data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the maximum value of\n\\[\\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}.\\]The answer is in the form \frac{m}{n}, where gcd(m, n)... | math | {
"ground_truth": "5121",
"style": "rule"
} | {
"answer": "5121",
"difficulty": "4/8",
"index": 0,
"problem": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the maximum value of\n\\[\\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}.\\]The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "145",
"style": "rule"
} | {
"answer": "145",
"difficulty": "6/8",
"index": 1,
"problem": "In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the ancient Chinese mathematical masterpiece \"Nine Chapters on the Mathematical Art,\" there is a problem called \"Division of Grains in a Granary\": When a granary is opened to re... | math | {
"ground_truth": "169.1",
"style": "rule"
} | {
"answer": "169.1",
"difficulty": "6/8",
"index": 2,
"problem": "In the ancient Chinese mathematical masterpiece \"Nine Chapters on the Mathematical Art,\" there is a problem called \"Division of Grains in a Granary\": When a granary is opened to receive grain, a person brings 1534 stones of rice. Upon inspect... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "How many ordered triples (x,y,z) of integers satisfy the system of equations below?\n\\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \\end{array} Let's t... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "0/8",
"index": 3,
"problem": "How many ordered triples (x,y,z) of integers satisfy the system of equations below?\n\\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \\end{array}"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The ... | math | {
"ground_truth": "84",
"style": "rule"
} | {
"answer": "84",
"difficulty": "0/8",
"index": 4,
"problem": "A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections t... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the smallest value of $k$ for which it is possible to mark $k$ cells on a $9 \\times 9$ board such that any placement of a three-cell corner touches at least two marked cells? ... | math | {
"ground_truth": "56",
"style": "rule"
} | {
"answer": "56",
"difficulty": "1/8",
"index": 5,
"problem": "What is the smallest value of $k$ for which it is possible to mark $k$ cells on a $9 \\times 9$ board such that any placement of a three-cell corner touches at least two marked cells?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Two consecutive sides of a quadrilateral measure $10 \\mathrm{~cm}$ and $15 \\mathrm{~cm}$. If each diagonal divides the quadrilateral into two regions of equal area, calculate its per... | math | {
"ground_truth": "50\\,",
"style": "rule"
} | {
"answer": "50\\,",
"difficulty": "1/8",
"index": 6,
"problem": "Two consecutive sides of a quadrilateral measure $10 \\mathrm{~cm}$ and $15 \\mathrm{~cm}$. If each diagonal divides the quadrilateral into two regions of equal area, calculate its perimeter."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that the first term of a geometric sequence $\\{{a_n}\\}$ is $\\frac{3}{2}$ and the sum of its first $n$ terms is $S_n$ $(n \\in \\mathbb{N^*})$, and $-2S_2, S_3, 4S_4$ form an a... | math | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"answer": "\\frac{3}{4}",
"difficulty": "7/8",
"index": 7,
"problem": "Given that the first term of a geometric sequence $\\{{a_n}\\}$ is $\\frac{3}{2}$ and the sum of its first $n$ terms is $S_n$ $(n \\in \\mathbb{N^*})$, and $-2S_2, S_3, 4S_4$ form an arithmetic sequence.\n\n(I) Find the general term formul... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If set $A=\\{-4, 2a-1, a^2\\}$, $B=\\{a-5, 1-a, 9\\}$, and $A \\cap B = \\{9\\}$, then the value of $a$ is. Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"answer": "-3",
"difficulty": "5/8",
"index": 8,
"problem": "If set $A=\\{-4, 2a-1, a^2\\}$, $B=\\{a-5, 1-a, 9\\}$, and $A \\cap B = \\{9\\}$, then the value of $a$ is."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $a,b$ be positive reals such that $\\frac{1}{a}+\\frac{1}{b}\\leq2\\sqrt2$ and $(a-b)^2=4(ab)^3$ . Find $\\log_a b$ . Let's think step by step and output the final answer wit... | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"answer": "-1",
"difficulty": "6/8",
"index": 9,
"problem": "Let $a,b$ be positive reals such that $\\frac{1}{a}+\\frac{1}{b}\\leq2\\sqrt2$ and $(a-b)^2=4(ab)^3$ . Find $\\log_a b$ ."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "0/8",
"index": 10,
"problem": "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: `Er... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "1. Let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding the real number $x$. Find all positive integers $M$ such that the sequence defined by\n$$\n\\begin{array}{l}\na_{0... | math | {
"ground_truth": "M \\geq 2",
"style": "rule"
} | {
"answer": "M \\geq 2",
"difficulty": "5/8",
"index": 11,
"problem": "1. Let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding the real number $x$. Find all positive integers $M$ such that the sequence defined by\n$$\n\\begin{array}{l}\na_{0}=\\frac{2 M+1}{2}, \\\\\na_{k+1}=a_{k}\\left\\lfloor a_{... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(\\Omega_{1}\\) be a circle tangent internally to another circle \\(\\Omega_{2}\\) at point \\(A\\). Let \\(P\\) be a point on \\(\\Omega_{2}\\). The first tangent to \\(\\Omega_... | math | {
"ground_truth": "\\widehat{QAR} = 2 \\widehat{XAY}",
"style": "rule"
} | {
"answer": "\\widehat{QAR} = 2 \\widehat{XAY}",
"difficulty": "0/8",
"index": 12,
"problem": "Let \\(\\Omega_{1}\\) be a circle tangent internally to another circle \\(\\Omega_{2}\\) at point \\(A\\). Let \\(P\\) be a point on \\(\\Omega_{2}\\). The first tangent to \\(\\Omega_{1}\\) passing through \\(P\\) in... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. Let's think step by step and output the final answer ... | math | {
