├── AMC ├── AMC_10_217.jsonl └── AMC_12_138.jsonl ├── Example └── example-prm800k.jsonl ├── LICENSE ├── README.md ├── check_answer-4o-option.py ├── example.py ├── img ├── case.png ├── example.png ├── method.png └── radar.png ├── reasoning.py └── utils.py /AMC/AMC_10_217.jsonl: -------------------------------------------------------------------------------- 1 | {"question": "When $7$ fair standard $6$ -sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\\frac{n}{6^{7}},$ where $n$ is a positive integer. What is $n$ （ ）", "options": ["42", "49", "56", "63", "84"], "answer": "E", "source": "2018 AMC 10A Problems"} 2 | {"question": "What is the greatest integer less than or equal to $\\frac{3^{100}+2^{100}}{3^{96}+2^{96}}（ ）$ ", "options": ["80", "81 ", "96 ", "97 ", "625"], "answer": "B", "source": "2018 AMC 10A Problems"} 3 | {"question": "Let $S$ be a set of $6$ integers taken from $\\{1,2,\\dots,12\\}$ with the property that if $a$ and $b$ are elements of $S$ with $a8."], "answer": "B", "source": "2020 AMC 10B Problems"} 83 | {"question": "There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other（ ）", "options": ["11 ", "12 ", "13 ", "14 ", "15"], "answer": "C", "source": "2020 AMC 10B Problems"} 84 | {"question": "An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color（ ）", "options": [" \\frac16 ", "\\frac15 ", " \\frac14 ", " \\frac13 ", " \\frac12"], "answer": "B", "source": "2020 AMC 10B Problems"} 85 | {"question": "In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$ . What is the digit $A$ （ ）", "options": [" 2 ", " 3 ", " 4 ", " 6 ", " 7"], "answer": "A", "source": "2020 AMC 10B Problems"} 86 | {"question": "Carl has $5$ cubes each having side length $1$ , and Kate has $5$ cubes each having side length $2$ . What is the total volume of these $10$ cubes（ ）", "options": ["24 ", "25 ", "28 ", "40 ", "45"], "answer": "E", "source": "2020 AMC 10B Problems"} 87 | {"question": "Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$ , together with its interior. For real $r\\geq0$ , let $S(r)$ be the set of points in $3$ -dimensional space that lie within a distance $r$ of some point in $B$ . The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$ , where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\\frac{bc}{ad}（ ）$ ", "options": [" 6 ", " 19 ", " 24 ", " 26 ", " 38"], "answer": "B", "source": "2020 AMC 10B Problems"} 88 | {"question": "What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$ （ ）", "options": [" 100 ", " 101 ", " 200 ", " 201 ", " 202"], "answer": "D", "source": "2020 AMC 10B Problems"} 89 | {"question": "How many positive integers $n$ satisfy $\\dfrac{n+1000}{70} = \\lfloor \\sqrt{n} \\rfloor（ ）$ (Recall that $\\lfloor x\\rfloor$ is the greatest integer not exceeding $x$ .)", "options": [" 2 ", " 4 ", " 6 ", " 30 ", " 32"], "answer": "C", "source": "2020 AMC 10B Problems"} 90 | {"question": "Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product $n = f_1\\cdot f_2\\cdots f_k,$ where $k\\ge1$ , the $f_i$ are integers strictly greater than $1$ , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$ , $2\\cdot 3$ , and $3\\cdot2$ , so $D(6) = 3$ . What is $D(96)$ （ ）", "options": [" 112 ", " 128 ", " 144 ", " 172 ", " 184"], "answer": "A", "source": "2020 AMC 10B Problems"} 91 | {"question": "The ratio of $w$ to $x$ is $4:3$ , the ratio of $y$ to $z$ is $3:2$ , and the ratio of $z$ to $x$ is $1:6$ . What is the ratio of $w$ to $y（ ）$ ", "options": ["4:3 ", "3:2 ", "8:3 ", "4:1 ", "16:3"], "answer": "E", "source": "2020 AMC 10B Problems"} 92 | {"question": "The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$ , where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$ （ ）", "options": ["2 ", "3 ", "5 ", "7 ", "11"], "answer": "D", "source": "2020 AMC 10B Problems"} 93 | {"question": "How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right（ ） (Tiles of the same color are indistinguishable.)", "options": ["210 ", "420 ", "630 ", "840 ", "1050"], "answer": "B", "source": "2020 AMC 10B Problems"} 94 | {"question": "Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$ -hour period（ ）", "options": ["50 ", "55 ", "60 ", "65 ", "70"], "answer": "B", "source": "2020 AMC 10B Problems"} 95 | {"question": "How many positive even multiples of $3$ less than $2020$ are perfect squares（ ）", "options": ["7 ", "8 ", "9 ", "10 ", "12"], "answer": "A", "source": "2020 AMC 10B Problems"} 96 | {"question": "What is the value of $(2^2-2)-(3^2-3)+(4^2-4)$ ", "options": [" 1 ", " 2 ", " 5 ", " 8 ", " 12"], "answer": "D", "source": "2021 AMC 10A Problems"} 97 | {"question": "For which of the following integers $b$ is the base- $b$ number $2021_b - 221_b$ not divisible by $3$ （ ）", "options": [" ~3 ", " ~4", " ~6", " ~7", " ~8"], "answer": "A", "source": "2021 AMC 10A Problems"} 98 | {"question": "What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$ , $AC = 3$ , $AD = 4$ , $BC = \\sqrt{13}$ , $BD = 2\\sqrt{5}$ , and $CD = 5$  （ ）", "options": [" ~3 ", " ~2\\sqrt{3} ", " ~4", " ~3\\sqrt{3}", " ~6"], "answer": "C", "source": "2021 AMC 10A Problems"} 99 | {"question": "Values for $A,B,C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect（ ） (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$ )", "options": ["30 ", "60 ", "90 ", "180 ", "360"], "answer": "C", "source": "2021 AMC 10A Problems"} 100 | {"question": "Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD},BC=CD=43$ , and $\\overline{AD}\\perp\\overline{BD}$ . Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$ , and let $P$ be the midpoint of $\\overline{BD}$ . Given that $OP=11$ , the length of $AD$ can be written in the form $m\\sqrt{n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$ （ ）", "options": ["65 ", "132 ", "157 ", "194", "215"], "answer": "D", "source": "2021 AMC 10A Problems"} 101 | {"question": "Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$ . Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$ . For which of the following numbers $x$ is $f(x)<0$ （ ）", "options": ["\\frac{17}{32} ", "\\frac{11}{16} ", "\\frac79 ", "\\frac76", "\\frac{25}{11}"], "answer": "E", "source": "2021 AMC 10A Problems"} 102 | {"question": "Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have（ ）", "options": [" ~600 ", " ~650 ", " ~1950 ", " ~2000", " ~2050"], "answer": "B", "source": "2021 AMC 10A Problems"} 103 | {"question": "Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\\sqrt{3}$ , and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\\sqrt{3}$ . The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\\sqrt{p}$ , where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$ （ ）", "options": [" ~47", " ~52", " ~55", " ~58", " ~63"], "answer": "C", "source": "2021 AMC 10A Problems"} 104 | {"question": "Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$ , the second sheet contains pages $3$ and $4$ , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$ . How many sheets were borrowed（ ）", "options": [" ~10", " ~13", " ~15", " ~17", " ~20"], "answer": "B", "source": "2021 AMC 10A Problems"} 105 | {"question": "The sum of two natural numbers is $17{,}402$ . One of the two numbers is divisible by $10$ . