Datasets:
Tasks:
Text Generation
Modalities:
Text
Formats:
parquet
Languages:
English
Size:
10K - 100K
License:
embedding large listlengths 0 0 | prompt large listlengths 1 1 | data_source large_stringclasses 1
value | ability large_stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a class, there are $\\mathrm{n}$ students. Any two students are either friends or not friends, and the friendship is mutual. In this class, any $\\mathrm{n}-1$ students can be arranged in a circ... | Polaris-Dataset-53K | math | {
"ground_truth": "10",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In a class, there are $\\mathrm{n}$ students. Any two students are either friends or not friends, and the friendship is mutual. In this class, any $\\mathrm{n}-1$ students can be arranged in a circle so that each pair of adjacent students in the circle are friends, but it i... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nSuppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "0",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Suppose that $p$ is prime and $1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p$. How many possible values of $p$ are there? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "19868",
"problem": "Solve the following math problem. Make sure to ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n3. (1b, 8-11) The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be bought to ensure that with a probability of 0.99, there are at least 5 working ones amon... | Polaris-Dataset-53K | math | {
"ground_truth": "7",
"style": "math"
} | {
"achievement_prior": 0,
"description": "3. (1b, 8-11) The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be bought to ensure that with a probability of 0.99, there are at least 5 working ones among them? The solution will be evaluated in a math environment.",
"elo": "-",... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that $α \\in (0,π)$, if $\\sin α + \\cos α = \\frac{\\sqrt{3}}{3}$, find the value of $\\cos^2 α - \\sin^2 α$.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{\\sqrt{5}}{3}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given that $α \\in (0,π)$, if $\\sin α + \\cos α = \\frac{\\sqrt{3}}{3}$, find the value of $\\cos^2 α - \\sin^2 α$. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "22902",
"problem": "Solve the following math problem. Make sure to put th... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor which positive integers $n$ , one can find real numbers $x_1,x_2,\\cdots ,x_n$ such that $$ \\dfrac{x_1^2+x_2^2+\\cdots+x_n^2}{\\left(x_1+2x_2+\\cdots+nx_n\\right)^2}=\\dfrac{27}{4n(n+1)(2... | Polaris-Dataset-53K | math | {
"ground_truth": "n\\equiv0,4,8\\pmod{9}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "For which positive integers $n$ , one can find real numbers $x_1,x_2,\\cdots ,x_n$ such that $$ \\dfrac{x_1^2+x_2^2+\\cdots+x_n^2}{\\left(x_1+2x_2+\\cdots+nx_n\\right)^2}=\\dfrac{27}{4n(n+1)(2n+1)} $$ and $i\\leq x_i\\leq 2i$ for all $i=1,2,\\cdots ,n$ ? The solu... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nTwo equal cones with a common vertex touch each other and some plane $\\alpha$. Let $l$ be the line along which the planes of the bases of the cones intersect. Find the angle between the line $l$ a... | Polaris-Dataset-53K | math | {
"ground_truth": "60",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Two equal cones with a common vertex touch each other and some plane $\\alpha$. Let $l$ be the line along which the planes of the bases of the cones intersect. Find the angle between the line $l$ and the plane $\\alpha$ if the height of each cone is 2 and the radius of the ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them.",
"role": "u... | Polaris-Dataset-53K | math | {
"ground_truth": "50",
"style": "math"
} | {
"achievement_prior": 0,
"description": "How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "28802",
"... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nAlice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same s... | Polaris-Dataset-53K | math | {
"ground_truth": "3895",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( a \\) and \\( b \\) be positive integers. If \\( a^{2} + a b + 1 \\) is divisible by \\( b^{2} + a b + 1 \\), prove that \\( a = b \\).",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "b",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( a \\) and \\( b \\) be positive integers. If \\( a^{2} + a b + 1 \\) is divisible by \\( b^{2} + a b + 1 \\), prove that \\( a = b \\). The solution will be evaluated in a math environment.",
"elo": "-",
"index": "26478",
"problem": "Solve the following math p... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "fish",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Peter`, `Arnold`, `Carol`, ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "3",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Bob`, `Peter`, `Arnold`, `E... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitu... | Polaris-Dataset-53K | math | {
"ground_truth": "15",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nTim is participating in the following three math contests. On each contest, his score is the number of correct answers.\n- The Local Area Inspirational Math Exam consists of 15 problems.\n- The Fur... | Polaris-Dataset-53K | math | {
"ground_truth": "50",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Tim is participating in the following three math contests. On each contest, his score is the number of correct answers.\n- The Local Area Inspirational Math Exam consists of 15 problems.\n- The Further Away Regional Math League has 10 problems.\n- The Distance-Optimized Mat... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n4. Two odd natural numbers $k$ and $n$ are given. Martin wrote the fraction $i / j$ on the board for each pair of natural numbers $i, j$ satisfying $1 \\leqq i \\leqq k$ and $1 \\leqq j \\leqq n$. ... | Polaris-Dataset-53K | math | {
"ground_truth": "\\dfrac{k+1}{n+1}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "4. Two odd natural numbers $k$ and $n$ are given. Martin wrote the fraction $i / j$ on the board for each pair of natural numbers $i, j$ satisfying $1 \\leqq i \\leqq k$ and $1 \\leqq j \\leqq n$. Determine the median of all these fractions, that is, such a real number $q$ ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n33. Let $\\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right|=2001, y_{k}=\\frac{1}{k} \\sum_{i=1}^{k} x_{i}, k=1,2, \\cdots, 2001$. Find $\\max \\sum_{k=1}^{2000}\\left|y_{k}-y_{k+1}\\right|$",
"rol... | Polaris-Dataset-53K | math | {
"ground_truth": "2000",
"style": "math"
} | {
"achievement_prior": 0,