"ground_truth": "581",
"style": "rule"
} | {
"answer": "581",
"difficulty": "1/8",
"index": 13,
"problem": "Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "fish",
"style": "rule"
} | {
"answer": "fish",
"difficulty": "2/8",
"index": 14,
"problem": "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: ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each st... | math | {
"ground_truth": "2500",
"style": "rule"
} | {
"answer": "2500",
"difficulty": "0/8",
"index": 15,
"problem": "Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\\frac{n}{1... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be a subset of the natural numbers such that $0 \\in S$, and for all $n \\in \\mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number... | math | {
"ground_truth": "47",
"style": "rule"
} | {
"answer": "47",
"difficulty": "1/8",
"index": 16,
"problem": "Let $S$ be a subset of the natural numbers such that $0 \\in S$, and for all $n \\in \\mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\\{0,1,\\ldots, 2019\\}$... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Shown below are rows 1, 2, and 3 of Pascal's triangle.\n\n\\[\n\\begin{array}{ccccccc}\n& & 1 & & 1 & & \\\\\n& 1 & & 2 & & 1 & \\\\\n1 & & 3 & & 3 & & 1\n\\end{array}\n\\]Let $(a_i),$... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\frac{1}{2}",
"difficulty": "7/8",
"index": 17,
"problem": "Shown below are rows 1, 2, and 3 of Pascal's triangle.\n\n\\[\n\\begin{array}{ccccccc}\n& & 1 & & 1 & & \\\\\n& 1 & & 2 & & 1 & \\\\\n1 & & 3 & & 3 & & 1\n\\end{array}\n\\]Let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right,... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Two circles of radius \\( r \\) are externally tangent to each other and internally tangent to the ellipse \\( x^2 + 4y^2 = 8 \\). Find \\( r \\). Let's think step by step and output t... | math | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{6}}{2}",
"difficulty": "7/8",
"index": 18,
"problem": "Two circles of radius \\( r \\) are externally tangent to each other and internally tangent to the ellipse \\( x^2 + 4y^2 = 8 \\). Find \\( r \\)."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If \\( a_{1}, a_{2}, \\cdots, a_{n} \\) are any real or complex numbers, they satisfy the equation\n$$\nx^{n}-n a_{1} x^{n-1}+c_{n}^{2} a_{2}^{2} x^{n-2}+\\cdots+(-1)^{i} c_{n}^{i} a_{... | math | {
"ground_truth": "a_{1}=a_{2}=\\cdots=a_{n}",
"style": "rule"
} | {
"answer": "a_{1}=a_{2}=\\cdots=a_{n}",
"difficulty": "2/8",
"index": 19,
"problem": "If \\( a_{1}, a_{2}, \\cdots, a_{n} \\) are any real or complex numbers, they satisfy the equation\n$$\nx^{n}-n a_{1} x^{n-1}+c_{n}^{2} a_{2}^{2} x^{n-2}+\\cdots+(-1)^{i} c_{n}^{i} a_{i}^{i} x^{n-i}+\\cdots+(-1)^{n} a_{n}^{n}... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice ro... | math | {
"ground_truth": "\\frac{21}{64}",
"style": "rule"
} | {
"answer": "\\frac{21}{64}",
"difficulty": "7/8",
"index": 20,
"problem": "Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice roll. Each time the dice is rolled, the chess piece is... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the greatest integer less than or equal to \\[\\frac{5^{80} + 3^{80}}{5^{75} + 3^{75}}?\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": ... | math | {
"ground_truth": "3124",
"style": "rule"
} | {
"answer": "3124",
"difficulty": "5/8",
"index": 21,
"problem": "What is the greatest integer less than or equal to \\[\\frac{5^{80} + 3^{80}}{5^{75} + 3^{75}}?\\]"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a right triangle with hypotenuse $AC$ . A square is inscribed in the triangle such that points $D,E$ are on $AC$ , $F$ is on $BC$ , and $G$ is on $A... | math | {
"ground_truth": "\\frac{500}{841}",
"style": "rule"
} | {
"answer": "\\frac{500}{841}",
"difficulty": "1/8",
"index": 22,
"problem": "Let $\\triangle ABC$ be a right triangle with hypotenuse $AC$ . A square is inscribed in the triangle such that points $D,E$ are on $AC$ , $F$ is on $BC$ , and $G$ is on $AB$ . Given that $AG=2$ and $CF=5$ , what is the... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the quadrilateral \\( ABCD \\), \\( AB = a \\), and the ratio of the measures of the four angles \\( A, B, C, D \\) is \\( 3:7:4:10 \\). Additionally, \\(\\angle CDB = 30^{\\circ}\\... | math | {
"ground_truth": "\\frac{\\sqrt{2}}{3}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{2}}{3}",
"difficulty": "4/8",
"index": 23,
"problem": "In the quadrilateral \\( ABCD \\), \\( AB = a \\), and the ratio of the measures of the four angles \\( A, B, C, D \\) is \\( 3:7:4:10 \\). Additionally, \\(\\angle CDB = 30^{\\circ}\\). Find the length of \\( BC \\)."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The sequence $b_1, b_2, b_3, \\dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \\ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$. Let's think step by step... | math | {
"ground_truth": "625",
"style": "rule"
} | {
"answer": "625",
"difficulty": "7/8",
"index": 24,
"problem": "The sequence $b_1, b_2, b_3, \\dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \\ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "8.5. What is the maximum number of members that can be in a sequence of non-zero integers, for which the sum of any seven consecutive numbers is positive, and the sum of any eleven con... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"answer": "16",
"difficulty": "3/8",
"index": 25,
"problem": "8.5. What is the maximum number of members that can be in a sequence of non-zero integers, for which the sum of any seven consecutive numbers is positive, and the sum of any eleven consecutive numbers is negative?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "8. Given that $\\triangle A B C$ is an inscribed triangle in the ellipse $\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1$, and $A B$ passes through the point $P(1,0)$, then the maximum value of t... | math | {