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers（ ）", "options": [" ~10{,}272", " ~11{,}700", " ~13{,}362", " ~14{,}238", " ~15{,}426"], "answer": "D", "source": "2021 AMC 10A Problems"} 106 | {"question": "A cart rolls down a hill, travelling $5$ inches the first second and accelerating so that during each successive $1$ -second time interval, it travels $7$ inches more than during the previous $1$ -second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel（ ）", "options": [" ~215 ", " ~360", " ~2992", " ~3195", " ~3242"], "answer": "D", "source": "2021 AMC 10A Problems"} 107 | {"question": "The quiz scores of a class with $k > 12$ students have a mean of $8$ . The mean of a collection of $12$ of these quiz scores is $14$ . What is the mean of the remaining quiz scores in terms of $k$ （ ）", "options": [" ~\\frac{14-8}{k-12} ", " ~\\frac{8k-168}{k-12} ", " ~\\frac{14}{12} - \\frac{8}{k} ", " ~\\frac{14(k-12)}{k^2} ", " ~\\frac{14(k-12)}{8k}"], "answer": "B", "source": "2021 AMC 10A Problems"} 108 | {"question": "Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet（ ）", "options": [" ~\\frac{12}{13} ", " ~1 ", " ~\\frac{13}{12} ", " ~\\frac{24}{13} ", " ~2"], "answer": "A", "source": "2021 AMC 10A Problems"} 109 | {"question": "Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that", "options": [" Purple snakes can add.\n\n \\textbf{(B) } Purple snakes are happy.\n\n \\textbf{(C) } Snakes that can add are purple.\n\n \\textbf{(D) } Happy snakes are not purple.\n\n \\textbf{(E) } Happy snakes can't subtract."], "answer": "A", "source": "2021 AMC 10A Problems"} 110 | {"question": "When a student multiplied the number $66$ by the repeating decimal, ", "options": ["15 ", "30 ", "45 ", "60 ", "75"], "answer": "E", "source": "2021 AMC 10A Problems"} 111 | {"question": "What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$ （ ）", "options": [" ~0", " ~\\frac{1}{4}", " ~\\frac{1}{2} ", " ~1 ", " ~2"], "answer": "D", "source": "2021 AMC 10A Problems"} 112 | {"question": "How many integer values of $x$ satisfy $|x|<3\\pi$ （ ）", "options": [" ~9 ", " ~10 ", " ~18 ", " ~19 ", " ~20"], "answer": "A", "source": "2021 AMC 10B Problems"} 113 | {"question": "An inverted cone with base radius $12 \\mathrm{cm}$ and height $18 \\mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \\mathrm{cm}$ . What is the height in centimeters of the water in the cylinder（ ）", "options": [" ~1.5 ", " ~3 ", " ~4 ", " ~4.5 ", " ~6"], "answer": "A", "source": "2021 AMC 10B Problems"} 114 | {"question": "Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce（ ）", "options": [" ~24 ", " ~30 ", " ~48 ", " ~60 ", " ~64"], "answer": "D", "source": "2021 AMC 10B Problems"} 115 | {"question": "Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$ . What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ （ ）", "options": [" ~1 : 16 ", " ~1 : 15 ", " ~1 : 14 ", " ~1 : 8 ", " ~1 : 3"], "answer": "C", "source": "2021 AMC 10B Problems"} 116 | {"question": "Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\\underline{32d}$ in base $n$ equals $263$ , and the value of the numeral $\\underline{324}$ in base $n$ equals the value of the numeral $\\underline{11d1}$ in base six. What is $n + d （ ）$ ", "options": [" ~10 ", " ~11 ", " ~13 ", " ~15 ", " ~16"], "answer": "B", "source": "2021 AMC 10B Problems"} 117 | {"question": "The real number $x$ satisfies the equation $x+\\frac{1}{x} = \\sqrt{5}$ . What is the value of $x^{11}-7x^{7}+x^3（ ）$ ", "options": [" ~-1 ", " ~0 ", " ~1 ", " ~2 ", " ~\\sqrt{5}"], "answer": "B", "source": "2021 AMC 10B Problems"} 118 | {"question": "Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. How many uphill integers are divisible by $15$ （ ）", "options": [" ~4 ", " ~5 ", " ~6 ", " ~7 ", " ~8"], "answer": "C", "source": "2021 AMC 10B Problems"} 119 | {"question": "Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \\dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon-- $11,$ Oscar-- $4,$ Aditi-- $7,$ Tyrone-- $16,$ Kim-- $17.$ Which of the following statements is true（ ）", "options": ["\\text{Ravon was given card 3.}", "\\text{Aditi was given card 3.}", "\\text{Ravon was given card 4.}", "\\text{Aditi was given card 4.}", "\\text{Tyrone was given card 7.}"], "answer": "C", "source": "2021 AMC 10B Problems"} 120 | {"question": "A fair $6$ -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number（ ）", "options": [" ~\\frac{1}{120} ", " ~\\frac{1}{32} ", " ~\\frac{1}{20} ", " ~\\frac{3}{20} ", " ~\\frac{1}{6}"], "answer": "C", "source": "2021 AMC 10B Problems"} 121 | {"question": "Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then returned to the set, the average value of the integers rises to $40$ . The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$ . What is the average value of all the integers in the set $S$ （ ）", "options": ["36.2 ", "36.4 ", "36.6", "36.8 ", "37"], "answer": "D", "source": "2021 AMC 10B Problems"} 122 | {"question": "What is the value of $\\sqrt{\\left(3-2\\sqrt{3}\\right)^2}+\\sqrt{\\left(3+2\\sqrt{3}\\right)^2}$ （ ）", "options": [" ~0 ", " ~4\\sqrt{3}-6 ", " ~6 ", " ~4\\sqrt{3} ", " ~4\\sqrt{3}+6"], "answer": "D", "source": "2021 AMC 10B Problems"} 123 | {"question": "Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n （ ）$ ", "options": [" ~47 ", " ~94 ", " ~227 ", " ~471 ", " ~542"], "answer": "D", "source": "2021 AMC 10B Problems"} 124 | {"question": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts（ ）", "options": [" ~23 ", " ~32 ", " ~37 ", " ~41 ", " ~64"], "answer": "B", "source": "2021 AMC 10B Problems"} 125 | {"question": "The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$ , while the other two multiply to $30$ . What is the sum of the ages of Jonie's four cousins（ ）", "options": [" ~21 ", " ~22 ", " ~23 ", " ~24 ", " ~25"], "answer": "B", "source": "2021 AMC 10B Problems"} 126 | {"question": "Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$ , and the afternoon class's mean score is $70$ . The ratio of the number of students in the morning class to the number of students in the afternoon class is $\\frac{3}{4}$ . What is the mean of the scores of all the students（ ）", "options": [" ~74 ", " ~75 ", " ~76 ", " ~77 ", " ~78"], "answer": "C", "source": "2021 AMC 10B Problems"} 127 | {"question": "The point $P(a,b)$ in the $xy$ -plane is first rotated counterclockwise by $90^\\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$ . The image of $P$ after these two transformations is at $(-6,3)$ . What is $b - a （ ）$ ", "options": [" ~1 ", " ~3 ", " ~5 ", " ~7 ", " ~9"], "answer": "A", "source": "2021 AMC 10B Problems"} 128 | {"question": "What is the value of $\\frac{(2112-2021)^2}{169}$ （ ）", "options": [" 7 ", " 21 ", " 49 ", " 64 ", " 91"], "answer": "C", "source": "2021 Fall AMC 10A Problems"} 129 | {"question": "A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$ . Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$ （ ）", "options": ["{-}18.5 ", "{-}13.5 ", "0 ", "13.5 ", "18.5"], "answer": "B", "source": "2021 Fall AMC 10A Problems"} 130 | {"question": "Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship（ ）", "options": ["70", "84", "98", "105", "126"], "answer": "A", "source": "2021 Fall AMC 10A Problems"} 131 | {"question": "The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\\text{nine}}.