data_source stringclasses 1 value | prompt listlengths 1 1 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|
math_dapo | [
{
"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\nGiven that $x$ and $y$ are positive real numbers such that $\\frac{5}{x} = \\frac{y}{13} = \\frac{x}{y}$, find the val... | MATH | {
"ground_truth": "1170",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "34ee05f8-dbaa-4084-bc63-8950901d7de2"
} |
math_dapo | [
{
"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\n16 chess players held a tournament among themselves where every two players played exactly one game. The scoring syste... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "36118016-16ae-4821-af0d-52b63553b9e7"
} |
math_dapo | [
{
"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 $a$, $b$, and $c$ be real numbers such that $0 \\le a, b, c \\le 5$ and $2a + b + c = 10$. Over all possible value... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "98617593-6ed7-445b-a1f6-6c40f8a4db9e"
} |
math_dapo | [
{
"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\nFind all 4-digit numbers $\\overline{abcd}$ that satisfy the following conditions:\n\n1. The number is a multiple of $... | MATH | {
"ground_truth": "3454",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1a6d2be5-54ee-4cb1-8d62-2628640b4e1d"
} |
math_dapo | [
{
"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\nShenelle has some square tiles. Some of the tiles have side length $5\\text{ cm}$ while the others have side length $3... | MATH | {
"ground_truth": "94",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "727c33a3-a561-41dd-9ec0-5675c9bcb5a6"
} |
math_dapo | [
{
"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 $a, b, c$ be real nonzero numbers such that $a + b + c = 12$ and \\[ \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + ... | MATH | {
"ground_truth": "56",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "45a96338-f6c2-4187-a320-5fa311f459ca"
} |
math_dapo | [
{
"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 positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $... | MATH | {
"ground_truth": "90",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "30bc6301-4409-4265-a28f-1824f8201795"
} |
math_dapo | [
{
"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 $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1) = 91$ and $P(-1) = -121$, compute the maxim... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "97a2f899-ab10-4d6c-81e7-9732717458e5"
} |
math_dapo | [
{
"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\nAlexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measur... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7e43a05f-7997-4efc-968a-343048eb91b1"
} |
math_dapo | [
{
"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\nComplex number $\\omega$ satisfies $\\omega^5 = 2$. Find the sum of all possible values of $\\omega^4 + \\omega^3 + \\... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fb98d9ba-f5f2-46de-ad9b-18f8bf7757b3"
} |
math_dapo | [
{
"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 remainder when $2^{1023}$ is divided by $1023$?\n\nRemember to put your answer on its own line after \"Ans... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28163743-5416-4d50-95dc-4ff71965d1ad"
} |
math_dapo | [
{
"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\nBob's Rice ID number consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any number of ti... | MATH | {
"ground_truth": "324",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d91386bb-d54f-4c4c-aeb2-2d19c2f21b63"
} |
math_dapo | [
{
"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 grid of points $X = \\{(m,n) \\mid 0 \\leq m,n \\leq 4 \\}$. We say a pair of points $\\{(a,b),(c,d)\\}$ ... | MATH | {
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f594d1f3-dc05-4f0d-ab0b-e00c54f4f72d"
} |
math_dapo | [
{
"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 denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum,... | MATH | {
"ground_truth": "168",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9e0eaec9-a2c1-42b7-b6f8-ef392670729e"
} |
math_dapo | [
{
"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\nAna has \\(22\\) coins. She can take from her friends either \\(6\\) coins or \\(18\\) coins, or she can give \\(12\\)... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3aa18105-9190-4611-a6cc-6463336b44f0"
} |
math_dapo | [
{
"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\nSix people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability t... | MATH | {
"ground_truth": "4825",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cbd5eb4f-17cf-4cf7-b502-50e1b40bfea9"
} |
math_dapo | [
{