"ground_truth": "\\dfrac{16\\sqrt{2}}{3}",
"style": "rule"
} | {
"answer": "\\dfrac{16\\sqrt{2}}{3}",
"difficulty": "3/8",
"index": 26,
"problem": "8. Given that $\\triangle A B C$ is an inscribed triangle in the ellipse $\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1$, and $A B$ passes through the point $P(1,0)$, then the maximum value of the area of $\\triangle A B C$ is $\\qquad$ ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a $4 \\times 4$ chessboard composed of 16 small squares, 8 of the squares are painted black such that each row and each column contains exactly 2 black squares. How many different w... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"answer": "90",
"difficulty": "2/8",
"index": 27,
"problem": "In a $4 \\times 4$ chessboard composed of 16 small squares, 8 of the squares are painted black such that each row and each column contains exactly 2 black squares. How many different ways are there to paint the squares?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The mean of the set of numbers $\\{106, 102, 95, 103, 100, y, x\\}$ is 104. What is the median of this set of seven numbers? Let's think step by step and output the final answer within... | math | {
"ground_truth": "103",
"style": "rule"
} | {
"answer": "103",
"difficulty": "1/8",
"index": 28,
"problem": "The mean of the set of numbers $\\{106, 102, 95, 103, 100, y, x\\}$ is 104. What is the median of this set of seven numbers?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "1. Observing the temperatures recorded in Cesenatico in the last months of December and January, Stefano noticed a strange coincidence: on all the days of this period (excluding the fi... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"answer": "-3",
"difficulty": "6/8",
"index": 29,
"problem": "1. Observing the temperatures recorded in Cesenatico in the last months of December and January, Stefano noticed a strange coincidence: on all the days of this period (excluding the first and last day), the minimum temperature was the sum of the mi... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these s... | math | {
"ground_truth": "\\frac{3 (\\sqrt{3} - 1)}{2}",
"style": "rule"
} | {
"answer": "\\frac{3 (\\sqrt{3} - 1)}{2}",
"difficulty": "6/8",
"index": 30,
"problem": "Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these semicircles. Find the radius of this ce... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .\n\n\n*2021 CCA Math Bonanza Lightning Round #3.4* Let's... | math | {
"ground_truth": "164",
"style": "rule"
} | {
"answer": "164",
"difficulty": "7/8",
"index": 31,
"problem": "Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .\n\n\n*2021 CCA Math Bonanza Lightning Round #3.4*"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Problem 3 (MAR CP 1992) :\nFrom the digits $1,2,...,9$ , we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order... | math | {
"ground_truth": "358926471",
"style": "rule"
} | {
"answer": "358926471",
"difficulty": "3/8",
"index": 32,
"problem": "Problem 3 (MAR CP 1992) :\nFrom the digits $1,2,...,9$ , we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$ , $123456798$ , ..., $987654... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "How many $6$ -tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true? Let's think step by step and output ... | math | {
"ground_truth": "364",
"style": "rule"
} | {
"answer": "364",
"difficulty": "6/8",
"index": 33,
"problem": "How many $6$ -tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "XLVI OM - III - Problem 2\n\nThe diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal area... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "0/8",
"index": 34,
"problem": "XLVI OM - III - Problem 2\n\nThe diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal areas?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "2. In the convex quadrilateral $ABCD$, $AB=CD$, $AC$ is a diagonal, $\\angle DAC>\\angle BCA$, and $\\angle DAC$ and $\\angle BCA$ are supplementary, $\\angle BAC>\\angle ACD$, and $\\... | math | {
"ground_truth": "45^\\circ",
"style": "rule"
} | {
"answer": "45^\\circ",
"difficulty": "4/8",
"index": 35,
"problem": "2. In the convex quadrilateral $ABCD$, $AB=CD$, $AC$ is a diagonal, $\\angle DAC>\\angle BCA$, and $\\angle DAC$ and $\\angle BCA$ are supplementary, $\\angle BAC>\\angle ACD$, and $\\angle BAC$ and $\\angle ACD$ are complementary. Then $\\a... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Problem 4. For an infinite sequence of numbers $x_{1}, x_{2}, x_{3}, \\ldots$, for all natural $n \\geq 4$, the relation $x_{n}=x_{n-1} \\cdot x_{n-3}$ holds. It is known that $x_{1}=1... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "6/8",
"index": 36,
"problem": "Problem 4. For an infinite sequence of numbers $x_{1}, x_{2}, x_{3}, \\ldots$, for all natural $n \\geq 4$, the relation $x_{n}=x_{n-1} \\cdot x_{n-3}$ holds. It is known that $x_{1}=1, x_{2}=1, x_{3}=-1$. Find $x_{2022}$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar... | math | {
"ground_truth": "111",
"style": "rule"
} | {
"answer": "111",
"difficulty": "3/8",
"index": 37,
"problem": "The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar cheese, and white bread are each offered at the deli. If Al ne... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$ . Find the smallest $n$ for which \r\n\r\n\\[ P(P(P(n))) > 10^{12} \\] Let's th... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "3/8",
"index": 38,
