data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$\\frac{m}{n}$ is the Irreducible fraction value of \\[3+\\frac{1}{3+\\frac{1}{3+\\frac13}}\\], what is the value of $... | MATH | {
"ground_truth": "142",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 0,
"raw_problem": "$\\frac{m}{n}$ is the Irreducible fraction value of \\[3+\\frac{1}{3+\\frac{1}{3+\\frac13}}\\], what is the value of $m+n$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater numb... | MATH | {
"ground_truth": "144",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 1,
"raw_problem": "How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the product of all real numbers $x$ such that the distance on the number line between $\\log_6x$ and $\\log_69... | MATH | {
"ground_truth": "81",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 2,
"raw_problem": "What is the product of all real numbers $x$ such that the distance on the number line between $\\log_6x$ and $\\log_69$ is twice the distance on the number line between $\\log_610$ and $1$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $M$ be the midpoint of $\\overline{AB}$ in regular tetrahedron $ABCD$. $\\frac{p}{q}=\\cos(\\angle CMD)$ is irred... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 3,
"raw_problem": "Let $M$ be the midpoint of $\\overline{AB}$ in regular tetrahedron $ABCD$. $\\frac{p}{q}=\\cos(\\angle CMD)$ is irreducible fraction, what is the value of $p+q$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 4,
"raw_problem": "Let $\\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to th... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the value of \\[(\\log 5)^{3}+(\\log 20)^{3}+(\\log 8)(\\log 0.25)\\] where $\\log$ denotes the base-ten logar... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 5,
"raw_problem": "What is the value of \\[(\\log 5)^{3}+(\\log 20)^{3}+(\\log 8)(\\log 0.25)\\] where $\\log$ denotes the base-ten logarithm?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right re... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 6,
"raw_problem": "The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$\nunits. What is the volume of the new box?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA $\\emph{triangular number}$ is a positive integer that can be expressed in the form $t_n = 1+2+3+\\cdots+n$, for som... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 7,
"raw_problem": "A $\\emph{triangular number}$ is a positive integer that can be expressed in the form $t_n = 1+2+3+\\cdots+n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are\n$t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose $a$ is a real number such that the equation \\[a\\cdot(\\sin{x}+\\sin{(2x)}) = \\sin{(3x)}\\]\nhas more than o... | MATH | {
"ground_truth": "-4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 8,
"raw_problem": "Suppose $a$ is a real number such that the equation \\[a\\cdot(\\sin{x}+\\sin{(2x)}) = \\sin{(3x)}\\]\nhas more than one solution in the interval $(0, \\pi)$. The set of all such $a$ that can be written\nin the form \\[(p,q) \\cup (q,r),\\]\nwhere $p, q,$ and $r$ are real numbers with ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise arou... | MATH | {
"ground_truth": "359",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 9,
"raw_problem": "Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive\ninteger $n$ such that performing the sequence of transformations $T_1, T_2, T... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $13$ cards numbered $1, 2, 3, \\ldots, 13$ are arranged in a row. The task is to pick them up in numerica... | MATH | {
"ground_truth": "8178",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 10,
"raw_problem": "Suppose that $13$ cards numbered $1, 2, 3, \\ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less th... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 11,
"raw_problem": "The sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIsosceles trapezoid $ABCD$ has parallel sides $\\overline{AD}$ and $\\overline{BC},$ with $BC < AD$ and $AB = CD.$ The... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 12,
"raw_problem": "Isosceles trapezoid $ABCD$ has parallel sides $\\overline{AD}$ and $\\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ Let $\\frac{r}{s}=frac{BC}{AD}$ is irreducible fraction, what is the value of $r+s$?",
"spli... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation\n$z^2 - cz + 10 =... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 13,
"raw_problem": "Let $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation\n$z^2 - cz + 10 = 0$. Points $z_1$, $z_2$, $\\frac{1}{z_1}$, and $\\frac{1}{z_2}$ are the vertices of (convex) quadrilateral $\\mathcal{Q}$ in the complex plane. When the area of $\\ma... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $h_n$ and $k_n$ be the unique relatively prime positive integers such that \\[\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 14,
"raw_problem": "Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \\[\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}=\\frac{h_n}{k_n}.\\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3, \\ldots, n$. For how many integers with $1\\le{n}\\le{... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many strings of length $5$ formed from the digits $0$, $1$, $2$, $3$, $4$ are there such that for each $j \\in \\{... | MATH | {
"ground_truth": "1296",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 15,
"raw_problem": "How many strings of length $5$ formed from the digits $0$, $1$, $2$, $3$, $4$ are there such that for each $j \\in \\{1,2,3,4\\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies this condition\nbecause it contains at least $1$ digit less than $1$, at lea... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 16,
"raw_problem": "A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \\le i \\le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \\le c_2 \\le \\cdots \\le c_... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 17,
"raw_problem": "The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe $\\textit{taxicab distance}$ between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \\[|... | MATH | {
"ground_truth": "841",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 18,
"raw_problem": "The $\\textit{taxicab distance}$ between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \\[|x_1 - x_2| + |y_1 - y_2|.\\]\nFor how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?",
"spli... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 19,
