 5 | noncomputable abbrev putnam_2005_a5_solution : ℝ := sorry
 6 | -- Real.pi * (Real.log 2) / 8
 7 | /--
 8 | Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$.
 9 | -/
10 | theorem putnam_2005_a5 :
11 |   ∫ x in (0:ℝ)..1, (Real.log (x+1))/(x^2 + 1) = putnam_2005_a5_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2006_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | noncomputable abbrev putnam_2006_a1_solution : ℝ := sorry
 4 | -- 6 * Real.pi ^ 2
 5 | /--
 6 | Find the volume of the region of points $(x,y,z)$ such that
 7 | \[
 8 | (x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2).
 9 | \]
10 | -/
11 | theorem putnam_2006_a1
12 | : ((MeasureTheory.volume {(x, y, z) : ℝ × ℝ × ℝ | (x ^ 2 + y ^ 2 + z ^ 2 + 8) ^ 2 ≤ 36 * (x ^ 2 + y ^ 2)}).toReal = putnam_2006_a1_solution) :=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2006_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
 5 | \[
 6 | \left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.
 7 | \]
 8 | -/
 9 | theorem putnam_2006_b2
10 | (n : ℕ)
11 | (npos : n > 0)
12 | (X : Finset ℝ)
13 | (hXcard : X.card = n)
14 | : (∃ S ⊆ X, S ≠ ∅ ∧ ∃ m : ℤ, |m + ∑ s ∈ S, s| ≤ 1 / (n + 1)) :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2007_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Nat
 4 | 
 5 | /--
 6 | Suppose that a finite group has exactly $n$ elements of order $p$, where $p$ is a prime. Prove that either $n = 0$ or $p$ divides $n+1$.
 7 | -/
 8 | theorem putnam_2007_a5
 9 | (G : Type*) [Group G] [Fintype G]
10 | (p n : ℕ)
11 | (hp : Nat.Prime p)
12 | (hn : n = {g : G | orderOf g = p}.encard)
13 | : n = 0 ∨ p ∣ (n + 1) :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2007_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Set Nat
 4 | 
 5 | /--
 6 | Let $f$ be a nonconstant polynomial with positive integer coefficients. Prove that if $n$ is a positive integer, then $f(n)$ divides $f(f(n) + 1)$ if and only if $n = 1$
 7 | -/
 8 | theorem putnam_2007_b1
 9 | (f : Polynomial ℤ)
10 | (hf : ∀ n : ℕ, f.coeff n ≥ 0)
11 | (hfnconst : ∃ n : ℕ, n > 0 ∧ f.coeff n > 0)
12 | (n : ℤ)
13 | (hn : n > 0)
14 | : f.eval n ∣ f.eval (f.eval n + 1) ↔ n = 1 :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2008_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y)+f(y,z)+f(z,x)=0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g:\mathbb{R} \to \mathbb{R}$ such that $f(x,y)=g(x)-g(y)$ for all real numbers $x$ and $y$.
 5 | -/
 6 | theorem putnam_2008_a1
 7 | (f : ℝ → ℝ → ℝ)
 8 | (hf : ∀ x y z : ℝ, f x y + f y z + f z x = 0)
 9 | : ∃ g : ℝ → ℝ, ∀ x y : ℝ, f x y = g x - g y :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2009_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology MvPolynomial Filter Set
 4 | 
 5 | abbrev putnam_2009_a5_solution : Prop := sorry
 6 | -- False
 7 | /--
 8 | Is there a finite abelian group $G$ such that the product of the orders of all its elements is 2^{2009}?
 9 | -/
10 | theorem putnam_2009_a5
11 | : (∃ (G : Type*) (_ : CommGroup G) (_ : Fintype G), ∏ g : G, orderOf g = 2^2009) ↔ putnam_2009_a5_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2010_a4.lean:
--------------------------------------------------------------------------------
1 | import Mathlib
2 | 
3 | /--
4 | Prove that for each positive integer $n$, the number $10^{10^{10^n}} + 10^{10^n} + 10^n - 1$ is not prime.
5 | -/
6 | theorem putnam_2010_a4
7 | : ∀ n : ℕ, n > 0 → ¬Nat.Prime (10^10^10^n + 10^10^n + 10^n - 1) :=
8 | sorry
9 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2010_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology Set
 4 | 
 5 | abbrev putnam_2010_b1_solution : Prop := sorry
 6 | -- False
 7 | /--
 8 | Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$?
 9 | -/
10 | theorem putnam_2010_b1
11 | : (∃ a : ℕ → ℝ, ∀ m : ℕ, m > 0 → ∑' i : ℕ, (a i)^m = m) ↔ putnam_2010_b1_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2010_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology Set
 4 | 
 5 | abbrev putnam_2010_b5_solution : Prop := sorry
 6 | -- False
 7 | /--
 8 | Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?
