├── .github └── workflows │ └── pr.yml ├── .gitignore ├── LICENSE ├── README.md ├── leanpkg.toml ├── scripts ├── build_new_files.sh ├── detect_errors.py ├── fetch_olean_cache.sh ├── lint-style.py ├── lint-style.sh ├── port_mathlib_imo_problems.py ├── port_minif2f.py └── update_mathlib.py └── src └── mathzoo ├── imports ├── README.md └── miniF2F.lean ├── misc ├── README.md └── miniF2F │ ├── algebra │ ├── 2complexrootspoly_xsqp49eqxp7itxpn7i.lean │ ├── 2rootsintpoly_am10tap11eqasqpam110.lean │ ├── 2rootspoly_apatapbeq2asqp2ab.lean │ ├── 2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7.lean │ ├── 2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4.lean │ ├── 3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta.lean │ ├── 9onxpypzleqsum2onxpy.lean │ ├── abpbcpcageq3_sumaonsqrtapbgeq3onsqrt2.lean │ ├── absapbon1pabsapbleqsumabsaon1pabsa.lean │ ├── absxm1pabsxpabsxp1eqxp2_0leqxleq1.lean │ ├── amgm_faxinrrp2msqrt2geq2mxm1div2x.lean │ ├── amgm_prod1toneq1_sum1tongeqn.lean │ ├── amgm_sqrtxymulxmyeqxpy_xpygeq4.lean │ ├── amgm_sum1toneqn_prod1tonleq1.lean │ ├── amgm_sumasqdivbgeqsuma.lean │ ├── amgm_sumasqdivbsqgeqsumbdiva.lean │ ├── apb4leq8ta4pb4.lean │ ├── apbmpcneq0_aeq0anbeq0anceq0.lean │ ├── apbon2pownleqapownpbpowon2.lean │ ├── apbpceq2_abpbcpcaeq1_aleq1on3anbleq1ancleq4on3.lean │ ├── binomnegdiscrineq_10alt28asqp1.lean │ ├── bleqa_apbon2msqrtableqambsqon8b.lean │ ├── cubrtrp1oncubrtreq3_rcubp1onrcubeq5778.lean │ ├── ineq_nto1onlt2m1on.lean │ ├── manipexpr_2erprsqpesqeqnrpnesq.lean │ ├── manipexpr_apbeq2cceqiacpbceqm2.lean │ ├── others_exirrpowirrrat.lean │ ├── sqineq_2at2pclta2c2p41pc.lean │ ├── sqineq_2unitcircatblt1.lean │ ├── sqineq_36azm9asqle36zsq.lean │ ├── sqineq_4bap1lt4bsqpap1sq.lean │ ├── sqineq_at2malt1.lean │ ├── sqineq_unitcircatbpabsamblt1.lean │ ├── sqineq_unitcircatbpamblt1.lean │ ├── sum1onsqrt2to1onsqrt10000lt198.lean │ └── xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3.lean │ ├── induction │ ├── 11div10tonmn1ton.lean │ ├── 12dvd4expnp1p20.lean │ ├── 1pxpownlt1pnx.lean │ ├── divisibility_3div2tooddnp1.lean │ ├── divisibility_3divnto3m2n.lean │ ├── divisibility_9div10tonm1.lean │ ├── ineq_nsqlefactn.lean │ ├── nfactltnexpnm1ngt3.lean │ ├── pord1p1on2powklt5on2.lean │ ├── pprime_pdvdapowpma.lean │ ├── prod1p1onk3le3m1onn.lean │ ├── seq_mul2pnp1.lean │ ├── sum2kp1npqsqm1.lean │ ├── sum_1oktkp1.lean │ ├── sum_odd.lean │ └── sumkexp3eqsumksq.lean │ └── numbertheory │ ├── 2dvd4expn.lean │ ├── 2pownm1prime_nprime.lean │ ├── 3pow2pownm1mod2pownp3eq2pownp2.lean │ ├── 4x3m7y3neq2003.lean │ ├── aneqprodakp4_anmsqrtanp1eq2.lean │ ├── aoddbdiv4asqpbsqmod8eq1.lean │ ├── exk2powkeqapb2mulbpa2_aeq1.lean │ ├── fxeq4powxp6powxp9powx_f2powmdvdf2pown.lean │ ├── nckeqnm1ckpnm1ckm1.lean │ ├── notequiv2i2jasqbsqdiv8.lean │ ├── prmdvsneqnsqmodpeq0.lean │ ├── sqmod3in01d.lean │ ├── sqmod4in01d.lean │ ├── sumkmulnckeqnmul2pownm1.lean │ ├── x5neqy2p4.lean │ └── xsqpysqintdenomeq.lean └── olympiads ├── README.md ├── aime ├── 1983 │ ├── p1.lean │ ├── p2.lean │ ├── p3.lean │ └── p9.lean ├── 1984 │ ├── p1.lean │ ├── p15.lean │ ├── p5.lean │ └── p7.lean ├── 1987 │ ├── p5.lean │ └── p8.lean ├── 1988 │ ├── p3.lean │ ├── p4.lean │ └── p8.lean ├── 1989 │ └── p8.lean ├── 1990 │ ├── p15.lean │ ├── p2.lean │ └── p4.lean ├── 1991 │ ├── p1.lean │ ├── p6.lean │ └── p9.lean ├── 1994 │ ├── p3.lean │ └── p4.lean ├── 1995 │ └── p7.lean ├── 1996 │ └── p5.lean ├── 1997 │ ├── p12.lean │ └── p9.lean ├── 1999 │ └── p11.lean ├── 2000 │ └── i │ │ └── p7.lean ├── 2001 │ └── i │ │ └── p3.lean ├── 2020 │ └── ii │ │ └── p6.lean └── README.md ├── amc ├── 8 │ └── 2020 │ │ ├── p22.lean │ │ ├── p23.lean │ │ ├── p6.lean │ │ └── p7.lean ├── 10 │ └── 2021 │ │ └── b │ │ └── p5.lean ├── 12 │ ├── 2000 │ │ ├── p1.lean │ │ ├── p11.lean │ │ ├── p12.lean │ │ ├── p15.lean │ │ ├── p20.lean │ │ ├── p5.lean │ │ └── p6.lean │ ├── 2001 │ │ ├── p2.lean │ │ ├── p21.lean │ │ ├── p5.lean │ │ └── p9.lean │ ├── 2002 │ │ ├── a │ │ │ ├── p1.lean │ │ │ ├── p12.lean │ │ │ ├── p13.lean │ │ │ ├── p21.lean │ │ │ └── p6.lean │ │ └── b │ │ │ ├── p11.lean │ │ │ ├── p19.lean │ │ │ ├── p2.lean │ │ │ ├── p3.lean │ │ │ ├── p4.lean │ │ │ ├── p6.lean │ │ │ └── p7.lean │ ├── 2003 │ │ ├── a │ │ │ ├── p1.lean │ │ │ ├── p23.lean │ │ │ ├── p24.lean │ │ │ ├── p25.lean │ │ │ └── p5.lean │ │ └── b │ │ │ ├── p17.lean │ │ │ ├── p6.lean │ │ │ └── p9.lean │ ├── 2004 │ │ └── b │ │ │ └── p3.lean │ ├── 2008 │ │ └── a │ │ │ ├── p15.lean │ │ │ ├── p2.lean │ │ │ ├── p25.lean │ │ │ ├── p4.lean │ │ │ └── p8.lean │ ├── 2009 │ │ └── a │ │ │ ├── p15.lean │ │ │ ├── p2.lean │ │ │ ├── p25.lean │ │ │ ├── p5.lean │ │ │ ├── p6.lean │ │ │ ├── p7.lean │ │ │ └── p9.lean │ ├── 2010 │ │ └── a │ │ │ ├── p10.lean │ │ │ ├── p11.lean │ │ │ └── p22.lean │ ├── 2011 │ │ └── a │ │ │ └── p18.lean │ ├── 2013 │ │ └── a │ │ │ ├── p4.lean │ │ │ ├── p7.lean │ │ │ └── p8.lean │ ├── 2015 │ │ └── a │ │ │ └── p10.lean │ ├── 2016 │ │ └── a │ │ │ ├── p2.lean │ │ │ └── p3.lean │ ├── 2017 │ │ └── a │ │ │ ├── p2.lean │ │ │ └── p7.lean │ ├── 2019 │ │ └── a │ │ │ ├── p12.lean │ │ │ ├── p21.lean │ │ │ └── p9.lean │ ├── 2020 │ │ ├── a │ │ │ ├── p10.lean │ │ │ ├── p13.lean │ │ │ ├── p15.lean │ │ │ ├── p22.lean │ │ │ ├── p25.lean │ │ │ ├── p4.lean │ │ │ ├── p7.lean │ │ │ └── p9.lean │ │ └── b │ │ │ ├── p13.lean │ │ │ ├── p2.lean │ │ │ ├── p21.lean │ │ │ ├── p22.lean │ │ │ ├── p5.lean │ │ │ └── p6.lean │ └── 2021 │ │ ├── a │ │ ├── p12.lean │ │ ├── p14.lean │ │ ├── p18.lean │ │ ├── p19.lean │ │ ├── p22.lean │ │ ├── p25.lean │ │ ├── p3.lean │ │ ├── p7.lean │ │ ├── p8.lean │ │ └── p9.lean │ │ └── b │ │ ├── p1.lean │ │ ├── p13.lean │ │ ├── p18.lean │ │ ├── p21.lean │ │ ├── p3.lean │ │ ├── p4.lean │ │ └── p9.lean └── README.md ├── imo ├── 1959 │ └── p1.lean ├── 1960 │ ├── p1.lean │ └── p2.lean ├── 1961 │ └── p1.lean ├── 1962 │ ├── p1.lean │ ├── p2.lean │ └── p4.lean ├── 1963 │ └── p5.lean ├── 1964 │ ├── p1.lean │ └── p2.lean ├── 1965 │ ├── p1.lean │ └── p2.lean ├── 1966 │ ├── p4.lean │ └── p5.lean ├── 1967 │ └── p3.lean ├── 1968 │ └── p5.lean ├── 1969 │ ├── p1.lean │ └── p2.lean ├── 1972 │ ├── p5.lean │ └── p5_alt1.lean ├── 1973 │ └── p3.lean ├── 1974 │ ├── p3.lean │ └── p5.lean ├── 1977 │ ├── p5.lean │ └── p6.lean ├── 1978 │ └── p5.lean ├── 1979 │ └── p1.lean ├── 1981 │ ├── p3.lean │ └── p6.lean ├── 1982 │ └── p1.lean ├── 1983 │ └── p6.lean ├── 1984 │ ├── p2.lean │ └── p6.lean ├── 1985 │ └── p6.lean ├── 1987 │ ├── p1.lean │ ├── p4.lean │ └── p6.lean ├── 1988 │ └── p6.lean ├── 1990 │ └── p3.lean ├── 1992 │ └── p1.lean ├── 1993 │ └── p5.lean ├── 1994 │ └── p1.lean ├── 1997 │ └── p5.lean ├── 1998 │ └── p2.lean ├── 2001 │ ├── p2.lean │ └── p6.lean ├── 2005 │ ├── p3.lean │ └── p4.lean ├── 2006 │ ├── p3.lean │ └── p6.lean ├── 2007 │ └── p6.lean ├── 2008 │ ├── p2.lean │ ├── p3.lean │ └── p4.lean ├── 2011 │ ├── p3.lean │ └── p5.lean ├── 2013 │ ├── p1.lean │ └── p5.lean ├── 2019 │ ├── p1.lean │ └── p4.lean ├── 2020 │ └── p2.lean ├── 2021 │ └── p1.lean └── README.md └── mathd ├── README.md ├── algebra ├── p10.lean ├── p101.lean ├── p104.lean ├── p107.lean ├── p109.lean ├── p11.lean ├── p110.lean ├── p113.lean ├── p114.lean ├── p116.lean ├── p119.lean ├── p123.lean ├── p125.lean ├── p126.lean ├── p129.lean ├── p13.lean ├── p131.lean ├── p132.lean ├── p137.lean ├── p139.lean ├── p140.lean ├── p141.lean ├── p142.lean ├── p143.lean ├── p144.lean ├── p148.lean ├── p149.lean ├── p15.lean ├── p151.lean ├── p153.lean ├── p156.lean ├── p158.lean ├── p159.lean ├── p160.lean ├── p17.lean ├── p170.lean ├── p171.lean ├── p176.lean ├── p181.lean ├── p182.lean ├── p184.lean ├── p185.lean ├── p188.lean ├── p190.lean ├── p192.lean ├── p196.lean ├── p206.lean ├── p208.lean ├── p209.lean ├── p214.lean ├── p215.lean ├── p22.lean ├── p224.lean ├── p234.lean ├── p24.lean ├── p245.lean ├── p246.lean ├── p247.lean ├── p251.lean ├── p263.lean ├── p267.lean ├── p270.lean ├── p275.lean ├── p276.lean ├── p28.lean ├── p282.lean ├── p288.lean ├── p289.lean ├── p293.lean ├── p296.lean ├── p302.lean ├── p304.lean ├── p31.lean ├── p313.lean ├── p314.lean ├── p320.lean ├── p323.lean ├── p327.lean ├── p329.lean ├── p33.lean ├── p332.lean ├── p338.lean ├── p342.lean ├── p346.lean ├── p35.lean ├── p354.lean ├── p359.lean ├── p362.lean ├── p37.lean ├── p388.lean ├── p392.lean ├── p393.lean ├── p398.lean ├── p400.lean ├── p405.lean ├── p410.lean ├── p412.lean ├── p419.lean ├── p421.lean ├── p422.lean ├── p427.lean ├── p43.lean ├── p432.lean ├── p433.lean ├── p437.lean ├── p44.lean ├── p440.lean ├── p441.lean ├── p451.lean ├── p452.lean ├── p455.lean ├── p459.lean ├── p462.lean ├── p478.lean ├── p48.lean ├── p480.lean ├── p482.lean ├── p484.lean ├── p487.lean ├── p493.lean ├── p509.lean ├── p51.lean ├── p510.lean ├── p513.lean ├── p536.lean ├── p547.lean ├── p55.lean ├── p568.lean ├── p59.lean ├── p598.lean ├── p616.lean ├── p67.lean ├── p69.lean ├── p73.lean ├── p756.lean ├── p76.lean ├── p77.lean ├── p80.lean ├── p89.lean └── p96.lean └── numbertheory ├── p100.lean ├── p101.lean ├── p102.lean ├── p109.lean ├── p110.lean ├── p1124.lean ├── p12.lean ├── p126.lean ├── p127.lean ├── p13.lean ├── p132.lean ├── p135.lean ├── p136.lean ├── p149.lean ├── p150.lean ├── p155.lean ├── p156.lean ├── p169.lean ├── p175.lean ├── p185.lean ├── p188.lean ├── p198.lean ├── p200.lean ├── p202.lean ├── p207.lean ├── p211.lean ├── p212.lean ├── p22.lean ├── p221.lean ├── p222.lean ├── p227.lean ├── p229.lean ├── p232.lean ├── p233.lean ├── p234.lean ├── p235.lean ├── p236.lean ├── p237.lean ├── p239.lean ├── p24.lean ├── p247.lean ├── p252.lean ├── p254.lean ├── p257.lean ├── p269.lean ├── p277.lean ├── p284.lean ├── p293.lean ├── p296.lean ├── p299.lean ├── p3.lean ├── p30.lean ├── p301.lean ├── p303.lean ├── p314.lean ├── p32.lean ├── p320.lean ├── p321.lean ├── p326.lean ├── p328.lean ├── p33.lean ├── p335.lean ├── p34.lean ├── p341.lean ├── p342.lean ├── p343.lean ├── p345.lean ├── p35.lean ├── p353.lean ├── p37.lean ├── p370.lean ├── p403.lean ├── p405.lean ├── p412.lean ├── p42.lean ├── p427.lean ├── p43.lean ├── p430.lean ├── p435.lean ├── p447.lean ├── p45.lean ├── p451.lean ├── p457.lean ├── p458.lean ├── p461.lean ├── p466.lean ├── p48.lean ├── p483.lean ├── p495.lean ├── p5.lean ├── p517.lean ├── p521.lean ├── p530.lean ├── p541.lean ├── p543.lean ├── p551.lean ├── p552.lean ├── p559.lean ├── p582.lean ├── p618.lean ├── p629.lean ├── p64.lean ├── p640.lean ├── p66.lean ├── p668.lean ├── p690.lean ├── p709.lean ├── p711.lean ├── p728.lean ├── p739.lean ├── p764.lean ├── p765.lean ├── p769.lean ├── p780.lean ├── p81.lean ├── p84.lean ├── p85.lean ├── p92.lean ├── p961.lean └── p99.lean /.gitignore: -------------------------------------------------------------------------------- 1 | *.olean 2 | /_target 3 | /leanpkg.path 4 | _cache 5 | __pycache__ 6 | all.lean 7 | *.depend 8 | /src/.noisy_files 9 | *~ 10 | .DS_Store 11 | *.lock -------------------------------------------------------------------------------- /leanpkg.toml: -------------------------------------------------------------------------------- 1 | [package] 2 | name = "mathzoo" 3 | version = "0.1" 4 | lean_version = "leanprover-community/lean:3.42.0" 5 | path = "src" 6 | 7 | [dependencies] 8 | mathlib = { git = "https://github.com/leanprover-community/mathlib", rev = "cd942870af69a837bde360337bee1a88eb1611c2" } -------------------------------------------------------------------------------- /scripts/build_new_files.sh: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env bash 2 | 3 | git diff --name-only origin/main | while read FILE ; do 4 | if [ -f ${FILE} ]; 5 | then 6 | NAME=$(basename ${FILE}) 7 | if [[ ${NAME} =~ (.*)?