"raw_problem": "A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all possible values of $X$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe infinite product\n\\[\\sqrt[3]{10} \\cdot \\sqrt[3]{\\sqrt[3]{10}} \\cdot \\sqrt[3]{\\sqrt[3]{\\sqrt[3]{10}}} \\cd... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 20,
"raw_problem": "The infinite product\n\\[\\sqrt[3]{10} \\cdot \\sqrt[3]{\\sqrt[3]{10}} \\cdot \\sqrt[3]{\\sqrt[3]{\\sqrt[3]{10}}} \\cdots\\]\nevaluates to a real number $\\sqrt{m}$. What is the value of m?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOn Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three type... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 21,
"raw_problem": "On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or t... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine $x\\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \\[((1\\diamond2)\\diamond3... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 22,
"raw_problem": "Define $x\\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \\[((1\\diamond2)\\diamond3)-(1\\diamond(2\\diamond3))?\\]",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n) = \\left( \\frac{-1+i\\sqrt{3}}{2} \\right)^n + \\left( \\frac{-1-i\\sqrt{3}}{2} \\right)^n$, where $i = \\s... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 23,
"raw_problem": "Let $f(n) = \\left( \\frac{-1+i\\sqrt{3}}{2} \\right)^n + \\left( \\frac{-1-i\\sqrt{3}}{2} \\right)^n$, where $i = \\sqrt{-1}$. What is $f(2022)$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nKayla rolls four fair $6$-sided dice. What is the denominator minus the numerator of the probability that at least one... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 24,
"raw_problem": "Kayla rolls four fair $6$-sided dice. What is the denominator minus the numerator of the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the numbers she rolls are greater than 2?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 25,
"raw_problem": "The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the numerator of the simplest fraction that represents the area of the region inside both ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is the nu... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 26,
"raw_problem": "The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is the numerator of the simplest fraction that represents $\\tan(\\angle ABC)$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many $4 \\times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries... | MATH | {
"ground_truth": "76",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 27,
"raw_problem": "How many $4 \\times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1, 2, 3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1, 2, 3,$ and $4,$ in some order? Output the... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEach square in a $5 \\times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where... | MATH | {
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 28,
"raw_problem": "Each square in a $5 \\times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:\n\nAny filled square with two or three filled neighbors remains f... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn $\\triangle{ABC}$ medians $\\overline{AD}$ and $\\overline{BE}$ intersect at $G$ and $\\triangle{AGE}$ is equilater... | MATH | {
"ground_truth": "44",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 29,
"raw_problem": "In $\\triangle{ABC}$ medians $\\overline{AD}$ and $\\overline{BE}$ intersect at $G$ and $\\triangle{AGE}$ is equilateral. Then $\\cos(C)$ can be written as $\\frac{m\\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn rhombus $ABCD$, point $P$ lies on segment $\\overline{AD}$ so that $\\overline{BP}$ $\\perp$ $\\overline{AD}$, $AP ... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 30,
"raw_problem": "In rhombus $ABCD$, point $P$ lies on segment $\\overline{AD}$ so that $\\overline{BP}$ $\\perp$ $\\overline{AD}$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? (Note: The figure is not drawn to scale.)\n",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial\n$x^2 + x + 1... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 31,
"raw_problem": "Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial\n$x^2 + x + 1$, the remainder is $x+2$, and when $P(x)$ is divided by the polynomial $x^2+1$, the remainder\nis $2x+1$. There is a unique polynomial of least degree with these two ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 32,
"raw_problem": "Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?Output the remainder when the answer is divided by 100.\n\n... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAnt Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time d... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 33,
"raw_problem": "Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x_0,x_1,x_2,\\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 34,
"raw_problem": "Let $x_0,x_1,x_2,\\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \n\\[S_n = \\sum_{k=0}^{n-1} x_k 2^k\\]\nSuppose $7S_n \\equiv 1 \\pmod{2^n}$ for all $n \\geq 1$. What is the value of the sum \n\\[x_{2019} + 2x_{2020} +... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe figure below depicts a regular $7$-gon inscribed in a unit circle.\n[asy] import geometry; unitsize(3cm); ... | MATH | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 35,
"raw_problem": "The figure below depicts a regular $7$-gon inscribed in a unit circle.\n[asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } fo... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFour regular hexagons surround a square with side length 1, each one sharing an edge with the square,\nas shown in the... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 36,
"raw_problem": "Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,\nas shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be\nwritten as $m \\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not div... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many of the first ten numbers of the sequence $121, 11211, 1112111, \\ldots$ are prime numbers?\n\nRemember to put... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 37,
"raw_problem": "How many of the first ten numbers of the sequence $121, 11211, 1112111, \\ldots$ are prime numbers?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?\n\nRemember... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 38,