"problem": "For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$ . Find the smallest $n$ for which \r\n\r\n\\[ P(P(P(n))) > 10^{12} \\]"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$? Let's th... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"answer": "7",
"difficulty": "0/8",
"index": 39,
"problem": "Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The triangles $ABC$ and $BDE$ are congruent isosceles triangles with bases $AB$ and $BD$ lying on the same line, and $C$ and $E$ are on the same side of this line. Let $O_{1}$ and $O_{... | math | {
"ground_truth": "parallelogram",
"style": "rule"
} | {
"answer": "parallelogram",
"difficulty": "0/8",
"index": 40,
"problem": "The triangles $ABC$ and $BDE$ are congruent isosceles triangles with bases $AB$ and $BD$ lying on the same line, and $C$ and $E$ are on the same side of this line. Let $O_{1}$ and $O_{2}$ be the centers of the circumscribed circles of tr... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "To what power must the root \\( x_{0} \\) of the equation \\( x^{11} + x^{7} + x^{3} = 1 \\) be raised to obtain the number \\( x_{0}^{4} + x_{0}^{3} - 1 \\)? Let's think step by step ... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"answer": "15",
"difficulty": "4/8",
"index": 41,
"problem": "To what power must the root \\( x_{0} \\) of the equation \\( x^{11} + x^{7} + x^{3} = 1 \\) be raised to obtain the number \\( x_{0}^{4} + x_{0}^{3} - 1 \\)?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A local hunter named Alex is interested in participating in wildlife conservation efforts. To help with the conservation of deer in the nearby forest, Alex decides to monitor the deer ... | math | {
"ground_truth": "128",
"style": "rule"
} | {
"answer": "128",
"difficulty": "1/8",
"index": 42,
"problem": "A local hunter named Alex is interested in participating in wildlife conservation efforts. To help with the conservation of deer in the nearby forest, Alex decides to monitor the deer population over several months. In January, Alex counts 120 dee... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of $[\\sqrt{1}] + [\\sqrt{2}] + [\\sqrt{3}] + \\cdots + [\\sqrt{1989 \\cdot 1990}] + [-\\sqrt{1}] + [-\\sqrt{2}] + [-\\sqrt{3}] + \\cdots + [-\\sqrt{1989 \\cdot 1990}... | math | {
"ground_truth": "-3956121",
"style": "rule"
} | {
"answer": "-3956121",
"difficulty": "7/8",
"index": 43,
"problem": "What is the value of $[\\sqrt{1}] + [\\sqrt{2}] + [\\sqrt{3}] + \\cdots + [\\sqrt{1989 \\cdot 1990}] + [-\\sqrt{1}] + [-\\sqrt{2}] + [-\\sqrt{3}] + \\cdots + [-\\sqrt{1989 \\cdot 1990}]$?\n(The 1st \"Hope Cup\" Mathematics Contest, 1990)"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 5 people standing in a line numbered 1 through 5 in a left to right order.\nEach person has the following attributes: Job, Food, Beverage, Music-Genre, Sport, Transport.\nTh... | math | {
"ground_truth": "tennis",
"style": "rule"
} | {
"answer": "tennis",
"difficulty": "0/8",
"index": 44,
"problem": "There are 5 people standing in a line numbered 1 through 5 in a left to right order.\nEach person has the following attributes: Job, Food, Beverage, Music-Genre, Sport, Transport.\nThe attributes have the following possible values:\n- Job: bar... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "X OM - I - Task 8\n\nThe sides of a triangle are the medians of another triangle. Calculate the ratio of the areas of both triangles. Can a triangle be constructed from the medians of ... | math | {
"ground_truth": "\\dfrac{3}{4}",
"style": "rule"
} | {
"answer": "\\dfrac{3}{4}",
"difficulty": "6/8",
"index": 45,
"problem": "X OM - I - Task 8\n\nThe sides of a triangle are the medians of another triangle. Calculate the ratio of the areas of both triangles. Can a triangle be constructed from the medians of any triangle?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A circle lies inside the parabola defined by the equation \\( y = 4x^2 \\) and is tangent to the parabola at two points. Determine how much higher the center of the circle is compared ... | math | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"answer": "\\frac{1}{8}",
"difficulty": "6/8",
"index": 46,
"problem": "A circle lies inside the parabola defined by the equation \\( y = 4x^2 \\) and is tangent to the parabola at two points. Determine how much higher the center of the circle is compared to the points of tangency."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "soccer",
"style": "rule"
} | {
"answer": "soccer",
"difficulty": "0/8",
"index": 47,
"problem": "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... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Olave sold 108 apples at a constant rate over 6 hours. If she continues to sell apples at the same rate, how many apples will she sell in the next 1 hour and 30 minutes?\n(A) 27\n(B) 3... | math | {
"ground_truth": "27",
"style": "rule"
} | {
"answer": "27",
"difficulty": "7/8",
"index": 48,
"problem": "Olave sold 108 apples at a constant rate over 6 hours. If she continues to sell apples at the same rate, how many apples will she sell in the next 1 hour and 30 minutes?\n(A) 27\n(B) 33\n(C) 45\n(D) 36\n(E) 21"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ .\n$\\textb... | math | {
"ground_truth": "\\textbf{(D)}\\28",
"style": "rule"
} | {
"answer": "\\textbf{(D)}\\28",
"difficulty": "0/8",
"index": 49,
"problem": "For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ .\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 20 \\qquad \\... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, mo... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"answer": "21",
"difficulty": "4/8",
"index": 50,
"problem": "Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines with... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that \\( a, b, c \\) are real numbers such that if \\( p_1, p_2 \\) are the roots of \\( ax^2 + bx + c = 0 \\) and \\( q_1, q_2 \\) are the roots of \\( cx^2 + bx + a = 0 \\), th... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "4/8",