\.lean ]]; 8 | then 9 | lean --json -T100000 ${FILE} | python3 scripts/detect_errors.py 10 | fi 11 | fi 12 | done 13 | -------------------------------------------------------------------------------- /src/mathzoo/imports/README.md: -------------------------------------------------------------------------------- 1 | # Imports 2 | 3 | This directory contains simple files that may be imported by other gems. The main use case is to bundle many mathlib imports together into one convenient import. -------------------------------------------------------------------------------- /src/mathzoo/misc/README.md: -------------------------------------------------------------------------------- 1 | # Misc 2 | 3 | This directory contains miscellaneous problems that do not (yet?) fit into the core taxonomy of the repository. -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/2rootsintpoly_am10tap11eqasqpam110.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 11 | (a : ℂ) : 12 | (a - 10) * (a + 11) = a^2 + a - 110 := 13 | begin 14 | ring, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/2rootspoly_apatapbeq2asqp2ab.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem algebra_2rootspoly_apatapbeq2asqp2ab 11 | (a b : ℂ) : 12 | (a + a) * (a + b) = 2 * a^2 + 2 * (a * b) := 13 | begin 14 | ring, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7 11 | (f z: ℂ) 12 | (h₀ : f + 3*z = 11) 13 | (h₁ : 3*(f - 1) - 5*z = -68) : 14 | f = -10 ∧ z = 7 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 11 | (x e : ℂ) 12 | (h₀ : x + e = 7) 13 | (h₁ : 2 * x + e = 3) : 14 | e = 11 ∧ x = -4 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/9onxpypzleqsum2onxpy.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_9onxpypzleqsum2onxpy 11 | (x y z : ℝ) 12 | (h₀ : 0 < x ∧ 0 < y ∧ 0 < z) : 13 | 9 / (x + y + z) ≤ 2 / (x + y) + 2 / (y + z) + 2 / (z + x) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/absapbon1pabsapbleqsumabsaon1pabsa.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_absapbon1pabsapbleqsumabsaon1pabsa 11 | (a b : ℝ) : 12 | abs (a + b) / (1 + abs (a + b)) ≤ abs a / (1 + abs a) + abs b / (1 + abs b) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/absxm1pabsxpabsxp1eqxp2_0leqxleq1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_absxm1pabsxpabsxp1eqxp2_0leqxleq1 11 | (x : ℝ) 12 | (h₀ : abs (x - 1) + abs x + abs (x + 1) = x + 2) : 13 | 0 ≤ x ∧ x ≤ 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/amgm_prod1toneq1_sum1tongeqn.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_amgm_prod1toneq1_sum1tongeqn 11 | (a : ℕ → nnreal) 12 | (n : ℕ) 13 | (h₀ : finset.prod (finset.range(n)) a = 1) : 14 | finset.sum (finset.range(n)) a ≥ n -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/amgm_sqrtxymulxmyeqxpy_xpygeq4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4 11 | (x y : ℝ) 12 | (h₀ : 0 < x ∧ 0 < y) 13 | (h₁ : y ≤ x) 14 | (h₂ : real.sqrt (x * y) * (x - y) = (x + y)) : 15 | x + y ≥ 4 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/amgm_sum1toneqn_prod1tonleq1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_amgm_sum1toneqn_prod1tonleq1 11 | (a : ℕ → nnreal) 12 | (n : ℕ) 13 | (h₀ : ∑ x in finset.range n, a x = n) : 14 | ∏ x in finset.range n, a x ≤ 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/amgm_sumasqdivbgeqsuma.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_amgm_sumasqdivbgeqsuma 11 | (a b c d : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d) : 13 | a^2 / b + b^2 / c + c^2 / d + d^2 / a ≥ a + b + c + d -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/amgm_sumasqdivbsqgeqsumbdiva.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_amgm_sumasqdivbsqgeqsumbdiva 11 | (a b c : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) : 13 | a^2 / b^2 + b^2 / c^2 + c^2 / a^2 ≥ b / a + c / b + a / c -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/apb4leq8ta4pb4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_apb4leq8ta4pb4 11 | (a b : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b) : 13 | (a + b)^4 ≤ 8 * (a^4 + b^4) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/apbon2pownleqapownpbpowon2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_apbon2pownleqapownpbpowon2 11 | (a b : ℝ) 12 | (n : ℕ) 13 | (h₀ : 0 < a ∧ 0 < b) 14 | (h₁ : 0 < n) : 15 | ((a + b) / 2)^n ≤ (a^n + b^n) / 2 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/bleqa_apbon2msqrtableqambsqon8b.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_bleqa_apbon2msqrtableqambsqon8b 11 | (a b : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b) 13 | (h₁ : b ≤ a) : 14 | (a + b) / 2 - real.sqrt (a * b) ≤ (a - b)^2 / (8 * b) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/cubrtrp1oncubrtreq3_rcubp1onrcubeq5778.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_cubrtrp1oncubrtreq3_rcubp1onrcubeq5778 11 | (r : ℝ) 12 | (h₀ : r^((1:ℝ) / 3) + 1 / r^((1:ℝ) / 3) = 3) : 13 | r^3 + 1 / r^3 = 5778 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/ineq_nto1onlt2m1on.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_ineq_nto1onlt2m1on 11 | (n : ℕ) : 12 | (n:ℝ)^((1:ℝ) / n) < 2 - 1 / n -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/manipexpr_2erprsqpesqeqnrpnesq.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem algebra_manipexpr_2erprsqpesqeqnrpnesq 11 | (e r : ℂ) : 12 | 2 * (e * r) + (e^2 + r^2) = (-r + (-e))^2 := 13 | begin 14 | ring, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/sqineq_2unitcircatblt1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem algebra_sqineq_2unitcircatblt1 11 | (a b : ℝ) 12 | (h₀ : a^2 + b^2 = 2) : 13 | a * b ≤ 1 := 14 | begin 15 | nlinarith [sq_nonneg (b - a)], 16 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/sqineq_at2malt1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem algebra_sqineq_at2malt1 11 | (a : ℝ) : 12 | a * (2 - a) ≤ 1 := 13 | begin 14 | suffices: 0 ≤ a^2 - 2*a + 1, nlinarith, 15 | suffices: 0 ≤ (a - 1)^2, nlinarith, 16 | nlinarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/sqineq_unitcircatbpabsamblt1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_sqineq_unitcircatbpabsamblt1 11 | (a b: ℝ) 12 | (h₀ : a^2 + b^2 = 1) : 13 | a * b + ∥a - b∥ ≤ 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/sqineq_unitcircatbpamblt1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem algebra_sqineq_unitcircatbpamblt1 11 | (a b: ℝ) 12 | (h₀ : a^2 + b^2 = 1) : 13 | a * b + (a - b) ≤ 1 := 14 | begin 15 | nlinarith [sq_nonneg (a - b)], 16 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/sum1onsqrt2to1onsqrt10000lt198.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_sum1onsqrt2to1onsqrt10000lt198 : 11 | ∑ k in finset.range 10001 \ finset.range 2, (1 / real.sqrt k) < 198 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/algebra/xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 11 | (x y z : ℤ) 12 | (h₀ : (x - y)^2 + (y - z)^2 + (z - x)^2 = x * y * z) : 13 | (x + y + z + 6) ∣ (x^3 + y^3 + z^3) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/11div10tonmn1ton.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_11div10tonmn1ton 11 | (n : ℕ) : 12 | 11 ∣ (10^n - (-1 : ℤ)^n) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/1pxpownlt1pnx.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_1pxpownlt1pnx 11 | (x : ℝ) 12 | (n : ℕ+) 13 | (h₀ : -1 < x) : 14 | (1 + ↑n*x) ≤ (1 + x)^(n:ℕ) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/divisibility_3div2tooddnp1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_divisibility_3div2tooddnp1 11 | (n : ℕ) : 12 | 3 ∣ (2^(2 * n + 1) + 1) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/divisibility_3divnto3m2n.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_divisibility_3divnto3m2n 11 | (n : ℕ) : 12 | 3 ∣ n^3 + 2 * n -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/divisibility_9div10tonm1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_divisibility_9div10tonm1 11 | (n : ℕ+) : 12 | 9 ∣(10^(n:ℕ) - 1) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/pord1p1on2powklt5on2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_pord1p1on2powklt5on2 11 | (n : ℕ) 12 | (h₀ : 0 < n) : 13 | ∏ k in finset.range (n + 1) \ finset.range 1, (1 + (1:ℝ) / 2^k) < 5 / 2 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/pprime_pdvdapowpma.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_pprime_pdvdapowpma 11 | (p a : ℕ) 12 | (h₀ : 0 < a) 13 | (h₁ : nat.prime p) : 14 | p ∣ (a^p - a) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/prod1p1onk3le3m1onn.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_prod1p1onk3le3m1onn 11 | (n : ℕ) 12 | (h₀ : 0 < n) : 13 | ∏ k in finset.range (n + 1) \ finset.range 1, (1 + (1:ℝ) / k^3) ≤ (3:ℝ) - 1 / ↑n -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/sum2kp1npqsqm1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_sum2kp1npqsqm1 11 | (n : ℕ) : 12 | ↑∑ k in (finset.range n), 2 * k + 3 = ↑(n + 1)^2 - (1:ℤ) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/sum_1oktkp1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_sum_1oktkp1 11 | (n : ℕ) : 12 | ∑ k in (finset.range n), (1:ℝ) / ((k + 1) * (k + 2)) = n / (n + 1) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/induction/sum_odd.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom induction_sum_odd 11 | (n : ℕ) : 12 | ∑ k in (finset.range n), 2 * k + 1 = n^2 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/2dvd4expn.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem numbertheory_2dvd4expn 11 | (n : ℕ) 12 | (h₀ : n ≠ 0) : 13 | 2 ∣ 4^n := 14 | begin 15 | apply dvd_pow _ h₀, 16 | norm_num, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/2pownm1prime_nprime.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_2pownm1prime_nprime 11 | (n : ℕ) 12 | (h₀ : 0 < n) 13 | (h₁ : nat.prime (2^n - 1)) : 14 | nat.prime n -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/3pow2pownm1mod2pownp3eq2pownp2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_3pow2pownm1mod2pownp3eq2pownp2 11 | (n : ℕ) 12 | (h₀ : 0 < n) : 13 | (3^(2^n) - 1) % (2^(n + 3)) = 2^(n + 2) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/aoddbdiv4asqpbsqmod8eq1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_aoddbdiv4asqpbsqmod8eq1 11 | (a : ℤ) 12 | (b : ℕ) 13 | (h₀ : odd a) 14 | (h₁ : 4 ∣ b) : 15 | (a^2 + b^2) % 8 = 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/exk2powkeqapb2mulbpa2_aeq1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_exk2powkeqapb2mulbpa2_aeq1 11 | (a b : ℕ+) 12 | (h₀ : ∃ k > 0, 2^k = (a + b^2) * (b + a^2)) : 13 | a = 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/nckeqnm1ckpnm1ckm1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_nckeqnm1ckpnm1ckm1 11 | (n k : ℕ) 12 | (h₀ : 0 < n ∧ 0 < k) 13 | (h₁ : k ≤ n) : 14 | nat.choose n k = nat.choose (n - 1) k + nat.choose (n - 1) (k - 1) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/notequiv2i2jasqbsqdiv8.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_notequiv2i2jasqbsqdiv8 11 | (a b : ℤ) : 12 | ¬ ((∃ i j, a = 2*i ∧ b=2*j) ↔ (∃ k, a^2 + b^2 = 8*k)) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/sqmod3in01d.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_sqmod3in01d 11 | (a : ℤ) : 12 | (a^2) % 3 = 0 ∨ (a^2) % 3 = 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/sqmod4in01d.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_sqmod4in01d 11 | (a : ℤ) : 12 | (a^2 % 4) = 0 ∨ (a^2 % 4) = 1 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/sumkmulnckeqnmul2pownm1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_sumkmulnckeqnmul2pownm1 11 | (n : ℕ) 12 | (h₀ : 0 < n) : 13 | ∑ k in finset.range (n + 1) \ finset.range 1, (k * nat.choose n k) = n * 2^(n - 1) -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/x5neqy2p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_x5neqy2p4 11 | (x y : ℤ) : 12 | x^5 ≠ y^2 + 4 -------------------------------------------------------------------------------- /src/mathzoo/misc/miniF2F/numbertheory/xsqpysqintdenomeq.