data_source stringclasses 5 values | prompt listlengths 1 1 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|
dapo_zero/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": "6.0",
"style": "rule"
} | {
"index": 200,
"split": "test"
} |
dapo_zero/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": "47.0",
"style": "rule"
} | {
"index": 201,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 202,
"split": "test"
} |
dapo_zero/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": "0.0",
"style": "rule"
} | {
"index": 203,
"split": "test"
} |
dapo_zero/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": "8.0",
"style": "rule"
} | {
"index": 204,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 205,
"split": "test"
} |
dapo_zero/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": "42.0",
"style": "rule"
} | {
"index": 206,
"split": "test"
} |
dapo_zero/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": "11.0",
"style": "rule"
} | {
"index": 207,
"split": "test"
} |
dapo_zero/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": "12.0",
"style": "rule"
} | {
"index": 208,
"split": "test"
} |
dapo_zero/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": "27.0",
"style": "rule"
} | {
"index": 209,
"split": "test"
} |
dapo_zero/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": "36.0",
"style": "rule"
} | {
"index": 210,
"split": "test"
} |
dapo_zero/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": "45.0",
"style": "rule"
} | {
"index": 211,
"split": "test"
} |
dapo_zero/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": "3159.0",
"style": "rule"
} | {
"index": 212,
"split": "test"
} |
dapo_zero/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": "36.0",
"style": "rule"
} | {
"index": 213,
"split": "test"
} |
dapo_zero/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": "7.0",
"style": "rule"
} | {
"index": 214,
"split": "test"
} |
dapo_zero/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": "21.0",
"style": "rule"
} | {
"index": 215,
"split": "test"
} |
dapo_zero/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": "3.0",
"style": "rule"
} | {
"index": 216,
"split": "test"
} |
dapo_zero/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": "1.0",
"style": "rule"
} | {
"index": 217,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 218,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 219,
"split": "test"
} |
dapo_zero/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": "29.0",
"style": "rule"
} | {
"index": 220,
"split": "test"
} |
dapo_zero/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": "96.0",
"style": "rule"
} | {
"index": 221,
"split": "test"
} |
dapo_zero/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": "1.0",
"style": "rule"
} | {
"index": 222,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 223,
"split": "test"
} |
dapo_zero/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": "-1.0",
"style": "rule"
} | {
"index": 224,
"split": "test"
} |
dapo_zero/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": "8.0",
"style": "rule"
} | {
"index": 225,
"split": "test"
} |
dapo_zero/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": "18.0",
"style": "rule"
} | {
"index": 226,
"split": "test"
} |
dapo_zero/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": "265.0",
"style": "rule"
} | {
"index": 227,
"split": "test"
} |
dapo_zero/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": "9.0",
"style": "rule"
} | {
"index": 228,
"split": "test"
} |
dapo_zero/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": "9.0",
"style": "rule"
} | {
"index": 229,
"split": "test"
} |
dapo_zero/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": "7.0",
"style": "rule"
} | {
"index": 230,
"split": "test"
} |
dapo_zero/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": "7.0",
"style": "rule"
} | {
"index": 231,
"split": "test"
} |
dapo_zero/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": "7.0",
"style": "rule"
} | {
"index": 232,
"split": "test"
} |
dapo_zero/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": "13.0",
"style": "rule"
} | {
"index": 233,
"split": "test"
} |
dapo_zero/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": "50.0",
"style": "rule"
} | {
"index": 234,
"split": "test"
} |
dapo_zero/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": "13.0",
"style": "rule"
} | {
"index": 235,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 236,
"split": "test"
} |
dapo_zero/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": "11.0",
"style": "rule"
} | {
"index": 237,
"split": "test"
} |
dapo_zero/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": "18.0",
"style": "rule"
} | {
"index": 238,
"split": "test"
} |
dapo_zero/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": "50.0",
"style": "rule"
} | {
"index": 239,
"split": "test"
} |
dapo_zero/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": "11.0",
"style": "rule"
} | {
"index": 240,
"split": "test"
} |
dapo_zero/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": "3.0",
"style": "rule"
} | {
"index": 241,
"split": "test"
} |
dapo_zero/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": "194.0",
"style": "rule"
} | {
"index": 242,
"split": "test"
} |
dapo_zero/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": "1625.0",
"style": "rule"
} | {
"index": 243,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 244,
"split": "test"
} |
dapo_zero/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": "6.0",
"style": "rule"
} | {
"index": 245,
"split": "test"
} |
dapo_zero/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": "901.0",
"style": "rule"
} | {
"index": 246,
"split": "test"
} |
dapo_zero/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": "144.0",
"style": "rule"
} | {
"index": 247,
"split": "test"
} |
dapo_zero/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": "8.0",
"style": "rule"
} | {
"index": 248,
"split": "test"
} |
dapo_zero/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": "142.0",
"style": "rule"
} | {
"index": 249,
"split": "test"
} |
