data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict | shard int64 0 7 | global_idx int64 0 799 |
|---|---|---|---|---|---|---|
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 possible median for the six number set $\\{x, 2x, 3, 2, 5, 4x\\}$ if $x$ can be any positive integer? Let's think step by step and output the final answer within \... | math | {
"ground_truth": "2.5",
"style": "rule"
} | {
"answer": "2.5",
"difficulty": "0/8",
"index": 0,
"problem": "What is the smallest possible median for the six number set $\\{x, 2x, 3, 2, 5, 4x\\}$ if $x$ can be any positive integer?"
} | 0 | 0 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). ... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"answer": "20",
"difficulty": "7/8",
"index": 1,
"problem": "Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). From these $n^2$ handshakes, two are randomly selected. Let event... | 1 | 1 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In any (not necessarily convex) quadrilateral \\(ABCD\\), where \\(AB = a\\), \\(BC = b\\), \\(CD = c\\), \\(DA = d\\), \\(AC = e\\), \\(BD = f\\), and \\(AC\\) intersects \\(BD\\) at ... | math | {
"ground_truth": "2ef\\cosP=-^2+b^2-^2+^2",
"style": "rule"
} | {
"answer": "2ef\\cosP=-^2+b^2-^2+^2",
"difficulty": "4/8",
"index": 3,
"problem": "In any (not necessarily convex) quadrilateral \\(ABCD\\), where \\(AB = a\\), \\(BC = b\\), \\(CD = c\\), \\(DA = d\\), \\(AC = e\\), \\(BD = f\\), and \\(AC\\) intersects \\(BD\\) at \\(P\\) with \\(\\angle APB = P\\), prove th... | 3 | 3 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"answer": "12",
"difficulty": "4/8",
"index": 4,
"problem": "Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are the m... | 4 | 4 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the circumcenter \\(O\\) of \\(\\triangle ABC\\), and \\(3 \\overrightarrow{OA} + 4 \\overrightarrow{OB} + 5 \\overrightarrow{OC} = \\mathbf{0}\\), find the value of \\(\\cos \\a... | math | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{5}}{5}",
"difficulty": "7/8",
"index": 5,
"problem": "Given the circumcenter \\(O\\) of \\(\\triangle ABC\\), and \\(3 \\overrightarrow{OA} + 4 \\overrightarrow{OB} + 5 \\overrightarrow{OC} = \\mathbf{0}\\), find the value of \\(\\cos \\angle ABC\\)."
} | 5 | 5 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "1. N1 (TWN) ${ }^{1 \\mathrm{MO} 04}$ Find all pairs of positive integers $(x, p)$ such that $p$ is a prime, $x \\leq 2 p$, and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$. Let's think ste... | math | {
"ground_truth": "(1, p)",
"style": "rule"
} | {
"answer": "(1, p)",
"difficulty": "0/8",
"index": 7,
"problem": "1. N1 (TWN) ${ }^{1 \\mathrm{MO} 04}$ Find all pairs of positive integers $(x, p)$ such that $p$ is a prime, $x \\leq 2 p$, and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$."
} | 7 | 7 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An isosceles triangle \\( ABC \\) is given (\\( AB = BC \\)). On the ray \\( BA \\) beyond the point \\( A \\), point \\( E \\) is marked, and on the side \\( BC \\), point \\( D \\) i... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "2/8",
"index": 8,
"problem": "An isosceles triangle \\( ABC \\) is given (\\( AB = BC \\)). On the ray \\( BA \\) beyond the point \\( A \\), point \\( E \\) is marked, and on the side \\( BC \\), point \\( D \\) is marked. It is known that\n\n\\[\n\\angle ADC = \\angle AEC = 60^... | 0 | 8 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given is a $n \\times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing ... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"answer": "30",
"difficulty": "3/8",
"index": 9,
"problem": "Given is a $n \\times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\\lim_{n \\to \\infty} np_n$ ."
} | 1 | 9 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On the radius \\( AO \\) of a circle with center \\( O \\), a point \\( M \\) is chosen. On one side of \\( AO \\) on the circle, points \\( B \\) and \\( C \\) are chosen such that \\... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"answer": "12",
"difficulty": "2/8",
"index": 11,
"problem": "On the radius \\( AO \\) of a circle with center \\( O \\), a point \\( M \\) is chosen. On one side of \\( AO \\) on the circle, points \\( B \\) and \\( C \\) are chosen such that \\( \\angle AMB = \\angle OMC = \\alpha \\). Find the length of \\... | 3 | 11 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the value of \\(2 - 13 \\cos 2\\alpha + \\sin^{-1}(2\\alpha)\\), given that \\(\\operatorname{ctg} \\alpha = -\\frac{1}{5}\\). Let's think step by step and output the final answer... | math | {
"ground_truth": "\\frac{57}{5}",
"style": "rule"
} | {
"answer": "\\frac{57}{5}",
"difficulty": "2/8",
"index": 12,
"problem": "Find the value of \\(2 - 13 \\cos 2\\alpha + \\sin^{-1}(2\\alpha)\\), given that \\(\\operatorname{ctg} \\alpha = -\\frac{1}{5}\\)."
} | 4 | 12 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For every non-empty subset of the natural number set $N^*$, we define the \"alternating sum\" as follows: arrange the elements of the subset in descending order, then start with the la... | math | {
"ground_truth": "448",
"style": "rule"
} | {
"answer": "448",
"difficulty": "6/8",
"index": 13,
"problem": "For every non-empty subset of the natural number set $N^*$, we define the \"alternating sum\" as follows: arrange the elements of the subset in descending order, then start with the largest number and alternately add and subtract each number. For ... | 5 | 13 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "7. The integer solutions $(x, y)$ of the indeterminate equation $x^{2}+y^{2}=x y+2 x+2 y$ are in total groups.\n\nThe integer solutions $(x, y)$ of the indeterminate equation $x^{2}+y^... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "1/8",
"index": 15,
"problem": "7. The integer solutions $(x, y)$ of the indeterminate equation $x^{2}+y^{2}=x y+2 x+2 y$ are in total groups.\n\nThe integer solutions $(x, y)$ of the indeterminate equation $x^{2}+y^{2}=x y+2 x+2 y$ are in total groups."