"description": "33. Let $\\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right|=2001, y_{k}=\\frac{1}{k} \\sum_{i=1}^{k} x_{i}, k=1,2, \\cdots, 2001$. Find $\\max \\sum_{k=1}^{2000}\\left|y_{k}-y_{k+1}\\right|$ The solution will be evaluated in a math environment.",
"elo": "-",
"index": "39448... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 5 people standing in a line numbered 1 through 5 in a left to right order.\nEach person has the following attributes: Hobby, Music-Genre, Pet, Sport.\nThe attributes have the following p... | Polaris-Dataset-53K | math | {
"ground_truth": "indie",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 5 people standing in a line numbered 1 through 5 in a left to right order.\nEach person has the following attributes: Hobby, Music-Genre, Pet, Sport.\nThe attributes have the following possible values:\n- Hobby: camping, collecting, gardening, reading, rock-climb... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "tennis",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Alice`, `Peter`, `A... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the function $f(x)=\\sqrt{3}\\sin^{2}(x+\\frac{\\pi}{4})-\\cos^{2}x-\\frac{1+\\sqrt{3}}{2} (x\\in\\mathbb{R})$.\n(1) Find the minimum value and the minimum positive period of the function $f(... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{4\\sqrt{3}+3}{10}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given the function $f(x)=\\sqrt{3}\\sin^{2}(x+\\frac{\\pi}{4})-\\cos^{2}x-\\frac{1+\\sqrt{3}}{2} (x\\in\\mathbb{R})$.\n(1) Find the minimum value and the minimum positive period of the function $f(x)$;\n(2) If $A$ is an acute angle, and vector $\\overrightarrow{m}=(1,5)$ is... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a circular arrangement of 101 natural numbers, it is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be among... | Polaris-Dataset-53K | math | {
"ground_truth": "41",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In a circular arrangement of 101 natural numbers, it is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be among the listed numbers? The solution will be evaluated in a math environment.",
... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n26. In the following diagram, $\\angle A C B=90^{\\circ}, D E \\perp B C, B E=A C, B D=\\frac{1}{2} \\mathrm{~cm}$, and $D E+B C=1 \\mathrm{~cm}$. Suppose $\\angle A B C=x^{\\circ}$. Find the value... | Polaris-Dataset-53K | math | {
"ground_truth": "30",
"style": "math"
} | {
"achievement_prior": 0,
"description": "26. In the following diagram, $\\angle A C B=90^{\\circ}, D E \\perp B C, B E=A C, B D=\\frac{1}{2} \\mathrm{~cm}$, and $D E+B C=1 \\mathrm{~cm}$. Suppose $\\angle A B C=x^{\\circ}$. Find the value of $x$. The solution will be evaluated in a math environment.",
"elo": "-"... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet set $M$ consist of an odd number of elements. For each element $x$ in set $M$, there is a uniquely determined subset $H_{x} \\subseteq M$ corresponding to $x$, satisfying the conditions:\n(1) F... | Polaris-Dataset-53K | math | {
"ground_truth": "1",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let set $M$ consist of an odd number of elements. For each element $x$ in set $M$, there is a uniquely determined subset $H_{x} \\subseteq M$ corresponding to $x$, satisfying the conditions:\n(1) For each $x \\in M$, $x \\in H_{x}$;\n(2) For any $x, y \\in M$, $y \\in H_{x}... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe base of a pyramid inscribed in a cone is a quadrilateral with pairs of adjacent sides equal, and the angle between one pair of adjacent sides is $\\alpha$. Find the ratio of the volume of the p... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{2\\sin\\alpha}{\\pi}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "The base of a pyramid inscribed in a cone is a quadrilateral with pairs of adjacent sides equal, and the angle between one pair of adjacent sides is $\\alpha$. Find the ratio of the volume of the pyramid to the volume of the cone. The solution will be evaluated in a math en... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHomer started peeling a pile of 60 potatoes at a rate of 4 potatoes per minute. Five minutes later, Christen joined him peeling at a rate of 6 potatoes per minute. After working together for 3 minu... | Polaris-Dataset-53K | math | {
"ground_truth": "23",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Homer started peeling a pile of 60 potatoes at a rate of 4 potatoes per minute. Five minutes later, Christen joined him peeling at a rate of 6 potatoes per minute. After working together for 3 minutes, Christen took a 2-minute break, then resumed peeling at a rate of 4 pota... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA circular piece of paper with a radius of 5 inches has a section cut out, forming a \\(225^\\circ\\) sector. This sector is then used to create a right circular cone by gluing points A and B toget... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{25\\pi}{4}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "A circular piece of paper with a radius of 5 inches has a section cut out, forming a \\(225^\\circ\\) sector. This sector is then used to create a right circular cone by gluing points A and B together. Determine the circumference of the cone's base, expressed in terms of \\... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nHow many different integers can be written as ⌊n^2 / 1998⌋ for n = 1, 2, ..., 1997?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "1498",
"style": "math"
} | {
"achievement_prior": 0,
"description": "How many different integers can be written as ⌊n^2 / 1998⌋ for n = 1, 2, ..., 1997? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "9736",
"problem": "Solve the following math problem. Make sure to put the answer (and only answer) inside ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFive friends were comparing how much scrap iron they brought to the collection. On average, it was $55 \\mathrm{~kg}$, but Ivan brought only $43 \\mathrm{~kg}$.\n\nWhat is the average amount of iro... | Polaris-Dataset-53K | math | {
"ground_truth": "58",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Five friends were comparing how much scrap iron they brought to the collection. On average, it was $55 \\mathrm{~kg}$, but Ivan brought only $43 \\mathrm{~kg}$.\n\nWhat is the average amount of iron brought without Ivan?\n\n(Note: By how many kilograms does Ivan's contribut... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and stroll... | Polaris-Dataset-53K | math | {
"ground_truth": "52",
"style": "math"
} | {
"achievement_prior": 0,