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom numbertheory_xsqpysqintdenomeq 11 | (x y : ℚ) 12 | (h₀ : (x^2 + y^2).denom = 1) : 13 | x.denom = y.denom -------------------------------------------------------------------------------- /src/mathzoo/olympiads/README.md: -------------------------------------------------------------------------------- 1 | # Olympiads 2 | 3 | This directory contains gems based on problems from mathematical olympiads. There are hundreds of different olympiads of varying difficulty, and in total they include tens of thousands of problems. See the [Art of Problem Solving contest collection page](https://artofproblemsolving.com/community/c13_contest_collections) for an extensive list. -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1983/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1983_p2 11 | (x p : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : 0 < p ∧ p < 15) 14 | (h₁ : p ≤ x ∧ x ≤ 15) 15 | (h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) : 16 | 15 ≤ f x -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1983/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1983_p3 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = (x^2 + (18 * x + 30) - 2 * real.sqrt (x^2 + (18 * x + 45)))) 13 | (h₁ : fintype (f⁻¹' {0})) : 14 | ∏ x in (f⁻¹' {0}).to_finset, x = 20 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1984/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1984_p5 11 | (a b : ℝ) 12 | (h₀ : real.log a / real.log 8 + real.log (b^2) / real.log 4 = 5) 13 | (h₁ : real.log b / real.log 8 + real.log (a^2) / real.log 4 = 7) : 14 | a * b = 512 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1984/p7.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1984_p7 11 | (f : ℕ+ → ℕ+) 12 | (h₀ : ∀ n, 1000 ≤ n → f n = n - 3) 13 | (h₁ : ∀ n, n < 1000 → f n = f (f (n + 5))) : 14 | f 84 = 997 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1987/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1987_p5 11 | (x y : ℤ) 12 | (h₀ : y^2 + 3 * (x^2 * y^2) = 30 * x^2 + 517): 13 | 3 * (x^2 * y^2) = 588 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1987/p8.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1987_p8 11 | (n : ℕ) 12 | (h₀ : 0 < n) 13 | (h₁ : ∃! k, (8:ℝ) / 15 < n / (n + k) ∧ (n:ℝ) / (n + k) < 7 / 13) : 14 | n ≤ 112 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1988/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1988_p3 11 | (x : ℝ) 12 | (h₀ : 0 < x) 13 | (h₁ : real.log (real.log x / real.log 8) / real.log 2 = real.log (real.log x / real.log 2) / real.log 8) : 14 | (real.log x / real.log 2)^2 = 27 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1988/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1988_p4 11 | (n : ℕ) 12 | (a : ℕ → ℝ) 13 | (h₀ : ∀ n, abs (a n) < 1) 14 | (h₁ : ∑ k in finset.range n, (abs (a k)) = 19 + abs (∑ k in finset.range n, a k)) : 15 | 20 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1988/p8.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1988_p8 11 | (f : ℕ+ → ℕ+ → ℝ) 12 | (h₀ : ∀ x, f x x = x) 13 | (h₁ : ∀ x y, f x y = f y x) 14 | (h₂ : ∀ x y, (↑x + ↑y) * f x y = y * (f x (x + y))) : 15 | f 14 52 = 364 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1990/p15.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1990_p15 11 | (a b x y : ℝ) 12 | (h₀ : a * x + b * y = 3) 13 | (h₁ : a * x^2 + b * y^2 = 7) 14 | (h₂ : a * x^3 + b * y^3 = 16) 15 | (h₃ : a * x^4 + b * y^4 = 42) : 16 | a * x^5 + b * y^5 = 20 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1990/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1990_p2 : 11 | (52 + 6 * real.sqrt 43)^((3:ℝ) / 2) - (52 - 6 * real.sqrt 43)^((3:ℝ) / 2) = 828 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1991/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1991_p1 11 | (x y : ℕ) 12 | (h₀ : 0 < x ∧ 0 < y) 13 | (h₁ : x * y + (x + y) = 71) 14 | (h₂ : x^2 * y + x * y^2 = 880) : 15 | x^2 + y^2 = 146 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1991/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1991_p6 11 | (r : ℝ) 12 | (h₀ : ∑ k in finset.range 92 \ finset.range 19, (int.floor (r + k / 100)) = 546) : 13 | int.floor (100 * r) = 743 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1991/p9.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1991_p9 11 | (x : ℝ) 12 | (m : ℚ) 13 | (h₀ : 1 / real.cos x + real.tan x = 22 / 7) 14 | (h₁ : 1 / real.sin x + 1 / real.tan x = m) : 15 | ↑m.denom + m.num = 44 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1994/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1994_p3 11 | (x : ℤ) 12 | (f : ℤ → ℤ) 13 | (h0 : f x + f (x-1) = x^2) 14 | (h1 : f 19 = 94): 15 | f (94) % 1000 = 561 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1994/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1994_p4 11 | (n : ℕ) 12 | (h₀ : 0 < n) 13 | (h₀ : ∑ k in finset.range (n + 1) \ finset.range 1, int.floor (real.log k / real.log 2) = 1994) : 14 | n = 312 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1997/p12.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1997_p12 11 | (x : ℝ) 12 | (h₀ : x = (∑ n in finset.range 45 \ finset.range 1, real.cos (n * π / 180)) / (∑ n in finset.range 45 \ finset.range 1, real.sin (n * π / 180))) : 13 | int.floor (100 * x) = 241 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1997/p9.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1997_p9 11 | (a : ℝ) 12 | (h₀ : 0 < a) 13 | (h₁ : 1 / a - int.floor (1 / a) = a^2 - int.floor (a^2)) 14 | (h₂ : 2 < a^2) 15 | (h₃ : a^2 < 3) : 16 | a^12 - 144 * (1 / a) = 233 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/1999/p11.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aime_1999_p11 11 | (m : ℚ) 12 | (h₀ : ∑ k in finset.erase (finset.range 36) 0, real.sin (5 * k * π / 180) = real.tan (m * π / 180)) 13 | (h₁ : (m.denom:ℝ) / m.num < 90) : 14 | ↑m.denom + m.num = 177 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/2020/ii/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom aimeII_2020_p6 11 | (t : ℕ+ → ℚ) 12 | (h₀ : t 1 = 20) 13 | (h₁ : t 2 = 21) 14 | (h₂ : ∀ n ≥ 3, t n = (5 * t (n - 1) + 1) / (25 * t (n - 2))) : 15 | ↑(t 2020).denom + (t 2020).num = 626 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/aime/README.md: -------------------------------------------------------------------------------- 1 | # American Invitational Mathematics Examination - AIME 2 | 3 | Official site: https://www.maa.org/math-competitions/american-invitational-mathematics-examination-aime -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2000/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2000_p1 11 | (i m o : ℕ) 12 | (h₀ : i ≠ 0 ∧ m ≠ 0 ∧ o ≠ 0) 13 | (h₁ : i*m*o = 2001) : 14 | i+m+o ≤ 671 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2000/p12.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2000_p12 11 | (a m c : ℕ) 12 | (h₀ : a + m + c = 12) : 13 | a*m*c + a*m + m*c + a*c ≤ 112 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2000/p15.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2000_p15 11 | (f : ℂ → ℂ) 12 | (h₀ : ∀ x, f (x / 3) = x^2 + x + 1) 13 | (h₁ : fintype (f ⁻¹' {7})) : 14 | ∑ y in (f⁻¹' {7}).to_finset, y / 3 = - 1 / 9 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2000/p20.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2000_p20 11 | (x y z : ℝ) 12 | (h₀ : 0 < x ∧ 0 < y ∧ 0 < z) 13 | (h₁ : x + 1/y = 4) 14 | (h₂ : y + 1/z = 1) 15 | (h₃ : z + 1/x = 7/3) : 16 | x*y*z = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2001/p21.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2001_p21 11 | (a b c d : ℕ) 12 | (h₀ : a*b*c*d = nat.factorial 8) 13 | (h₁ : a*b + a + b = 524) 14 | (h₂ : b*c + b + c = 146) 15 | (h₃ : c*d + c + d = 104) : 16 | ↑a - ↑d = (10:ℤ) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2001/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2001_p5 : 11 | finset.prod (finset.filter (λ x, ¬ even x) (finset.range 10000)) (id : ℕ → ℕ) = (10000!) / ((2^5000) * 5000!) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2001/p9.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12_2001_p9 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x > 0, ∀ y > 0, f (x * y) = f x / y) 13 | (h₁ : f 500 = 3) : f 600 = 5 / 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/a/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2002_p1 11 | (f : ℂ → ℂ) 12 | (h₀ : ∀ x, f x = (2 * x + 3) * (x - 4) + (2 * x + 3) * (x - 6)) 13 | (h₁ : fintype (f ⁻¹' {0})) : 14 | ∑ y in (f⁻¹' {0}).to_finset, y = 7 / 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/a/p12.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2002_p12 11 | (f : ℝ → ℝ) 12 | (k : ℝ) 13 | (h₀ : ∀ x, f x = x^2 - 63 * x + k) 14 | (h₁ : set.subset (f ⁻¹' {0}) { x : ℝ | ∃ n : ℕ, ↑n = x ∧ nat.prime n}) : 15 | k = 122 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/a/p13.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2002_p13 11 | (a b : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b) 13 | (h₁ : a ≠ b) 14 | (h₂ : abs (a - 1/a) = 1) 15 | (h₃ : abs (b - 1/b) = 1) : 16 | a + b = real.sqrt 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/a/p21.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2002_p21 11 | (u : ℕ → ℕ) 12 | (h₀ : u 0 = 4) 13 | (h₁ : u 1 = 7) 14 | (h₂ : ∀ n ≥ 2, u (n + 2) = (u n + u (n + 1)) % 10) : 15 | ∀ n, ∑ k in finset.range(n), u k > 10000 → 1999 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/b/p11.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2002_p11 11 | (a b : ℕ) 12 | (h₀ : nat.prime a) 13 | (h₁ : nat.prime b) 14 | (h₂ : nat.prime (a + b)) 15 | (h₃ : nat.prime (a - b)) : 16 | nat.prime (a + b + (a - b + (a + b))) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/b/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12b_2002_p2 11 | (x : ℤ) 12 | (h₀ : x = 4) : 13 | (3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 11 := 14 | begin 15 | rw h₀, 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/b/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2002_p4 11 | (n : ℕ+) 12 | (h₀ : (1 /. 2 + 1 /. 3 + 1 /. 7 + 1 /. ↑n).denom = 1) : 13 | n = 42 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2002/b/p7.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2002_p7 11 | (a b c : ℕ+) 12 | (h₀ : b = a + 1) 13 | (h₁ : c = b + 1) 14 | (h₂ : a * b * c = 8 * (a + b + c)) : 15 | a^2 + (b^2 + c^2) = 77 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2003/a/p24.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2003_p24 11 | (a b : ℝ) 12 | (h₀ : b ≤ a) 13 | (h₁ : 1 < b) : 14 | real.log (a / b) / real.log a + real.log (b / a) / real.log b ≤ 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2003/a/p25.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2003_p25 11 | (a b : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : 0 < b) 14 | (h₁ : ∀ x, f x = real.sqrt (a * x^2 + b * x)) 15 | (h₂ : {x | 0 ≤ f x} = f '' {x | 0 ≤ f x}) : 16 | a = 0 ∨ a = -4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2003/a/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2003_p5 11 | (a m c : ℕ) 12 | (h₀ : a ≤ 9 ∧ m ≤ 9 ∧ c ≤ 9) 13 | (h₁ : 10*(10*(10*(10*a + m) + c) + 1) + 0 + (10*(10*(10*(10*a + m) + c) + 1) + 2) = 123422) : 14 | a + m + c = 14 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2003/b/p17.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2003_p17 11 | (x y : ℝ) 12 | (h₀ : 0 < x ∧ 0 < y) 13 | (h₁ : real.log (x * y^3) = 1) 14 | (h₂ : real.log (x^2 * y) = 1) : 15 | real.log (x * y) = 3 / 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2003/b/p9.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12b_2003_p9 11 | (a b : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : ∀ x, f x = a * x + b) 14 | (h₁ : f 6 - f 2 = 12) : 15 | f 12 - f 2 = 30 := 16 | begin 17 | simp [*] at *, 18 | linarith, 19 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2004/b/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2004_p3 11 | (x y : ℕ) 12 | (h₀ : 2^x * 3^y = 1296) : 13 | x + y = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2008/a/p15.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2008_p15 11 | (k : ℕ) 12 | (h₀ : k = 2008^2 + 2^2008) : 13 | (k^2 + 2^k) % 10 = 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2008/a/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12a_2008_p2 11 | (x : ℝ) 12 | (h₀ : x * (1 / 2 + 2 / 3) = 1) : 13 | x = 6 / 7 := 14 | begin 15 | linarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2008/a/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2008_p4 : 11 | ∏ k in finset.range 502 \ finset.range 1, ((4:ℝ) * k + 4) / (4 * k) = 502 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2008/a/p8.