} | 7 | 15 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Let's think step by step and output the final answer within \\bo... | math | {
"ground_truth": "1440",
"style": "rule"
} | {
"answer": "1440",
"difficulty": "4/8",
"index": 16,
"problem": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number."
} | 0 | 16 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( x_{1}, x_{2}, \\cdots, x_{n} \\) be \\( n \\) non-negative real numbers \\((n>2, n \\in \\mathbb{N}^{*})\\) such that \\( \\sum_{i=1}^{n} x_{i} = n \\) and \\( \\sum_{i=1}^{n} ... | math | {
"ground_truth": "n^2-2",
"style": "rule"
} | {
"answer": "n^2-2",
"difficulty": "4/8",
"index": 17,
"problem": "Let \\( x_{1}, x_{2}, \\cdots, x_{n} \\) be \\( n \\) non-negative real numbers \\((n>2, n \\in \\mathbb{N}^{*})\\) such that \\( \\sum_{i=1}^{n} x_{i} = n \\) and \\( \\sum_{i=1}^{n} i x_{i} = 2n - 2 \\). Find the maximum value of the algebraic... | 1 | 17 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "15. If there exists a positive integer $m$ such that $m!$ ends with exactly $n$ zeros, then the positive integer $n$ is called a \"factorial tail number.\" How many non-\"factorial tai... | math | {
"ground_truth": "396",
"style": "rule"
} | {
"answer": "396",
"difficulty": "4/8",
"index": 19,
"problem": "15. If there exists a positive integer $m$ such that $m!$ ends with exactly $n$ zeros, then the positive integer $n$ is called a \"factorial tail number.\" How many non-\"factorial tail number\" positive integers are there less than 1992?"
} | 3 | 19 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In quadrilateral \\(ABCD\\), \\(\\angle DAB = \\angle DBC = 90^\\circ\\). Additionally, \\(DB = a\\) and \\(DC = b\\).\n\nFind the distance between the centers of two circles, one pass... | math | {
"ground_truth": "\\frac{\\sqrt{b^2-^2}}{2}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{b^2-^2}}{2}",
"difficulty": "4/8",
"index": 20,
"problem": "In quadrilateral \\(ABCD\\), \\(\\angle DAB = \\angle DBC = 90^\\circ\\). Additionally, \\(DB = a\\) and \\(DC = b\\).\n\nFind the distance between the centers of two circles, one passing through points \\(D, A, B\\), and the... | 4 | 20 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the largest possible k such that \\( p(x) \\geq k(x - a)^3 \\) for all non-negative \\( x \\) and all real polynomials \\( p(x) = x^3 + ax^2 + bx + c \\) which have real coeffi... | math | {
"ground_truth": "-\\frac{1}{27}",
"style": "rule"
} | {
"answer": "-\\frac{1}{27}",
"difficulty": "4/8",
"index": 21,
"problem": "What is the largest possible k such that \\( p(x) \\geq k(x - a)^3 \\) for all non-negative \\( x \\) and all real polynomials \\( p(x) = x^3 + ax^2 + bx + c \\) which have real coefficients and real, non-negative roots? For which polyn... | 5 | 21 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that \\( x_{1} \\) and \\( x_{2} \\) are distinct complex numbers and satisfy the following three equations:\n$$\n\\begin{array}{l}\nx_{1}(x_{1}+1)=A, \\quad x_{2}(x_{2}+1)=A, \\... | math | {
"ground_truth": "-7",
"style": "rule"
} | {
"answer": "-7",
"difficulty": "7/8",
"index": 23,
"problem": "Given that \\( x_{1} \\) and \\( x_{2} \\) are distinct complex numbers and satisfy the following three equations:\n$$\n\\begin{array}{l}\nx_{1}(x_{1}+1)=A, \\quad x_{2}(x_{2}+1)=A, \\\\\nx_{1}^{4}+3 x_{1}^{3}+5 x_{1}=x_{2}^{4}+3 x_{2}^{3}+5 x_{2}.... | 7 | 23 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ and $k$ be positive integers. Find all monic polynomials $f\\in \\mathbb{Z}[X]$ , of degree $n$ , such that $f(a)$ divides $f(2a^k)$ for $a\\in \\mathbb{Z}$ with $... | math | {
"ground_truth": "X^n",
"style": "rule"
} | {
"answer": "X^n",
"difficulty": "0/8",
"index": 24,
"problem": "Let $n$ and $k$ be positive integers. Find all monic polynomials $f\\in \\mathbb{Z}[X]$ , of degree $n$ , such that $f(a)$ divides $f(2a^k)$ for $a\\in \\mathbb{Z}$ with $f(a)\\neq 0$ ."
} | 0 | 24 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The integer part and fractional part of $(\\sqrt{5}+2)^{2n+1}$ (where $n \\in \\mathbf{N}$) are denoted as $P$ and $Q$, respectively. Find $Q(P+Q)$. Let's think step by step and output... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "3/8",
"index": 25,
"problem": "The integer part and fractional part of $(\\sqrt{5}+2)^{2n+1}$ (where $n \\in \\mathbf{N}$) are denoted as $P$ and $Q$, respectively. Find $Q(P+Q)$."
} | 1 | 25 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Solve the equation \\(2 x^{x} = \\sqrt{2}\\) for positive numbers. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\frac{1}{2}",
"difficulty": "7/8",
"index": 27,
"problem": "Solve the equation \\(2 x^{x} = \\sqrt{2}\\) for positive numbers."