"description": "A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seco... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n1. Task: Compare the numbers $\\left(\\frac{2}{3}\\right)^{2016}$ and $\\left(\\frac{4}{3}\\right)^{-1580}$.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "\\left(\\frac{2}{3}\\right)^{2016} < \\left(\\frac{4}{3}\\right)^{-1580}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "1. Task: Compare the numbers $\\left(\\frac{2}{3}\\right)^{2016}$ and $\\left(\\frac{4}{3}\\right)^{-1580}$. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "45889",
"problem": "Solve the following math problem. Make sure to put the answer... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n15. Given the function $f(x)=x^{2}+x-2$.\n(1) Try to find the analytical expression of the function $g(x)=\\frac{|f(x)|-f(x)}{2}$;\n(2) If $a>0$, when the line $y=a x+b$ intersects the curve $y=g(x... | Polaris-Dataset-53K | math | {
"ground_truth": "g(x) = \\begin{cases} 0 & \\text{if } x \\leq -2 \\text{ or } x \\geq 1, \\\\ -x^2 - x + 2 & \\text{if } -2 < x < 1. \\end{cases}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "15. Given the function $f(x)=x^{2}+x-2$.\n(1) Try to find the analytical expression of the function $g(x)=\\frac{|f(x)|-f(x)}{2}$;\n(2) If $a>0$, when the line $y=a x+b$ intersects the curve $y=g(x)$ at three different points, try to determine the relationship between $a$ a... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe triangle with vertices at \\((0,0)\\), \\((0,1)\\), \\((2,0)\\) is repeatedly reflected in the three lines \\(AB\\), \\(BC\\), \\(CA\\) where \\(A\\) is \\((0,0)\\), \\(B\\) is \\((3,0)\\), \\(... | Polaris-Dataset-53K | math | {
"ground_truth": "(24,36),(24,37),(26,36)",
"style": "math"
} | {
"achievement_prior": 0,
"description": "The triangle with vertices at \\((0,0)\\), \\((0,1)\\), \\((2,0)\\) is repeatedly reflected in the three lines \\(AB\\), \\(BC\\), \\(CA\\) where \\(A\\) is \\((0,0)\\), \\(B\\) is \\((3,0)\\), \\(C\\) is \\((0,3)\\). Show that one of the images has vertices \\((24,36)\\), ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n3 [ Constructions on a Projection Drawing ]\n\nIn a regular quadrilateral pyramid with a lateral edge of 20, the angle between the lateral edges lying in the same face is $\\frac{\\pi}{3}$. A line ... | Polaris-Dataset-53K | math | {
"ground_truth": "17",
"style": "math"
} | {
"achievement_prior": 0,
"description": "3 [ Constructions on a Projection Drawing ]\n\nIn a regular quadrilateral pyramid with a lateral edge of 20, the angle between the lateral edges lying in the same face is $\\frac{\\pi}{3}$. A line perpendicular to one of the lateral edges and passing through a point on this... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven triangle \\( \\triangle ABC \\) with its incenter \\( I \\). Circles \\( \\odot O_1 \\), \\( \\odot O_2 \\), and \\( \\odot O_3 \\) pass through \\( B, C \\); \\( A, C \\); and \\( A, B \\) r... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{r}{2}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given triangle \\( \\triangle ABC \\) with its incenter \\( I \\). Circles \\( \\odot O_1 \\), \\( \\odot O_2 \\), and \\( \\odot O_3 \\) pass through \\( B, C \\); \\( A, C \\); and \\( A, B \\) respectively, and each intersect \\( \\odot I \\) orthogonally. Circle \\( \\o... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( F_{1}, F_{2}, F_{3}, \\ldots \\) be the Fibonacci sequence, the sequence of positive integers satisfying\n\\[ F_{1}=F_{2}=1 \\quad \\text { and } \\quad F_{n+2}=F_{n+1}+F_{n} \\text { for a... | Polaris-Dataset-53K | math | {
"ground_truth": "Yes",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( F_{1}, F_{2}, F_{3}, \\ldots \\) be the Fibonacci sequence, the sequence of positive integers satisfying\n\\[ F_{1}=F_{2}=1 \\quad \\text { and } \\quad F_{n+2}=F_{n+1}+F_{n} \\text { for all } n \\geq 1. \\]\n\nDoes there exist an \\( n \\geq 1 \\) for which \\( F_... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWe consider \\( n \\) distinct points in a plane. For each segment formed by a pair of these points, we color the midpoint of the segment blue. How many blue points are needed, at minimum?\n\nIf tw... | Polaris-Dataset-53K | math | {
"ground_truth": "2n-3",
"style": "math"
} | {
"achievement_prior": 0,
"description": "We consider \\( n \\) distinct points in a plane. For each segment formed by a pair of these points, we color the midpoint of the segment blue. How many blue points are needed, at minimum?\n\nIf two midpoints coincide, they count as a single point. The solution will be eval... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIf $m$ men can do a job in $d$ days, then $m+r$ men can do the job in:\n$\\textbf{(A)}\\ d+r \\text{ days}\\qquad\\textbf{(B)}\\ d-r\\text{ days}\\qquad\\textbf{(C)}\\ \\frac{md}{m+r}\\text{ days}\... | Polaris-Dataset-53K | math | {
"ground_truth": "\\textbf{(C)}\\\\frac{md}{+r}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "If $m$ men can do a job in $d$ days, then $m+r$ men can do the job in:\n$\\textbf{(A)}\\ d+r \\text{ days}\\qquad\\textbf{(B)}\\ d-r\\text{ days}\\qquad\\textbf{(C)}\\ \\frac{md}{m+r}\\text{ days}\\qquad\\\\ \\textbf{(D)}\\ \\frac{d}{m+r}\\text{ days}\\qquad\\textbf{(E)}\\ ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nStar lists the whole numbers $1$ through $30$ once. Emilio copies Star's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Star adds her numbers and Emilio adds his numbers. How... | Polaris-Dataset-53K | math | {
"ground_truth": "103",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Star lists the whole numbers $1$ through $30$ once. Emilio copies Star's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's? The solution will be evaluated in a... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( f(x)=\\frac{1010 x+1009}{1009 x+1010} \\) be defined as:\n\\[ f^{(1)}(x)=f(x), \\]\n\\[ f^{(i)}(x)=f\\left(f^{(i-1)}(x)\\right) \\quad \\text{for } i = 2, 3, \\cdots. \\]\n\nFind \\( f^{(n)... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{(2019^n+1)x+2019^n-1}{(2019^n-1)x+2019^n+1}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( f(x)=\\frac{1010 x+1009}{1009 x+1010} \\) be defined as:\n\\[ f^{(1)}(x)=f(x), \\]\n\\[ f^{(i)}(x)=f\\left(f^{(i-1)}(x)\\right) \\quad \\text{for } i = 2, 3, \\cdots. \\]\n\nFind \\( f^{(n)}(x) \\). The solution will be evaluated in a math environment.",
