data_source stringclasses 10
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | raw_problem stringlengths 19 3.19k | raw_problem_id stringlengths 32 32 | __index_level_0__ int64 0 17.9k |
|---|---|---|---|---|---|---|---|
MATH##Aops | [
{
"content": "In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point outside triangle $ABC$ such that $\\angle BAD = \\angle DAC$ and $\\angle BDC = 90^\\circ$. Suppose that $AD = 1$ and that $\\frac{BD}{CD} = \\frac{3}{2}$. If $AB + AC$ can be expressed in the form ... | MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9a9b6eb4-a1cb-49d1-8c1e-62eaf2f74079"
} | In triangle $ABC$, $\sin \angle A = \frac{4}{5}$ and $\angle A < 90^\circ$. Let $D$ be a point outside triangle $ABC$ such that $\angle BAD = \angle DAC$ and $\angle BDC = 90^\circ$. Suppose that $AD = 1$ and that $\frac{BD}{CD} = \frac{3}{2}$. If $AB + AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a, b... | 25220d51175a69005bf4516c17d57645 | 0 |
MATH##Aops | [
{
"content": "Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\\triangle AXY$ can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m... | MATH | {
"ground_truth": "113",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b426d104-244d-4831-a2c4-cd756b61700a"
} | Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. | 5341461d9839137481e277e6420b991b | 1 |
MATH##Aops | [
{
"content": "There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then start working back in ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "186c94eb-3f6a-43b1-b2a4-f7dec58b7318"
} | There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then start working back in the other direction, p... | ae3f312808f1483fa87816eda256de53 | 3 |
MATH##Aops | [
{
"content": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ \\sum_{(a,b,c) \\in S} \\frac{1}{2^a 3^b 5^c} \\]\ncan be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.\n\nSolve the problem... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "485b39f5-e71b-42b3-931f-219becbee2a3"
} | Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | ba85830a2780505458ccc0231bc03327 | 4 |
MATH##Aops | [
{
"content": "$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?\n\nSolve the problem step by step... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "29f6a98f-0730-48f6-b637-2e902c24d9bd"
} | $n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations? | cf9a953832d0c89a6781307f089d0e60 | 7 |
MATH##Aops | [
{
"content": "Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.\n\nSolve the problem ste... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e4eede97-5bed-4a2a-9b64-5de4e8cde3e8"
} | Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$. | 40efbd095f9de76990c9342ea59659e4 | 9 |
MATH##Aops | [
{
"content": "There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maxi... | MATH | {
"ground_truth": "1007",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bcdfd731-b3bb-4f5a-b775-a97e88c908b6"
} | There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for w... | b32e9ebb516b46ead19a0b3e9ac3e9c9 | 10 |
MATH##Aops | [
{
"content": "Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight), a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus lea... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "06f17afd-4946-4188-8c5f-0d372bc3b07a"
} | Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight), a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at n... | 5c7fdd13e1cc1a24e50308abdfa1c8c8 | 13 |
MATH##Aops | [
{
"content": "Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all fo... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a2731ba7-b002-4c81-ab9c-9ca08675a19d"
} | Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all four numbers appear in t... | cc8ceff50b62f088e185f892fa1d6793 | 16 |
MATH##Aops | [
{
"content": "Find all integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b7119f82-4161-4e79-8194-93596b4f2fa3"
} | Find all integers $k$ such that all roots of the following polynomial are also integers:
$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$ | b3dccad009b09f73c8223e0222e72ffc | 19 |
MATH##Aops | [
{
"content": "A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer t... | MATH | {
"ground_truth": "300",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0e3e66e1-4eba-4aeb-b480-c1a2fea08d0f"
} | A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer tasks receive fewer can... | d86dc3bed95a9d30063f0e65bd91f03d | 20 |
MATH##Aops | [
{
"content": "If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0ae83368-3a58-4faa-9ab6-91e989ac15e9"
} | If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$. | 43a922015367ff504119a9832ef0121d | 21 |
MATH##Aops | [
{
"content": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:\n\\[\n\\sum_{(a,b,c)\\in S}\\frac{1}{2^a3^b5^c}\n\\]\nThis sum can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.\n\... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5d21ea89-7190-4dc8-b6c2-540ec349219b"
} | Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:
\[
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
\]
This sum can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | be54ce3e1c6eb86b5d6d932b135e63ff | 22 |
MATH##Aops | [
{
"content": "Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves: \n1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$. \n2. If there are 2 numbers (written on the board... | MATH | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c37bdb6f-39ba-48c8-896f-e4e1760bd040"
} | Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves:
1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$.