} | 3 | 27 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "iris",
"style": "rule"
} | {
"answer": "iris",
"difficulty": "0/8",
"index": 28,
"problem": "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: ... | 4 | 28 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\\angle BAF = 18^\\circ$ . $EF$ and $GH$ meet a... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"answer": "45",
"difficulty": "5/8",
"index": 29,
"problem": "Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\\angle BAF = 18^\\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of r... | 5 | 29 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the minimum value for \\(a, b > 0\\) of the expression\n\n$$\n\\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\\sqrt{4a^2 + b^2}}\n$$ Let's think step by step and output the f... | math | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{5}}{5}",
"difficulty": "1/8",
"index": 31,
"problem": "Find the minimum value for \\(a, b > 0\\) of the expression\n\n$$\n\\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\\sqrt{4a^2 + b^2}}\n$$"
} | 7 | 31 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\alpha$ be a positive real number. Determine the largest real number $C$ such that for all positive real numbers $x, y,$ and $z$ with $x y + y z + z x = \\alpha$, the inequality ... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"answer": "16",
"difficulty": "4/8",
"index": 32,
"problem": "Let $\\alpha$ be a positive real number. Determine the largest real number $C$ such that for all positive real numbers $x, y,$ and $z$ with $x y + y z + z x = \\alpha$, the inequality \n\n$$\n\\left(1+\\frac{\\alpha}{x^{2}}\\right)\\left(1+\\frac{\... | 0 | 32 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Points $A$, $B$, and $C$ lie on a circle $\\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\\ell$ tangent to the incircle of $\\triangle ABC$ at $T$ inters... | math | {
"ground_truth": "460",
"style": "rule"
} | {
"answer": "460",
"difficulty": "4/8",
"index": 33,
"problem": "Points $A$, $B$, and $C$ lie on a circle $\\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\\ell$ tangent to the incircle of $\\triangle ABC$ at $T$ intersects $\\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC... | 1 | 33 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Once, I decided to take a chairlift ride. At a certain moment, I noticed that the chairlift coming towards me had the number 95, and the next one had the number 0, followed by 1, 2, an... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"answer": "18",
"difficulty": "5/8",
"index": 35,
"problem": "Once, I decided to take a chairlift ride. At a certain moment, I noticed that the chairlift coming towards me had the number 95, and the next one had the number 0, followed by 1, 2, and so on. I looked at the number of my chair, which turned out to... | 3 | 35 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "C2. We call a positive integer sunny if it consists of four digits and if, moreover, each of the two outer digits is exactly 1 greater than the digit next to it. The numbers 8723 and 1... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"answer": "16",
"difficulty": "3/8",
"index": 36,
"problem": "C2. We call a positive integer sunny if it consists of four digits and if, moreover, each of the two outer digits is exactly 1 greater than the digit next to it. The numbers 8723 and 1001 are, for example, sunny, but 1234 and 87245 are not.\n\n(a) ... | 4 | 36 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is 0.3 less than 83.45 more than 29.7? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "112.85",
"style": "rule"
} | {
"answer": "112.85",
"difficulty": "7/8",
"index": 37,
"problem": "What is 0.3 less than 83.45 more than 29.7?"
} | 5 | 37 |
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": "teacher",
"style": "rule"
} | {
"answer": "teacher",
"difficulty": "2/8",
"index": 39,
"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... | 7 | 39 |
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$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible val... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "6/8",
"index": 40,
"problem": "Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?"
} | 0 | 40 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the decimal system, what is the smallest natural number, whose digits are only 0 or 1, that is divisible by 225? Let's think step by step and output the final answer within \\boxed{... | math | {
"ground_truth": "11,111,111,100",
"style": "rule"
} | {
"answer": "11,111,111,100",
"difficulty": "6/8",
"index": 41,
"problem": "In the decimal system, what is the smallest natural number, whose digits are only 0 or 1, that is divisible by 225?"
} | 1 | 41 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum o... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"answer": "34",
"difficulty": "7/8",
"index": 43,
"problem": "A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combi... | 3 | 43 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A group of schoolchildren arrived at a camp, among them Petya, Vasya, and Timofey, who were not acquainted with each other but each had acquaintances among the other children. Petya no... | math | {
"ground_truth": "\\frac{7}{20}",
"style": "rule"
} | {
"answer": "\\frac{7}{20}",
"difficulty": "7/8",
"index": 44,
"problem": "A group of schoolchildren arrived at a camp, among them Petya, Vasya, and Timofey, who were not acquainted with each other but each had acquaintances among the other children. Petya noticed that exactly \\( \\frac{1}{2} \\) of his acquai... | 4 | 44 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given an integer \\( n \\) (where \\( n \\geq 2 \\)). For a \\( 2n \\)-element ordered array \\( T = \\left(a_1, b_1, a_2, b_2, \\cdots, a_n, b_n \\right) \\), if each component of \\(... | math | {
"ground_truth": "3^n + \\frac{n(n+1)}{2}",
"style": "rule"
} | {
"answer": "3^n + \\frac{n(n+1)}{2}",
"difficulty": "0/8",
"index": 45,
"problem": "Given an integer \\( n \\) (where \\( n \\geq 2 \\)). For a \\( 2n \\)-element ordered array \\( T = \\left(a_1, b_1, a_2, b_2, \\cdots, a_n, b_n \\right) \\), if each component of \\( T \\) is either 0 or 1, and for any \\( p,... | 5 | 45 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form one large equilateral triangular corral. What is the ratio of t... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"answer": "\\frac{1}{4}",
"difficulty": "2/8",
"index": 47,
"problem": "The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form one large equilateral triangular corral. What is the ratio of the total area of the four small corrals to the area of... | 7 | 47 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 1... | math | {
"ground_truth": "1221",
"style": "rule"
} | {
"answer": "1221",
"difficulty": "7/8",
"index": 48,
"problem": "On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\\overline{2011 \\mathrm{ABCD... | 0 | 48 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sh... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"answer": "12",
"difficulty": "3/8",
"index": 49,
"problem": "A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence. \n\nHow many... | 1 | 49 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A regular octagon is inscribed in a square such that each side of the square is trisected by the vertices of the octagon. The perimeter of the square is now 144 centimeters. What is th... | math | {
"ground_truth": "1008",
"style": "rule"
} | {
"answer": "1008",
"difficulty": "5/8",
"index": 51,
"problem": "A regular octagon is inscribed in a square such that each side of the square is trisected by the vertices of the octagon. The perimeter of the square is now 144 centimeters. What is the area of the octagon inscribed in the square?"