"index": 51,
"problem": "Given that \\( a, b, c \\) are real numbers such that if \\( p_1, p_2 \\) are the roots of \\( ax^2 + bx + c = 0 \\) and \\( q_1, q_2 \\) are the roots of \\( cx^2 + bx + a = 0 \\), then \\( p_1, q_1, p_2, q_2 \\) form an arithmetic progression with... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "$M$ is a subset of $\\{1,2,3, \\ldots, 15\\}$ such that the product of any three distinct elements of $M$ is not a square. Determine the maximum number of elements in $M$. Let's think ... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"answer": "10",
"difficulty": "1/8",
"index": 52,
"problem": "$M$ is a subset of $\\{1,2,3, \\ldots, 15\\}$ such that the product of any three distinct elements of $M$ is not a square. Determine the maximum number of elements in $M$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the universe of Pi Zone, points are labeled with $2 \\times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplyin... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"answer": "17",
"difficulty": "2/8",
"index": 53,
"problem": "In the universe of Pi Zone, points are labeled with $2 \\times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If $3 \\in \\{a, a^2 - 2a\\}$, then the value of the real number $a$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"answer": "-1",
"difficulty": "0/8",
"index": 54,
"problem": "If $3 \\in \\{a, a^2 - 2a\\}$, then the value of the real number $a$ is ______."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Along the side of an equilateral triangle, a circle with a radius equal to the height of the triangle rolls. Prove that the angular measure of the arc intercepted on the circle by the ... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"answer": "60",
"difficulty": "5/8",
"index": 55,
"problem": "Along the side of an equilateral triangle, a circle with a radius equal to the height of the triangle rolls. Prove that the angular measure of the arc intercepted on the circle by the sides of the triangle is always $60^{0}$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "4. (8 points) A cruise ship travels from upstream location $A$ to downstream location $B$ in 1 hour. On the return trip, the ship doubles its speed and still takes 1 hour. Therefore, i... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"answer": "36",
"difficulty": "0/8",
"index": 56,
"problem": "4. (8 points) A cruise ship travels from upstream location $A$ to downstream location $B$ in 1 hour. On the return trip, the ship doubles its speed and still takes 1 hour. Therefore, if the cruise ship also doubles its speed when departing from $A$... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a $12\\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\\times 4$ and $4\\times 3$ rectangle bou... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"answer": "12",
"difficulty": "4/8",
"index": 57,
"problem": "In a $12\\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\\times 4$ and $4\\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum numbe... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2? Let's think step by step and output the final answer within \\b... | math | {
"ground_truth": "800",
"style": "rule"
} | {
"answer": "800",
"difficulty": "7/8",
"index": 58,
"problem": "How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest solution to the equation \\[\\frac{1}{x-3} + \\frac{1}{x-5} = \\frac{5}{x-4}.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": ... | math | {
"ground_truth": "4 - \\frac{\\sqrt{15}}{3}",
"style": "rule"
} | {
"answer": "4 - \\frac{\\sqrt{15}}{3}",
"difficulty": "7/8",
"index": 59,
"problem": "Find the smallest solution to the equation \\[\\frac{1}{x-3} + \\frac{1}{x-5} = \\frac{5}{x-4}.\\]"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 3 houses, numbered 1 to 3 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 ... | math | {
"ground_truth": "ranch",
"style": "rule"
} | {
"answer": "ranch",
"difficulty": "2/8",
"index": 60,
"problem": "There are 3 houses, numbered 1 to 3 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:... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "6. Solve the equation $\\sqrt{3 x^{2}+5 x+7}-\\sqrt{3 x^{2}+5 x+2}=1$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-2",
"style": "rule"
} | {
"answer": "-2",
"difficulty": "0/8",
"index": 61,
"problem": "6. Solve the equation $\\sqrt{3 x^{2}+5 x+7}-\\sqrt{3 x^{2}+5 x+2}=1$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Alicia had two containers. The first was $\\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which p... | math | {
"ground_truth": "\\textbf{(D)}\\frac{9}{10}",
"style": "rule"
} | {
"answer": "\\textbf{(D)}\\frac{9}{10}",
"difficulty": "0/8",
"index": 62,
"problem": "Alicia had two containers. The first was $\\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\tfrac{3... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A high school math preparation group consists of six science teachers and two liberal arts teachers. During a three-day period of smog-related class suspensions, they need to arrange t... | math | {
"ground_truth": "540",
"style": "rule"
} | {
"answer": "540",
"difficulty": "7/8",
"index": 63,
"problem": "A high school math preparation group consists of six science teachers and two liberal arts teachers. During a three-day period of smog-related class suspensions, they need to arrange teachers to be on duty for question-answering sessions. The requ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "6. (10 points) Doudou's family has 4 people. This year, Doudou's brother is 3 years older than Doudou, and Doudou's mother is 2 years younger than Doudou's father. 5 years ago, the tot... | math | {
