prompt listlengths 2 2 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point outside triangle $... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"index": 0,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 tri... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the p... | math | {
"ground_truth": "113",
"style": "rule"
} | {
"index": 1,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": 2,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $1$... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 3,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ex... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ ... | math | {
"ground_truth": "37",
"style": "rule"
} | {
"index": 4,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?\n\nRemember to put your answer on its own line after... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 5,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such that the arithmetic mean of the numbers $a, b, c$ is ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFour positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 6,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$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 su... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coi... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 7,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.\n\nRemember to p... | math | {
"ground_truth": "509",
"style": "rule"
} | {
"index": 8,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is project... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 9,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line serv... | math | {
"ground_truth": "1007",
"style": "rule"
} | {
"index": 10,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$, and altitude $AF$ meets $BC$ at $F$.... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ i... | math | {
"ground_truth": "28",
"style": "rule"
} | {
"index": 11,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.",
"role": "u... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 12,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 (starti... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 13,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burri... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nBurrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 14,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and each papaya costs $5$ RM. He bought at least one of ea... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDanial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 15,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 an... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recor... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 16,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDenote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=... | math | {
"ground_truth": "402",
"style": "rule"
} | {
"index": 17,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such that there are infinitely many positive integers $n$ f... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 18,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 19,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 distribut... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies b... | math | {
"ground_truth": "300",
"style": "rule"
} | {
"index": 20,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $a, 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\nRe... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 21,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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\\]\n... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the... | math | {
"ground_truth": "37",
"style": "rule"
} | {
"index": 22,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSome positive integers are initially written on a board, where each 2 of them are different. Each time we can do the f... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": 23,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 24,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCharlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list ... | math | {
"ground_truth": "8556",
"style": "rule"
} | {
"index": 25,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 t... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 26,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$ ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJoe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes o... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 27,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 28,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find ... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 29,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$ ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that,... | math | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": 30,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 su... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $\\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\\triangle{ABC}$ intersect $BC$ and $AC$ ... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 31,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 32,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of ... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 33,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 choic... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPetya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, ... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 34,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.\n\nRemember to put your answer ... | math | {
"ground_truth": "64",
"style": "rule"
} | {
"index": 35,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of the sum of the erased numbers divided by 13 is written on th... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOn a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of ... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 36,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\\frac{s(n)}{n}$?",
... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, whic... | math | {
"ground_truth": "1099",
"style": "rule"
} | {
"index": 37,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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, ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any tw... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 38,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 at... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 can $8$ mutually non-attacking rooks be placed on a $9 \\times 9$ chessboard so that all $8$ rooks are o... | math | {
"ground_truth": "40320",
"style": "rule"
} | {
"index": 39,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x)^n = 0$ has a rational root.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 40,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$ an... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 41,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.\n\nRemember to ... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 42,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls, and the third box contains 600 identical bl... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identi... | math | {
"ground_truth": "270901",
"style": "rule"
} | {
"index": 43,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $8... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 three points on the edge of a circular chord such that $B$ is due west of $C$ and $\\triangle... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 44,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with ... | math | {
"ground_truth": "31",
"style": "rule"
} | {
"index": 45,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any real numbers $x$ and $y$ that satisfy the equations:\n\\[\nx + y - xy = 155\n\\]\nand\n\\[\nx^2 + y^2 = 325\n\\]\nFind $|x^3 - y^3|$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 real numbers $x$ and $y$ that satisfy the equations:\n\\[\nx + y - xy = 155\n\\]\nand\n\\[\nx^2 + y^2 = 325\n\... | math | {
"ground_truth": "4375",
"style": "rule"
} | {
"index": 46,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white p... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are ex... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 47,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $k$ for which the equation:\n$$ \\text{lcm}(m,n) - \\text{gcd}(m,n) = k(m-n) $$\nhas no solution in positive integers $(m,n)$ with $m \\neq n$.",
"... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $k$ for which the equation:\n$$ \\text{lcm}(m,n) - \\text{gcd}(m,n) = k(m-n) $$\nhas no sol... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 48,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 49,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ point... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 50,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 intege... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer po... | math | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": 51,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose integers $a < b < c$ satisfy the following conditions:\n\\[\n a + b + c = 95 \\\\\n a^2 + b^2 + c^2 = 3083\n\\]\nFind the value of $c$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 integers $a < b < c$ satisfy the following conditions:\n\\[\n a + b + c = 95 \\\\\n a^2 + b^2 + c^2 = 3083\n\\... | math | {
"ground_truth": "37",
"style": "rule"
} | {
"index": 52,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 +... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 53,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may... | math | {
"ground_truth": "132",
"style": "rule"
} | {
"index": 54,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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"
} | {
"index": 55,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nByan is playing a game called \"raven, raven, falcon\" with his three friends. His friends sit in a circle, and Byan w... | math | {
"ground_truth": "107",
"style": "rule"
} | {
"index": 56,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular table under the following constraints:\n- Each cell contains a... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular ta... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 57,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "One day, students in school organized an exchange among themselves with the following rates:\n- $11$ strawberries exchange for $14$ raspberries.\n- $22$ cherries exchange for $21... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOne day, students in school organized an exchange among themselves with the following rates:\n- $11$ strawberries exch... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 58,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Deter... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ i... | math | {