"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\nFind the number of ordered triplets $(a, b, c)$ of positive integers such that $abc = 2008$.\n\nRemember to put your a... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8c5757e0-6ad1-4347-9da1-e1ffd4b948a8"
} |
math_dapo | [
{
"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 $x$, $y$, and $z$ are integers that satisfy the system of equations: \n\\[ x^2y + y^2z + z^2x = 2186 \\] \n\... | MATH | {
"ground_truth": "245",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "75900852-4d04-4690-872a-7495c760dd26"
} |
math_dapo | [
{
"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 a regular $2n + 1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of t... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cb5b64bb-dd94-47fa-9af3-7e1ccc16f403"
} |
math_dapo | [
{
"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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}... | MATH | {
"ground_truth": "8000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "141399c6-e07a-49ac-aefd-8c267b621c6a"
} |
math_dapo | [
{
"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 how many ways can Irena arrange the integers $1$ through $7$ in a line such that for any three consecutive numbers,... | MATH | {
"ground_truth": "232",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9acd4f36-6578-49f7-8854-81b46d4c8fb9"
} |
math_dapo | [
{
"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 $a_n$ be the integer closest to $\\sqrt{n}$. Find the sum $$\\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{... | MATH | {
"ground_truth": "88",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fe2ad430-534a-4bdd-a1da-a104018ade82"
} |
math_dapo | [
{
"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 each of the $9$ small circles of the following figure, write positive integers less than $10$, without repetitions.... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "347408b0-ae54-4285-b9e3-bd7cfc7b5a68"
} |
math_dapo | [
{
"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 $x^{x^4} = 4$, what is the value of $x^{x^2} + x^{x^8}$?\n\nRemember to put your answer on its own line after \"Ans... | MATH | {
"ground_truth": "258",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6d85c27e-3cca-469b-a7eb-2bee13d66b2a"
} |
math_dapo | [
{
"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 $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying the following system of equations:\n\\[\n\\begin{alig... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "010a5338-0919-4f46-b66a-9e13d63eaf32"
} |
math_dapo | [
{
"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 five-letter \"words\" can you spell using the letters $S$, $I$, and $T$, if a \"word\" is defined as any sequ... | MATH | {
"ground_truth": "123",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5066b664-1645-4b5c-b6a1-0feb432fed4e"
} |
math_dapo | [
{
"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\nTalithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Sat... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "045bbc03-700e-41ef-9407-8024765d83af"
} |
math_dapo | [
{
"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 $ABC$, $\\angle A = 45^\\circ$ and $M$ is the midpoint of $\\overline{BC}$. The line $\\overline{AM}$ inte... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e7533969-fd87-4196-a427-2ab85ac64384"
} |
math_dapo | [
{
"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\nPatrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left,... | MATH | {
"ground_truth": "277",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a380db75-a4cc-484c-b235-fcce050b7516"
} |
math_dapo | [
{
"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\nMoor owns $3$ shirts, one each of black, red, and green. Moor also owns $3$ pairs of pants, one each of white, red, an... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "58eeea7a-710e-48d1-a83b-7c6b9e95e8d6"
} |
math_dapo | [
{
"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 for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first t... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2e1c632c-eb15-4b58-a1c5-bfe72bbb8e17"
} |
math_dapo | [
{
"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\nFind a number with $3$ digits, knowing that:\n1. The sum of its digits is $9$.\n2. The product of its digits is $24$.\... | MATH | {
"ground_truth": "342",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4f8068dc-f561-43cc-b15b-15583c24c8ca"
} |
math_dapo | [
{
"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 trinomial $f(x)$ satisfies the equation $(f(x))^3 - f(x) = 0$, which has three real roots. Determine the y-coordin... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c3760854-37ec-4262-b895-e118c61465e3"
} |
math_dapo | [
{
"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 $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two part... | MATH | {