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2008_p8 11 | (x y : ℝ) 12 | (h₀ : 0 < x ∧ 0 < y) 13 | (h₁ : y^3 = 1) 14 | (h₂ : 6 * x^2 = 2 * (6 * y^2)) : 15 | x^3 = 2 * real.sqrt 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2009/a/p15.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2009_p15 11 | (n : ℕ) 12 | (h₀ : 0 < n) 13 | (h₁ : ∑ k in finset.erase (finset.range (n + 1)) 0, (↑k * (complex.I^k)) = 48 + 49 * complex.I) : 14 | n = 97 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2009/a/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12a_2009_p2 : 11 | 1 + (1 / (1 + (1 / (1 + 1)))) = (5:ℝ) / 3 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2009/a/p25.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2009_p25 11 | (a : ℕ+ → ℝ) 12 | (h₀ : a 1 = 1) 13 | (h₁ : a 2 = 1 / real.sqrt 3) 14 | (h₂ : ∀ n, a (n + 2) = (a n + a (n + 1)) / (1 - a n * a (n + 1))) : 15 | abs (a 2009) = 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2009/a/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12a_2009_p5 11 | (x : ℝ) 12 | (h₀ : x^3 - (x + 1) * (x - 1) * x = 5) : 13 | x^3 = 125 := 14 | begin 15 | nlinarith [pow_two x], 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2009/a/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2009_p6 11 | (m n p q : ℝ) 12 | (h₀ : p = 2 ^ m) 13 | (h₁ : q = 3 ^ n) : 14 | p^(2 * n) * (q^m) = 12^(m * n) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2010/a/p11.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2010_p11 11 | (x b : ℝ) 12 | (h₀ : 0 < b) 13 | (h₁ : (7:ℝ)^(x + 7) = 8^x) 14 | (h₂ : x = real.log (7^7) / real.log b) : 15 | b = 8 / 7 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2010/a/p22.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2010_p22 11 | (x : ℝ) : 12 | 49 ≤ ∑ k in finset.range 120 \ finset.range 1, abs (↑k * x - 1) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2013/a/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2013_p4 : 11 | (2^2014 + 2^2012) / (2^2014 - 2^2012) = (5:ℝ) / 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2013/a/p8.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2013_p8 11 | (x y : ℝ) 12 | (h₀ : x ≠ 0) 13 | (h₁ : y ≠ 0) 14 | (h₂ : x + 2 / x = y + 2 / y) : 15 | x * y = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2015/a/p10.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2015_p10 11 | (x y : ℤ) 12 | (h₀ : 0 < y) 13 | (h₁ : y < x) 14 | (h₂ : x + y + (x * y) = 80) : 15 | x = 26 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2016/a/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2016_p2 11 | (x : ℝ) 12 | (h₀ : (10:ℝ)^x * 100^(2 * x) = 1000^5) : 13 | x = 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2016/a/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2016_p3 11 | (f : ℝ → ℝ → ℝ) 12 | (h₀ : ∀ x, ∀ y ≠ 0, f x y = x - y * int.floor (x / y)) : 13 | f (3 / 8) (-(2 / 5)) = -(1 / 40) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2017/a/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12a_2017_p2 11 | (x y : ℝ) 12 | (h₀ : x ≠ 0) 13 | (h₁ : y ≠ 0) 14 | (h₂ : x + y = 4 * (x * y)) : 15 | 1 / x + 1 / y = 4 := 16 | begin 17 | field_simp, 18 | linarith, 19 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2017/a/p7.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2017_p7 11 | (f : ℕ → ℝ) 12 | (h₀ : f 1 = 2) 13 | (h₁ : ∀ n, 1 < n ∧ even n → f n = f (n - 1) + 1) 14 | (h₂ : ∀ n, 1 < n ∧ odd n → f n = f (n - 2) + 2) : 15 | f 2017 = 2018 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2019/a/p12.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2019_p12 11 | (x y : ℝ) 12 | (h₀ : x ≠ 1 ∧ y ≠ 1) 13 | (h₁ : real.log x / real.log 2 = real.log 16 / real.log y) 14 | (h₂ : x * y = 64) : 15 | real.log (x / y) / real.log 2 = 20 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2019/a/p21.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2019_p21 11 | (z : ℂ) 12 | (h₀ : z = (1 + complex.I) / real.sqrt 2) : 13 | (∑ k in finset.range 13 \ finset.range 1, (z^(k^2))) * (∑ k in finset.range 13 \ finset.range 1, (1 / z^(k^2))) = 36 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2019/a/p9.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2019_p9 11 | (a : ℕ+ → ℚ) 12 | (h₀ : a 1 = 1) 13 | (h₁ : a 2 = 3 / 7) 14 | (h₂ : ∀ n, a (n + 2) = (a n * a (n + 1)) / (2 * a n - a (n + 1))) : 15 | ↑(a 2019).denom + (a 2019).num = 8078 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/a/p10.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2020_p10 11 | (n : ℕ+) 12 | (h₀ : real.log (real.log n / real.log 16) / real.log 2 = real.log (real.log n / real.log 4) / real.log 4) : 13 | n = 256 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/a/p13.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2020_p13 11 | (a b c : ℕ) 12 | (n : nnreal) 13 | (h₀ : n ≠ 1) 14 | (h₁ : 1 < a ∧ 1 < b ∧ 1 < c) 15 | (h₂ : (n * ((n * (n^(1 / c)))^(1 / b)))^(1 / a) = (n^25)^(1 / 36)) : 16 | b = 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/a/p15.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2020_p15 11 | (a b : ℂ) 12 | (h₀ : a^3 - 8 = 0) 13 | (h₁ : b^3 - 8 * b^2 - 8 * b + 64 = 0) : 14 | complex.abs (a - b) ≤ 2 * real.sqrt 21 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/a/p25.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2020_p25 11 | (a : ℚ) 12 | (h₀ : fintype {x : ℝ | ↑⌊x⌋ * (x - ↑⌊x⌋) = ↑a * x ^ 2}) 13 | (h₁ : ∑ k in {x : ℝ | ↑⌊x⌋ * (x - ↑⌊x⌋) = ↑a * x^2}.to_finset, k = 420) : 14 | ↑a.denom + a.num = 929 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/b/p13.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2020_p13 : 11 | real.sqrt (real.log 6 / real.log 2 + real.log 6 / real.log 3) = real.sqrt (real.log 3 / real.log 2) + real.sqrt (real.log 2 / real.log 3) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/b/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12b_2020_p2 : 11 | ((100 ^ 2 - 7 ^ 2):ℝ) / (70 ^ 2 - 11 ^ 2) * ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) = 1 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/b/p21.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2020_p21 11 | (h₀ : fintype {n : ℕ+ | (↑n + (1000:ℝ)) / (70:ℝ) = int.floor (real.sqrt n)}) : 12 | finset.card {n : ℕ+ | (↑n + (1000:ℝ)) / (70:ℝ) = int.floor (real.sqrt n)}.to_finset = 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/b/p22.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2020_p22 11 | (t : ℝ) : 12 | ((2^t - 3 * t) * t) / (4^t) ≤ 1 / 12 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/b/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2020_p5 11 | (a b : ℕ+) 12 | (h₀ : (5:ℝ) / 8 * b - 2 / 3 * a = 7) 13 | (h₁ : ↑b - (5:ℝ) / 8 * b - (a - 2 / 3 * a) = 7) : 14 | a = 42 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2020/b/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2020_p6 11 | (n : ℕ) 12 | (h₀ : 9 ≤ n) : 13 | ∃ x : ℕ, (x:ℝ)^2 = (nat.factorial (n + 2) - nat.factorial (n + 1)) / nat.factorial n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/a/p14.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2021_p14 : 11 | (∑ k in (finset.erase (finset.range 21) 0), (real.log (3^(k^2)) / real.log (5^k))) * ∑ k in (finset.erase (finset.range 101) 0), (real.log (25^k) / real.log (9^k)) = 21000 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/a/p18.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2021_p18 11 | (f : ℚ → ℝ) 12 | (h₀ : ∀x>0, ∀y>0, f (x * y) = f x + f y) 13 | (h₁ : ∀p, nat.prime p → f p = p) : 14 | f (25 /. 11) < 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/a/p25.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2021_p25 11 | (n : ℕ+) 12 | (f : ℕ+ → ℝ) 13 | (h₀ : ∀ n, f n = (∑ k in (nat.divisors n), 1)/(n^((1:ℝ)/3))) 14 | (h₁ : ∀ p ≠ n, f p < f n) : 15 | n = 2520 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/a/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2021_p3 11 | (x y : ℕ) 12 | (h₀ : x + y = 17402) 13 | (h₁ : 10∣x) 14 | (h₂ : x / 10 = y) : 15 | ↑x - ↑y = (14238:ℤ) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/a/p7.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem amc12a_2021_p7 11 | (x : ℝ) 12 | (y : ℝ) : 13 | 1 ≤ ((x * y) - 1)^2 + (x + y)^2 := 14 | begin 15 | ring_nf, 16 | nlinarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/a/p8.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12a_2021_p8 11 | (d : ℕ → ℕ) 12 | (h₀ : d 0 = 0) 13 | (h₁ : d 1 = 0) 14 | (h₂ : d 2 = 1) 15 | (h₃ : ∀ n≥3, d n = d (n - 1) + d (n - 3)) : 16 | even (d 2021) ∧ odd (d 2022) ∧ even (d 2023) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/b/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2021_p1 11 | (h₀ : fintype {x : ℤ | ↑(abs x) < 3 * real.pi}): 12 | finset.card {x : ℤ | ↑(abs x) < 3 * real.pi}.to_finset = 19 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/b/p18.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2021_p18 11 | (z : ℂ) 12 | (h₀ : 12 * complex.norm_sq z = 2 * complex.norm_sq (z + 2) + complex.norm_sq (z^2 + 1) + 31) : 13 | z + 6 / z = -2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/b/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2021_p3 11 | (x : ℝ) 12 | (h₀ : 2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53) : 13 | x = 3 / 4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/b/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2021_p4 11 | (m a : ℕ+) 12 | (h₀ : ↑m / ↑a = (3:ℝ) / 4) : 13 | (84 * ↑m + 70 * ↑a) / (↑m + ↑a) = (76:ℝ) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/12/2021/b/p9.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom amc12b_2021_p9 : 11 | (real.log 80 / real.log 2) / (real.log 2 / real.log 40) - (real.log 160 / real.log 2) / (real.log 2 / real.log 20) = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/amc/README.md: -------------------------------------------------------------------------------- 1 | # American Mathematics Competitions (AMC) 2 | 3 | Official site: https://www.maa.org/math-competitions/about-amc 4 | 5 | Although AMC broadly construed includes the AIME and others, we use it to refer only to the specific competitions with AMC in the name: the AMC 8, AMC 10, and AMC 12. -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1960/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1960_p2 11 | (x : ℝ) 12 | (h₀ : 0 ≤ 1 + 2 * x) 13 | (h₁ : (1 - real.sqrt (1 + 2 * x))^2 ≠ 0) 14 | (h₂ : (4 * x^2) / (1 - real.sqrt (1 + 2*x))^2 < 2*x + 9) : 15 | -(1 / 2) ≤ x ∧ x < 45 / 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1962/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1962_p2 11 | (x : ℝ) 12 | (h₀ : 0 ≤ 3 - x) 13 | (h₁ : 0 ≤ x + 1) 14 | (h₂ : 1 / 2 < real.sqrt (3 - x) - real.sqrt (x + 1)) : 15 | -1 ≤ x ∧ x < 1 - real.sqrt 31 / 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1963/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1963_p5 : 11 | real.cos (π / 7) - real.cos (2 * π / 7) + real.cos (3 * π / 7) = 1 / 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1968/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1968_p5_1 11 | (a : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : 0 < a) 14 | (h₁ : ∀ x, f (x + a) = 1 / 2 + real.sqrt (f x - (f x)^2)) : 15 | ∃ b > 0, ∀ x, f (x + b) = f x -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1973/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1973_p3 11 | (a b : ℝ) 12 | (h₀ : ∃ x, x^4 + a * x^3 + b * x^2 + a * x + 1 = 0) : 13 | 4 / 5 ≤ a^2 + b^2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1974/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1974_p3 11 | (n : ℕ) : 12 | ¬ 5∣∑ k in finset.range n, (nat.choose (2 * n + 1) (2 * k + 1)) * (2^(3 * k)) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1974/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1974_p5 11 | (a b c d s : ℝ) 12 | (h₀ : s = a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)) : 13 | 1 < s ∧ s < 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1979/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1979_p1 11 | (p q : ℕ) 12 | (h₀ : 0 < q) 13 | (h₁ : ∑ k in finset.range 1320 \ finset.range 1, ((-1)^(k + 1) * ((1:ℝ) / k)) = p / q) : 14 | 1979 ∣ p -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1981/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1981_p6 11 | (f : ℕ → ℕ → ℕ) 12 | (h₀ : ∀ y, f 0 y = y + 1) 13 | (h₁ : ∀ x, f (x + 1) 0 = f x 1) 14 | (h₂ : ∀ x y, f (x + 1) (y + 1) = f x (f (x + 1) y)) : 15 | ∀ y, f 4 (y + 1) = 2^(f 4 y + 3) - 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1982/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1982_p1 11 | (f : ℕ+ → ℕ) 12 | (h₀ : ∀ m n, f (m + n) - f m - f n = 0 ∨ f (m + n) - f m - f n = 1) 13 | (h₁ : f 2 = 0) 14 | (h₂ : 0 < f 3) 15 | (h₃ : f 9999 = 3333) : 16 | f 1982 = 660 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1983/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1983_p6 11 | (a b c : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) 13 | (h₁ : c < a + b) 14 | (h₂ : b < a + c) 15 | (h₃ : a < b + c) : 16 | 0 ≤ a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1984/p2.