dapo_zero/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": "144.0",
"style": "rule"
} | {
"index": 250,
"split": "test"
} |
dapo_zero/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": "81.0",
"style": "rule"
} | {
"index": 251,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 252,
"split": "test"
} |
dapo_zero/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": "13.0",
"style": "rule"
} | {
"index": 253,
"split": "test"
} |
dapo_zero/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": "2.0",
"style": "rule"
} | {
"index": 254,
"split": "test"
} |
dapo_zero/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": "30.0",
"style": "rule"
} | {
"index": 255,
"split": "test"
} |
dapo_zero/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": "18.0",
"style": "rule"
} | {
"index": 256,
"split": "test"
} |
dapo_zero/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": "-4.0",
"style": "rule"
} | {
"index": 257,
"split": "test"
} |
dapo_zero/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": "359.0",
"style": "rule"
} | {
"index": 258,
"split": "test"
} |
dapo_zero/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": "8178.0",
"style": "rule"
} | {
"index": 259,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 260,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 261,
"split": "test"
} |
dapo_zero/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": "20.0",
"style": "rule"
} | {
"index": 262,
"split": "test"
} |
dapo_zero/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": "8.0",
"style": "rule"
} | {
"index": 263,
"split": "test"
} |
dapo_zero/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": "1296.0",
"style": "rule"
} | {
"index": 264,
"split": "test"
} |
dapo_zero/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": "17.0",
"style": "rule"
} | {
"index": 265,
"split": "test"
} |
dapo_zero/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": "6.0",
"style": "rule"
} | {
"index": 266,
"split": "test"
} |
dapo_zero/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": "841.0",
"style": "rule"
} | {
"index": 267,
"split": "test"
} |
dapo_zero/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": "36.0",
"style": "rule"
} | {
"index": 268,
"split": "test"
} |
dapo_zero/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": "10.0",
"style": "rule"
} | {
"index": 269,
"split": "test"
} |
dapo_zero/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": "7.0",
"style": "rule"
} | {
"index": 270,
"split": "test"
} |
dapo_zero/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": "2.0",
"style": "rule"
} | {
"index": 271,
"split": "test"
} |
dapo_zero/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": "2.0",
"style": "rule"
} | {
"index": 272,
"split": "test"
} |
dapo_zero/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": "20.0",
"style": "rule"
} | {
"index": 273,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 274,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 275,
"split": "test"
} |
dapo_zero/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": "76.0",
"style": "rule"
} | {
"index": 276,
"split": "test"
} |
dapo_zero/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": "22.0",
"style": "rule"
} | {
"index": 277,
"split": "test"
} |
dapo_zero/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": "44.0",
"style": "rule"
} | {
"index": 278,
"split": "test"
} |
dapo_zero/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": "20.0",
"style": "rule"
} | {
"index": 279,
"split": "test"
} |
dapo_zero/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": "23.0",
"style": "rule"
} | {
"index": 280,
"split": "test"
} |
dapo_zero/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": "36.0",
"style": "rule"
} | {
"index": 281,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 282,
"split": "test"
} |
dapo_zero/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": "6.0",
"style": "rule"
} | {
"index": 283,
"split": "test"
} |
dapo_zero/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": "47.0",
"style": "rule"
} | {
"index": 284,
"split": "test"
} |
dapo_zero/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": "4.0",
"style": "rule"
} | {
"index": 285,
"split": "test"
} |
dapo_zero/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": "0.0",
"style": "rule"
} | {
"index": 286,
"split": "test"
} |
dapo_zero/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": "8.0",
"style": "rule"
} | {
"index": 287,
"split": "test"
} |
dapo_zero/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": "5.0",
"style": "rule"
} | {
"index": 288,
"split": "test"
} |
dapo_zero/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": "42.0",
"style": "rule"
} | {
"index": 289,
"split": "test"
} |
dapo_zero/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": "11.0",
"style": "rule"
} | {
"index": 290,
"split": "test"
} |
dapo_zero/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": "12.0",
"style": "rule"
} | {
"index": 291,
"split": "test"
} |
dapo_zero/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": "27.0",
"style": "rule"
} | {
"index": 292,
"split": "test"
} |
dapo_zero/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": "36.0",
"style": "rule"
} | {
"index": 293,
"split": "test"
} |
dapo_zero/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": "45.0",
"style": "rule"
} | {
"index": 294,
"split": "test"
} |
dapo_zero/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": "3159.0",
"style": "rule"
} | {
"index": 295,
"split": "test"
} |
dapo_zero/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": "36.0",
"style": "rule"
} | {
"index": 296,
"split": "test"
} |
dapo_zero/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": "7.0",
"style": "rule"
} | {
"index": 297,
"split": "test"
} |
dapo_zero/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": "21.0",
"style": "rule"
} | {
"index": 298,
"split": "test"
} |
dapo_zero/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": "3.0",
"style": "rule"
} | {
"index": 299,
"split": "test"
} |
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