"ground_truth": "33",
"style": "rule"
} | {
"answer": "33",
"difficulty": "1/8",
"index": 64,
"problem": "6. (10 points) Doudou's family has 4 people. This year, Doudou's brother is 3 years older than Doudou, and Doudou's mother is 2 years younger than Doudou's father. 5 years ago, the total age of the family was 59 years, and 5 years later, the total ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "1. Provide an example of a natural number that is divisible by 2019, and the sum of its digits is also divisible by 2019. Don't forget to show that your example meets the condition. Le... | math | {
"ground_truth": "20192019\\ldots2019",
"style": "rule"
} | {
"answer": "20192019\\ldots2019",
"difficulty": "0/8",
"index": 65,
"problem": "1. Provide an example of a natural number that is divisible by 2019, and the sum of its digits is also divisible by 2019. Don't forget to show that your example meets the condition."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a sequence of numbers \\(1, 4, 7, 10, \\cdots, 697, 700\\), the first number is 1, and each subsequent number is equal to the previous number plus 3, until 700. Calculate the number... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"answer": "60",
"difficulty": "6/8",
"index": 66,
"problem": "In a sequence of numbers \\(1, 4, 7, 10, \\cdots, 697, 700\\), the first number is 1, and each subsequent number is equal to the previous number plus 3, until 700. Calculate the number of trailing zeroes in the product of all these numbers. For exa... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The brother says to his sister: \"When Kolya was as old as we both are together now, you were as old as I am now. And when Kolya was as old as you are now, you were then...\" How old w... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "7/8",
"index": 67,
"problem": "The brother says to his sister: \"When Kolya was as old as we both are together now, you were as old as I am now. And when Kolya was as old as you are now, you were then...\" How old was the sister then?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A certain commodity has a cost price of 200 yuan and a marked price of 400 yuan. What is the maximum discount that can be offered to ensure that the profit margin is not less than 40%?... | math | {
"ground_truth": "30\\%",
"style": "rule"
} | {
"answer": "30\\%",
"difficulty": "2/8",
"index": 68,
"problem": "A certain commodity has a cost price of 200 yuan and a marked price of 400 yuan. What is the maximum discount that can be offered to ensure that the profit margin is not less than 40%?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal se... | math | {
"ground_truth": "11",
"style": "rule"
} | {
"answer": "11",
"difficulty": "4/8",
"index": 69,
"problem": "9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "mystery",
"style": "rule"
} | {
"answer": "mystery",
"difficulty": "0/8",
"index": 70,
"problem": "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 nam... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "\\section*{Problem 2 - 261032}\n\nIn a domino game with the numbers \\(0,1, \\ldots, 6\\), each tile is divided into two halves, each half bearing one of the numbers.\n\nIn a domino ga... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "0/8",
"index": 71,
"problem": "\\section*{Problem 2 - 261032}\n\nIn a domino game with the numbers \\(0,1, \\ldots, 6\\), each tile is divided into two halves, each half bearing one of the numbers.\n\nIn a domino game, all combinations of two of the numbers \\(0,1, \\ldots, 6\\) ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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\\), \... | math | {
"ground_truth": "540",
"style": "rule"
} | {
"answer": "540",
"difficulty": "1/8",
"index": 72,
"problem": "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}\\) ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "10. On the side $AC$ of $\\triangle ABC$, take a point $D$ such that $\\angle BDC = \\angle ABC$. If $BC = 1$, what is the minimum possible distance between the circumcenter of $\\tria... | math | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule"
} | {
"answer": "\\dfrac{1}{2}",
"difficulty": "3/8",
"index": 73,
"problem": "10. On the side $AC$ of $\\triangle ABC$, take a point $D$ such that $\\angle BDC = \\angle ABC$. If $BC = 1$, what is the minimum possible distance between the circumcenter of $\\triangle ABC$ and the circumcenter of $\\triangle ABD$?"
... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but ... | math | {
"ground_truth": "\\textbf{(D)}\\;4(\\pi-\\sqrt{3})",
"style": "rule"
} | {
"answer": "\\textbf{(D)}\\;4(\\pi-\\sqrt{3})",
"difficulty": "1/8",
"index": 74,
"problem": "A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(\\sqrt{3}\\cos10°-\\sin10°)\\cos(B+35°)=\\sin80°$.\n$(1)$... | math | {
"ground_truth": "\\sqrt{6}+\\sqrt{2}",
"style": "rule"
} | {
"answer": "\\sqrt{6}+\\sqrt{2}",
"difficulty": "4/8",
"index": 75,
"problem": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(\\sqrt{3}\\cos10°-\\sin10°)\\cos(B+35°)=\\sin80°$.\n$(1)$ Find angle $B$.\n$(2)$ If $2b\\cos \\angle BAC... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindric... | math | {
"ground_truth": "1.5",
"style": "rule"
} | {
"answer": "1.5",
"difficulty": "2/8",
"index": 76,
"problem": "An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of t... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $a, b, c, d$ are real numbers such that $$|a-b|+|c-d|=99 ; \\quad|a-c|+|b-d|=1$$ Determine all possible values of $|a-d|+|b-c|$. Let's think step by step and output the final a... | math | {
"ground_truth": "99",
"style": "rule"
} | {
"answer": "99",
"difficulty": "3/8",
"index": 77,