"ground_truth": "1003",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "99892480-e568-4342-821f-c4d9a2a2f58b"
} |
math_dapo | [
{
"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\nRichard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ pro... | MATH | {
"ground_truth": "1100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "27892eb2-c6a5-407d-be8f-5d887140cfce"
} |
math_dapo | [
{
"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\nSet \n\\[A_n=\\sum_{k=1}^n \\frac{k^6}{2^k}.\\]\nFind \\(\\lim_{n\\to\\infty} A_n.\\)\n\nRemember to put your answer o... | MATH | {
"ground_truth": "9366",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3f887984-f347-4f47-9d4b-1b38b225e01c"
} |
math_dapo | [
{
"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\nKevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?\n\nRemember to... | MATH | {
"ground_truth": "63",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c36bd46e-e92c-46cc-a0bc-c1f31eab2b19"
} |
math_dapo | [
{
"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 choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$... | MATH | {
"ground_truth": "1225",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5abec22b-dc9c-41ab-85f1-320e1e8da297"
} |
math_dapo | [
{
"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 $x$, $y$, $z$ be real numbers that satisfy the following equations:\n\n\\[\n(x-y)^2 + (y-z)^2 + (z-x)^2 = 8, \\\\\... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9eae5078-9ddf-460b-a6b0-f19b7c07b137"
} |
math_dapo | [
{
"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\nFind the smallest positive integer $k$ which can be expressed in the form $k = 19^n - 5^m$ for some positive integers ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a02df7bb-e02a-4675-bde4-442f539425fe"
} |
math_dapo | [
{
"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\nPoint $P$ is in the interior of $\\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The thre... | MATH | {
"ground_truth": "600",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bbe33e70-b80c-4a9f-9cd6-3e303e0a265d"
} |
math_dapo | [
{
"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 **doubling sum** function is defined by:\n\\[ D(a,n) = a + 2a + 4a + 8a + \\ldots \\] \nThis sum contains \\( n \\... | MATH | {
"ground_truth": "9765",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8cba10eb-7d6b-4e60-891e-a9d8e76512f6"
} |
math_dapo | [
{
"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 triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $... | MATH | {
"ground_truth": "2713",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "33ac46e2-edd9-410d-95d6-fbe967062e8b"
} |
math_dapo | [
{
"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\nFind the smallest positive integer $k$ such that $k!$ ends in at least 43 zeroes.\n\nRemember to put your answer on it... | MATH | {
"ground_truth": "175",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "54615ae1-e91e-42a7-b37e-bfc2a4d85ae0"
} |
math_dapo | [
{
"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\nAs in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ ... | MATH | {
"ground_truth": "98",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49d10c1c-56bc-40cd-9791-3d416872aa6d"
} |
math_dapo | [
{
"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\nWill has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "542e66a8-c40c-4ede-afe9-505551eed5c0"
} |
math_dapo | [
{
"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 integers $x$ satisfy the following conditions?\n\n1. $|x| + 5 < 7$\n2. $|x - 3| > 2$\n\nRemember to put your ... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d67a5ead-0ac9-4357-b114-ad68aa11592c"
} |
math_dapo | [
{
"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 come across an ancient mathematical manuscript. It reads: \n\n\"To determine if a number is divisible by seventeen... | MATH | {
"ground_truth": "106",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2e637830-1da6-4eaf-8c93-e39aeefd55bf"
} |
math_dapo | [
{
"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\nFind the $2019$th strictly positive integer $n$ such that $\\binom{2n}{n}$ is not divisible by $5$.\n\nRemember to put... | MATH | {
"ground_truth": "37805",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "775cd03a-13cd-4dc9-ba03-27c7c33d0596"
} |
math_dapo | [
{
"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\nTwo circles have radii $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of the... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "297294d8-1c49-422b-a813-38bb1ae6e131"
} |
math_dapo | [
{