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1984_p2 11 | (a b : ℕ) 12 | (h₀ : 0 < a ∧ 0 < b) 13 | (h₁ : ¬ 7 ∣ a) 14 | (h₂ : ¬ 7 ∣ b) 15 | (h₃ : ¬ 7 ∣ (a + b)) 16 | (h₄ : (7^7) ∣ ((a + b)^7 - a^7 - b^7)) : 17 | 19 ≤ a + b -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1985/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1985_p6 11 | (f : ℕ+ → nnreal → ℝ) 12 | (h₀ : ∀ x, f 1 x = x) 13 | (h₁ : ∀ x n, f (n + 1) x = f n x * (f n x + 1 / n)) : 14 | ∃! a, ∀ n, 0 < f n a ∧ f n a < f (n + 1) a ∧ f (n + 1) a < 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1987/p4.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1987_p4 11 | (f : ℕ → ℕ) : 12 | ∃ n, f (f n) ≠ n + 1987 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1987/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1987_p6 11 | (p : ℕ) 12 | (f : ℕ → ℕ) 13 | (h₀ : ∀ x, f x = x^2 + x + p) 14 | (h₀ : ∀ k : ℕ, k ≤ nat.floor (real.sqrt (p / 3)) → nat.prime (f k)) : 15 | ∀ i ≤ p - 2, nat.prime (f i) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1990/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1990_p3 11 | (n : ℕ) 12 | (h₀ : 2 ≤ n) 13 | (h₁ : n^2 ∣ 2^n + 1) : 14 | n = 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1992/p1.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1992_p1 11 | (p q r : ℤ) 12 | (h₀ : 1 < p ∧ p < q ∧ q < r) 13 | (h₁ : (p - 1) * (q - 1) * (r - 1)∣(p * q * r - 1)) : 14 | (p, q, r) = (2, 4, 8) ∨ (p, q, r) = (3, 5, 15) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1993/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1993_p5 : 11 | ∃ f : ℕ+ → ℕ+, (∀ a b, (a < b) ↔ f a < f b) ∧ f 1 = 2 ∧ ∀ n, f (f n) = f n + n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/1997/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_1997_p5 11 | (x y : ℕ) 12 | (h₀ : 0 < x ∧ 0 < y) 13 | (h₁ : x^(y^2) = y^x) : 14 | (x, y) = (1, 1) ∨ (x, y) = (16, 2) ∨ (x, y) = (27, 3) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/2006/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_2006_p6 11 | (a b c : ℝ) : 12 | (a * b * (a^2 - b^2)) + (b * c * (b^2 - c^2)) + (c * a * (c^2 - a^2)) ≤ (9 * real.sqrt 2) / 32 * (a^2 + b^2 + c^2)^2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/2007/p6.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom imo_2007_p6 11 | (a : ℕ → nnreal) 12 | (h₀ : ∑ x in finset.range 100, ((a (x + 1))^2) = 1) : 13 | ∑ x in finset.range 99, ((a (x + 1))^2 * a (x + 2)) + (a 100)^2 * a 1 < 12 / 25 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/imo/README.md: -------------------------------------------------------------------------------- 1 | # The International Mathematical Olympiad (IMO) 2 | 3 | Official site: https://www.imo-official.org/ 4 | 5 | -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/README.md: -------------------------------------------------------------------------------- 1 | # The MATH Dataset 2 | 3 | This directory consists of problems taken from the [MATH dataset](https://github.com/hendrycks/math). This dataset consists of problems from mathematics competitions but unfortunately the provenance of these problems are not included in the dataset. We include problems from this dataset for historical reasons only, and strongly encourage contributors to submit gems directly to the directory that reflects their original source whenever possible. -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p10.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_10 : 11 | abs ((120:ℝ) / 100 * 30 - 130 / 100 * 20) = 10 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p101.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_101 11 | (x : ℝ) 12 | (h₀ : x^2 - 5 * x - 4 ≤ 10) : 13 | x ≥ -2 ∧ x ≤ 7 := 14 | begin 15 | split; nlinarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p104.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_104 11 | (x : ℝ) 12 | (h₀ : 125 / 8 = x / 12) : 13 | x = 375 / 2 := 14 | begin 15 | linarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p107.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_107 11 | (x y : ℝ) 12 | (h₀ : x^2 + 8 * x + y^2 - 6 * y = 0) : 13 | (x + 4)^2 + (y-3)^2 = 5^2 := 14 | begin 15 | linarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p109.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_109 11 | (a b : ℝ) 12 | (h₀ : 3 * a + 2 * b = 12) 13 | (h₁ : a = 4) : 14 | b = 0 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p113.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_113 11 | (x : ℝ) : 12 | x^2 - 14 * x + 3 ≥ 7^2 - 14 * 7 + 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p119.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_119 11 | (d e : ℝ) 12 | (h₀ : 2 * d = 17 * e - 8) 13 | (h₁ : 2 * e = d - 9) : 14 | e = 2 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p125.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_125 11 | (x y : ℕ+) 12 | (h₀ : 5 * x = y) 13 | (h₁ : (↑x - (3:ℤ)) + (y - (3:ℤ)) = 30) : 14 | x = 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p126.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_126 11 | (x y : ℝ) 12 | (h₀ : 2 * 3 = x - 9) 13 | (h₁ : 2 * (-5) = y + 1) : 14 | x = 15 ∧ y = -11 := 15 | begin 16 | split; linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p129.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_129 11 | (a : ℝ) 12 | (h₀ : a ≠ 0) 13 | (h₁ : 8⁻¹ / 4⁻¹ - a⁻¹ = 1) : 14 | a = -2 := 15 | begin 16 | field_simp at h₁, 17 | linarith, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p131.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_131 11 | (a b : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : ∀ x, f x = 2 * x^2 - 7 * x + 2) 14 | (h₁ : f a = 0) 15 | (h₂ : f b = 0) : 16 | 1 / (a - 1) + 1 / (b - 1) = -1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p137.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_137 11 | (x : ℕ) 12 | (h₀ : ↑x + (4:ℝ) / (100:ℝ) * ↑x = 598) : 13 | x = 575 := 14 | begin 15 | have h₁ : ↑x = (575:ℝ), linarith, 16 | assumption_mod_cast, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p139.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_139 11 | (s : ℝ → ℝ → ℝ) 12 | (h₀ : ∀ x≠0, ∀y≠0, s x y = (1/y - 1/x) / (x-y)) : 13 | s 3 11 = 1/33 := 14 | begin 15 | norm_num [h₀], 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p141.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_141 11 | (a b : nnreal) 12 | (h₁ : (a * b)=180) 13 | (h₂ : 2 * (a + b)=54) : 14 | nnreal.sqrt (a^2 + b^2) = 369 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p142.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_142 11 | (m b : ℝ) 12 | (h₀ : m * 7 + b = -1) 13 | (h₁ : m * (-1) + b = 7) : 14 | m + b = 5 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p143.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_143 11 | (f g : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = x + 1) 13 | (h₁ : ∀ x, g x = x^2 + 3) : 14 | f (g 2) = 8 := 15 | begin 16 | rw [h₀, h₁], 17 | norm_num, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p144.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_144 11 | (a b c d : ℕ+) 12 | (h₀ : (c:ℤ) - (b:ℤ) = (d:ℤ)) 13 | (h₁ : (b:ℤ) - (a:ℤ) = (d:ℤ)) 14 | (h₂ : a + b + c = 60) 15 | (h₃ : a + b > c) : 16 | d < 10 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p148.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_148 11 | (c : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : ∀ x, f x = c * x^3 - 9 * x + 3) 14 | (h₁ : f 2 = 9) : 15 | c = 3 := 16 | begin 17 | rw h₀ at h₁, 18 | linarith, 19 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p149.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_149 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x < -5, f x = x^2 + 5) 13 | (h₁ : ∀ x ≥ -5, f x = 3 * x -8) 14 | (h₂ : fintype (f⁻¹' {10})) : 15 | ∑ k in (f⁻¹' {10}).to_finset, k = 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p15.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_15 11 | (s : ℕ+ → ℕ+ → ℕ+) 12 | (h₀ : ∀ a b, s a b = a^(b:ℕ) + b^(a:ℕ)) : 13 | s 2 6 = 100 := 14 | begin 15 | rw h₀, 16 | refl, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p151.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_151 : 11 | int.ceil (real.sqrt 27) - int.floor (real.sqrt 26) = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p153.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_153 11 | (n : ℝ) 12 | (h₀ : n = 1 / 3) : 13 | int.floor (10 * n) + int.floor (100 * n) + int.floor (1000 * n) + int.floor (10000 * n) = 3702 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p158.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_158 11 | (a : ℕ) 12 | (h₀ : even a) 13 | (h₁ : ↑∑ k in finset.range 8, (2 * k + 1) - ↑∑ k in finset.range 5, (a + 2 * k) = (4:ℤ)) : 14 | a = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p160.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_160 11 | (n x : ℝ) 12 | (h₀ : n + x = 97) 13 | (h₁ : n + 5 * x = 265) : 14 | n + 2 * x = 139 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p17.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_17 11 | (a : ℝ) 12 | (h₀ : real.sqrt (4 + real.sqrt (16 + 16 * a)) + real.sqrt (1 + real.sqrt (1 + a)) = 6) : 13 | a = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p170.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_170 11 | (h₀ : fintype {n : ℤ | abs (n - 2) ≤ 5 + 6 / 10}) : 12 | finset.card { n : ℤ | abs (n - 2) ≤ 5 + 6 / 10}.to_finset = 11 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p171.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_171 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀x, f x = 5 * x + 4) : 13 | f 1 = 9 := 14 | begin 15 | rw h₀, 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p176.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_176 11 | (x : ℝ) : 12 | (x + 1)^2 * x = x^3 + 2 * x^2 + x := 13 | begin 14 | ring_nf, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p181.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_181 11 | (n : ℝ) 12 | (h₀ : n ≠ 3) 13 | (h₁ : (n + 5) / (n - 3) = 2) : n = 11 := 14 | begin 15 | rw div_eq_iff at h₁, 16 | linarith, 17 | exact sub_ne_zero.mpr h₀, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p182.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_182 11 | (y : ℂ) : 12 | 7 * (3 * y + 2) = 21 * y + 14 := 13 | begin 14 | ring_nf, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p185.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_185 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = abs (x + 4)) 13 | (h₁ : fintype {x : ℝ | f x < 9}) : 14 | finset.card { x : ℝ | f x < 9 }.to_finset = 17 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p188.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_188 11 | (σ : equiv ℝ ℝ) 12 | (h : σ.1 2 = σ.2 2) : 13 | σ.1 (σ.1 2) = 2 := 14 | begin 15 | simp [h] 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p190.