2. If there are 2 numbers (written on the board) in the form $n, n+4$, ... | 9086b9b982df68571e6ebd359db3bb66 | 23 |
MATH##Aops | [
{
"content": "Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integers. He first rolls the die, and if the number showing on top is $k$, he then uniformly and randomly selects a $k$-element subset from his list and... | MATH | {
"ground_truth": "8556",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "983e41c8-805d-4669-9be6-11c1d5938192"
} | Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integers. He first rolls the die, and if the number showing on top is $k$, he then uniformly and randomly selects a $k$-element subset from his list and calculates the sum of... | d6548fef04dca8937cef6d922a186c98 | 25 |
MATH##Aops | [
{
"content": "The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e4b51fc2-4672-4b39-bcbb-85270800a0e7"
} | The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
... | e7b9d057ac3aeb78df7f16185b64e984 | 26 |
MATH##Aops | [
{
"content": "Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and ... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6fe5bcfc-af5a-42d7-80c9-bcae028c835f"
} | Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from t... | 2d73a30c9e6a252a190859317525e5f5 | 27 |
MATH##Aops | [
{
"content": "Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "879b93ad-c7f5-4795-bc0a-3e9fe3c57566"
} | Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$. | 4eb9502118450a2ecf078abdfe5f4b69 | 29 |
MATH##Aops | [
{
"content": "Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. ... | MATH | {
"ground_truth": "1010",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "55d21e01-08c9-4d43-b828-c0c670e4710c"
} | Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. The winner is the one ... | 56acbd28ce818b93370b234f4d0bc7b8 | 30 |
MATH##Aops | [
{
"content": "Triangle $\\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \\neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\\triangle{AED}$ is $5$, find the area of $\\trian... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bf2cddfc-c4c0-4067-b320-3bb506875cf5"
} | Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. | eb10b2f304dc9cff3e2f624e1bb6c191 | 31 |
MATH##Aops | [
{
"content": "Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equatio... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7c88d931-3bab-40c3-b474-f66e1910035b"
} | Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x) = a$. The game... | ec178f4d86cd49477e50b3e3c84d5f55 | 34 |
MATH##Aops | [
{
"content": "Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "af9089ec-686b-4471-ae02-8d196030763b"
} | Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$. | 228b4317c109e0acb14e88ae5d0be12e | 35 |
MATH##Aops | [
{
"content": "Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.\n\nSolve the ... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "13b92cd1-6a1c-4212-9de2-a652f660cc44"
} | Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$. | 73c55e17e4f5129363a6f9d5f5276269 | 38 |
MATH##Aops | [
{
"content": "How many ways can $8$ mutually non-attacking rooks be placed on a $9 \\times 9$ chessboard so that all $8$ rooks are on squares of the same color?\n\n(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)\n\nSolve the problem step by step and pro... | MATH | {
"ground_truth": "40320",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8d76b16e-88ca-450d-83dc-f9fe14a6dd7e"
} | How many ways can $8$ mutually non-attacking rooks be placed on a $9 \times 9$ chessboard so that all $8$ rooks are on squares of the same color?
(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.) | 347fae8f3ef10d395da112d18d36a38f | 39 |
MATH##Aops | [
{
"content": "Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.\n\nSolve the problem step by step and provide the f... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "22c41ff8-45bf-4489-8e53-031ee4f78ce7"
} | Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$. | 352058ac69fa97f30de63a2aef623b83 | 41 |
MATH##Aops | [
{
"content": "Determine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3da20697-ce17-4435-a17b-1e71485c8c5b"
} | Determine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational. | 32e4fee6a58c4ab97e2595d3640b502c | 42 |
MATH##Aops | [
{
"content": "Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\\triangle ABC$ is an equilateral triangle with each side measuring $86$ meters. A boy swam from $A$ directly towards $B$. After covering a distance of $x$ meters, he turned and swam westwar... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b921bada-7372-4658-9150-b16e3d5298b0"
} | Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\triangle ABC$ is an equilateral triangle with each side measuring $86$ meters. A boy swam from $A$ directly towards $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore a... | d3523f79dc02b36e16e839f2794e3bdc | 44 |
MATH##Aops | [
{
"content": "Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected.\n\nSolve the problem step by step and provide the final answer in \"\\boxed... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b3d6095c-5105-43bd-ac97-15d3fb5fa1cf"
} | Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected. | 86f8fa28aab2140d4a430f8cce9438db | 49 |
MATH##Aops | [
{
"content": "Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:\n- Those that contain the faces of the tetrahedron, and\n- Each of the planes parallel to a face o... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "252e5683-3a33-46d1-ad78-918c2f76fc93"
} | Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
- Those that contain the faces of the tetrahedron, and