"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\nDetermine the positive integers expressible in the form $\\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of posit... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f318b887-f273-44bb-9969-4fdb43966bcc"
} |
math_dapo | [
{
"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\nCompute the number of triangles of different sizes which contain the gray triangle in the figure below.\n\n\\[ \\text{... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f5fb58de-e35f-445b-bf30-b4092503ebdf"
} |
math_dapo | [
{
"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\nFind the number of all unordered pairs $\\{A,B \\}$ of subsets of an $8$-element set, such that $A \\cap B \\neq \\emp... | MATH | {
"ground_truth": "23606",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a54ed7b8-d0f1-464f-92dd-6c8463c192a3"
} |
math_dapo | [
{
"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 any positive integer $a$, let $\\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest ... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "532c9de0-0c56-4de3-866e-790b25de0697"
} |
math_dapo | [
{
"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\nJonathan discovers that his ideal match is Sara Lark. To increase his chances of finding a girlfriend, he is open to d... | MATH | {
"ground_truth": "23520",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e8c0571e-8399-4e92-9f8e-fe7d74796a13"
} |
math_dapo | [
{
"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\nDetermine the value of the sum \\( \\left| \\sum_{1 \\leq i < j \\leq 50} ij(-1)^{i+j} \\right|.\\)\n\nRemember to put... | MATH | {
"ground_truth": "21150",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0cff3dc6-e51f-4b36-a85b-097181d5272f"
} |
math_dapo | [
{
"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 Vila Par, all the truth coins weigh an even number of grams, while the false coins weigh an odd number of grams. Th... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "949f49c1-5854-4951-95bc-48907dbfa628"
} |
math_dapo | [
{
"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\nFind the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.\n\nRemember... | MATH | {
"ground_truth": "104",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d58f63c4-34eb-454e-ac16-d122cf9cae78"
} |
math_dapo | [
{
"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\nGorf the frog is standing on the first lily pad in a row of lily pads numbered from $1$ to $20$ from left to right. On... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e2e7ce3e-8bd5-419b-b9db-e6817dfffa51"
} |
math_dapo | [
{
"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\nFind the smallest natural number $n$ for which there exists a natural number $x$ such that\n\\[(x+1)^3 + (x+2)^3 + (x+... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ffe5892c-c7b0-488d-919e-08c516a5652e"
} |
math_dapo | [
{
"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 a function such that $f(x + y) = f(x) + f(y)$ for all $x, y \\in \\mathbb{R}$ and $f(1) = 100$. Calculate $... | MATH | {
"ground_truth": "403791300",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "52123277-4c25-491b-a30c-316690eface6"
} |
math_dapo | [
{
"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\nDetermine the leftmost three digits of the number \\[1^1 + 2^2 + 3^3 + \\ldots + 999^{999} + 1000^{1000}.\\]\n\nRememb... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fe0d5bc5-0c40-45e0-8722-3a3dd7ef8618"
} |
math_dapo | [
{
"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 $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\ma... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5b5665ef-cc5c-43f0-aec2-e846dd0758f9"
} |
math_dapo | [
{
"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 positive integers $v, w, x, y$, and $z$ satisfy the equation \\[v + \\frac{1}{w + \\frac{1}{x + \\frac{1}{y + \\fr... | MATH | {
"ground_truth": "12354",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bd6c8fa7-1ffa-442a-805a-d2eba03f8b4b"
} |
math_dapo | [
{
"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\nCompute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.\n\nRemembe... | MATH | {
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d47140c4-868d-4622-a302-eb9c13fadaa1"
} |
math_dapo | [
{
"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 rectangular field that measures $20\\,\\text{m}$ by $15\\,\\text{m}$. Xiaoyu the butterfly is sitting at th... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0c7597b6-8977-4b5e-987b-4090c96fa41f"
} |
math_dapo | [
{
"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 $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a,... | MATH | {
"ground_truth": "648",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b3e974b4-fdf4-407f-a2eb-bb9166130b0e"
} |
math_dapo | [
{