$ What is the remainder when $N$ is divided by $5（ ）$ ", "options": [" 0", " 1", " 2", " 3", "4"], "answer": "D", "source": "2021 Fall AMC 10A Problems"} 132 | {"question": "Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls（ ）", "options": [" \\frac{1}{64}", " \\frac{1}{6}", " \\frac{1}{4}", " \\frac{5}{16}", "\\frac{1}{2}"], "answer": "D", "source": "2021 Fall AMC 10A Problems"} 133 | {"question": "A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$ . A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$ . The value of $s$ can be written as $a+\\frac{b\\pi}{c}$ , where $a,b$ , and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$ （ ）", "options": [" ~10", " ~11", " ~12", " ~13", " ~14"], "answer": "A", "source": "2021 Fall AMC 10A Problems"} 134 | {"question": "Menkara has a $4 \\times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch（ ）", "options": [" 16 ", " 17 ", " 18 ", " 19 ", " 20"], "answer": "E", "source": "2021 Fall AMC 10A Problems"} 135 | {"question": "For each positive integer $n$ , let $f_1(n)$ be twice the number of positive integer divisors of $n$ , and for $j \\ge 2$ , let $f_j(n) = f_1(f_{j-1}(n))$ . For how many values of $n \\le 50$ is $f_{50}(n) = 12（ ）$ ", "options": ["7", "8", "9", "10", "11"], "answer": "D", "source": "2021 Fall AMC 10A Problems"} 136 | {"question": "A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\\tilde{p}(1)$ （ ）", "options": [" \\dfrac{5}{16} ", " \\dfrac{1}{2} ", " \\dfrac{5}{8} ", " 1 ", " \\dfrac{9}{8}"], "answer": "A", "source": "2021 Fall AMC 10A Problems"} 137 | {"question": "What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube（ ）", "options": ["3", "4", "5", "6", "7"], "answer": "D", "source": "2021 Fall AMC 10A Problems"} 138 | {"question": "Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\\frac{1}{2}$ -mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A（ ）", "options": ["2 \\frac{3}{4} ", "3 \\frac{3}{4} ", "4 \\frac{1}{2} ", "5 \\frac{1}{2} ", "6 \\frac{3}{4}"], "answer": "B", "source": "2021 Fall AMC 10A Problems"} 139 | {"question": "The six-digit number $\\underline{2}\\,\\underline{0}\\,\\underline{2}\\,\\underline{1}\\,\\underline{0}\\,\\underline{A}$ is prime for only one digit $A.$ What is $A（ ）$ ", "options": ["1 ", "3 ", "5 ", "7 ", "9"], "answer": "E", "source": "2021 Fall AMC 10A Problems"} 140 | {"question": "Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$ st pole along this road is exactly one mile ( $5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride（ ）", "options": ["6", "8", "10", "11", "15"], "answer": "B", "source": "2021 Fall AMC 10A Problems"} 141 | {"question": "A two-digit positive integer is said to be $\\emph{cuddly}$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly（ ）", "options": ["0", "1", "2", "3", "4"], "answer": "B", "source": "2021 Fall AMC 10A Problems"} 142 | {"question": "When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even（ ）", "options": ["\\frac{3}{8} ", "\\frac{4}{9} ", "\\frac{5}{9} ", "\\frac{9}{16} ", "\\frac{5}{8}"], "answer": "E", "source": "2021 Fall AMC 10A Problems"} 143 | {"question": "What is the value of $1234 + 2341 + 3412 + 4123$ ", "options": ["\\: 10{,}000", ": 10{,}010", ": 10{,}110", ": 11{,}000", ": 11{,}110"], "answer": "E", "source": "2021 Fall AMC 10B Problems"} 144 | {"question": "Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, \"I can't tell who has the larger number.\" Then Bob says, \"I know who has the larger number.\" Alice says, \"You do（ ） Is your number prime（ ）\" Bob replies, \"Yes.\" Alice says, \"In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. \" What is the sum of the two numbers drawn from the hat（ ）", "options": ["27", "37", "47", "57", "67"], "answer": "A", "source": "2021 Fall AMC 10B Problems"} 145 | {"question": "Which of the following conditions is sufficient to guarantee that integers $x$ , $y$ , and $z$ satisfy the equation", "options": ["x>y and y=z \n\n ", "x=y-1 and y=z-1 \n\n ", "x=z+1 and y=x+1 \n\n ", "x=z and y-1=x \n\n ", "x+y+z=1"], "answer": "D", "source": "2021 Fall AMC 10B Problems"} 146 | {"question": "Una rolls $6$ standard $6$ -sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4（ ）$ ", "options": ["\\: \\frac34", ": \\frac{57}{64}", ": \\frac{59}{64}", ": \\frac{187}{192}", ": \\frac{63}{64}"], "answer": "C", "source": "2021 Fall AMC 10B Problems"} 147 | {"question": "Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions（ ）", "options": ["1.6", "1.8", "2.0", "2.2", "2.4"], "answer": "D", "source": "2021 Fall AMC 10B Problems"} 148 | {"question": "In a particular game, each of $4$ players rolls a standard $6{ }$ -sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game（ ）", "options": ["\\frac{61}{216}", "\\frac{367}{1296}", "\\frac{41}{144}", "\\frac{185}{648}", "\\frac{11}{36}"], "answer": "C", "source": "2021 Fall AMC 10B Problems"} 149 | {"question": "Regular polygons with $5,6,7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect（ ）", "options": ["52", "56", "60", "64", "68"], "answer": "E", "source": "2021 Fall AMC 10B Problems"} 150 | {"question": "For each integer $n\\geq 2$ , let $S_n$ be the sum of all products $jk$ , where $j$ and $k$ are integers and $1\\leq j d(R, S)$ （ ）", "options": [" \\frac{7}{22} ", " \\frac{1}{3} ", " \\frac{3}{8} ", " \\frac{5}{12} ", " \\frac{1}{2}"], "answer": "A", "source": "2023 AMC 10A Problems"} 199 | {"question": "How many positive perfect squares less than $2023$ are divisible by $5$ （ ）", "options": [" 8 ", "9 ", "10 ", "11 ", " 12"], "answer": "A", "source": "2023 AMC 10A Problems"} 200 | {"question": "How many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$ （ ）", "options": ["~14", "~15", "~16", "~17", "~18"], "answer": "E", "source": "2023 AMC 10A Problems"} 201 | {"question": "An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$ . What is the value of the cube（ ）", "options": [" 42 ", " 63 ", " 84 ", " 126 ", " 252"], "answer": "D", "source": "2023 AMC 10A Problems"} 202 | {"question": "Janet rolls a standard $6$ -sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$ （ ）", "options": ["\\frac{2}{9}", "\\frac{49}{216}", "\\frac{25}{108}", "\\frac{17}{72}", "\\frac{13}{54}"], "answer": "B", "source": "2023 AMC 10A Problems"} 203 | {"question": "Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at $110$ degrees Fahrenheit, which is $0$ degrees on the Breadus scale. Bread is baked at $350$ degrees Fahrenheit, which is $100$ degrees on the Breadus scale. Bread is done when its internal temperature is $200$ degrees Fahrenheit. What is this in degrees on the Breadus scale（ ）", "options": ["33", "34.5", "36", "37.5", "39"], "answer": "D", "source": "2023 AMC 10A Problems"} 204 | {"question": "A digital display shows the current date as an $8$ -digit integer consisting of a $4$ -digit year, followed by a $2$ -digit month, followed by a $2$ -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date（ ）", "options": ["~5", "~6", "~7", "~8", "~9"], "answer": "E", "source": "2023 AMC 10A Problems"} 205 | {"question": "Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice（ ）", "options": [" \\frac{1}{12} ", " \\frac{1}{4} ", " \\frac{1}{6} ", " \\frac{1}{8} ", " \\frac{2}{9}"], "answer": "C", "source": "2023 AMC 10B Problems"} 206 | {"question": "Suzanne went to the bank and withdrew $\\$800$ . The teller gave her this amount using $\\$20$ bills, $\\$50$ bills, and $\\$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received（ ）", "options": [" 45 ", " 21 \\qquad \\text{(C) } 36 \\qquad \\text{(D) } 28 \\qquad \\text{(E) } 32"], "answer": "B", "source": "2023 AMC 10B Problems"} 207 | {"question": "How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$ （ ）", "options": ["7", "1", "3", "6", "5"], "answer": "C", "source": "2023 AMC 10B Problems"} 208 | {"question": "Define an $\\textit{upno}$ to be a positive integer of $2$ or more digits where the digits are strictly", "options": ["~512", "~10", "~0", "~9", "~511"], "answer": "E", "source": "2023 AMC 10B Problems"} 209 | {"question": "Suppose $a$ , $b$ , and $c$ are positive integers such that $\\frac{a}{14}+\\frac{b}{15}=\\frac{c}{210}.