 9 | -/
10 | theorem putnam_2010_b5 :
11 |   (∃ f : ℝ → ℝ, StrictMono f ∧ Differentiable ℝ f ∧ (∀ x : ℝ, deriv f x = f (f x))) ↔ putnam_2010_b5_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2011_b1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Matrix
 4 | 
 5 | /--
 6 | Let $h$ and $k$ be positive integers. Prove that for every $\epsilon>0$, there are positive integers $m$ and $n$ such that $\epsilon<|h\sqrt{m}-k\sqrt{n}|<2\epsilon$.
 7 | -/
 8 | theorem putnam_2011_b1
 9 | (h k : ℤ)
10 | (hkpos : h > 0 ∧ k > 0)
11 | : ∀ ε > 0, ∃ m n : ℤ, m > 0 ∧ n > 0 ∧ ε < |h * Real.sqrt m - k * Real.sqrt n| ∧ |h * Real.sqrt m - k * Real.sqrt n| < 2 * ε :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2011_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Matrix Set
 4 | 
 5 | /--
 6 | Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\}$,
 7 | \[
 8 | \sum_{k=0}^{p-1} k! n^k \qquad \mbox{is not divisible by $p$.}
 9 | \]
10 | -/
11 | theorem putnam_2011_b6
12 | (p : ℕ)
13 | (hp : Odd p ∧ Nat.Prime p)
14 | : {n ∈ Finset.range p | ¬ p ∣ ∑ k : Finset.range p, Nat.factorial k * n^(k : ℕ)}.card ≥ (p + 1)/2 :=
15 | sorry
16 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2014_a5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Nat
 4 | 
 5 | /--
 6 | Let \[ P_n(x) = 1 + 2 x  + 3 x^2 + \cdots + n x^{n-1}.\] Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j \neq k$.
 7 | -/
 8 | theorem putnam_2014_a5
 9 | (P : ℕ → Polynomial ℂ)
10 | (hP : ∀ n, P n = ∑ i ∈ Finset.Icc 1 n, i * Polynomial.X ^ (i - 1))
11 | : ∀ (j k : ℕ), (j > 0 ∧ k > 0) → j ≠ k → IsCoprime (P j) (P k) :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2014_b4.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter Nat
 4 | 
 5 | /--
 6 | Show that for each positive integer \( n \), all the roots of the polynomial $ \sum_{k=0}^{n} 2^k(n-k)x^k $ are real numbers.
 7 | -/
 8 | theorem putnam_2014_b4
 9 | (n : ℕ)
10 | (P: Polynomial ℂ)
11 | (npos : n > 0)
12 | (Px : P.degree = n ∧ ∀ k ∈ Set.Icc 0 n, P.coeff k = 2 ^ (k * (n - k)))
13 | : ∀ r ∈ P.roots, r.im = 0 :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2015_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_2015_a3_solution : ℂ := sorry
 4 | -- 13725
 5 | /--
 6 | Compute $\log_2 \left( \prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right)$. Here $i$ is the imaginary unit (that is, $i^2=-1$).
 7 | -/
 8 | theorem putnam_2015_a3 :
 9 |     Complex.log (∏ a : Fin 2015, ∏ b : Fin 2015, (1 + Complex.exp (2 * Real.pi * Complex.I * (a.1 + 1) * (b.1 + 1) / 2015))) / Complex.log 2 = putnam_2015_a3_solution :=
10 | sorry
11 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2016_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Polynomial Filter Topology Real Set Nat List
 4 | 
 5 | abbrev putnam_2016_b6_solution : ℝ := sorry
 6 | -- 1
 7 | /--
 8 | Evaluate $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n+1}$.
 9 | -/
10 | theorem putnam_2016_b6 :
11 |   ∑' k : ℕ, ((-1 : ℝ) ^ ((k + 1 : ℤ) - 1) / (k + 1 : ℝ)) * ∑' n : ℕ, (1 : ℝ) / ((k + 1) * (2 ^ n) + 1) = putnam_2016_b6_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2017_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Topology Filter
 4 | 
 5 | /--
 6 | Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.
 7 | -/
 8 | theorem putnam_2017_b3
 9 | (f : ℝ → ℝ)
10 | (c : ℕ → ℝ)
11 | (hc : ∀ n, c n = 0 ∨ c n = 1)
12 | (hf : ∀ x, f x = ∑' n : ℕ, (c n) * x^n)
13 | : f (2/3) = 3/2 → Irrational (f 1/2) :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2018_a3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | noncomputable abbrev putnam_2018_a3_solution : ℝ := sorry
 4 | -- 480/49
 5 | /--
 6 | Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.