} | 3 | 51 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who i... | math | {
"ground_truth": "60.316",
"style": "rule"
} | {
"answer": "60.316",
"difficulty": "0/8",
"index": 52,
"problem": "Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who is 172 cm tall."
} | 4 | 52 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "How many positive numbers are there among the first 100 terms of the sequence: $\\sin 1^{\\circ}, \\sin 10^{\\circ}, \\sin 100^{\\circ}, \\sin 1000^{\\circ}, \\ldots ?$ Let's think ste... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "7/8",
"index": 53,
"problem": "How many positive numbers are there among the first 100 terms of the sequence: $\\sin 1^{\\circ}, \\sin 10^{\\circ}, \\sin 100^{\\circ}, \\sin 1000^{\\circ}, \\ldots ?$"
} | 5 | 53 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started ... | math | {
"ground_truth": "69",
"style": "rule"
} | {
"answer": "69",
"difficulty": "5/8",
"index": 55,
"problem": "Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What... | 7 | 55 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a circular circus arena with a radius of 10 meters, a lion runs. Moving along a broken line, the lion runs 30 kilometers.\n\nProve that the sum of all the angles by which the lion t... | math | {
"ground_truth": "2998",
"style": "rule"
} | {
"answer": "2998",
"difficulty": "4/8",
"index": 56,
"problem": "In a circular circus arena with a radius of 10 meters, a lion runs. Moving along a broken line, the lion runs 30 kilometers.\n\nProve that the sum of all the angles by which the lion turns is at least 2998 radians."
} | 0 | 56 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a $4\\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the firs... | math | {
"ground_truth": "\\frac{43}{3}",
"style": "rule"
} | {
"answer": "\\frac{43}{3}",
"difficulty": "4/8",
"index": 57,
"problem": "Given a $4\\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and t... | 1 | 57 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Write in ascending order the multiples of 3 which, when 1 is added, are perfect squares, i.e., $3, 15, 24, 48, \\ldots$ What is the multiple of 3 in the $2006^{\\mathrm{th}}$ position?... | math | {
"ground_truth": "9060099",
"style": "rule"
} | {
"answer": "9060099",
"difficulty": "7/8",
"index": 59,
"problem": "Write in ascending order the multiples of 3 which, when 1 is added, are perfect squares, i.e., $3, 15, 24, 48, \\ldots$ What is the multiple of 3 in the $2006^{\\mathrm{th}}$ position?"
} | 3 | 59 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $M = \\{(a,b,c,d)|a,b,c,d \\in \\{1,2,3,4\\} \\text{ and } abcd > 1\\}$ . For each $n\\in \\{1,2,\\dots, 254\\}$ , the sequence $(a_1, b_1, c_1, d_1)$ , $(a_2, b_2, c_2, d_2)$ ... | math | {
"ground_truth": "(1,2), (1,4), (2,1), (2,3), (3,2), (3,4), (4,1), (4,3)",
"style": "rule"
} | {
"answer": "(1,2), (1,4), (2,1), (2,3), (3,2), (3,4), (4,1), (4,3)",
"difficulty": "1/8",
"index": 60,
"problem": "Let $M = \\{(a,b,c,d)|a,b,c,d \\in \\{1,2,3,4\\} \\text{ and } abcd > 1\\}$ . For each $n\\in \\{1,2,\\dots, 254\\}$ , the sequence $(a_1, b_1, c_1, d_1)$ , $(a_2, b_2, c_2, d_2)$ , $\\dots$ ... | 4 | 60 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "10. For the curve $C: x^{4}+y^{2}=1$, consider the following statements:\n(1) The curve $C$ is symmetric with respect to the origin;\n(2) The curve $C$ is symmetric with respect to the... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "0/8",
"index": 61,
"problem": "10. For the curve $C: x^{4}+y^{2}=1$, consider the following statements:\n(1) The curve $C$ is symmetric with respect to the origin;\n(2) The curve $C$ is symmetric with respect to the line $y=x$;\n(3) The area enclosed by the curve $C$ is less than... | 5 | 61 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{0}, a_{1}, \\ldots$ and $b_{0}, b_{1}, \\ldots$ be geometric sequences with common ratios $r_{a}$ and $r_{b}$, respectively, such that $$\\sum_{i=0}^{\\infty} a_{i}=\\sum_{i=0}... | math | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"answer": "\\frac{4}{3}",
"difficulty": "5/8",
"index": 63,
"problem": "Let $a_{0}, a_{1}, \\ldots$ and $b_{0}, b_{1}, \\ldots$ be geometric sequences with common ratios $r_{a}$ and $r_{b}$, respectively, such that $$\\sum_{i=0}^{\\infty} a_{i}=\\sum_{i=0}^{\\infty} b_{i}=1 \\quad \\text { and } \\quad\\left(... | 7 | 63 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\{x\\}$ be a sequence of positive reals $x_1, x_2, \\ldots, x_n$ , defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$ . And for $n \\geq 1$ we have:\r\n\r\n\\[x_{n+4} = \\sqrt[4]... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "2/8",
"index": 64,
"problem": "Let $\\{x\\}$ be a sequence of positive reals $x_1, x_2, \\ldots, x_n$ , defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$ . And for $n \\geq 1$ we have:\r\n\r\n\\[x_{n+4} = \\sqrt[4]{x_{n} \\cdot x_{n+1} \\cdot x_{n+2} \\cdot x_{n+3}}.\\]\r\n\r\nS... | 0 | 64 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from th... | math | {
"ground_truth": "761",
"style": "rule"
} | {
"answer": "761",
"difficulty": "0/8",
"index": 65,
"problem": "There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one o... | 1 | 65 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Prime number $p>3$ is congruent to $2$ modulo $3$ . Let $a_k = k^2 + k +1$ for $k=1, 2, \\ldots, p-1$ . Prove that product $a_1a_2\\ldots a_{p-1}$ is congruent to $3$ modul... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "5/8",
"index": 67,
"problem": "Prime number $p>3$ is congruent to $2$ modulo $3$ . Let $a_k = k^2 + k +1$ for $k=1, 2, \\ldots, p-1$ . Prove that product $a_1a_2\\ldots a_{p-1}$ is congruent to $3$ modulo $p$ ."