$ Which of the following statements are necessarily true（ ）", "options": ["~\\text{I, II, and III}", "~\\text{I only}", "~\\text{I and II only}", "~\\text{III only}", "~\\text{II and III only}"], "answer": "E", "source": "2023 AMC 10B Problems"} 210 | {"question": "Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\\%$ sales tax on the discounted price. He had $\\$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy（ ） ", "options": ["46", "50", "48", "47", "49"], "answer": "B", "source": "2023 AMC 10B Problems"} 211 | {"question": "Each of $2023$ balls is randomly placed into one of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls（ ）", "options": [" \\frac{2}{3} ", " \\frac{3}{10} ", " \\frac{1}{2} ", " \\frac{1}{3} ", " \\frac{1}{4}"], "answer": "A", "source": "2023 AMC 10B Problems"} 212 | {"question": "An arithmetic sequence of positive integers has $\\text{n} \\ge 3$ terms, initial term $a$ , and common difference $d > 1$ . Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$ . The sum of the terms he wrote was $222$ . What is $a + d + n$ （ ）", "options": [" 24 ", " 20 ", " 22 ", " 28 ", " 26"], "answer": "B", "source": "2023 AMC 10B Problems"} 213 | {"question": "A $3-4-5$ right triangle is inscribed in circle $A$ , and a $5-12-13$ right triangle is inscribed in circle $B$ . What is the ratio of the area of circle $A$ to the area of circle $B$ （ ）", "options": ["\\frac{9}{25}", "\\frac{1}{9}", "\\frac{1}{5}", "\\frac{25}{169}", "\\frac{4}{25}"], "answer": "D", "source": "2023 AMC 10B Problems"} 214 | {"question": "Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint（ ） ", "options": [" 162,500 ", " 162.5 ", "1,625 ", "1,625,000 ", " 16,250"], "answer": "C", "source": "2023 AMC 10B Problems"} 215 | {"question": "Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$ . Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$ . How many numbers are written on the blackboard（ ）", "options": ["10", "5", "6", "8", "9"], "answer": "A", "source": "2023 AMC 10B Problems"} 216 | {"question": "Let $L_{1}=1, L_{2}=3$ , and $L_{n+2}=L_{n+1}+L_{n}$ for $n\\geq 1$ . How many terms in the sequence $L_{1}, L_{2}, L_{3},...,L_{2023}$ are even（ ）", "options": ["673", "1011", "675", "1010", "674"], "answer": "E", "source": "2023 AMC 10B Problems"} 217 | {"question": "Square $ABCD$ is rotated $20^{\\circ}$ clockwise about its center to obtain square $EFGH$ , as shown below.", "options": ["24^{\\circ} ", "35^{\\circ} ", "30^{\\circ} ", "32^{\\circ} ", "20^{\\circ}"], "answer": "B", "source": "2023 AMC 10B Problems"} 218 | -------------------------------------------------------------------------------- /AMC/AMC_12_138.jsonl: -------------------------------------------------------------------------------- 1 | {"id": 0, "question": "Which of the following is equivalent to\n $(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$", "options": [" ~3^{128} + 2^{128} ", " ~3^{127} + 2^{127} + 2 \\cdot 3^{63} + 3 \\cdot 2^{63} ", " ~3^{127} + 2^{127} ", " ~3^{128}-2^{128} "], "answer": "D", "source": "2021 AMC 12A Problems/Problem 9", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2021_AMC_12A_Problems/Problem_9"} 2 | {"id": 1, "question": "How many integer values of $x$ satisfy $|x|<3\\pi$ ?", "options": [" ~9 ", " ~10 ", " ~19 ", " ~20"], "answer": "C", "source": "2021 AMC 12B Problems/Problem 1", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_1"} 3 | {"id": 2, "question": "Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?", "options": ["8", "10", "5 ", "9 "], "answer": "B", "source": "2021 AMC 12B Problems/Problem 10", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_10"} 4 | {"id": 3, "question": "Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$ . Let $P$ be the point on $\\overline{AC}$ such that $PC=10$ . There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$", "options": ["6\\sqrt2 ", "\\frac{84}5", "18", "12\\sqrt2 "], "answer": "D", "source": "2021 AMC 12B Problems/Problem 11", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_11"} 5 | {"id": 4, "question": "Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then returned to the set, the average value of the integers rises to $40$ . The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$ . What is the average value of all the integers in the set $S$ ?", "options": ["36.6", "36.4 ", "36.8 ", "36.2 "], "answer": "C", "source": "2021 AMC 12B Problems/Problem 12", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_12"} 6 | {"id": 5, "question": "Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1 d(R, S)$ ?", "options": [" \\frac{1}{3} ", " \\frac{5}{12} ", "frac{7}{22} ", " \\frac{1}{2}"], "answer": "C", "source": "2023 AMC 12A Problems/Problem 21", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_21"} 103 | {"id": 102, "question": "Let $f$ be the unique function defined on the positive integers such that $\\sum_{d\\mid n}d\\cdot f\\left(\\frac{n}{d}\\right)=1$ for all positive integers $n$ . What is $f(2023)$ ?", "options": ["~96", "~116", "~108", "~-1536"], "answer": "A", "source": "2023 AMC 12A Problems/Problem 22", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_22"} 104 | {"id": 103, "question": "How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n $(1+2a)(2+2b)(2a+b) = 32ab?$", "options": ["1", "3", "0", "\\text{an infinite number}"], "answer": "A", "source": "2023 AMC 12A Problems/Problem 23", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_23"} 105 | {"id": 104, "question": "Let $K$ be the number of sequences $A_1$ , $A_2$ , $\\dots$ , $A_n$ such that $n$ is a positive integer less than or equal to $10$ , each $A_i$ is a subset of $\\{1, 2, 3, \\dots, 10\\}$ , and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$ , inclusive. For example, $\\{\\}$ , $\\{5, 7\\}$ , $\\{2, 5, 7\\}$ , $\\{2, 5, 7\\}$ , $\\{2, 5, 6, 7, 9\\}$ is one such sequence, with $n = 5$ .What is the remainder when $K$ is divided by $10$ ?", "options": [" 1 ", " 9", " 3 ", " 5 "], "answer": "D", "source": "2023 AMC 12A Problems/Problem 24", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_24"} 106 | {"id": 105, "question": "There is a unique sequence of integers $a_1, a_2, \\cdots a_{2023}$ such that\n $\\tan2023x = \\frac{a_1 \\tan x + a_3 \\tan^3 x + a_5 \\tan^5 x + \\cdots + a_{2023} \\tan^{2023} x}{1 + a_2 \\tan^2 x + a_4 \\tan^4 x \\cdots + a_{2022} \\tan^{2022} x}$ whenever $\\tan 2023x$ is defined. What is $a_{2023}?$", "options": [" -1 ", " 1 ", " 2023", " -2022 "], "answer": "A", "source": "2023 AMC 12A Problems/Problem 25", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_25"} 107 | {"id": 106, "question": "How many positive perfect squares less than $2023$ are divisible by $5$ ?", "options": ["11 ", " 12", "10 ", " 8 "], "answer": "D", "source": "2023 AMC 12A Problems/Problem 3", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_3"} 108 | {"id": 107, "question": "How many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$ ?", "options": ["~16", "~18", "~14", "~17"], "answer": "B", "source": "2023 AMC 12A Problems/Problem 4", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_4"} 109 | {"id": 108, "question": "Janet rolls a standard $6$ -sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$ ?", "options": ["\\frac{49}{216}", "\\frac{13}{54}", "\\frac{17}{72}", "\\frac{2}{9}"], "answer": "A", "source": "2023 AMC 12A Problems/Problem 5", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_5"} 110 | {"id": 109, "question": "Points $A$ and $B$ lie on the graph of $y=\\log_{2}x$ . The midpoint of $\\overline{AB}$ is $(6, 2)$ . What is the positive