"ground_truth": "80",
"style": "rule"
} | {
"index": 59,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all integers $n$ such that $\\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 integers $n$ such that $\\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.\n\nRemember to put ... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 60,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.\n\nRemember t... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 61,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $1$... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 numbers $1, 2, \\ldots, 10000$ are placed in the cells of a $100 \\times 100$ square grid, each number appearing e... | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 62,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incircles of triangles $ABD$ and $ACD$ touch $AD$ at the same poi... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incir... | math | {
"ground_truth": "152",
"style": "rule"
} | {
"index": 63,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean o... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 64,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ represents the number of divisors of the number $n$, i... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 65,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural number.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural n... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 66,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 maximu... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total o... | math | {
"ground_truth": "200",
"style": "rule"
} | {
"index": 67,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $y$.... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIt is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where... | math | {
"ground_truth": "1987",
"style": "rule"
} | {
"index": 68,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \\neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \\neq -f(-1)$. Suppose furthermore ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 69,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of the numbers $a^3 - 1$, $a^2 - 1$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of ... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 70,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that nonzero real numbers $x$ and $y$ satisfy the equations $x + \\frac{1}{y} = 3$ and $y + \\frac{1}{x} = 4$, find the value of $xy + \\frac{1}{xy}$.",
"role": "user"
... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 nonzero real numbers $x$ and $y$ satisfy the equations $x + \\frac{1}{y} = 3$ and $y + \\frac{1}{x} = 4$, f... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 71,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backward.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $k + 25973$ is a palindrome, a number that reads the same forward and... | math | {
"ground_truth": "89",
"style": "rule"
} | {
"index": 72,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real c... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 73,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last 30 digits all equal to 9.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last 30 d... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"index": 74,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 the number of ways for a rook to return to its original square on a $4 \\times 4$ chessboard in 8 moves, starting ... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 75,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\\( \\triangle ABC \\) is an isosceles triangle with \\( AB = AC \\). Point \\( P \\) is located inside \\( \\triangle ABC \\) such that \\( \\angle BCP = 30^\\circ \\), \\( \\an... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n\\( \\triangle ABC \\) is an isosceles triangle with \\( AB = AC \\). Point \\( P \\) is located inside \\( \\triangle... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 76,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFive congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to it... | math | {
"ground_truth": "720",
"style": "rule"
} | {
"index": 77,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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)}\\ ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of ... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 78,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:\n\n1. $f(n)$ is a po... | math | {
"ground_truth": "3816",
"style": "rule"
} | {
"index": 79,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and exten... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = ... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 80,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nReal nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:\n\\[\nA = \\frac{x}{y} + \\frac... | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": 81,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.\n\nRemember to put your answer ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 82,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $x$ such that $3 \\times 2^x = n^2 - 1$ for some positive integer $n$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 all positive integers $x$ such that $3 \\times 2^x = n^2 - 1$ for some positive integer $n$.\n\nRememb... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 83,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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)&=7... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 satisfying the simultaneous equations:\n\\[\n\\begin{align*}\nx(y^2+yz+z^2)... | math | {
"ground_truth": "31",
"style": "rule"
} | {
"index": 84,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, a... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 a pentagon $ABCDE$ with the following properties:\n\n1. Angles $\\angle A$ and $\\angle C$ are right angles.\n2.... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"index": 85,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.",
"role": "user"
}
] | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.\n\nR... | math | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": 86,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $AC$ ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $\\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\\triangle ABC$ with its center on segment... | math | {
"ground_truth": "415",
"style": "rule"
} | {
"index": 87,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ respectively. Given that $BD = 20$ and... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally ... | math | {
"ground_truth": "936",
"style": "rule"
} | {
"index": 88,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Petya gave Vasya a number puzzle. Petya chose a digit $X$ and said, \"I am thinking of a three-digit number that is divisible by 11. The hundreds digit is $X$ and the tens digit ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPetya gave Vasya a number puzzle. Petya chose a digit $X$ and said, \"I am thinking of a three-digit number that is di... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 89,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 sh... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nAda has been told to wr... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 90,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additionally, \\( AB = 18 \\) and \\( BC = 24 \\). Equilateral tria... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additio... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 91,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We are given a coin of diameter $\\frac{1}{2}$ and a checkerboard of $1 \\times 1$ squares with an area of $2010 \\times 2010$. We toss the coin such that it lands completely on ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWe are given a coin of diameter $\\frac{1}{2}$ and a checkerboard of $1 \\times 1$ squares with an area of $2010 \\tim... | math | {
"ground_truth": "6029",
"style": "rule"
} | {
"index": 92,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ at points $A$ and $B$. The intersect... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGraph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadra... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 93,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:\n- $... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 94,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, with $p, q$ being prime numbers and $m, n$ being positive int... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, w... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 95,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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-negat... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nUsing each of the digits $1, 2, 3, \\ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit num... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 96,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 $n$ such that the following statement holds:\n\nIf $a_1, a_2, a_3, a_4, a_5, a_... | math | {
"ground_truth": "107",
"style": "rule"
} | {
"index": 97,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve for $x$:\n\\[\nv - w + x - y + z = 79 \\\\\nv + w + x + y + z = -1 \\\\\nv + 2w + 4x + 8y + 16z = -2 \\\\\nv + 3w + 9x + 27y + 81z = -1 \\\\\nv + 5w + 25x + 125y + 625z = 7... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSolve for $x$:\n\\[\nv - w + x - y + z = 79 \\\\\nv + w + x + y + z = -1 \\\\\nv + 2w + 4x + 8y + 16z = -2 \\\\\nv + 3... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 98,
"split": "test"
} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:\n\n1. Add 1 to the u... | lighteval/MATH | [
{
"content": "Solve the following math problem step by step. 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 have a 4-digit whole number that is a perfect square. Another number is constructed by making the following change... | math | {
"ground_truth": "3025",
"style": "rule"
} | {
"index": 99,
"split": "test"
} |
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