} | 3 | 67 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of degrees in the measure of angle $x$.\n\n[asy]\nimport markers;\nsize (5cm,5cm);\npair A,B,C,D,F,H;\n\nA=(0,0);\nB=(5,0);\nC=(9,0);\nD=(3.8,7);\n\nF=(2.3,7.2);\nH=(5.... | math | {
"ground_truth": "82^\\circ",
"style": "rule"
} | {
"answer": "82^\\circ",
"difficulty": "0/8",
"index": 68,
"problem": "Find the number of degrees in the measure of angle $x$.\n\n[asy]\nimport markers;\nsize (5cm,5cm);\npair A,B,C,D,F,H;\n\nA=(0,0);\nB=(5,0);\nC=(9,0);\nD=(3.8,7);\n\nF=(2.3,7.2);\nH=(5.3,7.2);\n\ndraw((4.2,6.1){up}..{right}(5.3,7.2));\ndraw((... | 4 | 68 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "huawei p50",
"style": "rule"
} | {
"answer": "huawei p50",
"difficulty": "0/8",
"index": 69,
"problem": "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 ... | 5 | 69 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "Peter",
"style": "rule"
} | {
"answer": "Peter",
"difficulty": "1/8",
"index": 71,
"problem": "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:... | 7 | 71 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that \\(x_{1}, x_{2}, x_{3}, \\ldots\\) are positive real numbers for which\n\\[ x_{n}^{n}=\\sum_{j=0}^{n-1} x_{n}^{j} \\]\nfor \\( n=1,2,3, \\ldots \\). Prove that for all \\(... | math | {
"ground_truth": "2-\\frac{1}{2^{n-1}}\\lex_n<2-\\frac{1}{2^n}",
"style": "rule"
} | {
"answer": "2-\\frac{1}{2^{n-1}}\\lex_n<2-\\frac{1}{2^n}",
"difficulty": "2/8",
"index": 72,
"problem": "Suppose that \\(x_{1}, x_{2}, x_{3}, \\ldots\\) are positive real numbers for which\n\\[ x_{n}^{n}=\\sum_{j=0}^{n-1} x_{n}^{j} \\]\nfor \\( n=1,2,3, \\ldots \\). Prove that for all \\( n \\),\n\\[ 2-\\frac{... | 0 | 72 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "2. Find all such triples of prime numbers $p, q$ and $r$, that $\\frac{p}{q}=\\frac{8}{r-1}+1$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "us... | math | {
"ground_truth": "(3, 2, 17)",
"style": "rule"
} | {
"answer": "(3, 2, 17)",
"difficulty": "0/8",
"index": 73,
"problem": "2. Find all such triples of prime numbers $p, q$ and $r$, that $\\frac{p}{q}=\\frac{8}{r-1}+1$."
} | 1 | 73 |
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 such that $n \\geq 4$. Find the greatest positive integer $k$ such that there exists a triangle with integer side lengths all not greater than $n$, and th... | math | {
"ground_truth": "\\lfloor\\frac{n-1}{3}\\rfloor",
"style": "rule"
} | {
"answer": "\\lfloor\\frac{n-1}{3}\\rfloor",
"difficulty": "6/8",
"index": 75,
"problem": "Let $n$ be a positive integer such that $n \\geq 4$. Find the greatest positive integer $k$ such that there exists a triangle with integer side lengths all not greater than $n$, and the difference between any two sides (... | 3 | 75 |
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": "mountain",
"style": "rule"
} | {
"answer": "mountain",
"difficulty": "5/8",
"index": 76,
"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 na... | 4 | 76 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A $5 \\times 5$ square consisting of cells was divided into several parts of different areas, each part consisting of an integer number of cells. What is the maximum number of parts th... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "7/8",
"index": 77,
"problem": "A $5 \\times 5$ square consisting of cells was divided into several parts of different areas, each part consisting of an integer number of cells. What is the maximum number of parts that could result from such a division?"
} | 5 | 77 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find \\( n > 1 \\) such that using stamp denominations of \\( n \\) and \\( n+2 \\), it is possible to obtain any value \\( \\geq 2n + 2 \\). Let's think step by step and output the fi... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "7/8",
"index": 79,
"problem": "Find \\( n > 1 \\) such that using stamp denominations of \\( n \\) and \\( n+2 \\), it is possible to obtain any value \\( \\geq 2n + 2 \\)."
} | 7 | 79 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "1. Let $p$ be a prime number. Find all natural numbers $x$ and $y$ that satisfy the equation $p \\cdot(x-5)=x \\cdot y$. Let's think step by step and output the final answer within \\b... | math | {
"ground_truth": "(x, y) = (5p, p - 1)",
"style": "rule"
} | {
"answer": "(x, y) = (5p, p - 1)",
"difficulty": "1/8",
"index": 80,
"problem": "1. Let $p$ be a prime number. Find all natural numbers $x$ and $y$ that satisfy the equation $p \\cdot(x-5)=x \\cdot y$."
} | 0 | 80 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The area of the orthogonal projection of a circle with a radius of 1 on the plane $\\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to... | math | {
"ground_truth": "\\frac{2\\sqrt{\\pi^2 - 1}}{\\pi}",
"style": "rule"
} | {
"answer": "\\frac{2\\sqrt{\\pi^2 - 1}}{\\pi}",
"difficulty": "6/8",
"index": 81,
"problem": "The area of the orthogonal projection of a circle with a radius of 1 on the plane $\\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\\alpha$."