"elo": "-... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition:\n(i) $a_1=b_1=1$ (ii) $b_n=a_n b_{n-1}-2$ (iii) $n$ is an integer larger than $1$ .\nLet ${b_n}$ b... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{3}{2}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition:\n(i) $a_1=b_1=1$ (ii) $b_n=a_n b_{n-1}-2$ (iii) $n$ is an integer larger than $1$ .\nLet ${b_n}$ be a bounded sequence. Prove that $\\sum_{n=1}^{\\infty} \\frac{1}{a_1a_2\\cd... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor what real values of $k>0$ is it possible to dissect a $1 \\times k$ rectangle into two similar, but noncongruent, polygons?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "$k \\ne 1$",
"style": "math"
} | {
"achievement_prior": 0,
"description": "For what real values of $k>0$ is it possible to dissect a $1 \\times k$ rectangle into two similar, but noncongruent, polygons? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "27645",
"problem": "Solve the following math problem. Make sur... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet the real numbers \\(a_1, a_2, \\cdots, a_{100}\\) satisfy the following conditions: (i) \\(a_1 \\geq a_2 \\geq \\cdots \\geq a_{100} \\geq 0\\); (ii) \\(a_1 + a_2 \\leq 100\\); (iii) \\(a_3 + a... | Polaris-Dataset-53K | math | {
"ground_truth": "10000",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let the real numbers \\(a_1, a_2, \\cdots, a_{100}\\) satisfy the following conditions: (i) \\(a_1 \\geq a_2 \\geq \\cdots \\geq a_{100} \\geq 0\\); (ii) \\(a_1 + a_2 \\leq 100\\); (iii) \\(a_3 + a_4 + \\cdots + a_{100} \\leq 100\\). Find the maximum value of \\(a_1^2 + a_2... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\\pi$ square inches, excluding the area enclosed by the rectangle in its starting positi... | Polaris-Dataset-53K | math | {
"ground_truth": "12",
"style": "math"
} | {
"achievement_prior": 0,
"description": "A $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. The solution will be evaluated in ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven a circle circumscribed around triangle \\(FDC\\), a tangent \\(FK\\) is drawn such that \\(\\angle KFC = 58^\\circ\\). Points \\(K\\) and \\(D\\) lie on opposite sides of line \\(FC\\) as sho... | Polaris-Dataset-53K | math | {
"ground_truth": "61",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given a circle circumscribed around triangle \\(FDC\\), a tangent \\(FK\\) is drawn such that \\(\\angle KFC = 58^\\circ\\). Points \\(K\\) and \\(D\\) lie on opposite sides of line \\(FC\\) as shown in the diagram. Find the acute angle between the angle bisectors of \\(\\a... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nQuestion 236, Oleg wrote a real number above each column square (outside the table) and to the left of each row square (outside the table) on a blank $50 \\times 50$ grid. It is known that all 100 ... | Polaris-Dataset-53K | math | {
"ground_truth": "1250",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Question 236, Oleg wrote a real number above each column square (outside the table) and to the left of each row square (outside the table) on a blank $50 \\times 50$ grid. It is known that all 100 real numbers he wrote are distinct, and there are exactly 50 rational and 50 ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n89. The triangle is defined by points $A(5 ; 2), B(-1 ;-4), C(-5$; -3). Form the equation of the line passing through point $B$ and parallel to $A C$.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "x - 2y = 7",
"style": "math"
} | {
"achievement_prior": 0,
"description": "89. The triangle is defined by points $A(5 ; 2), B(-1 ;-4), C(-5$; -3). Form the equation of the line passing through point $B$ and parallel to $A C$. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "40456",
"problem": "Solve the following... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$ , $a^2=m_2^2+n_2^2$ , and $m_1-n_1=m_2-n... | Polaris-Dataset-53K | math | {
"ground_truth": "261",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$ , $a^2=m_2^2+n_2^2$ , and $m_1-n_1=m_2-n_2.$ The solution will be evaluated in a math environment.",
"elo": "-",
... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that the positive integer \\( n \\) does not exceed 2000 and can be expressed as the sum of at least 60 consecutive positive integers, find the number of such \\( n \\).",
"role": "user"
... | Polaris-Dataset-53K | math | {
"ground_truth": "6",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given that the positive integer \\( n \\) does not exceed 2000 and can be expressed as the sum of at least 60 consecutive positive integers, find the number of such \\( n \\). The solution will be evaluated in a math environment.",
"elo": "-",
"index": "8729",
"proble... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "3",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Peter`, `Arnold`, `... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nI bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of the ticket, given that my neighbor easily solved ... | Polaris-Dataset-53K | math | {
"ground_truth": "99999",
"style": "math"
} | {
"achievement_prior": 0,
"description": "I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of the ticket, given that my neighbor easily solved this problem. The solution will be evaluated in a math environment.",
"elo"... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n## Task B-3.7.\n\nIn the figure, a parallelogram $A B C D$ is shown, with the shorter side $\\overline{A B}$ being $2 \\mathrm{~cm}$ long, and the height to this side is $12 \\mathrm{~cm}$. A line ... | Polaris-Dataset-53K | math | {
"ground_truth": "264\\pi",
"style": "math"
} | {
"achievement_prior": 0,
"description": "## Task B-3.7.\n\nIn the figure, a parallelogram $A B C D$ is shown, with the shorter side $\\overline{A B}$ being $2 \\mathrm{~cm}$ long, and the height to this side is $12 \\mathrm{~cm}$. A line $p$ passes through the vertex $D$ of the parallelogram, which is perpendicula... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWrite $x^{10} + x^5 + 1$ as the product of two polynomials with integer coefficients.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "(x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Write $x^{10} + x^5 + 1$ as the product of two polynomials with integer coefficients. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "25609",
"problem": "Solve the following math problem. Make sure to put the answer (and only answer) insi... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nRegular hexagon \\( P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} \\) has side length 2. For \\( 1 \\leq i \\leq 6 \\), let \\( C_{i} \\) be a unit circle centered at \\( P_{i} \\) and \\( \\ell_{i} \\) be o... | Polaris-Dataset-53K | math | {