- Each of the planes parallel to a face of the tetrahedron and co... | 98d604f195cb09494a47416fbd08314c | 50 |
MATH##Aops | [
{
"content": "For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.\n\nSolve the problem step by step and provide the... | MATH | {
"ground_truth": "2021",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1606d6e5-a84f-41ac-a2e3-599c58dae42c"
} | For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$. | c7aa6221e849c7d1cbe0285d8ba48f05 | 51 |
MATH##Aops | [
{
"content": "Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "132",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7f660475-d091-4d5b-b852-464391ab7241"
} | Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct. | ebd4b66e0309f89d666c738fefffbc40 | 54 |
MATH##Aops | [
{
"content": "Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.\n\nSuppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which ha... | MATH | {
"ground_truth": "1003",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e1dbaa4f-0cd5-4b83-97b2-ce02106dca47"
} | Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the... | 98fc3c30f8ce1c738a5d82cadd1db392 | 55 |
MATH##Aops | [
{
"content": "Byan is playing a game called \"raven, raven, falcon\" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either \"raven\" or \"falcon,\" each with a probability of $\\frac{1}{2}$. The game ends when Byan ... | MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f233c131-de52-4a0c-84dc-7316a2aff5de"
} | Byan is playing a game called "raven, raven, falcon" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either "raven" or "falcon," each with a probability of $\frac{1}{2}$. The game ends when Byan has said "falcon" twice. The ... | f5142064e3b7316450133bd1e28f8e1b | 56 |
MATH##Aops | [
{
"content": "The numbers $1, 2, \\ldots, 10000$ are placed in the cells of a $100 \\times 100$ square grid, each number appearing exactly once. In this arrangement, numbers that differ by $1$ are located in adjacent cells. We then calculate the distances between the centers of every pair of cells where the num... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c64599b1-6466-4bb5-a2ab-d1bc6c8b4b03"
} | The numbers $1, 2, \ldots, 10000$ are placed in the cells of a $100 \times 100$ square grid, each number appearing exactly once. In this arrangement, numbers that differ by $1$ are located in adjacent cells. We then calculate the distances between the centers of every pair of cells where the numbers differ by exactly $... | a9ce91f2a44ef737c3bf9ee522054a87 | 62 |
MATH##Aops | [
{
"content": "An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c5508fb1-703c-4cef-a533-e4ac784e3c33"
} | An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second number is the same as... | 4aad7d9b04c2a580b618466f3f3da9a5 | 64 |
MATH##Aops | [
{
"content": "Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
... | MATH | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0b1af1b6-125d-45b6-a9e1-0d324815a135"
} | Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$. | 54d2ac110c24bfa9d5d08cffcaf81a92 | 67 |
MATH##Aops | [
{
"content": "It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "1987",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b4143738-861e-4f11-8f99-56ed5e08de28"
} | It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$. | ee3df60a21c352b1f949583849005be8 | 68 |
MATH##Aops | [
{
"content": "Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$.\n\nSolve the proble... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b274cff3-2bed-4b85-8826-0a798a94f3fb"
} | Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$. | 09d25c41f9a6df708b616aeef691ce67 | 73 |
MATH##Aops | [
{
"content": "Let the number of ways for a rook to return to its original square on a $4 \\times 4$ chessboard in 8 moves, starting from a corner, be $k$. Find the number of positive integers that are divisors of $k$. A \"move\" counts as shifting the rook by a positive number of squares on the board along a ro... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7180a4a0-2d91-48e1-9b42-154e531c6a74"
} | Let the number of ways for a rook to return to its original square on a $4 \times 4$ chessboard in 8 moves, starting from a corner, be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that th... | d0fb5ec52fdecc8baa5f974414d2b197 | 75 |
MATH##Aops | [
{
"content": "Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of ... | MATH | {
"ground_truth": "720",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fb0ef667-a1fd-4d1e-b1c1-fcb5fc7dbd44"
} | Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by t... | f760a3ccfd3fa05fd22d408322ccf06c | 77 |
MATH##Aops | [
{
"content": "The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?\n\n\\[\n\\textbf{a)}\\ 0 \\quad \\textbf{b)}\\ 2 \\quad \\textbf{c)}\\ 4 \\quad \\textbf{d)}\\ 6 \\quad \\textbf{e)}\\ 8\n\\]\n\nSolve the problem step by step and pro... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3fa5dd7f-22e1-4e9b-a158-ad08686cf480"
} | The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
\[
\textbf{a)}\ 0 \quad \textbf{b)}\ 2 \quad \textbf{c)}\ 4 \quad \textbf{d)}\ 6 \quad \textbf{e)}\ 8
\] | 0ce9545e69330b4129fb8a0df51dbb20 | 78 |
MATH##Aops | [
{
"content": "Let $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:\n\n1. $f(n)$ is a positive integer.\n2. $f(n + 1) > f(n)$.\n3. $f(f(n)) = 3n$.\n\nFind $f(2001)$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user... | MATH | {
"ground_truth": "3816",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "93d3fc9b-16fc-4313-a4a7-b5f44ef850c9"
} | Let $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:
1. $f(n)$ is a positive integer.