difference between the $x$ -coordinates of $A$ and $B$ ?", "options": ["~9", "~4\\sqrt{5}", "~8", "~4\\sqrt{3}"], "answer": "B", "source": "2023 AMC 12A Problems/Problem 6", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_6"} 111 | {"id": 110, "question": "A digital display shows the current date as an $8$ -digit integer consisting of a $4$ -digit year, followed by a $2$ -digit month, followed by a $2$ -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?", "options": ["~5", "~7", "~8", "~9"], "answer": "D", "source": "2023 AMC 12A Problems/Problem 7", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_7"} 112 | {"id": 111, "question": "Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$ . If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$ . What is the mean of her quiz scores currently?", "options": ["4", "8", "6", "7"], "answer": "D", "source": "2023 AMC 12A Problems/Problem 8", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_8"} 113 | {"id": 112, "question": "Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?", "options": [" \\frac{2}{9}", " \\frac{1}{6} ", "frac{1}{12} ", " \\frac{1}{8} "], "answer": "B", "source": "2023 AMC 12B Problems/Problem 1", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_1"} 114 | {"id": 113, "question": "In the $xy$ -plane, a circle of radius $4$ with center on the positive $x$ -axis is tangent to the $y$ -axis at the origin, and a circle with radius $10$ with center on the positive $y$ -axis is tangent to the $x$ -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?\n\n $", "options": ["\\dfrac{1}{\\sqrt{29}} ", "\\dfrac{2}{7} ", "\\dfrac{2}{5}", "\\dfrac{3}{7} "], "answer": "C", "source": "2023 AMC 12B Problems/Problem 10", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_10"} 115 | {"id": 114, "question": "For complex number $u = a+bi$ and $v = c+di$ (where $i=\\sqrt{-1}$ ), define the binary operation\n\n $u \\otimes v = ac + bdi$ \n\nSuppose $z$ is a complex number such that $z\\otimes z = z^{2}+40$ . What is $|z|$ ?", "options": ["2", "5\\sqrt{2}", "\\sqrt{10}", "\\sqrt{5}"], "answer": "B", "source": "2023 AMC 12B Problems/Problem 12", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_12"} 116 | {"id": 115, "question": "For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?\n\n $", "options": ["6 ", "4", "8 ", "5 "], "answer": "D", "source": "2023 AMC 12B Problems/Problem 14", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_14"} 117 | {"id": 116, "question": "Suppose $a$ , $b$ , and $c$ are positive integers such that $\\frac{a}{14}+\\frac{b}{15}=\\frac{c}{210}.$ Which of the following statements are necessarily true?\n\nI. If $\\gcd(a,14)=1$ or $\\gcd(b,15)=1$ or both, then $\\gcd(c,210)=1$ .\n\nII. If $\\gcd(c,210)=1$ , then $\\gcd(a,14)=1$ or $\\gcd(b,15)=1$ or both.\n\nIII. $\\gcd(c,210)=1$ if and only if $\\gcd(a,14)=\\gcd(b,15)=1$ .", "options": ["~\\text{I only}", "~\\text{III only}", "~\\text{II and III only}", "~\\text{I, II, and III}"], "answer": "C", "source": "2023 AMC 12B Problems/Problem 15", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_15"} 118 | {"id": 117, "question": "In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x?$", "options": ["11", "10", "8", "7"], "answer": "A", "source": "2023 AMC 12B Problems/Problem 16", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_16"} 119 | {"id": 118, "question": "Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\\circ,$ what is the area of $ABC$ ?", "options": ["15\\sqrt{3}", "14\\sqrt{2}", "20\\sqrt{2}", "12\\sqrt{3}"], "answer": "A", "source": "2023 AMC 12B Problems/Problem 17", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_17"} 120 | {"id": 119, "question": "Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was $3$ points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was $18$ points higher than her average for the first semester and was again $3$ points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?", "options": [" If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.", " Zelda's quiz average for the academic year equaled Yolanda's.\n\n ", " Yolanda's quiz average for the academic year was 3 points higher than Zelda's.\n", " Yolanda's quiz average for the academic year was 22 points higher than Zelda's.\n\n "], "answer": "D", "source": "2023 AMC 12B Problems/Problem 18", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_18"} 121 | {"id": 120, "question": "Each of $2023$ balls is randomly placed into one of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?", "options": [" \\frac{1}{2} ", " \\frac{1}{4}", " \\frac{1}{3} ", "frac{2}{3} "], "answer": "D", "source": "2023 AMC 12B Problems/Problem 19", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_19"} 122 | {"id": 121, "question": "Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\\%$ sales tax on the discounted price. He had $\\$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?", "options": ["47", "48", "50", "46"], "answer": "C", "source": "2023 AMC 12B Problems/Problem 2", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_2"} 123 | {"id": 122, "question": "Cyrus the frog jumps $2$ units in a direction, then $2$ more in another direction. What is the probability that he lands less than $1$ unit away from his starting position?", "options": ["~\\frac{2\\arcsin \\frac{1}{4}}{\\pi}", "~\\frac{\\sqrt{3}}{8}", "~\\frac{\\arctan \\frac{1}{2}}{\\pi}", "~\\frac{1}{5}"], "answer": "A", "source": "2023 AMC 12B Problems/Problem 20", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_20"} 124 | {"id": 123, "question": "A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\\sqrt3$ inches, its top diameter is $6$ inches, and its bottom diameter is $12$ inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?", "options": ["6 + 6\\pi", " 6\\sqrt5 ", " 6\\sqrt3 + \\pi", " 6 + 3\\pi"], "answer": "C", "source": "2023 AMC 12B Problems/Problem 21", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_21"} 125 | {"id": 124, "question": "A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $f(a + b) + f(a - b) = 2f(a) f(b).$ Which one of the following cannot be the value of $f(1)?$", "options": [" 2 ", " 1 ", " -2", " -1 "], "answer": "C", "source": "2023 AMC 12B Problems/Problem 22", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_22"} 126 | {"id": 125, "question": "When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$ ?", "options": ["~10", "~11", "~9", "~6"], "answer": "B", "source": "2023 AMC 12B Problems/Problem 23", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_23"} 127 | {"id": 126, "question": "Suppose that $a$ , $b$ , $c$ and $d$ are positive integers satisfying all of the following relations.