"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 $x, y, z$ be positive real numbers that satisfy at least one of the inequalities: $2xy > 1$, $yz > 1$. Find the le... | MATH | {
"ground_truth": "-8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5543ff8c-9d5d-4503-9aff-41bc9051aced"
} |
math_dapo | [
{
"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 $a_n = \\sum_{d|n} \\frac{1}{2^{d+ \\frac{n}{d}}}$. In other words, $a_n$ is the sum of $\\frac{1}{2^{d+ \\frac{n}... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d0fda6f8-3075-4798-bafc-1c3e9c64b179"
} |
math_dapo | [
{
"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\nGiven 10 real numbers $a_1, a_2, \\ldots, a_{10}$, 45 sums of two of these numbers are formed as $a_i + a_j$ for $1 \\... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9d809c67-4dac-45f0-ac9c-35cc08e0516e"
} |
math_dapo | [
{
"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 $12 \\times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on... | MATH | {
"ground_truth": "73",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "384732df-ef60-4b15-a6fe-cc330ca6a1c9"
} |
math_dapo | [
{
"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 Stromquist Comet is visible every 61 years. If the comet was visible in 2017, what is the next leap year when the ... | MATH | {
"ground_truth": "2444",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f562c7e3-8cd2-4a87-89e7-9959c702fe9f"
} |
math_dapo | [
{
"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 $f(x) = (x-1)(x-2)(x-3)^2(x-4)(x-5)(x-6)$. Compute $f''(3) - f'(3) + f(3)$.\n\nRemember to put your answer on its ... | MATH | {
"ground_truth": "-24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ed68ed19-4bc2-4f32-8b33-620349af5738"
} |
math_dapo | [
{
"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 few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its neighboring... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3f368730-1d4c-48fd-af19-005a57ae919b"
} |
math_dapo | [
{
"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 $ABC$ be a triangle with side lengths $13$, $14$, and $15$. The points on the interior of $\\triangle ABC$ that ar... | MATH | {
"ground_truth": "193",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "348e9ebb-aa8c-4131-a54b-8ec29f3c2e17"
} |
math_dapo | [
{
"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 $S = \\{1, 2, \\dots, 9\\}$. Compute the number of functions $f : S \\rightarrow S$ such that, for all $s \\in S$,... | MATH | {
"ground_truth": "288",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3d6f814a-141e-4707-933e-108973d8a276"
} |
math_dapo | [
{
"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\nAn equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \\in \\overline{AB}$, $Q \\in \\overline{A... | MATH | {
"ground_truth": "5073",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "808d8a12-b5cf-4a7f-9c37-aff94648da72"
} |
math_dapo | [
{
"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\nDetermine the sum of all possible positive integers $n$, such that the product of its digits equals $n^2 - 15n - 27$.\... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ba752ecb-9f63-40fd-b3af-38e3ad578186"
} |
math_dapo | [
{
"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 prime number $p$ is called a \\textbf{moderate} number if for every $2$ positive integers $k > 1$ and $m$, there exi... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d45cbe00-45ee-4516-8b66-b80258ccab60"
} |
math_dapo | [
{
"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 group of 6 students decided to form \"study groups\" and \"service activity groups\" based on the following principl... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cf3149da-64cb-491c-bf2e-bc34ff2599bc"
} |
math_dapo | [
{
"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 rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the large recta... | MATH | {
"ground_truth": "2000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3b688b7c-c6e1-4678-b9bb-d7a106af899e"
} |
math_dapo | [
{
"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 smallest positive odd integer that has the same number of positive divisors as $360$?\n\nRemember to put y... | MATH | {
"ground_truth": "3465",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a20fe409-e57b-40a3-a4e8-ab4c37008548"
} |
math_dapo | [
{
"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 a function such that $f(x) + f(x+1) = 2^x$ and $f(0) = 2010$. Find the last two digits of $f(2010)$.\n\nRem... | MATH | {
"ground_truth": "51",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5f4b34d0-5c90-413b-8ddb-ff58a1d77715"
} |
math_dapo | [
{
"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 the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}} \\rfloor$ f... | MATH | {