2. $f(n + 1) > f(n)$.
3. $f(f(n)) = 3n$.
Find $f(2001)$. | 17279727d7bc1a81353d2da323a5978d | 79 |
MATH##Aops | [
{
"content": "Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:\n\\[\nA = \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} = \\frac{x}{z} + \\frac{z}{y} + \\frac{y}{x} + 1\n\\]\nDetermine the value of $A$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed... | MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "062bf4f7-e84d-4dd7-9acb-496fba7ec282"
} | Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:
\[
A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1
\]
Determine the value of $A$. | c4959dbc8ba06a5703860443176ef993 | 81 |
MATH##Aops | [
{
"content": "Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations:\n\\[\n\\begin{align*}\nx(y^2+yz+z^2)&=3y+10z,\\\\\ny(z^2+zx+x^2)&=21z+24x,\\\\\nz(x^2+xy+y^2)&=7x+28y.\n\\end{align*}\n\\]\nFind $xy+yz+zx$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "02e748ed-58af-45bc-af14-f620ed890500"
} | Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations:
\[
\begin{align*}
x(y^2+yz+z^2)&=3y+10z,\\
y(z^2+zx+x^2)&=21z+24x,\\
z(x^2+xy+y^2)&=7x+28y.
\end{align*}
\]
Find $xy+yz+zx$. | b8a1ca6e1c0eb3117ebda10fcaeccef9 | 84 |
MATH##Aops | [
{
"content": "Given a pentagon $ABCDE$ with the following properties:\n\n1. Angles $\\angle A$ and $\\angle C$ are right angles.\n2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.\n3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segment $AE$.\n\nFind the... | MATH | {
"ground_truth": "90",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ded6f725-9a2b-4e18-aa01-65e862a37b61"
} | Given a pentagon $ABCDE$ with the following properties:
1. Angles $\angle A$ and $\angle C$ are right angles.
2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.
3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segment $AE$.
Find the measure of angle $\angle E$. | 996e3717eb958ee599a9f00252cf873b | 85 |
MATH##Aops | [
{
"content": "Let $\\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\\frac{m}{n}$ for relative... | MATH | {
"ground_truth": "415",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49322731-a106-4e27-a99c-b2403c96b0a4"
} | Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integer... | b19d17092a1140e06cb48e5220341bec | 87 |
MATH##Aops | [
{
"content": "A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nAda has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.\n\nAda has wr... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1ddc5b1e-6874-49af-b44c-012083a8f504"
} | A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Ada has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.
Ada has written forty haikus so far ... | 02b1f62913e4c37a749c092757f023ee | 90 |
MATH##Aops | [
{
"content": "The quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additionally, \\( AB = 18 \\) and \\( BC = 24 \\). Equilateral triangles \\( \\triangle APB \\), \\( \\triangle BQC \\), and \\( \\triangle CRD \\) are constructed outside the quadrilater... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e1d07660-04dd-4306-badc-8a63aefb15e2"
} | The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimet... | ceb3fd14e79276a1199e52b46cca7f90 | 91 |
MATH##Aops | [
{
"content": "Let $ABC$ be an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:\n- $A'B' \\parallel AB$, \n- $C'C$ is the bisector of angle $A'C'B'$, \n- $A'C' + B'C' = AB$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7b999c0b-f929-4519-9817-fdf790909912"
} | Let $ABC$ be an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:
- $A'B' \parallel AB$,
- $C'C$ is the bisector of angle $A'C'B'$,
- $A'C' + B'C' = AB$. | 768cc49995ad83e8b793b0dfc3774bbf | 94 |
MATH##Aops | [
{
"content": "Using each of the digits $1, 2, 3, \\ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Their sum ends in $n$ zeroes, where $n$ is a non-negative integer. Determine the maximum possible value of $n$.\n\nSolve the problem step by step and provide the final answer... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1dad8702-7b20-4528-b376-8a5d11135e4e"
} | Using each of the digits $1, 2, 3, \ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Their sum ends in $n$ zeroes, where $n$ is a non-negative integer. Determine the maximum possible value of $n$. | f6915919e037f57da8a54e39c73aba4d | 96 |
MATH##Aops | [
{
"content": "Determine the largest positive integer $n$ such that the following statement holds:\n\nIf $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > ... | MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4fe69060-c57f-4375-8eb8-3431c2c74a76"
} | Determine the largest positive integer $n$ such that the following statement holds:
If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$. | 243a8e499d407975f756ea0922dd7726 | 97 |
MATH##Aops | [
{
"content": "Three 12 cm $\\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the vol... | MATH | {
"ground_truth": "864",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fe84e2ab-5d14-453f-aa72-5e5b39100c55"
} | Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) ... | b7aa6b57a88276b2b01c423b7f675f26 | 100 |
MATH##Aops | [
{
"content": "Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:\n\n\\[\n\\begin{align*}\n700 \\cdot 712 \\cdot 718 + 320,\\\\\n701 \... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "36d35edc-b2d5-498f-acbb-bfe37efe1a1c"
} | Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:
\[
\begin{align*}
700 \cdot 712 \cdot 718 + 320,\\
701 \cdot 713 \cdot 719 + 320,\\
702 \... | 57c45ff187684f1fdb0c052efdfe5abe | 101 |
MATH##Aops | [
{
"content": "Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.\n\nSolve the problem step by step and provid... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c56e7e6b-658d-49f1-a41f-26cccc0be82c"
} | Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles. | 3aae3e7297e615f0526f67c10f3b5b3c | 102 |
MATH##Aops | [
{
"content": "In a bag, there are $1007$ black and $1007$ white balls, each randomly numbered from $1$ to $2014$. At each step, one ball is drawn and placed on the table. Additionally, if desired, two balls of different colors can be selected from the table and placed in a separate bag. This action earns points... | MATH | {
"ground_truth": "1007",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9ca7220b-3118-47e6-8654-4be1a9269b2b"
} | In a bag, there are $1007$ black and $1007$ white balls, each randomly numbered from $1$ to $2014$. At each step, one ball is drawn and placed on the table. Additionally, if desired, two balls of different colors can be selected from the table and placed in a separate bag. This action earns points equal to the absolute... | dda9e755e9dc6e4ba3b3485b39295f4f | 104 |
MATH##Aops | [
{
"content": "In the country Máxico, there are two islands: the island \"Mayor\" and the island \"Menor\". The island \"Mayor\" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island \"Menor\" has only one state with $31$ cities. \"Aeropapantla\" and \"Aerocenzontle\" are the airlines that of... | MATH | {
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5e0cbdd2-1475-414e-b171-6f5bed784551"
} | In the country Máxico, there are two islands: the island "Mayor" and the island "Menor". The island "Mayor" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapant... | 97aec1000b0dfa797319085a7ceec5cb | 106 |
MATH##Aops | [
{
"content": "Several cells are marked in a $100 \\times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will ce... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b19fb901-898d-426b-9416-72c39782e9ca"
} | Several cells are marked in a $100 \times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do t... | f6b2815f5818e1fee3b3e97839653e95 | 107 |
MATH##Aops | [
{
"content": "In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \\in \\{0,1,\\dots,a\\}$ is chosen uniformly at random. Similarly,... | MATH | {
"ground_truth": "102",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "09734e5f-d739-4545-b317-fdde985ef379"
} | In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. Similarly, some number of the partic... | 068c0ebee5ece74309ffc5cec0a9384e | 109 |
MATH##Aops | [
{
"content": "30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** m... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4cf2a4f5-48ac-4018-aa93-8ab86e8ca2c6"
} | 30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** matches. Find the maxim... | c49391fe5e33da80badc92bd6a624a33 | 110 |
MATH##Aops | [
{
"content": "In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to one of the following... | MATH | {
"ground_truth": "300",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b2434142-dbaa-40b2-b785-10d1e8e90cee"
} | In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to one of the following options:
\[ \textbf{... | 6ac2803a512f80faa2d411120c1ddccc | 111 |
MATH##Aops | [
{
"content": "For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c09b98ba-0c04-41e1-9c73-c520facf5d33"
} | For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$. | ae2ace863eee56be1baf3cc6bea433df | 113 |
MATH##Aops | [
{
"content": "Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.\n\nSolve the problem step by ... | MATH | {
"ground_truth": "1594",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0de60080-7630-487c-a278-ac27265f935e"
} | Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$. | 36b6053b4bf4db900a21ed3ed37d28aa | 117 |
MATH##Aops | [
{
"content": "10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from t... | MATH | {
"ground_truth": "828",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ef7e7024-d992-4f99-9213-e436702ad1c7"
} | 10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\f... | 7b23760a17000d939d9f6c9676312cec | 118 |
MATH##Aops | [
{
"content": "Let $\\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\\Gamma$ with diameter $\\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\\Gamma$ at a point $X$. The len... | MATH | {
"ground_truth": "76",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "de73c551-1c3d-4204-a09a-cbcac6651579"
} | Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in ... | 1feed0360250ea3ef3cf1ee6f974c822 | 122 |
MATH##Aops | [
{
"content": "A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:\n\n- It has $3$ digits.\n- It is a palindrome.\n- The middle digit is a multiple of $4$.\n- It is divisible by $11$.\n\nWhat is the sum of all possible... | MATH | {
"ground_truth": "726",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6315b083-49c0-4724-a2e6-313c0a3fe828"
} | A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:
- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$.