\n\n $abcd=2^6\\cdot 3^9\\cdot 5^7$ \n $\\text{lcm}(a,b)=2^3\\cdot 3^2\\cdot 5^3$ \n $\\text{lcm}(a,c)=2^3\\cdot 3^3\\cdot 5^3$ \n $\\text{lcm}(a,d)=2^3\\cdot 3^3\\cdot 5^3$ \n $\\text{lcm}(b,c)=2^1\\cdot 3^3\\cdot 5^2$ \n $\\text{lcm}(b,d)=2^2\\cdot 3^3\\cdot 5^2$ \n $\\text{lcm}(c,d)=2^2\\cdot 3^3\\cdot 5^2$ \n\nWhat is $\\text{gcd}(a,b,c,d)$ ?", "options": ["~45", "~3", "~30", "~15"], "answer": "B", "source": "2023 AMC 12B Problems/Problem 24", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_24"} 128 | {"id": 127, "question": "A $3-4-5$ right triangle is inscribed in circle $A$ , and a $5-12-13$ right triangle is inscribed in circle $B$ . What is the ratio of the area of circle $A$ to the area of circle $B$ ?", "options": ["\\frac{4}{25}", "\\frac{25}{169}", "\\frac{1}{5}", "\\frac{9}{25}"], "answer": "B", "source": "2023 AMC 12B Problems/Problem 3", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_3"} 129 | {"id": 128, "question": "Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?", "options": ["1,625 ", " 162.5 ", " 162,500 ", " 16,250"], "answer": "A", "source": "2023 AMC 12B Problems/Problem 4", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_4"} 130 | {"id": 129, "question": "When the roots of the polynomial \n\n $P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}$ \n\nare removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is $P(x)$ positive?", "options": ["~6", "~4", "~7", "~5"], "answer": "A", "source": "2023 AMC 12B Problems/Problem 6", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_6"} 131 | {"id": 130, "question": "For how many integers $n$ does the expression $\\sqrt{\\frac{\\log (n^2) - (\\log n)^2}{\\log n - 3}}$ represent a real number, where log denotes the base $10$ logarithm?", "options": ["900 ", "2", "901", " 2 "], "answer": "C", "source": "2023 AMC 12B Problems/Problem 7", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_7"} 132 | {"id": 131, "question": "How many nonempty subsets $B$ of ${0, 1, 2, 3, \\cdots, 12}$ have the property that the number of elements in $B$ is equal to the least element of $B$ ? For example, $B = {4, 6, 8, 11}$ satisfies the condition.", "options": [" 156", " 256 ", " 108 ", " 144 "], "answer": "D", "source": "2023 AMC 12B Problems/Problem 8", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_8"} 133 | {"id": 132, "question": "What is the area of the region in the coordinate plane defined by\n\n $| | x | - 1 | + | | y | - 1 | \\le 1$ ?", "options": ["8 ", "15 ", "12", "2 "], "answer": "A", "source": "2023 AMC 12B Problems/Problem 9", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_9"} 134 | {"id": 133, "question": "If $x$ and $y$ are positive integers such that their product is $64,$ what is the sum of all distinct values for $xy+x+y?$", "options": ["393", "391", "392", "395"], "answer": "B", "source": "FidgetBoss 4000's 2019 AMC 12B Problems/Problem 1", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/FidgetBoss_4000%27s_2019_AMC_12B_Problems/Problem_1"} 135 | {"id": 134, "question": "If $x$ and $y$ are positive integers such that their product is $64,$ what is the sum of all distinct values for $xy+x+y?$", "options": ["393", "392", "394", "391"], "answer": "D", "source": "FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 1", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/FidgetBoss_4000%27s_2019_Mock_AMC_12B_Problems/Problem_1"} 136 | {"id": 135, "question": "(fidgetboss_4000) In the diagram below, $ABC$ is an isosceles right triangle with a right angle at $B$ and with a hypotenuse of $40\\sqrt2$ units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside $\\triangle ABC$ .", "options": [" 27", " 26", " 30", " 28"], "answer": "D", "source": "FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 2", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/FidgetBoss_4000%27s_2019_Mock_AMC_12B_Problems/Problem_2"} 137 | {"id": 136, "question": "There are $\\binom{8}{4}=70$ distinct quadrilaterals that can be formed from the vertices of a regular octagon. Which of these statements must hold true for all those quadrilaterals?", "options": ["\\text{All of the 70 quadrilaterals are kites.}", "\\text{None of the above statements are true.}", "\\text{All of the 70 quadrilaterals are cyclic.}", "\\text{All of the 70 quadrilaterals are rectangles.}"], "answer": "C", "source": "FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/FidgetBoss_4000%27s_2019_Mock_AMC_12B_Problems/Problem_3"} 138 | {"id": 137, "question": "(fidgetboss_4000) Mark rolled two standard dice. Given that he rolled two distinct values, find the probability that he rolled two primes.", "options": ["\\frac{2}{5}", "\\frac{1}{5}", "\\frac{1}{12}", "\\frac{1}{7}"], "answer": "B", "source": "FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 4", "step": 5, "url": "https://artofproblemsolving.com/wiki/index.php/FidgetBoss_4000%27s_2019_Mock_AMC_12B_Problems/Problem_4"} 139 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Apache License 2 | Version 2.0, January 2004 3 | http://www.apache.org/licenses/ 4 | 5 | TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION 6 | 7 | 1. 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15 | 16 |

17 | 18 | ## 📜 News 19 | 🚀 [2025/1/21] The inference code of BoostStep is released 20 | 21 | 🚀 [2025/1/7] The example problem bank and the AMC test set are released 22 | 23 | 🚀 [2025/1/7] The paper is released! 24 | 25 | ## 🛠️ Usage 26 | 27 | The inference code is released, you can run our booststep on AMC-12 benchmark by the following command: 28 | ```shell 29 | python reasoning.py --api_key="Your_API_KEY" 30 | ``` 31 | 32 | Then, run `check_answer-4o-option.py` to judge your answer with GPT-4o. 33 | 34 | ## ⭐ Demos 35 | 36 |

37 | 38 |

39 | 40 | Here we demonstrate a specific example. Given the question, we first let the model to have a first-try on step one. Unfortunately, the model makes an error on tangent sum formula, therefore leading to a wrong step. Then, we find a similar step which correctly leverage tangent sum formula in the step-level example problem bank according to this first-try. Therefore, with the guidance provided, the model correctly applied the tangent sum formula during the second reasoning attempt and arrived at the correct answer. 41 | 42 | ## Citation 43 | If you find our work helpful for your research, please consider giving a star or a citation, thanks! 44 | ``` 45 | @article{zhang2025booststep, 46 | title={BoostStep: Boosting mathematical capability of Large Language Models via improved single-step reasoning}, 47 | author={Zhang, Beichen and Liu, Yuhong and Dong, Xiaoyi and Zang, Yuhang and Zhang, Pan and Duan, Haodong and Cao, Yuhang and Lin, Dahua and Wang, Jiaqi}, 48 | journal={arXiv preprint arXiv:2501.03226}, 49 | year={2025} 50 | } 51 | ``` 52 | -------------------------------------------------------------------------------- /check_answer-4o-option.py: -------------------------------------------------------------------------------- 1 | import json 2 | from openai import OpenAI 3 | import openai 4 | import pandas as pd 5 | import jsonlines 6 | import json 7 | 8 | hit = 0 9 | total = 0 10 | api_keys = "your_api_key" 11 | m_name = 'gpt-4o-2024-05-13' 12 | client = OpenAI(api_key=api_keys) 13 | prompt = "Compare whether the given answer with the groundtruth is the same. Reply with only one single number 1 (correct) or 0 (wrong). Minor expression difference can be ignored if the two mathematical expression is the same." 14 | 15 | i = 0 16 | total_option = [] 17 | 18 | gt_transfer = {"A":0, "B":1, "C":2, "D":3, "E":4} 19 | 20 | i = 0 21 | 22 | with jsonlines.open('1-shot-AMC=12_thrsd=0.7-answer.jsonl', mode='w') as writer: 23 | with open('1-shot-AMC=12_thrsd=0.7.jsonl', 'r', encoding='utf-8') as file: 24 | for item in jsonlines.Reader(file): 25 | solution = item["solution"] 26 | answer = item['answer'] 27 | choice = item['choice'] 28 | 29 | choice_list = choice.split(",") 30 | gt = item['gt'] 31 | 32 | gt_answer = str(choice_list[gt_transfer[gt]]) 33 | 34 | if answer == "A" or answer == "B" or answer == "C" or answer == "D" or answer == "E": 35 | gt_answer = gt 36 | 37 | text = "Answer: " + str(answer) + " Groundtruth: " + str(gt_answer) 38 | 39 | conversation_history = [{"role": "system", 40 | "content": [ 41 | {"type": "text", "text": prompt}, 42 | ] 43 | }, 44 | {"role": "user", 45 | "content": [ 46 | {"type": "text", "text": text}, 