"problem": "Suppose $a, b, c, d$ are real numbers such that $$|a-b|+|c-d|=99 ; \\quad|a-c|+|b-d|=1$$ Determine all possible values of $|a-d|+|b-c|$."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\\equiv 3\\pmod 4. $ Let's thin... | math | {
"ground_truth": "p\\equiv3\\pmod{4}",
"style": "rule"
} | {
"answer": "p\\equiv3\\pmod{4}",
"difficulty": "3/8",
"index": 78,
"problem": "Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\\equiv 3\\pmod 4. $"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( n \\) be a positive integer. Claudio has \\( n \\) cards, each labeled with a different number from 1 to \\( n \\). He takes a subset of these cards, and multiplies together th... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"answer": "17",
"difficulty": "0/8",
"index": 79,
"problem": "Let \\( n \\) be a positive integer. Claudio has \\( n \\) cards, each labeled with a different number from 1 to \\( n \\). He takes a subset of these cards, and multiplies together the numbers on the cards. He remarks that, given any positive inte... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent uni... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"answer": "32",
"difficulty": "4/8",
"index": 80,
"problem": "An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( T = \\left\\{9^{k} \\mid k \\right. \\) be integers, \\(\\left.0 \\leqslant k \\leqslant 4000\\right\\} \\). Given that \\(9^{4000}\\) has 3817 digits and its most significant ... | math | {
"ground_truth": "184",
"style": "rule"
} | {
"answer": "184",
"difficulty": "0/8",
"index": 81,
"problem": "Let \\( T = \\left\\{9^{k} \\mid k \\right. \\) be integers, \\(\\left.0 \\leqslant k \\leqslant 4000\\right\\} \\). Given that \\(9^{4000}\\) has 3817 digits and its most significant digit is 9, how many elements in \\(T\\) have 9 as their most s... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "(1) In an arithmetic sequence $\\{ a_n \\}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.\n\n(2) Determine the interval of decrease for the function $y = \\log_{\\f... | math | {
"ground_truth": "10\\pi",
"style": "rule"
} | {
"answer": "10\\pi",
"difficulty": "7/8",
"index": 82,
"problem": "(1) In an arithmetic sequence $\\{ a_n \\}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.\n\n(2) Determine the interval of decrease for the function $y = \\log_{\\frac{1}{2}}(x^2 + 2x - 3)$.\n\n(3) In triangle $ABC$, given $... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Show that the vector field \\( a=\\frac{2 \\cos \\theta}{r^{3}} \\mathbf{e}_{r}+\\frac{\\sin \\theta}{r^{3}} \\mathbf{e}_{\\theta} \\) is solenoidal. Let's think step by step and outpu... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "6/8",
"index": 83,
"problem": "Show that the vector field \\( a=\\frac{2 \\cos \\theta}{r^{3}} \\mathbf{e}_{r}+\\frac{\\sin \\theta}{r^{3}} \\mathbf{e}_{\\theta} \\) is solenoidal."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Set \\( A = \\{1, 2, \\cdots, n\\} \\). If there exist nonempty sets \\( B \\) and \\( C \\) such that \\( B \\cap C = \\emptyset \\), \\( B \\cup C = A \\), and the sum of the squares... | math | {
"ground_truth": "19",
"style": "rule"
} | {
"answer": "19",
"difficulty": "6/8",
"index": 84,
"problem": "Set \\( A = \\{1, 2, \\cdots, n\\} \\). If there exist nonempty sets \\( B \\) and \\( C \\) such that \\( B \\cap C = \\emptyset \\), \\( B \\cup C = A \\), and the sum of the squares of the elements in \\( B \\) is \\( M \\), and the sum of the s... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "3. Between the digits of the number 987654321, signs + should be placed so that the sum is 99. In how many ways can this be achieved? Let's think step by step and output the final answ... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "1/8",
"index": 85,
"problem": "3. Between the digits of the number 987654321, signs + should be placed so that the sum is 99. In how many ways can this be achieved?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "7. In a chess tournament, 8 players participated. They scored $7, 6, 4, 4, 3, 2, 1, 5$ and 0.5 points respectively. How many points did the players who took the first four places lose ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "1/8",
"index": 86,
"problem": "7. In a chess tournament, 8 players participated. They scored $7, 6, 4, 4, 3, 2, 1, 5$ and 0.5 points respectively. How many points did the players who took the first four places lose in matches against the others?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the first three figures after the decimal point in the decimal expression of the number \\(\\frac{0.123456789101112\\ldots495051}{0.515049\\ldots121110987654321}\\). Please provid... | math | {
"ground_truth": "239",
"style": "rule"
} | {
"answer": "239",
"difficulty": "2/8",
"index": 87,
"problem": "Find the first three figures after the decimal point in the decimal expression of the number \\(\\frac{0.123456789101112\\ldots495051}{0.515049\\ldots121110987654321}\\). Please provide the value of the first three digits after the decimal point a... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?\n$\\textbf{(A) }a=\\f... | math | {
"ground_truth": "\\textbf{(E)}>\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\textbf{(E)}>\\frac{1}{2}",
"difficulty": "0/8",
"index": 88,
"problem": "Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?\n$\\textbf{(A) }a=\\frac14 \\qquad \\textbf{(B) }\\frac14 < a... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "At a national day parade, there is a formation of 100 people. In the formation, each person's left hand either holds a red flower or a yellow flower. Each person's right hand either ho... | math | {
"ground_truth": "33",
"style": "rule"
} | {
"answer": "33",
"difficulty": "7/8",
"index": 89,