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_190 : 11 | ((3:ℝ) / 8 + 7 / 8) / (4 / 5) = 25 / 16 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p196.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_196 11 | (h₀ : fintype {x : ℝ | abs (2 - x) = 3}) : 12 | ∑ k in {x : ℝ | abs (2 - x) = 3}.to_finset, k = 4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p208.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_208 : 11 | real.sqrt 1000000 - 1000000^((1:ℝ)/3) = 900 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p209.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_209 11 | (σ : equiv ℝ ℝ) 12 | (h₀ : σ.2 2 = 10) 13 | (h₁ : σ.2 10 = 1) 14 | (h₂ : σ.2 1 = 2) : 15 | σ.1 (σ.1 10) = 1 := 16 | begin 17 | rw [← h₀, ← h₂], 18 | simp, 19 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p215.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_215 11 | (h₀ : fintype {x : ℝ | (x + 3)^2 = 121}) : 12 | ∑ k in {x : ℝ | (x + 3)^2 = 121}.to_finset, k = -6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p22.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_22 : 11 | real.log (5^4) / real.log (5^2) = 2 := 12 | begin 13 | field_simp [two_mul], 14 | rw div_eq_iff, 15 | ring, 16 | norm_num, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p224.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_224 11 | (h₀ : fintype { n : ℕ | real.sqrt n < 7 / 2 ∧ 2 < real.sqrt n}) : 12 | finset.card { n : ℕ | real.sqrt n < 7 / 2 ∧ 2 < real.sqrt n}.to_finset = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p24.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_24 11 | (x : ℝ) 12 | (h₀ : x / 50 = 40) : 13 | x = 2000 := 14 | begin 15 | nlinarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p245.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_245 11 | (x : ℝ) 12 | (h₀ : x ≠ 0) : 13 | (4 / x)⁻¹ * ((3 * x^3) / x)^2 * ((1 / (2 * x))⁻¹)^3 = 18 * x^8 := 14 | begin 15 | field_simp [h₀]; ring, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p247.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_247 11 | (t s : ℝ) 12 | (n : ℤ) 13 | (h₀ : t = 2 * s - s^2) 14 | (h₁ : s = n^2 - 2^n + 1) 15 | (n = 3) : 16 | t = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p251.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_251 11 | (x : ℝ) 12 | (h₀ : x ≠ 0) 13 | (h₁ : 3 + 1 / x = 7 / x) : 14 | x = 2 := 15 | begin 16 | field_simp [h₀] at h₁, 17 | linarith, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p275.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_275 11 | (x : ℝ) 12 | (h : ((11:ℝ)^(1 / 4))^(3 * x - 3) = 1 / 5) : 13 | ((11:ℝ)^(1 / 4))^(6 * x + 2) = 121 / 25 := 14 | begin 15 | revert x h, 16 | norm_num, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p276.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_276 11 | (a b : ℤ) 12 | (h₀ : ∀ x : ℝ, 10 * x^2 - x - 24 = (a * x - 8) * (b * x + 3)) : 13 | a + b = 12 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p28.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_28 11 | (c : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : ∀ x, f x = 2 * x^2 + 5 * x + c) 14 | (h₁ : ∃ x, f x ≤ 0) : 15 | c ≤ 25/8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p293.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_293 11 | (x : nnreal) : 12 | real.sqrt (60 * x) * real.sqrt (12 * x) * real.sqrt (63 * x) = 36 * x * real.sqrt (35 * x) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p296.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_296 : 11 | abs (((3491 - 60) * (3491 + 60) - 3491^2):ℤ) = 3600 := 12 | begin 13 | rw abs_of_nonpos, 14 | norm_num, 15 | norm_num, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p302.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_302 : 11 | (complex.I / 2)^2 = -(1 / 4) := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p304.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_304 : 11 | 91^2 = 8281 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p31.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_31 11 | (x : nnreal) 12 | (u : ℕ → nnreal) 13 | (h₀ : ∀ n, u (n + 1) = nnreal.sqrt (x + u n)) 14 | (h₁ : filter.tendsto u filter.at_top (𝓝 9)) : 15 | 9 = nnreal.sqrt (x + 9) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p313.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_313 11 | (v i z : ℂ) 12 | (h₀ : v = i * z) 13 | (h₁ : v = 1 + complex.I) 14 | (h₂ : z = 2 - complex.I) : 15 | i = 1/5 + 3/5 * complex.I -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p314.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_314 11 | (n : ℕ) 12 | (h₀ : n = 11) : 13 | (1 / 4)^(n + 1) * 2^(2 * n) = 1 / 4 := 14 | begin 15 | rw h₀, 16 | norm_num, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p320.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_320 11 | (x : nnreal) 12 | (a b c : ℕ+) 13 | (h₀ : 2 * x^2 = 4 * x + 9) 14 | (h₁ : x = (a + nnreal.sqrt b) / c) : 15 | a + b + c = 26 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p323.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_323 11 | (σ : equiv ℝ ℝ) 12 | (h : ∀ x, σ.1 x = x^3 - 8) : 13 | σ.2 (σ.1 (σ.2 19)) = 3 := 14 | begin 15 | simp, 16 | norm_num [*, equiv.symm_apply_eq] at *, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p329.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_329 11 | (x y : ℝ) 12 | (h₀ : 3 * y = x) 13 | (h₁ : 2 * x + 5 * y = 11) : 14 | x + y = 4 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p33.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_33 11 | (x y z : ℝ) 12 | (h₀ : x ≠ 0) 13 | (h₁ : 2 * x = 5 * y) 14 | (h₂ : 7 * y = 10 * z) : 15 | z / x = 7 / 25 := 16 | begin 17 | field_simp, 18 | nlinarith, 19 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p332.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_332 11 | (x y : nnreal) 12 | (h₀ : (x + y) / 2 = 7) 13 | (h₁ : real.sqrt (x * y) = real.sqrt 19) : 14 | x^2 * y^2 = 158 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p346.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_346 11 | (f g : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = 2 * x - 3) 13 | (h₁ : ∀ x, g x = x + 1) : 14 | g (f 5 - 1) = 7 := 15 | begin 16 | rw [h₀, h₁], 17 | norm_num, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p35.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_35 11 | (p q : ℝ → ℝ) 12 | (h₀ : ∀ x, p x = 2 - x^2) 13 | (h₁ : ∀ x≠0, q x = 6 / x) : 14 | p (q 2) = -7 := 15 | begin 16 | norm_num [h₀, h₁], 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p354.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_354 11 | (a d : ℝ) 12 | (h₀ : a + 6 * d = 30) 13 | (h₁ : a + 10 * d = 60) : 14 | a + 20 * d = 135 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p359.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_359 11 | (y : ℝ) 12 | (h₀ : y + 6 + y = 2 * 12) : 13 | y = 9 := 14 | begin 15 | linarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p362.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_362 11 | (a b : ℝ) 12 | (h₀ : a^2 * b^3 = 32 / 27) 13 | (h₁ : a / b^3 = 27 / 4) : 14 | a + b = 8 / 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p37.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_37 11 | (x y : ℝ) 12 | (h₀ : x + y = 7) 13 | (h₁ : 3 * x + y = 45) : 14 | x^2 - y^2 = 217 := 15 | begin 16 | nlinarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p388.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_388 11 | (x y z : ℝ) 12 | (h₀ : 3 * x + 4 * y - 12 * z = 10) 13 | (h₁ : -2 * x - 3 * y + 9 * z = -4) : 14 | x = 14 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p392.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_392 11 | (n : ℕ) 12 | (h₀ : even n) 13 | (h₁ : (↑n - 2)^2 + ↑n^2 + (↑n + 2)^2 = (12296:ℤ)) : 14 | ((↑n - 2) * ↑n * (↑n + 2)) / 8 = (32736:ℤ) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p393.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_393 11 | (σ : equiv ℝ ℝ) 12 | (h₀ : ∀ x, σ.1 x = 4 * x^3 + 1) : 13 | σ.2 33 = 2 := 14 | begin 15 | simp [*, sq] at *, 16 | norm_num [h₀, equiv.symm_apply_eq], 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p398.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_398 11 | (a b c : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) 13 | (h₁ : 9 * b = 20 * c) 14 | (h₂ : 7 * a = 4 * b) : 15 | 63 * a = 80 * c := 16 | begin 17 | linarith, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p400.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_400 11 | (x : ℝ) 12 | (h₀ : 5 + 500 / 100 * 10 = 110 / 100 * x) : 13 | x = 50 := 14 | begin 15 | linarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p410.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_410 11 | (x y : ℝ) 12 | (h₀ : y = x^2 - 6 * x + 13) : 13 | 4 ≤ y := 14 | begin 15 | cases le_total x 3 with h, 16 | all_goals { nlinarith }, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p412.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_412 11 | (x y : ℝ) 12 | (h₀ : x + y = 25) 13 | (h₁ : x - y = 11) : 14 | x = 18 := 15 | begin 16 | linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p419.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_419 11 | (a b : ℝ) 12 | (h₀ : a = -1) 13 | (h₁ : b = 5) : 14 | -a - b^2 + 3 * (a * b) = -39 := 15 | begin 16 | rw [h₀, h₁], 17 | norm_num, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p432.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_432 11 | (x : ℝ) : 12 | (x + 3) * (2 * x - 6) = 2 * x^2 - 18 := 13 | begin 14 | linarith, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p433.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_433 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = 3 * real.sqrt (2 * x - 7) - 8) : 13 | f 8 = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p437.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_437 11 | (x y : ℝ) 12 | (n : ℤ) 13 | (h₀ : x^3 = -45) 14 | (h₁ : y^3 = -101) 15 | (h₂ : x < n) 16 | (h₃ : ↑n < y) : 17 | n = -4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p44.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_44 11 | (s t : ℝ) 12 | (h₀ : s = 9 - 2 * t) 13 | (h₁ : t = 3 * s + 1) : 14 | s = 1 ∧ t = 4 := 15 | begin 16 | split; linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p440.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_440 11 | (x : ℝ) 12 | (h₀ : 2 / 3 / 3 = x / 10) : 13 | x = 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p441.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_441 11 | (x : ℝ) 12 | (h₀ : x ≠ 0) : 13 | 12 / (x * x) * (x^4 / (1 * x)) * (35 / (3 * x)) = 10 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p452.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_452 11 | (a : ℕ+ → ℝ) 12 | (h₀ : ∀ n, a (n + 2) - a (n + 1) = a (n + 1) - a n) 13 | (h₁ : a 1 = 2 / 3) 14 | (h₂ : a 2 = 4 / 5) : 15 | a 5 = 11 / 15 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p455.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_455 11 | (x : ℝ) 12 | (h₀ : 2 * (2 * (2 * (2 * x))) = 48) : 13 | x = 3 := 14 | begin 15 | linarith, 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p462.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_462 : 11 | (1 / 2 + 1 / 3) * (1 / 2 - 1 / 3) = 5 / 36 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p48.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_48 11 | (q e : ℂ) 12 | (h₀ : q = 9 - 4 * complex.I) 13 | (h₁ : e = -3 - 4 * complex.I) : q - e = 12 := 14 | begin 15 | rw [h₀, h₁], 16 | ring, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p480.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_480 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x < 0, f x = -(x^2) - 1) 13 | (h₁ : ∀ x, 0 ≤ x ∧ x < 4 → f x = 2) 14 | (h₂ : ∀ x ≥ 4, f x = real.sqrt x) : 15 | f π = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p487.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_487 11 | (a b c d : ℝ) 12 | (h₀ : b = a^2) 13 | (h₁ : a + b = 1) 14 | (h₂ : d = c^2) 15 | (h₃ : c + d = 1) : 16 | real.sqrt ((a - c)^2 + (b - d)^2)= real.sqrt 10 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p493.