47 | ] 48 | } ] 49 | 50 | response = client.chat.completions.create( 51 | model='gpt-4o-mini-2024-07-18', 52 | messages = conversation_history, 53 | max_tokens=4096 54 | ) 55 | result = response.choices[0].message.content 56 | 57 | save = {'number': i, "answer":answer, "gt": gt, "choice":choice, 'hit':result} 58 | writer.write(save) 59 | 60 | total += 1 61 | 62 | hit += int(result) 63 | 64 | i += 1 65 | 66 | #break 67 | 68 | print(hit) 69 | print(total) 70 | print(hit/total) -------------------------------------------------------------------------------- /example.py: -------------------------------------------------------------------------------- 1 | import os 2 | import json 3 | from sklearn.feature_extraction.text import TfidfVectorizer 4 | from sklearn.metrics.pairwise import cosine_similarity 5 | import sys 6 | import numpy as np 7 | import utils 8 | 9 | def convert_example(question, step_num): 10 | if step_num == 0: 11 | return 'Problem: ' + question['problem'] + '\nSolution: Step 1:' + question['Step 1'] 12 | example = 'Problem: ' + question['problem'] + '\n' 13 | step_key = 'Step '+str(step_num) 14 | for key, value in question.items(): 15 | if 'Step' not in key: 16 | continue 17 | if key==step_key: 18 | example = example + '\nKey Step: ' + key + ': ' + value 19 | else: 20 | example = example + key + ': ' + value 21 | return example 22 | 23 | def construct_example_bank(file_path="./Example/example-prm800k.jsonl"): 24 | id1 = 0 25 | id2 = 0 26 | data_example = [] 27 | example_problem = [] 28 | example_step = [] 29 | problem2step = {} #record a map between step to problem 30 | 31 | with open(file_path, 'r') as file: 32 | for line in file: 33 | # change string for each line into json format 34 | question = json.loads(line) 35 | data_example.append(question) 36 | 37 | example_problem.append(question['problem']) 38 | for key in question.keys(): 39 | if 'Step' in key: 40 | example_step.append(question[key]) 41 | problem2step[id2] = [id1, key] 42 | id2 += 1 43 | id1 += 1 44 | 45 | 46 | vectorizer_example_step = TfidfVectorizer() 47 | tfidf_matrix_example_step = vectorizer_example_step.fit_transform(example_step) 48 | example_step_embeddings = tfidf_matrix_example_step.toarray() 49 | 50 | return vectorizer_example_step, tfidf_matrix_example_step, example_step_embeddings, problem2step, data_example 51 | 52 | def retrieve_step(key, vectorizer_step, example_step_embeddings, problem2step, example_data, thrsd=0.7): 53 | new_step_embedding = vectorizer_step.transform([key]).toarray() 54 | similarities = cosine_similarity(new_step_embedding, example_step_embeddings).flatten() 55 | 56 | max_similarity = similarities.max() 57 | example_num = problem2step[similarities.argmax()][0] 58 | example_step = problem2step[similarities.argmax()][1] 59 | 60 | if max_similarity >= thrsd: 61 | has_example = True 62 | example = 'Example: ' + convert_example(example_data[example_num], int(example_step.rsplit(' ', 1)[-1])) 63 | print("Example for current step: " + example_step) 64 | 65 | else: 66 | has_example = False 67 | example = "" 68 | print ("No Example for this step") 69 | 70 | return has_example, example -------------------------------------------------------------------------------- /img/case.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/beichenzbc/BoostStep/8544d311fa816c7a96c67ed35927f64facaabfb7/img/case.png -------------------------------------------------------------------------------- /img/example.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/beichenzbc/BoostStep/8544d311fa816c7a96c67ed35927f64facaabfb7/img/example.png -------------------------------------------------------------------------------- /img/method.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/beichenzbc/BoostStep/8544d311fa816c7a96c67ed35927f64facaabfb7/img/method.png -------------------------------------------------------------------------------- /img/radar.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/beichenzbc/BoostStep/8544d311fa816c7a96c67ed35927f64facaabfb7/img/radar.png -------------------------------------------------------------------------------- /reasoning.py: -------------------------------------------------------------------------------- 1 | import os 2 | from openai import OpenAI 3 | import json 4 | import openai 5 | import time 6 | from sklearn.feature_extraction.text import TfidfVectorizer 7 | from sklearn.metrics.pairwise import cosine_similarity 8 | import time 9 | import sys 10 | import numpy as np 11 | import argparse 12 | import datetime 13 | 14 | import utils 15 | import example 16 | 17 | def first_try(client, problem, options, pre_solution, first_step, amc): 18 | example = "" 19 | has_example = False 20 | messages = utils.construct_message(problem, pre_solution, example, options, has_example, first_step, amc) 21 | response = client.chat.completions.create( 22 | model="gpt-4o", 23 | messages=messages, 24 | temperature=0, 25 | max_tokens=5000, 26 | top_p=1 27 | ) 28 | first_try_reasoning = response.choices[0].message.content 29 | print("first try for this step: " + first_try_reasoning) 30 | return first_try_reasoning 31 | 32 | def answer_problem_0_shot(client, problem, options, amc): 33 | if amc == 12: 34 | prompt = "You are a professional math problem solver. I will give you a problem and four options and only one is correct. Please give the solution(fewer than 10000 tokens) to the problem in steps starting by 'Step $i$:' where $i$ is step number. And output the final choice of option in the format: '\\boxed{A}'. Only 'A,B,C,D' can be chosen and you must choose one." 35 | else: 36 | prompt = "You are a professional math problem solver. I will give you a problem and five options and only one is correct. Please give the solution(fewer than 10000 tokens) to the problem in steps starting by 'Step $i$:' where $i$ is step number. And output the final choice of option in the format: '\\boxed{A}'. Only 'A,B,C,D,E' can be chosen and you must choose one." 37 | messages =[ 38 | { 39 | "role": "system", 40 | "content": prompt 41 | }, 42 | { 43 | "role": "user", 44 | "content": problem 45 | }, 46 | { 47 | "role": "user", 48 | "content": options 49 | } 50 | ] 51 | response = client.chat.completions.create( 52 | model="gpt-4o", 53 | messages=messages, 54 | temperature=0, 55 | max_tokens=11000, 56 | top_p=1 57 | ) 58 | answer = response.choices[0].message.content 59 | return answer 60 | 61 | 62 | def solve(client, whole_problem, vectorizer_step, example_step_embeddings, problem2step, example_data, thrsd=0.7, amc=12): 63 | flag = 0 64 | problem = 'Problem: ' + whole_problem['question'] 65 | options = 'Options: ' + str(whole_problem['options']) 66 | example_num = 0 67 | max_similarity = 0 68 | total_solution = "" 69 | 70 | #find example 71 | 72 | step_num = 0 73 | max_step = 20 74 | first_step = True 75 | while True: 76 | step_num += 1 77 | print(f"first try for step-{step_num}:") 78 | first_try_reasoning = first_try(client, problem, options, total_solution, first_step, amc) 79 | 80 | print(f"finding example step for step-{step_num}") 81 | has_example, example_step = example.retrieve_step(first_try_reasoning, vectorizer_step, example_step_embeddings, problem2step, example_data, thrsd) 82 | 83 | print(f"generating final step for step-{step_num}") 84 | new_message = utils.construct_message(problem, total_solution, example_step, options, has_example, first_step, amc) 85 | response = client.chat.completions.create( 86 | model="gpt-4o", 87 | messages=new_message, 88 | temperature=0, 89 | max_tokens=5000, 90 | top_p=1 91 | ) 92 | final_reasoning = response.choices[0].message.content 93 | print("final reason for this step: " + final_reasoning) 94 | pre_solution = total_solution 95 | total_solution += final_reasoning 96 | first_step = False 97 | 98 | if '\\boxed' in total_solution: 99 | break 100 | 101 | if step_num > max_step or final_reasoning in pre_solution: #get stuck repeating some content, happens very occasionally 102 | total_solution = answer_problem_0_shot(client, problem, options, amc) 103 | break 104 | 105 | answer = utils.extract_answer(total_solution) 106 | 107 | return total_solution, answer 108 | 109 | def main(args): 110 | 111 | thrsd = args.thrsd 112 | amc = args.amc 113 | example_file_path = args.example_file_path 114 | api_key = args.api_key 115 | 116 | client = OpenAI( 117 | api_key=api_key, 118 | ) 119 | 120 | data=[] 121 | if amc == 12: 122 | file_path = "./AMC/AMC_12_138.jsonl" 123 | output_file = '1-shot-AMC=12_thrsd=0.7-new.jsonl' 124 | elif amc == 10: 125 | file_path = "./AMC/AMC_10_217.jsonl" 126 | output_file = '1-shot-AMC=10_thrsd=0.7.jsonl' 127 | 128 | with open(file_path, 'r') as file: 129 | for line in file: 130 | data.append(json.loads(line)) 131 | 132 | vectorizer_example_step, tfidf_matrix_example_step, example_step_embeddings, problem2step, data_example = example.construct_example_bank(example_file_path) 133 | 134 | for i, whole_problem in enumerate(data): 135 | total_solution, answer = solve(client, whole_problem, vectorizer_example_step, example_step_embeddings, problem2step, data_example, thrsd, amc) 136 | save = {'number': i, "solution": total_solution, "answer":answer, "gt": whole_problem['answer'], 'choice':str(whole_problem['options'])} 137 | print(save) 138 | utils.save_result(save, output_file) 139 | 140 | if __name__ == '__main__': 141 | 142 | parser = argparse.ArgumentParser(description='Define hyper-parameters for reasoning') 143 | parser.add_argument('--thrsd', default=0.7, type=float, help='thrsd for filtering not similar examples') 144 | parser.add_argument('--amc', default=12, type=int, help='version of AMC dataset') 145 | parser.add_argument('--example_file_path', default="./Example/example-prm800k.jsonl", type=str, help='The file path of example problem bank') 146 | parser.add_argument('--api_key', default="your_api_key", type=str, help='API_KEY for OpenAI') 147 | args = parser.parse_args() 148 | 149 | t = str(datetime.datetime.now()) 150 | out_file = t[2:][:-7] + '.txt' 151 | sys.stdout = open(out_file, 'a', buffering=30000) 152 | sys.stderr = open(out_file, 'a', buffering=30000) 153 | main(args) 154 | -------------------------------------------------------------------------------- /utils.py: -------------------------------------------------------------------------------- 1 | import os 2 | import json 3 | import sys 4 | import numpy as np 5 | import argparse 6 | 7 | def extract_answer(response): 8 | position = response.find('\\boxed{') 9 | if position==-1: 10 | return 'wrong' 11 | num = 1 12 | position += 7 13 | start = position 14 | while num>0: 15 | if response[position]=='{': 16 | num+=1 17 | elif response[position]=='}': 18 | num-=1 19 | position+=1 20 | answer_extracted = response[start:position-1] 21 | return answer_extracted 22 | 23 | 24 | def construct_message(problem, solution, example, options, has_example, first_step, amc=12): 25 | if has_example: 26 | if amc==12: 27 | if first_step: 28 | prompt = "You are a professional math problem solver. I will give you a math problem and several options. And I will give you another one with its first step which you can refer to. Please output only the first step(fewer than 4000 tokens) to the first problem, starting with 'Step 1:'." 29 | else: 30 | prompt = "You are a professional math problem solver. I will give you a math problem and part of its solution. And you need to only output the next step of the solution(fewer than 4000 tokens), starting with 'Step $i$:', where $i$ is the step number. In case you don't know how to derive the correct content, an example with 'Key Step' will be given. You need to learn how 'Key Step' is derived, and implement similar strategy in your derivation procedure. If you think that the final step is derived, choose an option in the 4 options given. Output the final choice of option in the format: '\\boxed{A}'. Only 'A,B,C,D' can be chosen." 31 | else: 32 | if first_step: 33 | prompt = "You are a professional math problem solver. I will give you a math problem and several options. And I will give you another one with its first step which you can refer to. Please output only the first step(fewer than 4000 tokens) to the first problem, starting with 'Step 1:'." 34 | else: 35 | prompt = "You are a professional math problem solver. I will give you a math problem and part of its solution. And you need to only output the next step of the solution(fewer than 4000 tokens), starting with 'Step $i$:', where $i$ is the step number. In case you don't know how to derive the correct content, an example with 'Key Step' will be given. You need to learn how 'Key Step' is derived, and implement similar strategy in your derivation procedure. If you think that the final step is derived, choose an option in the 5 options given. Output the final choice of option in the format: '\\boxed{A}'. Only 'A,B,C,D,E' can be chosen." 36 | messages = [ 37 | { 38 | "role": "system", 39 | "content": prompt 40 | }, 41 | { 42 | "role": "user", 43 | "content": problem 44 | }, 45 | { 46 | "role": "user", 47 | "content": options 48 | }, 49 | { 50 | "role":"user", 51 | "content": solution 52 | }, 53 | { 54 | "role": "user", 55 | "content": example 56 | }, 57 | ] 58 | else: 59 | if amc == 12: 60 | if first_step: 61 | prompt = "You are a professional math problem solver. I will give you a math problem and several options. Please output only the first step(fewer than 4000 tokens) to the first problem, starting with 'Step 1:'." 62 | else: 63 | prompt = "You are a professional math problem solver. I will give you a math problem and part of its solution. And you need to output only the next step of the solution(fewer than 4000 tokens), starting with 'Step $i$:', where $i$ is the step number. If you think that the final step is derived, choose an option in the 4 options given. Output the final choice of option in the format: '\\boxed{A}'. Only 'A,B,C,D' can be chosen." 64 | else: 65 | if first_step: 66 | prompt = "You are a professional math problem solver. I will give you a math problem and several options. Please output only the first step(fewer than 4000 tokens) to the first problem, starting with 'Step 1:'." 67 | else: 68 | prompt = "You are a professional math problem solver. I will give you a math problem and part of its solution. And you need to output only the next step of the solution(fewer than 4000 tokens), starting with 'Step $i$:', where $i$ is the step number. If you think that the final step is derived, choose an option in the 5 options given. Output the final choice of option in the format: '\\boxed{A}'. Only 'A,B,C,D,E' can be chosen." 69 | messages = [ 70 | { 71 | "role": "system", 72 | "content": prompt 73 | }, 74 | { 75 | "role": "user", 76 | "content": problem 77 | }, 78 | { 79 | "role": "user", 80 | "content": options 81 | }, 82 | { 83 | "role":"user", 84 | "content":solution 85 | } 86 | ] 87 | return messages 88 | 89 | def save_result(result, output_file): 90 | with open(output_file, 'a') as f: 91 | json_str = json.dumps(result) 92 | f.write(json_str + '\n') --------------------------------------------------------------------------------