"raw_problem": "For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of the x and y coordinates of the new position of the point $(-1, -2)$ after rotating $270^{\\circ}$ c... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 39,
"raw_problem": "What is the sum of the x and y coordinates of the new position of the point $(-1, -2)$ after rotating $270^{\\circ}$ counterclockwise about the point $(3, 1)$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider the following $100$ sets of $10$ elements each:\n\\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\... | MATH | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 40,
"raw_problem": "Consider the following $100$ sets of $10$ elements each:\n\\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\\ldots,20\\},\\\\ &\\{21,22,23,\\ldots,30\\},\\\\ &\\vdots\\\\ &\\{991,992,993,\\ldots,1000\\}. \\end{align*}\nHow many of these sets contain exactly two multiples of ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCamila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 41,
"raw_problem": "Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sequence $a_0,a_1,a_2,\\cdots$ is a strictly increasing arithmetic sequence of positive integers such that \\[2^{a... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 42,
"raw_problem": "The sequence $a_0,a_1,a_2,\\cdots$ is a strictly increasing arithmetic sequence of positive integers such that \\[2^{a_7}=2^{27} \\cdot a_7.\\] What is the minimum possible value of $a_2$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 mil... | MATH | {
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 43,
"raw_problem": "Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPositive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$?\n\nRemember to put your answer ... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 44,
"raw_problem": "Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the degree measure of the acute angle formed by lines with slopes $2$ and $\\frac{1}{3}$?\n\nRemember to put y... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 45,
"raw_problem": "What is the degree measure of the acute angle formed by lines with slopes $2$ and $\\frac{1}{3}$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the value of\n\\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \\dots + 18^3 - 17^3?\\]\n\nRemember to put your answer o... | MATH | {
"ground_truth": "3159",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 46,
"raw_problem": "What is the value of\n\\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \\dots + 18^3 - 17^3?\\]",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn a table tennis tournament every participant played every other participant exactly once. Although there were twice ... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 47,
"raw_problem": "In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\\%$ more than the number of games won by right-handed playe... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many complex numbers satisfy the equation $z^5=\\overline{z}$, where $\\overline{z}$ is the conjugate of the compl... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 48,
"raw_problem": "How many complex numbers satisfy the equation $z^5=\\overline{z}$, where $\\overline{z}$ is the conjugate of the complex number $z$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 49,
"raw_problem": "Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\\tfrac{\\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFlora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 50,
"raw_problem": "Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\\frac{1}{2^m}$.\nWhat is the probability that Flora will eventually land at 10? Writ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the product of all solutions to the equation\n\\[\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023\\]\n\... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 51,
"raw_problem": "What is the product of all solutions to the equation\n\\[\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023\\]",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe weight of $\\frac{1}{3}$ of a large pizza together with $3 \\frac{1}{2}$ cups of orange slices is the same as the ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 52,
"raw_problem": "The weight of $\\frac{1}{3}$ of a large pizza together with $3 \\frac{1}{2}$ cups of orange slices is the same as the weight of $\\frac{3}{4}$ of a large pizza together with $\\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs $\\frac{1}{4}$ of a pound. What is the weigh... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.\n1\n1 1\n1 3 1\n1 5 5 1\n1 7 11 7 1\nEach ro... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 53,
"raw_problem": "Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.\n1\n1 1\n1 3 1\n1 5 5 1\n1 7 11 7 1\nEach row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the pol... | MATH | {
"ground_truth": "29",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 54,
"raw_problem": "If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\\overline{AC}$ a... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f$ be the unique function defined on the positive integers such that \\[\\sum_{d\\mid n}d\\cdot f\\left(\\frac{n}... | MATH | {
"ground_truth": "96",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 55,
"raw_problem": "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 $f(2023)$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\\[(1+2a)(2+2b)(2a+b) = 32ab?\\]\n\nReme... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 56,
"raw_problem": "How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\\[(1+2a)(2+2b)(2a+b) = 32ab?\\]",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $K$ be the number of sequences $A_1$, $A_2$, $\\dots$, $A_n$ such that $n$ is a positive integer less than or equa... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 57,
"raw_problem": "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, $\\{\\}$... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere is a unique sequence of integers $a_1, a_2, \\cdots a_{2023}$ such that\n\\[\\tan2023x = \\frac{a_1 \\tan x + a_... | MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 58,
"raw_problem": "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 defin... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive perfect squares less than $2023$ are divisible by $5$?\n\nRemember to put your answer on its own lin... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 59,
"raw_problem": "How many positive perfect squares less than $2023$ are divisible by $5$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$?\n\nRemember to put your answer ... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 60,