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_493 11 | (f : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = x^2 - 4 * real.sqrt x + 1) : 13 | f (f 4) = 70 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p509.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_509 : 11 | real.sqrt ((5 / real.sqrt 80 + real.sqrt 845 / 9 + real.sqrt 45) / real.sqrt 5) = 13 / 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p51.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_51 11 | (a b : ℝ) 12 | (h₀ : 0 < a ∧ 0 < b) 13 | (h₁ : a + b = 35) 14 | (h₂ : a = (2/5) * b) : 15 | b - a = 15 := 16 | begin 17 | linarith, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p510.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_510 11 | (x y : ℝ) 12 | (h₀ : x + y = 13) 13 | (h₁ : x * y = 24) : 14 | real.sqrt (x^2 + y^2) = 11 := 15 | begin 16 | norm_num [real.sqrt_eq_cases] at *, 17 | nlinarith, 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p513.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_513 11 | (a b : ℝ) 12 | (h₀ : 3 * a + 2 * b = 5) 13 | (h₁ : a + b = 2) : 14 | a = 1 ∧ b = 1 := 15 | begin 16 | split; linarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p536.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_536 : 11 | ↑3! * ((2:ℝ)^3 + real.sqrt 9) / 2 = (33:ℝ) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p547.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_547 11 | (x y : ℝ) 12 | (h₀ : x = 5) 13 | (h₁ : y = 2) : 14 | real.sqrt (x^3 - 2^y) = 11 := 15 | begin 16 | simp [h₀, h₁], 17 | norm_num [real.sqrt_eq_cases], 18 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p568.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_568 11 | (a : ℝ) : 12 | (a - 1) * (a + 1) * (a + 2) - (a - 2) * (a + 1) = a^3 + a^2 := 13 | begin 14 | linarith, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p598.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_598 11 | (a b c d : ℝ) 12 | (h₁ : ((4:ℝ)^a) = 5) 13 | (h₂ : ((5:ℝ)^b) = 6) 14 | (h₃ : ((6:ℝ)^c) = 7) 15 | (h₄ : ((7:ℝ)^d) = 8) : 16 | a * b * c * d = 3 / 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p67.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_algebra_67 11 | (f g : ℝ → ℝ) 12 | (h₀ : ∀ x, f x = 5 * x + 3) 13 | (h₁ : ∀ x, g x = x^2 - 2) : 14 | g (f (-1)) = 2 := 15 | begin 16 | norm_num [h₀, h₁], 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p73.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_73 11 | (p q r x : ℂ) 12 | (h₀ : (x - p) * (x - q) = (r - p) * (r - q)) 13 | (h₁ : x ≠ r) : 14 | x = p + q - r -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p756.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_756 11 | (a b : ℝ) 12 | (h₀ : (2:ℝ)^a = 32) 13 | (h₁ : a^b = 125) : 14 | b^a = 243 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/algebra/p77.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_algebra_77 11 | (a b : ℝ) 12 | (f : ℝ → ℝ) 13 | (h₀ : a ≠ 0 ∧ b ≠ 0) 14 | (h₁ : ∀ x, f x = x^2 + a * x + b) 15 | (h₂ : f a = 0) 16 | (h₃ : f b = 0) : 17 | a = 1 ∧ b = -2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p100.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_100 11 | (n : ℕ+) 12 | (h₀ : nat.gcd n 40 = 10) 13 | (h₁ : nat.lcm n 40 = 280) : 14 | n = 70 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p101.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_101 : 11 | (17 * 18) % 4 = 2 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p102.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_102 : 11 | (2^8) % 5 = 1 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p110.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_110 11 | (a b : ℕ) 12 | (h₀ : 0 < a ∧ 0 < b ∧ b ≤ a) 13 | (h₁ : (a + b) % 10 = 2) 14 | (h₂ : (2 * a + b) % 10 = 1) : 15 | (a - b) % 10 = 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p1124.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_1124 11 | (n : ℕ) 12 | (h₀ : n ≤ 9) 13 | (h₁ : 18∣374 * 10 + n) : 14 | n = 4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p12.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_12 : 11 | finset.card (finset.filter (λ x, 20∣x) (finset.range 86 \ finset.range 15)) = 4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p127.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_127 : 11 | (∑ k in (finset.range 101), 2^k) % 7 = 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p13.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_13 11 | (u v : ℕ+) 12 | (h₀ : 14 * ↑u % 100 = 46) 13 | (h₁ : 14 * ↑v % 100 = 46) 14 | (h₂ : u < 50) 15 | (h₃ : v < 100) 16 | (h₄ : 50 < v) : 17 | ((u + v):ℕ) / 2 = 64 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p132.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_132 : 11 | 2004 % 12 = 0 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p136.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_136 11 | (n : ℕ) 12 | (h₀ : 123 * n + 17 = 39500) : n = 321 := 13 | begin 14 | linarith, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p150.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_150 11 | (n : ℕ) 12 | (h₀ : ¬ nat.prime (7 + 30 * n)) : 13 | 6 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p155.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_155 : 11 | finset.card (finset.filter (λ x, x % 19 = 7) (finset.range 1000 \ finset.range 100)) = 52 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p156.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_156 11 | (n : ℕ+) : 12 | nat.gcd (n + 7) (2 * n + 1) ≤ 13 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p169.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_169 : 11 | nat.gcd (nat.factorial 20) 200000 = 40000 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p175.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_175 : 11 | (2^2010) % 10 = 4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p185.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_185 11 | (n : ℕ) 12 | (h₀ : n % 5 = 3) : 13 | (2 * n) % 5 = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p188.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_188 : 11 | nat.gcd 180 168 = 12 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p198.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_198 : 11 | (5^2005) % 100 = 25 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p200.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_200 : 11 | 139 % 11 = 7 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p202.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_202 : 11 | (19^19 + 99^99) % 10 = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p207.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_207 : 11 | 8 * 9^2 + 5 * 9 + 2 = 695 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p211.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_211 : 11 | finset.card (finset.filter (λ n, 6 ∣ (4 * ↑n - (2:ℤ))) (finset.range 60)) = 20 := 12 | begin 13 | norm_cast, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p212.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_212 : 11 | (16^17 * 17^18 * 18^19) % 10 = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p22.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_22 11 | (b : ℕ) 12 | (h₀ : b < 10) 13 | (h₁ : nat.sqrt (10 * b + 6) * nat.sqrt (10 * b + 6) = 10 * b + 6) : 14 | b = 3 ∨ b = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p221.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_221 11 | (h₀ : fintype {x : ℕ | 0 < x ∧ x < 1000 ∧ x.divisors.card = 3}) : 12 | finset.card {x : ℕ | 0 < x ∧ x < 1000 ∧ finset.card (nat.divisors x) = 3}.to_finset = 11 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p222.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_222 11 | (b : ℕ) 12 | (h₀ : nat.lcm 120 b = 3720) 13 | (h₁ : nat.gcd 120 b = 8) : 14 | b = 248 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p227.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_227 11 | (x y n : ℕ) 12 | (h₀ : ↑x / (4:ℝ) + y / 6 = (x + y) / n) : 13 | n = 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p232.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_232 11 | (x y z : zmod 31) 12 | (h₀ : x = 3⁻¹) 13 | (h₁ : y = 5⁻¹) 14 | (h₂ : z = (x + y)⁻¹) : 15 | z = 29 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p233.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_233 11 | (b : zmod (11^2)) 12 | (h₀ : b = 24⁻¹) : 13 | b = 116 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p234.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_234 11 | (a b : ℕ) 12 | (h₀ : 1 ≤ a ∧ a ≤ 9 ∧ b ≤ 9) 13 | (h₁ : (10 * a + b)^3 = 912673) : 14 | a + b = 16 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p235.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_235 : 11 | (29 * 79 + 31 * 81) % 10 = 2 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p236.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_236 : 11 | (1999^2000) % 5 = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p237.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_237 : 11 | (∑ k in (finset.range 101), k) % 6 = 4 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p239.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_239 : 11 | (∑ k in finset.erase (finset.range 13) 0, k) % 4 = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p24.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_24 : 11 | (∑ k in (finset.erase (finset.range 10) 0), 11^k) % 100 = 59 := 12 | begin 13 | norm_num [finset.sum], 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p247.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_247 11 | (n : ℕ) 12 | (h₀ : (3 * n) % 2 = 11) : 13 | n % 11 = 8 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p252.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_252 : 11 | (nat.factorial 7) % 23 = 3 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p254.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_254 : 11 | (239 + 174 + 83) % 10 = 6 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p257.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_257 11 | (x : ℕ) 12 | (h₀ : 1 ≤ x ∧ x ≤ 100) 13 | (h₁ : 77∣(∑ k in (finset.range 101), k - x)) : 14 | x = 45 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p269.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_269 : 11 | (2005^2 + 2005^0 + 2005^0 + 2005^5) % 100 = 52 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p277.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_277 11 | (m n : ℕ) 12 | (h₀ : nat.gcd m n = 6) 13 | (h₁ : nat.lcm m n = 126) : 14 | 60 ≤ m + n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p293.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_293 11 | (n : ℕ) 12 | (h₀ : n ≤ 9) 13 | (h₁ : 11∣20 * 100 + 10 * n + 7) : 14 | n = 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p296.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_296 11 | (n : ℕ) 12 | (h₀ : 2 ≤ n) 13 | (h₁ : ∃ x, x^3 = n) 14 | (h₂ : ∃ t, t^4 = n) : 15 | 4096 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p299.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_299 : 11 | (1 * 3 * 5 * 7 * 9 * 11 * 13) % 10 = 5 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p3.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_3 : 11 | (∑ x in finset.range 10, ((x + 1)^2)) % 10 = 5 := 12 | begin 13 | dec_trivial!, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p30.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_30 : 11 | (33818^2 + 33819^2 + 33820^2 + 33821^2 + 33822^2) % 17 = 0 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p301.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_301 11 | (j : ℕ+) : 12 | (3 * (7 * ↑j + 1)) % 7 = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p303.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_303 11 | (h₀ : fintype {n : ℕ | 2 ≤ n ∧ 171 ≡ 80 [MOD n] ∧ 468 ≡ 13 [MOD n]}) : 12 | ∑ k in {n : ℕ | 2 ≤ n ∧ 171 ≡ 80 [MOD n] ∧ 468 ≡ 13 [MOD n]}.to_finset, k = 111 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p314.