"ground_truth": "1603",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Regular hexagon \\( P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} \\) has side length 2. For \\( 1 \\leq i \\leq 6 \\), let \\( C_{i} \\) be a unit circle centered at \\( P_{i} \\) and \\( \\ell_{i} \\) be one of the internal common tangents of \\( C_{i} \\) and \\( C_{i+2} \\), wher... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn the right triangle \\( \\triangle ABC \\),\n\n\\[\n\\angle A = 90^\\circ, \\, AB = AC\n\\]\n\n\\( M \\) and \\( N \\) are the midpoints of \\( AB \\) and \\( AC \\) respectively. \\( D \\) is an... | Polaris-Dataset-53K | math | {
"ground_truth": "4\\sqrt{2}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In the right triangle \\( \\triangle ABC \\),\n\n\\[\n\\angle A = 90^\\circ, \\, AB = AC\n\\]\n\n\\( M \\) and \\( N \\) are the midpoints of \\( AB \\) and \\( AC \\) respectively. \\( D \\) is an arbitrary point on the segment \\( MN \\) (excluding points \\( M \\) and \\... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConvert $314_{10}$ to base 6.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "1242_6",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Convert $314_{10}$ to base 6. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "19820",
"problem": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConvert $314_{10}$ to base 6.",
"spli... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind the volume of a regular quadrangular prism if the angle between the prism's diagonal and a side face is \\(\\alpha\\), and the side of the base is \\(a\\).",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{^3\\sqrt{\\cos2\\alpha}}{\\sin\\alpha}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Find the volume of a regular quadrangular prism if the angle between the prism's diagonal and a side face is \\(\\alpha\\), and the side of the base is \\(a\\). The solution will be evaluated in a math environment.",
"elo": "-",
"index": "28484",
"problem": "Solve the... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle... | Polaris-Dataset-53K | math | {
"ground_truth": "13",
"style": "math"
} | {
"achievement_prior": 0,
"description": "A tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle. The solution will be evaluated in a math environment.",
"elo": "-",
"in... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nNine barrels. In how many ways can nine barrels be arranged in three tiers so that the numbers written on the barrels that are to the right of any barrel or below it are larger than the number writ... | Polaris-Dataset-53K | math | {
"ground_truth": "42",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Nine barrels. In how many ways can nine barrels be arranged in three tiers so that the numbers written on the barrels that are to the right of any barrel or below it are larger than the number written on the barrel itself? An example of a correct arrangement is having 123 i... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nTwo granaries, A and B, originally each stored whole bags of grain. If 90 bags are transferred from granary A to granary B, the number of bags in granary B will be twice the number of bags in grana... | Polaris-Dataset-53K | math | {
"ground_truth": "153",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Two granaries, A and B, originally each stored whole bags of grain. If 90 bags are transferred from granary A to granary B, the number of bags in granary B will be twice the number of bags in granary A. If a certain number of bags are transferred from granary B to granary A... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nWhat is the largest integer for which each pair of consecutive digits is a square?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "81649",
"style": "math"
} | {
"achievement_prior": 0,
"description": "What is the largest integer for which each pair of consecutive digits is a square? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "3537",
"problem": "Solve the following math problem. Make sure to put the answer (and only answer) inside \... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe nine interior intersection points on a 4 by 4 grid of squares are shown. How many interior intersection points are there on a 12 by 12 grid of squares?\n(A) 100\n(B) 121\n(C) 132\n(D) 144\n(E) ... | Polaris-Dataset-53K | math | {
"ground_truth": "121",
"style": "math"
} | {
"achievement_prior": 0,
"description": "The nine interior intersection points on a 4 by 4 grid of squares are shown. How many interior intersection points are there on a 12 by 12 grid of squares?\n(A) 100\n(B) 121\n(C) 132\n(D) 144\n(E) 169 The solution will be evaluated in a math environment.",
"elo": "-",
"... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nConsider a $ n \\times n$ checkerboard with $ n > 1, n \\in \\mathbb{N}.$ How many possibilities are there to put $ 2n \\minus{} 2$ identical pebbles on the checkerboard (each on a different ... | Polaris-Dataset-53K | math | {
"ground_truth": "2^n",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Consider a $ n \\times n$ checkerboard with $ n > 1, n \\in \\mathbb{N}.$ How many possibilities are there to put $ 2n \\minus{} 2$ identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "7.5",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "12384",
"problem": "Solve the following math problem. Make s... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "15",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "10210",
"problem": "Solve the following math problem. Make s... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nPositive real numbers \\( a, b, c \\) are in a geometric progression \\((q \\neq 1)\\), and \\( \\log _{a} b, \\log _{b} c, \\log _{c} a \\) are in an arithmetic progression. Find the common differ... | Polaris-Dataset-53K | math | {
"ground_truth": "-\\frac{3}{2}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Positive real numbers \\( a, b, c \\) are in a geometric progression \\((q \\neq 1)\\), and \\( \\log _{a} b, \\log _{b} c, \\log _{c} a \\) are in an arithmetic progression. Find the common difference \\( d \\). The solution will be evaluated in a math environment.",
"el... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor any integer $a$ , let $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of $f(a)$ that are prime?\n\n*Proposed by Alexander Wang*",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "22",
"style": "math"
} | {
"achievement_prior": 0,
"description": "For any integer $a$ , let $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of $f(a)$ that are prime?\n\n*Proposed by Alexander Wang* The solution will be evaluated in a math environment.",
"elo": "-",
"index": "29521",