"ground_truth": "1973",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7174f5d2-50f6-4673-b083-ef41125bc12a"
} |
math_dapo | [
{
"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\nDetermine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that: \n- Ea... | MATH | {
"ground_truth": "1958",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e35d1013-a6af-4ac7-9375-eb8a5b6b2569"
} |
math_dapo | [
{
"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 triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicul... | MATH | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bff2c6e1-989b-4b6f-854c-aa7367986d4b"
} |
math_dapo | [
{
"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 closest number to $169$ that is divisible by $9$?\n\nRemember to put your answer on its own line after \"A... | MATH | {
"ground_truth": "171",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c4fa767c-c471-462b-a648-b82e3b4cb9ee"
} |
math_dapo | [
{
"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 $T = \\{ 1, 2, 3, \\dots, 14, 15 \\}$. A subset $S$ of $T$ is defined as \\textit{handy} if the sum of all the ele... | MATH | {
"ground_truth": "6560",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "48b61bd5-8374-4873-a137-428c63de1890"
} |
math_dapo | [
{
"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 $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \\sqrt{3}$. We construct four semicircles $\\omega_1$, $\\omega_2$... | MATH | {
"ground_truth": "243",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "997d1704-03d8-4f76-9478-745c9db621c8"
} |
math_dapo | [
{
"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 $\\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Define $a_1 = 1997^... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d3156d4d-7ae1-4745-910a-915ab5cabaec"
} |
math_dapo | [
{
"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 complex number $K$ such that the quadratic polynomial $7x^2 + Kx + 12 - 5i$ has exactly one root, where $i ... | MATH | {
"ground_truth": "364",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5a685594-4bd1-425d-8f45-2ed5906d5256"
} |
math_dapo | [
{
"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 fair $6$-sided die is repeatedly rolled until a $1$, $4$, $5$, or $6$ is rolled. What is the expected value of the p... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4bc486f2-bda4-4860-97fb-7ec3ea0df0a7"
} |
math_dapo | [
{
"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\nKevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?\n\nRemember to... | MATH | {
"ground_truth": "63",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "43bf8a2b-7c1a-4f39-bcca-f30a50baaccf"
} |
math_dapo | [
{
"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\nLine segment $\\overline{AB}$ has a perpendicular bisector $\\overline{CD}$, where $C$ is the midpoint of $\\overline{... | MATH | {
"ground_truth": "1080",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fc3febf4-fda2-47c2-ad59-43a12fd6fdee"
} |
math_dapo | [
{
"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(x) = \\cos(x^3 - 4x^2 + 5x - 2). \\]\nIf we let \\( f^{(k)} \\) denote the \\( k \\)th derivative of \\( f \... | MATH | {
"ground_truth": "907200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "499a18ea-ab3c-4208-a103-52ba73b945a4"
} |
math_dapo | [
{
"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\n\\[ S = \\tan^{-1}(2020) + \\sum_{j = 0}^{2020} \\tan^{-1}(j^2 - j + 1). \\]\nThen $S$ can be written as $\\fr... | MATH | {
"ground_truth": "4045",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6281f4d0-5dbc-4647-a0fd-4d346f3b9095"
} |
math_dapo | [
{
"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 $N = \\prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divides evenly int... | MATH | {
"ground_truth": "624",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bb54abb1-3c5b-40d9-9d80-247ecd6032bc"
} |
math_dapo | [
{
"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\nFelix chooses a positive integer as the starting number and writes it on the board. He then repeats the following step... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0fd575a4-b360-4cc5-be09-becc99416fd2"
} |
math_dapo | [
{
"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\nFind the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1, \\dots, X_n$ in the plane wi... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a5fe12ac-779c-4af5-b695-de9db129841f"
} |
math_dapo | [
{
"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 permute the letters $\\{S,C,R,A,M,B,L,E\\}$ without the permutation containing the substrin... | MATH | {
"ground_truth": "40200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5f11702d-2c3d-4672-9518-725a18ec52d9"
} |
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