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_314 11 | (r n : ℕ) 12 | (h₀ : r = 1342 % 13) 13 | (h₁ : 0 < n) 14 | (h₂ : 1342∣n) 15 | (h₃ : n % 13 < r) : 16 | 6710 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p32.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_32 11 | (h₀ : fintype { n : ℕ | n ∣ 36}) : 12 | ∑ k in { n : ℕ | n ∣ 36}.to_finset, k = 91 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p320.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_320 11 | (n : ℕ) 12 | (h₀ : n < 101) 13 | (h₁ : 101 ∣ (123456 - n)) : 14 | n = 34 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p321.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_321 11 | (n : zmod 1399) 12 | (h₁ : n = 160⁻¹) : 13 | n = 1058 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p328.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_328 : 11 | (5^999999) % 7 = 6 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p33.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_33 11 | (n : ℕ) 12 | (h₀ : n < 398) 13 | (h₁ : (n * 7) % 398 = 1) : 14 | n = 57 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p335.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_335 11 | (n : ℕ) 12 | (h₀ : n % 7 = 5) : 13 | (5 * n) % 7 = 4 := 14 | begin 15 | norm_num [nat.mul_mod, h₀], 16 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p34.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_34 11 | (x: ℕ) 12 | (h₀ : x < 100) 13 | (h₁ : x*9 % 100 = 1) : 14 | x = 89 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p341.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_341 11 | (a b c : ℕ) 12 | (h₀ : a ≤ 9 ∧ b ≤ 9 ∧ c ≤ 9) 13 | (h₁ : (5^100) % 1000 = 10*(10*a + b) + c) : 14 | a + b + c = 13 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p342.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_342 : 11 | 54 % 6 = 0 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p343.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_343 : 11 | (∏ k in finset.range 6, (2 * k + 1)) % 10 = 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p345.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_345 : 11 | (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p35.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_35 11 | (h₀ : fintype { n : ℕ | n ∣ (nat.sqrt 196)}) : 12 | ∑ k in { n : ℕ | n ∣ (nat.sqrt 196)}.to_finset, k = 24 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p353.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_353 11 | (s : ℕ) 12 | (h₀ : s = ∑ k in finset.range 4019 \ finset.range 2010, k) : 13 | s % 2009 = 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p37.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_37 : 11 | (nat.lcm 9999 100001) = 90900909 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p370.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_370 11 | (n : ℕ) 12 | (h₀ : n % 7 = 3) : 13 | (2 * n + 1) % 7 = 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p403.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_403 : 11 | ∑ k in (nat.proper_divisors 198), k = 270 := 12 | begin 13 | refl, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p42.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_42 11 | (u v : ℕ+) 12 | (h₀ : 27 * ↑u % 40 = 17) 13 | (h₁ : 27 * ↑v % 40 = 17) 14 | (h₂ : u < 40) 15 | (h₃ : v < 80) 16 | (h₄ : 40 < v) : 17 | (u + v) = 62 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p427.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_427 11 | (a : ℕ) 12 | (h₀ : a = (∑ k in (nat.divisors 500), k)) : 13 | ∑ k in finset.filter (λ x, nat.prime x) (nat.divisors a), k = 25 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p43.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_43 11 | (n : ℕ+) 12 | (h₀ : 15^(n:ℕ) ∣ nat.factorial 942) 13 | (h₁ : ∀ m, 15^(m:ℕ) ∣ nat.factorial 942 → m ≤ n) : 14 | n = 233 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p447.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_447 : 11 | ∑ k in finset.filter (λ x, 3∣x) (finset.erase (finset.range 50) 0), (k % 10) = 78 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p45.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_45 : 11 | (nat.gcd 6432 132) + 11 = 23 := 12 | begin 13 | simp only [nat.gcd_comm], 14 | norm_num, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p457.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_457 11 | (n : ℕ) 12 | (h₀ : 0 < n) 13 | (h₁ : 80325∣(n!)) : 14 | 17 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p458.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_458 11 | (n : ℕ) 12 | (h₀ : n % 8 = 7) : 13 | n % 4 = 3 := 14 | begin 15 | conv_lhs { rw ← nat.mod_add_div n 8 }, 16 | norm_num [nat.mul_mod, h₀, nat.add_mod], 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p461.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_461 11 | (n : ℕ) 12 | (h₀ : n = finset.card (finset.filter (λ x, gcd x 8 = 1) (finset.range 8 \ finset.range 1))) : 13 | (3^n) % 8 = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p466.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_466 : 11 | (∑ k in (finset.range 11), k) % 9 = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p48.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_48 11 | (b : ℕ) 12 | (h₀ : 0 < b) 13 | (h₁ : 3 * b^2 + 2 * b + 1 = 57) : 14 | b = 4 := 15 | begin 16 | nlinarith, 17 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p483.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_483 11 | (a : ℕ+ → ℕ+) 12 | (h₀ : a 1 = 1) 13 | (h₁ : a 2 = 1) 14 | (h₂ : ∀ n, a (n + 2) = a (n + 1) + a n) : 15 | ((a 100):ℕ) % 4 = 3 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p495.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_495 11 | (a b : ℕ) 12 | (h₀ : 0 < a ∧ 0 < b) 13 | (h₁ : a % 10 = 2) 14 | (h₂ : b % 10 = 4) 15 | (h₃ : nat.gcd a b = 6) : 16 | 108 ≤ nat.lcm a b -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p5.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_5 11 | (n : ℕ) 12 | (h₀ : 10 ≤ n) 13 | (h₁ : ∃ x, x^2 = n) 14 | (h₂ : ∃ t, t^3 = n) : 15 | 64 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p517.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_517 : 11 | (121 * 122 * 123) % 4 = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p521.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_521 11 | (m n : ℕ) 12 | (h₀ : even m) 13 | (h₁ : even n) 14 | (h₂ : m - n = 2) 15 | (h₃ : m * n = 288) : 16 | m = 18 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p530.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_530 11 | (n k : ℕ+) 12 | (h₀ : ↑n / ↑k < (6:ℝ)) 13 | (h₁ : (5:ℝ) < ↑n / ↑k) : 14 | 22 ≤ (nat.lcm n k) / (nat.gcd n k) -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p541.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_541 11 | (m n : ℕ) 12 | (h₀ : 1 < m) 13 | (h₁ : 1 < n) 14 | (h₂ : m * n = 2005) : 15 | m + n = 406 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p543.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_543 : 11 | (∑ k in (nat.divisors (30^4)), 1) - 2 = 123 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p551.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_551 : 11 | 1529 % 6 = 5 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p559.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_559 11 | (x y : ℕ) 12 | (h₀ : x % 3 = 2) 13 | (h₁ : y % 5 = 4) 14 | (h₂ : x % 10 = y % 10) : 15 | 14 ≤ x -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p582.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_582 11 | (n : ℕ) 12 | (h₀ : 0 < n) 13 | (h₁ : 3∣n) : 14 | ((n + 4) + (n + 6) + (n + 8)) % 9 = 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p618.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_618 11 | (n : ℕ) 12 | (p : ℕ → ℕ) 13 | (h₀ : ∀ x, p x = x^2 - x + 41) 14 | (h₁ : 1 < nat.gcd (p n) (p (n+1))) : 15 | 41 ≤ n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p629.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_629 : 11 | is_least {t : ℕ+ | (nat.lcm 12 t)^3 = (12 * t)^2} 18 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p640.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_640 : 11 | (91145 + 91146 + 91147 + 91148) % 4 = 2 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p66.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_66 : 11 | 194 % 11 = 7 := 12 | begin 13 | exact rfl, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p668.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_668 11 | (l r : zmod 7) 12 | (h₀ : l = (2 + 3)⁻¹) 13 | (h₁ : r = 2⁻¹ + 3⁻¹) : 14 | l - r = 1 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p690.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_690 : 11 | is_least {a : ℕ+ | a ≡ 2 [MOD 3] ∧ a ≡ 4 [MOD 5] ∧ a ≡ 6 [MOD 7] ∧ a ≡ 8 [MOD 9]} 314 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p709.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_709 11 | (n : ℕ+) 12 | (h₀ : finset.card (nat.divisors (2*n)) = 28) 13 | (h₁ : finset.card (nat.divisors (3*n)) = 30) : 14 | finset.card (nat.divisors (6*n)) = 35 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p711.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_711 11 | (m n : ℕ) 12 | (h₀ : 0 < m ∧ 0 < n) 13 | (h₁ : gcd m n = 8) 14 | (h₂ : lcm m n = 112) : 15 | 72 ≤ m + n -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p728.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_728 : 11 | (29^13 - 5^13) % 7 = 0 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p739.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_739 : 11 | (nat.factorial 9) % 10 = 0 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p764.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_764 11 | (p : ℕ) 12 | (h₀ : nat.prime p) 13 | (h₁ : 7 ≤ p) : 14 | ∑ k in finset.erase (finset.range (p - 1)) 0, ((k:zmod p)⁻¹ * ((k:zmod p) + 1)⁻¹) = 2 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p765.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_765 11 | (x : ℤ) 12 | (h₀ : x < 0) 13 | (h₁ : (24 * x) % 1199 = 15) : 14 | x ≤ -449 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p769.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_769 : 11 | (129^34 + 96^38) % 11 = 9 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p780.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_780 11 | (m x : ℕ) 12 | (h₀ : 10 ≤ m) 13 | (h₁ : m ≤ 99) 14 | (h₂ : (6 * x) % m = 1) 15 | (h₃ : (x - 6^2) % m = 0) : 16 | m = 43 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p81.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_81 : 11 | 71 % 3 = 2 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p84.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_84 : 11 | int.floor ((9:ℝ) / 160 * 100) = 5 := 12 | begin 13 | rw int.floor_eq_iff, 14 | norm_num, 15 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p85.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_85 : 11 | 1 * 3^3 + 2 * 3^2 + 2*3 + 2 = 53 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p92.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_92 11 | (n : ℕ) 12 | (h₀ : (5 * n) % 17 = 8) : 13 | n % 17 = 5 -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p961.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | theorem mathd_numbertheory_961 : 11 | 2003 % 11 = 1 := 12 | begin 13 | norm_num, 14 | end -------------------------------------------------------------------------------- /src/mathzoo/olympiads/mathd/numbertheory/p99.lean: -------------------------------------------------------------------------------- 1 | /- 2 | Copyright (c) 2021 OpenAI. All rights reserved. 3 | Released under Apache 2.0 license as described in the file LICENSE. 4 | Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f 5 | -/ 6 | import mathzoo.imports.miniF2F 7 | 8 | open_locale nat rat real big_operators topological_space 9 | 10 | axiom mathd_numbertheory_99 11 | (n : ℕ) 12 | (h₀ : (2 * n) % 47 = 15) : 13 | n % 47 = 31 --------------------------------------------------------------------------------