 7 | -/
 8 | theorem putnam_2018_a3 :
 9 |     IsGreatest
10 |       {∑ i, Real.cos (3 * x i) | (x : Fin 10 → ℝ) (hx : ∑ i, Real.cos (x i) = 0)}
11 |       putnam_2018_a3_solution :=
12 |   sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2018_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | /--
 4 | Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \leq 1\}$.
 5 | -/
 6 | theorem putnam_2018_b2
 7 | (n : ℕ)
 8 | (hn : n > 0)
 9 | (f : ℕ → ℂ → ℂ)
10 | (hf : ∀ z : ℂ, f n z = ∑ i ∈ Finset.range n, (n - i) * z^i)
11 | : ∀ z : ℂ, ‖z‖ ≤ 1 → f n z ≠ 0 :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2018_b3.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_2018_b3_solution : Set ℕ := sorry
 4 | -- {2^2, 2^4, 2^16, 2^256}
 5 | /--
 6 | Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$.
 7 | -/
 8 | theorem putnam_2018_b3
 9 |   (n : ℕ) (hn : 0 < n) :
10 |   (n < 10^100 ∧ ((n : ℤ) ∣ (2^n : ℤ) ∧ (n - 1 : ℤ) ∣ (2^n - 1 : ℤ) ∧ (n - 2 : ℤ) ∣ (2^n - 2 : ℤ))) ↔ n ∈ putnam_2018_b3_solution :=
11 | sorry
12 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2020_a2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | abbrev putnam_2020_a2_solution : ℕ → ℕ := sorry
 4 | -- fun k ↦ 4 ^ k
 5 | /--
 6 | Let $k$ be a nonnegative integer. Evaluate
 7 | \[
 8 | \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
 9 | \]
10 | -/
11 | theorem putnam_2020_a2
12 | (k : ℕ)
13 | : (∑ j ∈ Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j = putnam_2020_a2_solution k) :=
14 | sorry
15 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2020_b5.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology Set
 4 | 
 5 | /--
 6 | For $j \in \{1, 2, 3, 4\}$, let $z_j$ be a complex number with $|z_j| = 1$ and $z_j \neq 1$. Prove that \[ 3 - z_1 - z_2 - z_3 - z_4 + z_1 z_2 z_3 z_4 \neq 0. \]
 7 | -/
 8 | theorem putnam_2020_b5
 9 | (z : Fin 4 → ℂ)
10 | (hzle1 : ∀ n, ‖z n‖ = 1)
11 | (hzne1 : ∀ n, z n ≠ 1)
12 | : 3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) ≠ 0:=
13 | sorry
14 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2020_b6.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Filter Topology Set
 4 | 
 5 | /--
 6 | Let $n$ be a positive integer. Prove that $\sum_{k=1}^n(-1)^{\lfloor k(\sqrt{2}-1) \rfloor} \geq 0$.
 7 | -/
 8 | theorem putnam_2020_b6
 9 | (n : ℕ)
10 | (npos : n > 0)
11 | : ∑ k ∈ Finset.Icc 1 n, ((-1) ^ Int.floor (k * (Real.sqrt 2 - 1)) : ℝ) ≥ 0 :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2023_b2.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | open Nat
 4 | 
 5 | abbrev putnam_2023_b2_solution : ℕ := sorry
 6 | -- 3
 7 | /--
 8 | For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 * n$. What is the minimum value of $k(n)$?
 9 | -/
10 | theorem putnam_2023_b2
11 | : sInf {(digits 2 (2023*n)).sum | n > 0} = putnam_2023_b2_solution :=
12 | sorry
13 | 


--------------------------------------------------------------------------------
/lean4/src/putnam_2024_a1.lean:
--------------------------------------------------------------------------------
 1 | import Mathlib
 2 | 
 3 | noncomputable abbrev putnam_2024_a1_solution : Set ℕ := sorry
 4 | --{1}
 5 | /--
 6 | Determine all positive integers $n$ for which there exist positive integers $a$, $b$ and $c$ satisfying $2a^n + 3b^n = 4c^n$.
 7 | -/
 8 | theorem putnam_2024_a1 :
 9 |     {n : ℕ | 0 < n ∧ ∃ (a b c : ℕ), 0 < a ∧ 0 < b ∧ 0 < c ∧ 2*a^n + 3*b^n = 4*c^n}
10 |       = putnam_2024_a1_solution :=
11 |   sorry
12 | 


--------------------------------------------------------------------------------