"raw_problem": "How many digits are in the base-ten representation of $8^5 \\cdot 5^{10} \\cdot 15^5$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJanet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probabi... | MATH | {
"ground_truth": "265",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 61,
"raw_problem": "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 $3$? The final answer can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPoints $A$ and $B$ lie on the graph of $y=\\log_{2}x$. The midpoint of $\\overline{AB}$ is $(6, 2)$. What is the posit... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 62,
"raw_problem": "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 $B$? The final answer can be written in the form $m \\sqrt{n}$, where $m$ and $n$ are relatively prime positive i... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-dig... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 63,
"raw_problem": "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... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMaureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 64,
"raw_problem": "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?",
... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses co... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 65,
"raw_problem": "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 in... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origi... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 66,
"raw_problem": "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 p... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCalculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the oth... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 67,
"raw_problem": "Calculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other. The final answer can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m^2+n^2$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor complex number $u = a+bi$ and $v = c+di$ (where $i=\\sqrt{-1}$), define the binary operation\n$u \\otimes v = ac +... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 68,
"raw_problem": "For complex number $u = a+bi$ and $v = c+di$ (where $i=\\sqrt{-1}$), define the binary operation\n$u \\otimes v = ac + bdi$\nSuppose $z$ is a complex number such that $z\\otimes z = z^{2}+40$. What is $|z|^2$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is ... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 69,
"raw_problem": "A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the areas of all $6$ faces of $P$ is $\\frac{11}{2}$, and the volume of $P$ is $\\frac{1}{2}$. Find the length of the longest interior diagonal connecting two ve... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?\n\... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 70,
"raw_problem": "For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expe... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 71,
"raw_problem": "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 $x?$",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has a... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 72,
"raw_problem": "Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\\circ,$ Find the area of $ABC$. The final answer can be simplified in the form $m \\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ without ... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCarlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\\%$ on ever... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 73,
"raw_problem": "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 o... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. Wh... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 74,
"raw_problem": "When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.\n\\[abcd=2^6\\cdot... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 75,
"raw_problem": "Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.\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\\[\... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find t... | MATH | {
"ground_truth": "194",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 76,
"raw_problem": "A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find the ratio of the area of circle $A$ to the area of circle $B$. The final answer can be written in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positi... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip ... | MATH | {
"ground_truth": "1625",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 77,
"raw_problem": "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?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nYou are playing a game. A $2 \\times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertica... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 78,
"raw_problem": "You are playing a game. A $2 \\times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \\times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle.... |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen the roots of the polynomial \n\\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}\\]\nare removed... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 79,
"raw_problem": "When the roots of the polynomial \n\\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}\\]\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?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor how many integers $n$ does the expression\\[\\sqrt{\\frac{\\log (n^2) - (\\log n)^2}{\\log n - 3}}\\]represent a r... | MATH | {
"ground_truth": "901",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 80,
"raw_problem": "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?",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many nonempty subsets $B$ of ${0, 1, 2, 3, \\cdots, 12}$ have the property that the number of elements in $B$ is e... | MATH | {
"ground_truth": "144",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 81,
"raw_problem": "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.",
"split": null
} |
dapo_amc | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the area of the region in the coordinate plane defined by\n$| | x | - 1 | + | | y | - 1 | \\le 1$?\n\nRemember... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": 82,
"raw_problem": "What is the area of the region in the coordinate plane defined by\n$| | x | - 1 | + | | y | - 1 | \\le 1$?",
"split": null
} |
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