} | 1 | 81 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "$M$ is the midpoint of the side $AB$ in an equilateral triangle $\\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and par... | math | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"answer": "120^\\circ",
"difficulty": "3/8",
"index": 83,
"problem": "$M$ is the midpoint of the side $AB$ in an equilateral triangle $\\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $... | 3 | 83 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find all real numbers $p$ such that the cubic equation\n$$\n5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p\n$$\nhas two roots that are both natural numbers. Let's think step by step and output... | math | {
"ground_truth": "76",
"style": "rule"
} | {
"answer": "76",
"difficulty": "1/8",
"index": 84,
"problem": "Find all real numbers $p$ such that the cubic equation\n$$\n5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p\n$$\nhas two roots that are both natural numbers."
} | 4 | 84 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A natural number \\( k \\) has the property: if \\( n \\) is divisible by \\( k \\), then the number formed by writing the digits of \\( n \\) in reverse order is also divisible by \\(... | math | {
"ground_truth": "99",
"style": "rule"
} | {
"answer": "99",
"difficulty": "4/8",
"index": 85,
"problem": "A natural number \\( k \\) has the property: if \\( n \\) is divisible by \\( k \\), then the number formed by writing the digits of \\( n \\) in reverse order is also divisible by \\( k \\). Prove that \\( k \\) is a factor of 99."
} | 5 | 85 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "swede",
"style": "rule"
} | {
"answer": "swede",
"difficulty": "0/8",
"index": 87,
"problem": "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:... | 7 | 87 |
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": "short",
"style": "rule"
} | {
"answer": "short",
"difficulty": "0/8",
"index": 88,
"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:... | 0 | 88 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the range of the function \\(f(x) = \\frac{1}{g\\left(\\frac{64 g\\left(16 g\\left(\\log_{2} x\\right)\\right)}{5}\\right)}\\), where \\(g(x) = \\sqrt[5]{x} + \\frac{1}{\\sqrt[5]{... | math | {
"ground_truth": "[-\\frac{2}{5},0)\\cup(0,\\frac{2}{5}]",
"style": "rule"
} | {
"answer": "[-\\frac{2}{5},0)\\cup(0,\\frac{2}{5}]",
"difficulty": "3/8",
"index": 89,
"problem": "Find the range of the function \\(f(x) = \\frac{1}{g\\left(\\frac{64 g\\left(16 g\\left(\\log_{2} x\\right)\\right)}{5}\\right)}\\), where \\(g(x) = \\sqrt[5]{x} + \\frac{1}{\\sqrt[5]{x}}\\)."
} | 1 | 89 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ . Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "2001",
"style": "rule"
} | {
"answer": "2001",
"difficulty": "7/8",
"index": 91,
"problem": "Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ ."
} | 3 | 91 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Julia possesses a deck of 18 cards, numbered from 1 to 18. After shuffling, she distributes the cards face down in 3 rows and 6 columns.\n\nJulia chooses a sum \\(X\\) and flips two ca... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "3/8",
"index": 92,
"problem": "Julia possesses a deck of 18 cards, numbered from 1 to 18. After shuffling, she distributes the cards face down in 3 rows and 6 columns.\n\nJulia chooses a sum \\(X\\) and flips two cards. If the sum is \\(X\\), she removes the pair from the table, ... | 4 | 92 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seat... | math | {
"ground_truth": "96",
"style": "rule"
} | {
"answer": "96",
"difficulty": "3/8",
"index": 93,
"problem": "Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each... | 5 | 93 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Prove that the number of triples $(A,B,C)$ where $A, B, C$ are subsets of $\\{ 1,2, \\ldots , n \\}$ such that $A \\cap B \\cap C = \\phi$ , $A \\cap B \\not= \\phi$ , $C \\c... | math | {
"ground_truth": "7^n-2\\cdot6^n+5^n",
"style": "rule"
} | {
"answer": "7^n-2\\cdot6^n+5^n",
"difficulty": "5/8",
"index": 95,
"problem": "Prove that the number of triples $(A,B,C)$ where $A, B, C$ are subsets of $\\{ 1,2, \\ldots , n \\}$ such that $A \\cap B \\cap C = \\phi$ , $A \\cap B \\not= \\phi$ , $C \\cap B \\not= \\phi$ is $7^n - 2 . 6^n + 5^n$ ."... | 7 | 95 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a right triangle \\( \\triangle ABC \\) with legs \\( AC=12 \\), \\( BC=5 \\). Point \\( D \\) is a moving point on the hypotenuse \\( AB \\). The triangle is folded along \\( CD... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"answer": "32",
"difficulty": "0/8",
"index": 96,
"problem": "Given a right triangle \\( \\triangle ABC \\) with legs \\( AC=12 \\), \\( BC=5 \\). Point \\( D \\) is a moving point on the hypotenuse \\( AB \\). The triangle is folded along \\( CD \\) to form a right dihedral angle \\( A-CD-B \\). When the len... | 0 | 96 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "7/8",
"index": 97,
"problem": "Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on o... | 1 | 97 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "We can find sets of 13 distinct positive integers that add up to 2142. Find the largest possible greatest common divisor of these 13 distinct positive integers. Let's think step by ste... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"answer": "21",
"difficulty": "7/8",
"index": 99,
"problem": "We can find sets of 13 distinct positive integers that add up to 2142. Find the largest possible greatest common divisor of these 13 distinct positive integers."