"problem": "Solve the follow... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven that the quadratic equation \\( (5a + 2b)x^2 + ax + b = 0 \\) has a unique solution for \\( x \\), find the value of \\( x \\).",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{5}{2}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given that the quadratic equation \\( (5a + 2b)x^2 + ax + b = 0 \\) has a unique solution for \\( x \\), find the value of \\( x \\). The solution will be evaluated in a math environment.",
"elo": "-",
"index": "12476",
"problem": "Solve the following math problem. Ma... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThree. (20 points) For the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $2$, $M$ is the midpoint of edge $A A_{1}$. A circle $\\Gamma$ is constructed through vertices $A$, $B_{1}$, and $... | Polaris-Dataset-53K | math | {
"ground_truth": "\\left[ \\sqrt{\\dfrac{15 - 8\\sqrt{3}}{3}}, \\sqrt{\\dfrac{15 + 8\\sqrt{3}}{3}} \\right]",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Three. (20 points) For the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $2$, $M$ is the midpoint of edge $A A_{1}$. A circle $\\Gamma$ is constructed through vertices $A$, $B_{1}$, and $C$. Let $P$ be any point on circle $\\Gamma$. Find the range of values for th... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( f(x) = ax^4 + bx^3 + cx^2 + dx + e \\). For any \\( x \\in [-1, 1] \\), \\( f(x) \\in [-1, 1] \\). Find the maximum value of \\( |c| \\).",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "8",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( f(x) = ax^4 + bx^3 + cx^2 + dx + e \\). For any \\( x \\in [-1, 1] \\), \\( f(x) \\in [-1, 1] \\). Find the maximum value of \\( |c| \\). The solution will be evaluated in a math environment.",
"elo": "-",
"index": "9134",
"problem": "Solve the following math ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 4 people standing in a line numbered 1 through 4 in a left to right order.\nEach person has the following attributes: Beverage, Food, Pet, Sport.\nThe attributes have the following possi... | Polaris-Dataset-53K | math | {
"ground_truth": "garlic",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 4 people standing in a line numbered 1 through 4 in a left to right order.\nEach person has the following attributes: Beverage, Food, Pet, Sport.\nThe attributes have the following possible values:\n- Beverage: fanta, juice, milk, water\n- Food: artichoke, cabbag... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $ABCD$ be a rhombus with angle $\\angle A = 60^\\circ$ . Let $E$ be a point, different from $D$ , on the line $AD$ . The lines $CE$ and $AB$ intersect at $F$ . The lines $DF$ and ... | Polaris-Dataset-53K | math | {
"ground_truth": "120",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let $ABCD$ be a rhombus with angle $\\angle A = 60^\\circ$ . Let $E$ be a point, different from $D$ , on the line $AD$ . The lines $CE$ and $AB$ intersect at $F$ . The lines $DF$ and $BE$ intersect at $M$ . Determine the angle $\\angle BMD$ as a function ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind all functions $ f: \\mathbb{Q}^{\\plus{}} \\mapsto \\mathbb{Q}^{\\plus{}}$ such that: \n\\[ f(x) \\plus{} f(y) \\plus{} 2xy f(xy) \\equal{} \\frac {f(xy)}{f(x\\plus{}y)}.\\]",
"role": "use... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{1}{x^2}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Find all functions $ f: \\mathbb{Q}^{\\plus{}} \\mapsto \\mathbb{Q}^{\\plus{}}$ such that: \n\\[ f(x) \\plus{} f(y) \\plus{} 2xy f(xy) \\equal{} \\frac {f(xy)}{f(x\\plus{}y)}.\\] The solution will be evaluated in a math environment.",
"elo": "-",
"index": "32996",
"pr... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "toyota camry",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Arnold`, `Peter`, `... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nCircle $\\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\\omega$ is tangent to $\\overline{H M_{1}}$ at $A, \\overline{M_{1} M_{2}}$ at $I, \\overline{M_{2} T}$ at $M$, and $\\overline{... | Polaris-Dataset-53K | math | {
"ground_truth": "540",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Circle $\\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\\omega$ is tangent to $\\overline{H M_{1}}$ at $A, \\overline{M_{1} M_{2}}$ at $I, \\overline{M_{2} T}$ at $M$, and $\\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area o... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn the diagram below, $AB = 30$ and $\\angle ADB = 90^\\circ$. If $\\sin A = \\frac{3}{5}$ and $\\sin C = \\frac{1}{4}$, what is the length of $DC$?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "18\\sqrt{15}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In the diagram below, $AB = 30$ and $\\angle ADB = 90^\\circ$. If $\\sin A = \\frac{3}{5}$ and $\\sin C = \\frac{1}{4}$, what is the length of $DC$? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "33482",
"problem": "Solve the following m... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "green",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Arnold`, `Peter`, `Alice`, ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nTo promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. For her small kite, Genevieve draws the kite on a grid with the points one ... | Polaris-Dataset-53K | math | {
"ground_truth": "21",
"style": "math"
} | {
"achievement_prior": 0,
"description": "To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. For her small kite, Genevieve draws the kite on a grid with the points one inch apart, as shown below.\n\n[asy]\nfor ( int x = 0; x <= 6; ++x )\n{\n\nfo... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the parametric equations of curve $C_1$ are $$\\begin{cases} x=2\\cos\\theta \\\\ y=\\sin\\theta\\end{cases}(\\theta \\text{ is the parameter}),$$ and the parametric equations of curve $C_2$ ... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac {12-2 \\sqrt {13}}{5}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given the parametric equations of curve $C_1$ are $$\\begin{cases} x=2\\cos\\theta \\\\ y=\\sin\\theta\\end{cases}(\\theta \\text{ is the parameter}),$$ and the parametric equations of curve $C_2$ are $$\\begin{cases} x=-3+t \\\\ y= \\frac {3+3t}{4}\\end{cases}(t \\text{ is... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nTriangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$.... | Polaris-Dataset-53K | math | {
"ground_truth": "5",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\\cdot CD =$ The solution will be evaluated... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n\nOne day, a conversation took place in a company:\n\n- \"We need to call Misha immediately!\" exclaimed Vanya.\n\nHowever, nobody remembered Misha's phone number.\n\n- \"I remember the last three ... | Polaris-Dataset-53K | math | {
"ground_truth": "7111765",