"problem": "At a national day parade, there is a formation of 100 people. In the formation, each person's left hand either holds a red flower or a yellow flower. Each person's right hand either holds a red balloon or a green balloon. It is known that 42 people... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On a paper with a Cartesian coordinate system, draw a circle with center $A(-1,0)$ and radius $2\\sqrt{2}$, with a fixed point $B(1,0)$. Fold the paper so that a point $P$ on the circu... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "1/8",
"index": 90,
"problem": "On a paper with a Cartesian coordinate system, draw a circle with center $A(-1,0)$ and radius $2\\sqrt{2}$, with a fixed point $B(1,0)$. Fold the paper so that a point $P$ on the circumference of the circle coincides with point $B$, and connect $A$ ... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "4. Solve the equation $\\sqrt{5-x}=x^{2}-5$.\n\n---\n\nNote: The translation maintains the original text's formatting and structure. Let's think step by step and output the final answe... | math | {
"ground_truth": "\\dfrac{-1 - \\sqrt{21}}{2}",
"style": "rule"
} | {
"answer": "\\dfrac{-1 - \\sqrt{21}}{2}",
"difficulty": "6/8",
"index": 91,
"problem": "4. Solve the equation $\\sqrt{5-x}=x^{2}-5$.\n\n---\n\nNote: The translation maintains the original text's formatting and structure."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let a square have a side length of \\(10^{0.2}\\) meters. Compute the area of the square and then multiply it by \\(10^{0.1}\\), \\(10^{-0.3}\\), and \\(10^{0.4}\\). Let's think step b... | math | {
"ground_truth": "10^{0.6}",
"style": "rule"
} | {
"answer": "10^{0.6}",
"difficulty": "1/8",
"index": 92,
"problem": "Let a square have a side length of \\(10^{0.2}\\) meters. Compute the area of the square and then multiply it by \\(10^{0.1}\\), \\(10^{-0.3}\\), and \\(10^{0.4}\\)."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the maximum number of rooks that can be placed on a $3n \\times 3n$ chessboard so that each of them is attacked by no more than one of the others? Let's think step by step and ... | math | {
"ground_truth": "4n",
"style": "rule"
} | {
"answer": "4n",
"difficulty": "2/8",
"index": 93,
"problem": "What is the maximum number of rooks that can be placed on a $3n \\times 3n$ chessboard so that each of them is attacked by no more than one of the others?"
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $A, B, C, D$ be four points in space such that among the segments $AB, AC, AD, BC, BD, CD$ at most one has a length greater than 1. Find the maximum possible sum of the lengths of ... | math | {
"ground_truth": "5+\\sqrt{3}",
"style": "rule"
} | {
"answer": "5+\\sqrt{3}",
"difficulty": "3/8",
"index": 94,
"problem": "Let $A, B, C, D$ be four points in space such that among the segments $AB, AC, AD, BC, BD, CD$ at most one has a length greater than 1. Find the maximum possible sum of the lengths of these six segments."
} |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A cube with sides 1 meter in length is filled with water and has a tiny hole through which the water drains into a cylinder of radius 1 meter. If the water level in the cube is falling... | math | {
"ground_truth": "\\frac{1}{\\pi}\\,",
"style": "rule"
} | {
"answer": "\\frac{1}{\\pi}\\,",
"difficulty": "1/8",
"index": 95,
"problem": "A cube with sides 1 meter in length is filled with water and has a tiny hole through which the water drains into a cylinder of radius 1 meter. If the water level in the cube is falling at a rate of 1 centimeter per second, at what r... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $S^{}_{}$ be a subset of $\\{1,2,3^{}_{},\\ldots,1989\\}$ such that no two members of $S^{}_{}$ differ by $4^{}_{}$ or $7^{}_{}$ . What is the largest number of elements $S^{}_{}$ ... | math | {
"ground_truth": "905",
"style": "rule"
} | {
"answer": "905",
"difficulty": "3/8",
"index": 96,
"problem": "Let $S^{}_{}$ be a subset of $\\{1,2,3^{}_{},\\ldots,1989\\}$ such that no two members of $S^{}_{}$ differ by $4^{}_{}$ or $7^{}_{}$ . What is the largest number of elements $S^{}_{}$ can have?\nPlease give the answer directly without any intermed... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( ABC \\) be a triangle and \\( D \\) the foot of the angle bisector from \\( A \\). Denote \\( O \\) (resp. \\( O_1 \\), resp. \\( O_2 \\)) as the circumcenter of \\( \\triangle... | math | {
"ground_truth": "OO_1=OO_2",
"style": "rule"
} | {
"answer": "OO_1=OO_2",
"difficulty": "0/8",
"index": 97,
"problem": "Let \\( ABC \\) be a triangle and \\( D \\) the foot of the angle bisector from \\( A \\). Denote \\( O \\) (resp. \\( O_1 \\), resp. \\( O_2 \\)) as the circumcenter of \\( \\triangle ABC \\) (resp. \\( \\triangle ABD \\), resp. \\( \\trian... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The distinct points \\(x_n\\) are dense in the interval \\((0, 1)\\). \\(x_1, x_2, \\ldots, x_{n-1}\\) divide \\((0, 1)\\) into \\(n\\) sub-intervals, one of which must contain \\(x_n\... | math | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"answer": "\\frac{1}{3}",
"difficulty": "0/8",
"index": 98,
"problem": "The distinct points \\(x_n\\) are dense in the interval \\((0, 1)\\). \\(x_1, x_2, \\ldots, x_{n-1}\\) divide \\((0, 1)\\) into \\(n\\) sub-intervals, one of which must contain \\(x_n\\). This part is divided by \\(x_n\\) into two sub-int... |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$\n$\\textbf{(A) } 8 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 16 ... | math | {
"ground_truth": "\\textbf{(C)}16",
"style": "rule"
} | {
"answer": "\\textbf{(C)}16",
"difficulty": "0/8",
"index": 99,
"problem": "For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$\n$\\textbf{(A) } 8 \\qquad \\textbf{(B) } 12 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 17 \\qquad \\textbf{(E) } 21... |
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