} | 3 | 99 |
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": "oneplus 9",
"style": "rule"
} | {
"answer": "oneplus 9",
"difficulty": "2/8",
"index": 100,
"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 ... | 4 | 100 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Determine all functions $f: \\mathbb{Z}\\to\\mathbb{Z}$ satisfying \\[f\\big(f(m)+n\\big)+f(m)=f(n)+f(3m)+2014\\] for all integers $m$ and $n$ .\n\n*Proposed by Netherlands* Let's... | math | {
"ground_truth": "f(n)=2n+1007",
"style": "rule"
} | {
"answer": "f(n)=2n+1007",
"difficulty": "1/8",
"index": 101,
"problem": "Determine all functions $f: \\mathbb{Z}\\to\\mathbb{Z}$ satisfying \\[f\\big(f(m)+n\\big)+f(m)=f(n)+f(3m)+2014\\] for all integers $m$ and $n$ .\n\n*Proposed by Netherlands*"
} | 5 | 101 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | math | {
"ground_truth": "swede",
"style": "rule"
} | {
"answer": "swede",
"difficulty": "0/8",
"index": 103,
"problem": "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... | 7 | 103 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a quadrilateral $MNQP$, there are two non-intersecting circles positioned such that one of them is tangent to the sides $MN$, $NP$, $PQ$, and the other is tangent to the sides $MN$,... | math | {
"ground_truth": "p",
"style": "rule"
} | {
"answer": "p",
"difficulty": "0/8",
"index": 104,
"problem": "In a quadrilateral $MNQP$, there are two non-intersecting circles positioned such that one of them is tangent to the sides $MN$, $NP$, $PQ$, and the other is tangent to the sides $MN$, $MQ$, $PQ$. Points $B$ and $A$ lie on the sides $MN$ and $PQ$ r... | 0 | 104 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A point \\( P \\) lies at the center of square \\( ABCD \\). A sequence of points \\( \\{P_n\\} \\) is determined by \\( P_0 = P \\), and given point \\( P_i \\), point \\( P_{i+1} \\)... | math | {
"ground_truth": "\\frac{1225}{16384}",
"style": "rule"
} | {
"answer": "\\frac{1225}{16384}",
"difficulty": "4/8",
"index": 105,
"problem": "A point \\( P \\) lies at the center of square \\( ABCD \\). A sequence of points \\( \\{P_n\\} \\) is determined by \\( P_0 = P \\), and given point \\( P_i \\), point \\( P_{i+1} \\) is obtained by reflecting \\( P_i \\) over on... | 1 | 105 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, point $M$ lies inside the triangle such that $\\angle MBA = 30^\\circ$ and $\\angle MAB = 10^\\circ$. Given that $\\angle ACB = 80^\\circ$ and $AC = BC$, find $\\a... | math | {
"ground_truth": "70",
"style": "rule"
} | {
"answer": "70",
"difficulty": "7/8",
"index": 107,
"problem": "In $\\triangle ABC$, point $M$ lies inside the triangle such that $\\angle MBA = 30^\\circ$ and $\\angle MAB = 10^\\circ$. Given that $\\angle ACB = 80^\\circ$ and $AC = BC$, find $\\angle AMC$."
} | 3 | 107 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutiv... | math | {
"ground_truth": "0.3072",
"style": "rule"
} | {
"answer": "0.3072",
"difficulty": "2/8",
"index": 108,
"problem": "The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point... | 4 | 108 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given \\( n \\), how many ways can we write \\( n \\) as a sum of one or more positive integers \\( a_1 \\leq a_2 \\leq \\cdots \\leq a_k \\) with \\( a_k - a_1 = 0 \\) or \\( 1 \\)? L... | math | {
"ground_truth": "n",
"style": "rule"
} | {
"answer": "n",
"difficulty": "3/8",
"index": 109,
"problem": "Given \\( n \\), how many ways can we write \\( n \\) as a sum of one or more positive integers \\( a_1 \\leq a_2 \\leq \\cdots \\leq a_k \\) with \\( a_k - a_1 = 0 \\) or \\( 1 \\)?"
} | 5 | 109 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For how many three-digit whole numbers does the sum of the digits equal $25$?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 8 \\qquad \\text{(... | math | {
"ground_truth": "(C)\\6",
"style": "rule"
} | {
"answer": "(C)\\6",
"difficulty": "1/8",
"index": 111,
"problem": "For how many three-digit whole numbers does the sum of the digits equal $25$?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 8 \\qquad \\text{(E)}\\ 10$"
} | 7 | 111 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Ava and Tiffany participate in a knockout tournament consisting of a total of 32 players. In each of 5 rounds, the remaining players are paired uniformly at random. In each pair, both ... | math | {
"ground_truth": "116",
"style": "rule"
} | {
"answer": "116",
"difficulty": "0/8",
"index": 112,
"problem": "Ava and Tiffany participate in a knockout tournament consisting of a total of 32 players. In each of 5 rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked ou... | 0 | 112 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An iguana writes the number 1 on the blackboard. Every minute afterwards, if the number $x$ is written, the iguana erases it and either writes $\\frac{1}{x}$ or $x+1$. Can the iguana e... | math | {
"ground_truth": "\\dfrac{20}{17}",
"style": "rule"
} | {
"answer": "\\dfrac{20}{17}",
"difficulty": "5/8",
"index": 113,
"problem": "An iguana writes the number 1 on the blackboard. Every minute afterwards, if the number $x$ is written, the iguana erases it and either writes $\\frac{1}{x}$ or $x+1$. Can the iguana eventually write the number $\\frac{20}{17}$ ?"
} | 1 | 113 |
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 graph of the equation $(x+y)^2=x^2+y^2$?\n$\\textbf{(A) } \\text{the\\,empty\\,set}\\qquad \\textbf{(B) } \\textrm{one\\,point}\\qquad \\textbf{(C)... | math | {
"ground_truth": "\\textbf{(C)}\\",
"style": "rule"
} | {
"answer": "\\textbf{(C)}\\",
"difficulty": "1/8",
"index": 115,
"problem": "Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?\n$\\textbf{(A) } \\text{the\\,empty\\,set}\\qquad \\textbf{(B) } \\textrm{one\\,point}\\qquad \\textbf{(C) } \\textrm{two\\,lines} \\qquad \\textbf{(D) } \\... | 3 | 115 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a class, there are \\( n \\) boys and \\( n \\) girls (\\( n \\geq 3 \\)). They sat around a round table such that no two boys and no two girls sit next to each other. The teacher h... | math | {
"ground_truth": "n \\text{ is odd}",
"style": "rule"
} | {
"answer": "n \\text{ is odd}",
"difficulty": "0/8",
"index": 116,
"problem": "In a class, there are \\( n \\) boys and \\( n \\) girls (\\( n \\geq 3 \\)). They sat around a round table such that no two boys and no two girls sit next to each other. The teacher has \\( 2n \\) cards with the numbers \\( 1, 2, 3... | 4 | 116 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\\angle AEB, \\angle BFC, \\angle CGD, \\angle AHD$ are all right. What is the la... | math | {
"ground_truth": "98",
"style": "rule"
} | {
"answer": "98",
"difficulty": "2/8",
"index": 117,
"problem": "Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\\angle AEB, \\angle BFC, \\angle CGD, \\angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?\n\n[i]Proposed by ... | 5 | 117 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a regular hexagon \\( A_6 \\) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon. Let's thi... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"answer": "13",
"difficulty": "2/8",
"index": 119,
"problem": "Given a regular hexagon \\( A_6 \\) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon."