"style": "math"
} | {
"achievement_prior": 0,
"description": "\nOne day, a conversation took place in a company:\n\n- \"We need to call Misha immediately!\" exclaimed Vanya.\n\nHowever, nobody remembered Misha's phone number.\n\n- \"I remember the last three digits of the phone number were consecutive natural numbers,\" said Nastya.\n... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFind a number that can be multiplied by 4 and then divided by 5 simply by moving its first digit to the end of the number.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "2173913043478260869565",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Find a number that can be multiplied by 4 and then divided by 5 simply by moving its first digit to the end of the number. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "27317",
"problem": "Solve the following math problem. Make sure to ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( P(X) \\) be a monic polynomial of degree 2017 such that \\( P(1) = 1 \\), \\( P(2) = 2 \\), ..., \\( P(2017) = 2017 \\). What is the value of \\( P(2018) \\)?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "2017!+2018",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( P(X) \\) be a monic polynomial of degree 2017 such that \\( P(1) = 1 \\), \\( P(2) = 2 \\), ..., \\( P(2017) = 2017 \\). What is the value of \\( P(2018) \\)? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "17518",
"problem": "Sol... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nFor any positive integer \\( q_{0} \\), consider the sequence \\( q_{i} = \\left( q_{i-1} - 1 \\right)^{3} + 3 \\) for \\( i = 1, 2, \\cdots, n \\). If each \\( q_{i} \\) \\((i = 1, 2, \\cdots, n) ... | Polaris-Dataset-53K | math | {
"ground_truth": "2",
"style": "math"
} | {
"achievement_prior": 0,
"description": "For any positive integer \\( q_{0} \\), consider the sequence \\( q_{i} = \\left( q_{i-1} - 1 \\right)^{3} + 3 \\) for \\( i = 1, 2, \\cdots, n \\). If each \\( q_{i} \\) \\((i = 1, 2, \\cdots, n) \\) is a power of a prime, determine the maximum possible value of \\( n \\).... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n3 [ Completing the square. Sums of squares ]\n\n Find all real solutions of the equation with 4 unknowns: $x^{2}+y^{2}+z^{2}+t^{2}=x(y+z+t)$.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "(0, 0, 0, 0)",
"style": "math"
} | {
"achievement_prior": 0,
"description": "3 [ Completing the square. Sums of squares ]\n\n Find all real solutions of the equation with 4 unknowns: $x^{2}+y^{2}+z^{2}+t^{2}=x(y+z+t)$. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "50417",
"problem": "Solve the following math pro... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( t \\) be a positive number greater than zero.\n\nQuadrilateral \\(ABCD\\) has vertices \\(A(0,3), B(0,k), C(t, 10)\\), and \\(D(t, 0)\\), where \\(k>3\\) and \\(t>0\\). The area of quadrila... | Polaris-Dataset-53K | math | {
"ground_truth": "13",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( t \\) be a positive number greater than zero.\n\nQuadrilateral \\(ABCD\\) has vertices \\(A(0,3), B(0,k), C(t, 10)\\), and \\(D(t, 0)\\), where \\(k>3\\) and \\(t>0\\). The area of quadrilateral \\(ABCD\\) is 50 square units. What is the value of \\(k\\)? The soluti... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nDetermine the smallest integer $k \\geq 1$ such that the value of the expression $\\sin(kx) \\cdot \\sin^k(x) + \\cos(kx) \\cdot \\cos^k(x) - \\cos^k(2x)$ does not depend on $x$.",
"role": "use... | Polaris-Dataset-53K | math | {
"ground_truth": "3",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Determine the smallest integer $k \\geq 1$ such that the value of the expression $\\sin(kx) \\cdot \\sin^k(x) + \\cos(kx) \\cdot \\cos^k(x) - \\cos^k(2x)$ does not depend on $x$. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "20443",
"pr... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whos... | Polaris-Dataset-53K | math | {
"ground_truth": "7",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true,... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n**p1.** Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$ . Given that $\\angle ABC$ is a right angle, determine the length of $AC$ .**p2.** Suppose $m$ and $n$ are integers suc... | Polaris-Dataset-53K | math | {
"ground_truth": "31",
"style": "math"
} | {
"achievement_prior": 0,
"description": "**p1.** Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$ . Given that $\\angle ABC$ is a right angle, determine the length of $AC$ .**p2.** Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$ . Find the largest possible value of $m-n$ .**p3.** S... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThe Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?... | Polaris-Dataset-53K | math | {
"ground_truth": "14400",
"style": "math"
} | {
"achievement_prior": 0,
"description": "The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs? The solution will be evaluated in a math environment.",
"elo": "-",
"ind... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $\\{ z_n \\}_{n \\ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$ , $|z_k z_{k+1}|=2^k$ . Denote $f_n... | Polaris-Dataset-53K | math | {
"ground_truth": "2^{1011}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let $\\{ z_n \\}_{n \\ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$ , $|z_k z_{k+1}|=2^k$ . Denote $f_n=|z_1+z_2+\\cdots+z_n|,$ for $n=1,2,\\cdots$ (1) Find the minimum of $f_{2... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet \\( \\triangle ABC \\) be a triangle with \\(BA > BC\\). Let \\( M \\) be the midpoint of \\( AC \\), and the internal bisector of angle \\( \\widehat{B} \\) intersects \\( AC \\) at \\( I \\).... | Polaris-Dataset-53K | math | {
"ground_truth": "90",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let \\( \\triangle ABC \\) be a triangle with \\(BA > BC\\). Let \\( M \\) be the midpoint of \\( AC \\), and the internal bisector of angle \\( \\widehat{B} \\) intersects \\( AC \\) at \\( I \\). The line parallel to \\( BC \\) passing through \\( I \\) intersects \\( AM ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn the diagram, $PQ$ and $RS$ are diameters of a circle with radius 4. If $PQ$ and $RS$ are perpendicular, what is the area of the shaded region?\n\n[asy]\nsize(120);\nimport graph;\nfill((-1,-1)-... | Polaris-Dataset-53K | math | {
"ground_truth": "16+8\\pi",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In the diagram, $PQ$ and $RS$ are diameters of a circle with radius 4. If $PQ$ and $RS$ are perpendicular, what is the area of the shaded region?\n\n[asy]\nsize(120);\nimport graph;\nfill((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray);\nfill(Arc((0,0),sqrt(2),45,135)--c... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nThere are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following cha... | Polaris-Dataset-53K | math | {