- It is divisible by $11$.
What is the sum of all possible numbers of cows that the her... | 1ef9bb79893bc2915b4787c548aeb9c0 | 123 |
MATH##Aops | [
{
"content": "All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?\n\nSolve the problem step by step and provide the fi... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "29ae2e4e-631e-47b3-83fb-77cea98575be"
} | All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string? | 91a9fd204de52c3eeed4ffa3a6c6460f | 124 |
MATH##Aops | [
{
"content": "Let $f(n)$ be a function that fulfills the following properties:\n\n- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.\n- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.\n- If $a$ is a divisor of $b$, then: $f\\left(\\frac{b}{a}\\right) = |f(b) - f(a)|$.\n\... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "210830fd-5398-4859-9874-4364fe85cdde"
} | Let $f(n)$ be a function that fulfills the following properties:
- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.
- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.
- If $a$ is a divisor of $b$, then: $f\left(\frac{b}{a}\right) = |f(b) - f(a)|$.
Find $\displaystyle f(2009^{20... | 84a381d5dc20d01c7b66acfbd493de8b | 125 |
MATH##Aops | [
{
"content": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:\n- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers o... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cf6640af-1d43-414e-a62c-593105292999"
} | Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
- In the ... | 162c229d531de3e4b8b708a4d3f0fc67 | 127 |
MATH##Aops | [
{
"content": "A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row... | MATH | {
"ground_truth": "2019",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "af26354d-1b6c-4153-a09f-6acbfebdefb0"
} | A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at ... | 63b72ed565538c47e560c281a9a7f30c | 134 |
MATH##Aops | [
{
"content": "Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2384be17-9019-4e38-afea-cc7adabca2eb"
} | Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get? | 50f196eb7e606f6bd7ef7a8dbfb8b255 | 135 |
MATH##Aops | [
{
"content": "Find the smallest positive $m$ such that there exist at least 11 even and 11 odd positive integers $n$ for which $\\frac{n^3 + m}{n + 2}$ is an integer.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "42e51739-2c27-460b-93df-a29d9740ecea"
} | Find the smallest positive $m$ such that there exist at least 11 even and 11 odd positive integers $n$ for which $\frac{n^3 + m}{n + 2}$ is an integer. | 5d324a08488d20855c5aaa58fa4a959f | 138 |
MATH##Aops | [
{
"content": "A $8 \\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maxim... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6f0fdfbf-e1bc-41ec-9ea1-ff5ffdb74566"
} | A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$. | 450110af6102ed59850f1f0a219b7007 | 140 |
MATH##Aops | [
{
"content": "Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k \\geq 2$ such that $3^n = x^k + y^k$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d11e2dfd-296e-474d-81da-532662f248f0"
} | Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k \geq 2$ such that $3^n = x^k + y^k$. | 198bb515b19e56eb655e3b6f82e480ba | 141 |
MATH##Aops | [
{
"content": "Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For ... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3c2719eb-d362-46f9-9b66-5edc5395ad91"
} | Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ ... | 24cc46a2149205b198f1a7106a60c4a2 | 142 |
MATH##Aops | [
{
"content": "There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:\n1) From each town, there are exactly $k$ direct routes to $k$ other towns.\n2) If two towns are not connected by a d... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28939f34-74cb-4fac-a3c4-f7948e86a020"
} | There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) From each town, there are exactly $k$ direct routes to $k$ other towns.