} | 7 | 119 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "118. A circle is inscribed in a square, and then a new square is inscribed in the circle. Find the ratio of the areas of these squares. Let's think step by step and output the final an... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "6/8",
"index": 120,
"problem": "118. A circle is inscribed in a square, and then a new square is inscribed in the circle. Find the ratio of the areas of these squares."
} | 0 | 120 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Through the center \\( O \\) of an equilateral triangle \\( ABC \\) with side length \\( a \\) runs a line \\( A_1 B_1 \\) which cuts segments \\( x \\) and \\( y \\) from sides \\( AC... | math | {
"ground_truth": "3xy-2a(x+y)+^2=0",
"style": "rule"
} | {
"answer": "3xy-2a(x+y)+^2=0",
"difficulty": "3/8",
"index": 121,
"problem": "Through the center \\( O \\) of an equilateral triangle \\( ABC \\) with side length \\( a \\) runs a line \\( A_1 B_1 \\) which cuts segments \\( x \\) and \\( y \\) from sides \\( AC \\) and \\( BC \\) respectively, where \\( x = A... | 1 | 121 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a $15 \\times 15$ board, some pairs of centers of adjacent cells are connected by segments such that a closed, non-self-intersecting polygonal chain is formed, symmetric with res... | math | {
"ground_truth": "200",
"style": "rule"
} | {
"answer": "200",
"difficulty": "0/8",
"index": 123,
"problem": "Given a $15 \\times 15$ board, some pairs of centers of adjacent cells are connected by segments such that a closed, non-self-intersecting polygonal chain is formed, symmetric with respect to one of the diagonals of the board. Prove that the leng... | 3 | 123 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "3. (50 points) Given that $p$ is a prime number, the fractional part of $\\sqrt{p}$ is $x$, and the fractional part of $\\frac{1}{x}$ is $\\frac{\\sqrt{p}-31}{75}$. Find all prime numb... | math | {
"ground_truth": "2011",
"style": "rule"
} | {
"answer": "2011",
"difficulty": "1/8",
"index": 124,
"problem": "3. (50 points) Given that $p$ is a prime number, the fractional part of $\\sqrt{p}$ is $x$, and the fractional part of $\\frac{1}{x}$ is $\\frac{\\sqrt{p}-31}{75}$. Find all prime numbers $p$ that satisfy the condition."
} | 4 | 124 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Lil writes one of the letters \\( \\text{P}, \\text{Q}, \\text{R}, \\text{S} \\) in each cell of a \\( 2 \\times 4 \\) table. She does this in such a way that, in each row and in each ... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"answer": "24",
"difficulty": "1/8",
"index": 125,
"problem": "Lil writes one of the letters \\( \\text{P}, \\text{Q}, \\text{R}, \\text{S} \\) in each cell of a \\( 2 \\times 4 \\) table. She does this in such a way that, in each row and in each \\( 2 \\times 2 \\) square, all four letters appear. In how man... | 5 | 125 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Compute the number of positive real numbers \\( x \\) that satisfy\n$$\n\\left(3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x\\right)^{16}=2022 x^{13} .\n$$ Let's think step ... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"answer": "9",
"difficulty": "2/8",
"index": 127,
"problem": "Compute the number of positive real numbers \\( x \\) that satisfy\n$$\n\\left(3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x\\right)^{16}=2022 x^{13} .\n$$"
} | 7 | 127 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the hyperbola $$\\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1$$ (a > 0, b > 0), a circle with center at point (b, 0) and radius a is drawn. The circle intersects with one of ... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"answer": "\\sqrt{2}",
"difficulty": "6/8",
"index": 128,
"problem": "Given the hyperbola $$\\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1$$ (a > 0, b > 0), a circle with center at point (b, 0) and radius a is drawn. The circle intersects with one of the asymptotes of the hyperbola at points M and N, and ∠... | 0 | 128 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The equation $AB \\times CD = EFGH$, where each letter $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$ represents a different digit, and the values of $A$, $C$, and $E$ are all nonzero, has... | math | {
"ground_truth": "1058",
"style": "rule"
} | {
"answer": "1058",
"difficulty": "1/8",
"index": 129,
"problem": "The equation $AB \\times CD = EFGH$, where each letter $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$ represents a different digit, and the values of $A$, $C$, and $E$ are all nonzero, has many solutions. For example, $46 \\times 85 = 3910$. Find th... | 1 | 129 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points ... | math | {
"ground_truth": "\\frac{349}{500}",
"style": "rule"
} | {
"answer": "\\frac{349}{500}",
"difficulty": "3/8",
"index": 131,
"problem": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and... | 3 | 131 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a $7 \\times 7$ grid, choose $k$ cells such that the centers of any 4 chosen cells do not form the vertices of a rectangle. Find the maximum value of $k$ that satisfies this conditi... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"answer": "21",
"difficulty": "6/8",
"index": 132,
"problem": "In a $7 \\times 7$ grid, choose $k$ cells such that the centers of any 4 chosen cells do not form the vertices of a rectangle. Find the maximum value of $k$ that satisfies this condition."
} | 4 | 132 |
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