"ground_truth": "lawyer",
"style": "math"
} | {
"achievement_prior": 0,
"description": "There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Alice`, `Eric`, `Peter`, `A... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $k$ be an integer and $k > 1$. Define a sequence $\\{a_n\\}$ as follows:\n\n$a_0 = 0$,\n\n$a_1 = 1$, and\n\n$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.\n\nDetermine, with proof, all possible ... | Polaris-Dataset-53K | math | {
"ground_truth": "2",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let $k$ be an integer and $k > 1$. Define a sequence $\\{a_n\\}$ as follows:\n\n$a_0 = 0$,\n\n$a_1 = 1$, and\n\n$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.\n\nDetermine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \\not= m)$ and positi... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$. A ray $OX$ intersects this circle at two points $C$ and $D$ such that $OC = CD = 1$. If $M$ is the point of i... | Polaris-Dataset-53K | math | {
"ground_truth": "\\frac{4}{3}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$. A ray $OX$ intersects this circle at two points $C$ and $D$ such that $OC = CD = 1$. If $M$ is the point of intersection of ray $OX$ and segment $AB$, what is the length of segment $OM$?... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "0.032",
"style": "math"
} | {
"achievement_prior": 0,
"description": "A shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "6716",
"problem": "Solve the following math problem... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn how many ways can we choose two different integers between -100 and 100 inclusive, so that their sum is greater than their product?",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "10199",
"style": "math"
} | {
"achievement_prior": 0,
"description": "In how many ways can we choose two different integers between -100 and 100 inclusive, so that their sum is greater than their product? The solution will be evaluated in a math environment.",
"elo": "-",
"index": "3642",
"problem": "Solve the following math problem. Ma... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nLet $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether $$ \\exp \\left( \\sum_{n=1}^{\\infty} \\frac{ B(n)}{n(n+1)} \\right) $$ is a ratio... | Polaris-Dataset-53K | math | {
"ground_truth": "4",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether $$ \\exp \\left( \\sum_{n=1}^{\\infty} \\frac{ B(n)}{n(n+1)} \\right) $$ is a rational number. The solution will be evaluated in a math environment.",
"elo": ... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven the sets\n$$\n\\begin{array}{l}\nA=\\{2,0,1,7\\}, \\\\\nB=\\{x \\mid x=a b, a, b \\in A\\} .\n\\end{array}\n$$\ndetermine the number of elements in set $B$.",
"role": "user"
}
] | Polaris-Dataset-53K | math | {
"ground_truth": "7",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given the sets\n$$\n\\begin{array}{l}\nA=\\{2,0,1,7\\}, \\\\\nB=\\{x \\mid x=a b, a, b \\in A\\} .\n\\end{array}\n$$\ndetermine the number of elements in set $B$. The solution will be evaluated in a math environment.",
"elo": "-",
"index": "21073",
"problem": "Solve t... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nAlice thinks of four positive integers $a\\leq b\\leq c\\leq d$ satisfying $\\{ab+cd,ac+bd,ad+bc\\}=\\{40,70,100\\}$ . What are all the possible tuples $(a,b,c,d)$ that Alice could be thinking... | Polaris-Dataset-53K | math | {
"ground_truth": "(1,4,6,16)",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Alice thinks of four positive integers $a\\leq b\\leq c\\leq d$ satisfying $\\{ab+cd,ac+bd,ad+bc\\}=\\{40,70,100\\}$ . What are all the possible tuples $(a,b,c,d)$ that Alice could be thinking of? The solution will be evaluated in a math environment.",
"elo": "-",
... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nJaviera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next, Javiera says a natural number $k$, where $1 \\leq k \\leq n... | Polaris-Dataset-53K | math | {
"ground_truth": "1010",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next, Javiera says a natural number $k$, where $1 \\leq k \\leq n$, and Claudio must move the token to the right or to the left by $k$ squares... |
[] | [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nGiven an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \\frac {3}{2})$$ on the ellipse ... | Polaris-Dataset-53K | math | {
"ground_truth": "- \\frac { \\sqrt {3}}{12}",
"style": "math"
} | {
"achievement_prior": 0,
"description": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \\frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpe... |
End of preview. Expand in Data Studio
Polaris 53K: conflict-free RL and GEPA splits
This dataset derives deterministic, mutually disjoint splits from
kushasareen/sdpo-datasets/polaris_5k
at source commit 51fd81230be64a06d77e7edc33d3c0d60b3186b5.
Splits
| Split | Rows | Intended use |
|---|---|---|
train (train_split.parquet) |
51,911 | RL actor training and GEPA trajectory training |
validation (val_split.parquet) |
100 | Fixed held-out GEPA prompt evaluation |
test (test.parquet) |
500 | Final model evaluation |
Validation is sampled from the cleaned source train data. Test is sampled from the cleaned source test data. Both use seed 42.
De-duplication and conflict policy
The normalized problem key applies Unicode NFKC normalization, normalizes line endings, and collapses whitespace while preserving case.
- If one normalized problem has multiple normalized ground-truth answers, every occurrence of that problem is removed globally.
- If repeated normalized problems have the same normalized answer, the first occurrence is retained.
- If a consistent problem occurs in both source train and source test, the test occurrence is preserved and the train occurrence is removed.
- The three published splits have no normalized-problem overlap.
The build removed 26 train rows belonging to 13 conflicting problem groups, 235 additional duplicate train rows, and 6 train rows overlapping source test. No duplicate or conflicting rows were found within the source test data.
See split_manifest.json for source and output SHA-256 hashes, exact audit
counts, normalization details, and split-key hashes.
Schema and provenance
The source parquet schema is preserved, including prompt, reward_model, and
extra_info. Only extra_info.split is updated to identify each output split.
The source dataset is derived from
POLARIS-Project/Polaris-Dataset-53K.
- Downloads last month
- 66