2) If two towns are not connected by a direct route, then there ... | 1cc39e7f6057fe1209cfa5cfced5b22d | 143 |
MATH##Aops | [
{
"content": "Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\\frac{n}{100}$ probability of $1$ meter forward, to meter $n+1... | MATH | {
"ground_truth": "2500",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3090ec86-8b98-41ec-9dd9-ac53dc968392"
} | Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\frac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\frac{100-n}{... | 24d9d4734b4b5119725df50eb5a5b63e | 146 |
MATH##Aops | [
{
"content": "A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?\n\nSolve the problem step by st... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "837399cf-67d9-4a3b-8b70-3dd641d129a6"
} | A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$? | 697036bcc2b413030d72b04c377cddbf | 148 |
MATH##Aops | [
{
"content": "25 checkers are placed on the 25 leftmost squares of a $1 \\times N$ board. A checker can either move to the empty adjacent square to its right or jump over an adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find the minimal $N$ such that all the checke... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6bea93b4-0507-404c-b57c-a85939ea4030"
} | 25 checkers are placed on the 25 leftmost squares of a $1 \times N$ board. A checker can either move to the empty adjacent square to its right or jump over an adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find the minimal $N$ such that all the checkers can be placed in a r... | e8e37a5188030da2c2381c78a489dc10 | 150 |
MATH##Aops | [
{
"content": "In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam.\n\nSolve the problem step by step an... | MATH | {
"ground_truth": "2000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "91eb4d06-4b68-4f48-a5c7-b07e88b4e020"
} | In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam. | 52bb0e259563cffd51843c4e8691e502 | 156 |
MATH##Aops | [
{
"content": "Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "179",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8111beff-b97f-4342-85dd-3265cb49275f"
} | Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square. | 3a3ce376c4560a71bca8dacd577822a6 | 157 |
MATH##Aops | [
{
"content": "It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a4d5e40f-1033-4390-9a83-2af9719afa68"
} | It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest. | 6945c7fef02fe1e0ac05d8e596967af9 | 158 |
MATH##Aops | [
{
"content": "There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines i... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d6d68efd-086f-4610-a586-78e8c8a4f7a2"
} | There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no t... | aade6f2abdbcf893f02ffb287fc753f7 | 161 |
MATH##Aops | [
{
"content": "There is a pile of eggs. Joan counted the eggs, but her count was off by $1$ in the $1$'s place. Tom counted the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs an... | MATH | {
"ground_truth": "439",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1203b437-bb39-4cda-a91b-0a79c80aa72d"
} | There is a pile of eggs. Joan counted the eggs, but her count was off by $1$ in the $1$'s place. Tom counted the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct coun... | 9641d99b67ff332a4447140c7b66a818 | 162 |
MATH##Aops | [
{
"content": "Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number o... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "69e8dac9-cb08-4a17-a7a9-10c98384a063"
} | Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games. | 18f05cdae1b939cb0bb3a155614e46aa | 163 |
MATH##Aops | [
{
"content": "Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8e3712a1-6d0f-4310-a8e4-ffc14b381d96"
} | Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$. | 900bed6f4da04c7d5b768747d2e0de97 | 164 |
MATH##Aops | [
{
"content": "In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$.\n\nSolve the problem step by step and provide... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e9ea24d5-87ac-451e-b769-671685ccc6e8"
} | In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$. | b78f2b0546653de3460c55a560cc6648 | 166 |
MATH##Aops | [
{
"content": "Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$.\n\nSolve the problem step by step and provi... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d969047d-3352-444f-91ed-ae94375dac23"
} | Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$. | 019785900496783647be7dcfffe0d7a8 | 167 |
MATH##Aops | [
{
"content": "A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "807c6510-7136-441a-8b15-6f966fd03d72"
} | A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number... | 0ee5865b8b779f2363424bf461c987f9 | 169 |
MATH##Aops | [
{
"content": "Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is the smallest possible value of $b$?\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d09dd7ff-462a-4a7c-9279-28e3ef14e146"
} | Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is the smallest possible value of $b$? | 372a65e88f5da97dfe733afe82f0d11b | 170 |
MATH##Aops | [
{
"content": "Find all positive integers $l$ for which the equation $$a^3+b^3+ab=(lab+1)(a+b)$$ has a solution over positive integers $a,b$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "422d765f-749c-4c29-aec5-d40365d7cbcc"
} | Find all positive integers $l$ for which the equation $$a^3+b^3+ab=(lab+1)(a+b)$$ has a solution over positive integers $a,b$. | de9ff52b3f00ae3e65d01f8398913923 | 172 |
MATH##Aops | [
{
"content": "Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:\n\n1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or \n2. ... | MATH | {
"ground_truth": "2179",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fbbdce62-ed37-40dd-adb0-48f69313346d"
} | Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:
1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or
2. From $P$ and $Q$ to $5P-4... | 3ce021b8dc24d87272a578fb2976c591 | 174 |
MATH##Aops | [
{
"content": "In quadrilateral $ABCD$, it is given that $AB = DB$ and $AD = BC$. If the angles are $\nabla ABD = 36^{\\circ}$ and $\nabla BCD = 54^{\\circ}$, find the measure of angle $\nabla ADC$ in degrees.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"... | MATH | {
"ground_truth": "102",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b6e58319-1a23-4349-aaff-d5741169c760"
} | In quadrilateral $ABCD$, it is given that $AB = DB$ and $AD = BC$. If the angles are $
abla ABD = 36^{\circ}$ and $
abla BCD = 54^{\circ}$, find the measure of angle $
abla ADC$ in degrees. | d34c3aa657bbadc95a155983c1bf8301 | 175 |
MATH##Aops | [
{
"content": "Find the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \\dots, n$ with three different colors, there exist two numbers of the same color, whose difference is a perfect square.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7da9c773-1120-480b-b7f0-7d3bc6ecc6b0"
} | Find the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \dots, n$ with three different colors, there exist two numbers of the same color, whose difference is a perfect square. | 734b8a25d64faddf7f397dce9176bad7 | 177 |
MATH##Aops | [
{
"content": "Find all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equal to $a$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a0d3594b-e952-4a6d-a255-95858a37ca21"
} | Find all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equal to $a$. | d484bc4fb4075d6fc05574a62a995cbe | 181 |
MATH##Aops | [
{
"content": "Draw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles by connecting the vertices of neighboring squares. If this pattern is extended symmetrically to infinity, what is the ratio of the number of triangles to the number of squares?\n\nSolve the pro... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "343af5d6-7730-4b2d-9093-5aac262ed5d1"
} | Draw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles by connecting the vertices of neighboring squares. If this pattern is extended symmetrically to infinity, what is the ratio of the number of triangles to the number of squares? | 5c478c1efc89191d2d1c01c2bdbc802f | 182 |
MATH##Aops | [
{
"content": "Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-complete if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n$ divides $d_1$ and gcd$(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there?\... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "73681417-78c1-438e-ba63-c51ab27d95d1"
} | Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-complete if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n$ divides $d_1$ and gcd$(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there? | aa169b522d4489eb10b5af6ad3d03c64 | 183 |
MATH##Aops | [
{
"content": "A time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle between the hour hand and minute hand on a clock is less than $\\frac{360}{11}$ degrees is $\\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$.\n\nSolve the probl... | MATH | {
"ground_truth": "211",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2cae6a18-2118-487d-a193-cccb7aca494e"
} | A time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle between the hour hand and minute hand on a clock is less than $\frac{360}{11}$ degrees is $\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$. | d5272e1325b48c7cf7b2e715f62975c6 | 186 |
MATH##Aops | [
{
"content": "Find all integers $n$ such that $1 < n < 10^6$ and $n^3 - 1$ is divisible by $10^6 n - 1$.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "1000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6ea68b67-aeb7-4673-980a-ae4f8054e7eb"
} | Find all integers $n$ such that $1 < n < 10^6$ and $n^3 - 1$ is divisible by $10^6 n - 1$. | a08df79c2afb7745db4596158b2daab0 | 187 |
MATH##Aops | [
{
"content": "Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, ... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e761cad8-e5e0-40a1-8b92-b499cb6354ce"
} | Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, how many arrangements ... | 2b40123868642d67a67439c55f3a190f | 192 |
MATH##Aops | [
{
"content": "Find the smallest positive integer $n$ such that $n$ is divisible by $100$ and has exactly $100$ divisors.\n\nSolve the problem step by step and provide the final answer in \"\\boxed{...}\"",
"role": "user"
}
] | MATH | {
"ground_truth": "162000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cc509606-0dd3-490e-9f8d-a441db2192de"
} | Find the smallest positive integer $n$ such that $n$ is divisible by $100$ and has exactly $100$ divisors. | 3c4d49ffaa5760244c85a17f2c234feb | 195 |
End of preview. Expand in Data Studio
YAML Metadata Warning:empty or missing yaml metadata in repo card
Check out the documentation for more information.
DAPO-Math-17k, Qwen3-8B non8 subset
Rows from tongyx361/DAPO-Math-Unique-Boxed-17k
that Qwen/Qwen3-8B did not solve 8/8 under repeated sampling
(single-turn, no-tool, 8
rollouts per prompt, criterion num_correct < 8).
- Rows: 6389 of 17398
- Schema: identical to the source dataset (
prompt,reward_model.ground_truth, ...) - Row order follows the source dataset; original row indices are recoverable by matching prompts against the source.
- Source num_correct histogram (8 = solved every rollout, excluded):
{
"0": 1683,
"1": 549,
"2": 409,
"3": 416,
"4": 445,
"5": 563,
"6": 815,
"7": 1509,
"8": 11009
}
Produced by research/math_with_code/scripts/push_non8_dataset_to_hf.py in
NVIDIA-NeMo/RL (math_with_code research project); selection.json in this repo
carries the full filter provenance.
- Downloads last month
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