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": "Find the number of pentominoes (5-square polyominoes) that span a 3-by-3 rectangle, where polyominoes that are flips or rotations of each other are considered the same polyomino. Let's... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "1/8",
"index": 400,
"problem": "Find the number of pentominoes (5-square polyominoes) that span a 3-by-3 rectangle, where polyominoes that are flips or rotations of each other are considered the same polyomino."
} | 0 | 400 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the parametric equation of line $l$ is $\\begin{cases} & x=1+3t \\\\ & y=2-4t \\end{cases}$ (where $t$ is the parameter), calculate the cosine of the inclination angle of line $l... | math | {
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
} | {
"answer": "-\\frac{3}{5}",
"difficulty": "7/8",
"index": 401,
"problem": "Given the parametric equation of line $l$ is $\\begin{cases} & x=1+3t \\\\ & y=2-4t \\end{cases}$ (where $t$ is the parameter), calculate the cosine of the inclination angle of line $l$."
} | 1 | 401 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volum... | math | {
"ground_truth": "216",
"style": "rule"
} | {
"answer": "216",
"difficulty": "2/8",
"index": 403,
"problem": "The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes?"
} | 3 | 403 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface... | math | {
"ground_truth": "\\frac{\\sqrt{3} - 1}{4}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{3} - 1}{4}",
"difficulty": "2/8",
"index": 404,
"problem": "The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface."
} | 4 | 404 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A point is randomly thrown on the segment [3, 8] and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\\) satis... | math | {
"ground_truth": "4/15",
"style": "rule"
} | {
"answer": "4/15",
"difficulty": "2/8",
"index": 405,
"problem": "A point is randomly thrown on the segment [3, 8] and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\)."
} | 5 | 405 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( n \\) (\\( \\geqslant 3 \\)) be a positive integer, and \\( M \\) be an \\( n \\)-element set. Find the largest positive integer \\( k \\) that satisfies the following conditio... | math | {
"ground_truth": "\\binom{n-1}{2}",
"style": "rule"
} | {
"answer": "\\binom{n-1}{2}",
"difficulty": "3/8",
"index": 407,
"problem": "Let \\( n \\) (\\( \\geqslant 3 \\)) be a positive integer, and \\( M \\) be an \\( n \\)-element set. Find the largest positive integer \\( k \\) that satisfies the following condition:\n\nThere exists a family \\( \\psi \\) of \\( k... | 7 | 407 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "## Task B-1.5.\n\nOn the way to the sports hall, Marin asked Ivan for the number of his seat. Wanting to reduce the tension before the important sports event and entertain his friend, ... | math | {
"ground_truth": "198",
"style": "rule"
} | {
"answer": "198",
"difficulty": "4/8",
"index": 408,
"problem": "## Task B-1.5.\n\nOn the way to the sports hall, Marin asked Ivan for the number of his seat. Wanting to reduce the tension before the important sports event and entertain his friend, Ivan enigmatically replied: \"All the digits of that number ar... | 0 | 408 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "## Task 23/89\n\nA sequence is given by the formation rule\n\n$$\na_{k}=p \\cdot k \\cdot(k+1)+1\n$$\n\nwhere $p$ is a prime number and $k>0$. The 7th term is the square of a prime num... | math | {
"ground_truth": "(3, 13)",
"style": "rule"
} | {
"answer": "(3, 13)",
"difficulty": "3/8",
"index": 409,
"problem": "## Task 23/89\n\nA sequence is given by the formation rule\n\n$$\na_{k}=p \\cdot k \\cdot(k+1)+1\n$$\n\nwhere $p$ is a prime number and $k>0$. The 7th term is the square of a prime number $P$. Determine all possible pairs $(p ; P)$.\n\n---"
} | 1 | 409 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the plane $Oxy$, the coordinates of point $A$ are defined by the equation $5a^{2} + 12ax + 4ay + 8x^{2} + 8xy + 4y^{2} = 0$, and a parabola with vertex at point $B$ is defined by th... | math | {
"ground_truth": "(-\\infty,-2)\\cup(0,\\frac{2}{3})\\cup(\\frac{8}{7},+\\infty)",
"style": "rule"
} | {
"answer": "(-\\infty,-2)\\cup(0,\\frac{2}{3})\\cup(\\frac{8}{7},+\\infty)",
"difficulty": "1/8",
"index": 411,
"problem": "In the plane $Oxy$, the coordinates of point $A$ are defined by the equation $5a^{2} + 12ax + 4ay + 8x^{2} + 8xy + 4y^{2} = 0$, and a parabola with vertex at point $B$ is defined by the e... | 3 | 411 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For each real number $p > 1$ , find the minimum possible value of the sum $x+y$ , where the numbers $x$ and $y$ satisfy the equation $(x+\\sqrt{1+x^2})(y+\\sqrt{1+y^2}) = p$ . L... | math | {
"ground_truth": "\\frac{p-1}{\\sqrt{p}}",
"style": "rule"
} | {
"answer": "\\frac{p-1}{\\sqrt{p}}",
"difficulty": "2/8",
"index": 412,
"problem": "For each real number $p > 1$ , find the minimum possible value of the sum $x+y$ , where the numbers $x$ and $y$ satisfy the equation $(x+\\sqrt{1+x^2})(y+\\sqrt{1+y^2}) = p$ ."
} | 4 | 412 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Six circles with radii equal to 1 are placed on a plane such that the distance between the centers of any two of them is greater than \\( d \\). What is the smallest \\( d \\) for whic... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "3/8",
"index": 413,
"problem": "Six circles with radii equal to 1 are placed on a plane such that the distance between the centers of any two of them is greater than \\( d \\). What is the smallest \\( d \\) for which there exists a line that does not intersect any of the circles... | 5 | 413 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ be a given quadratic polynomial. Find all functions $f : \\mathbb{R}\\to\\mathbb{R}$ such that $$ f(x+y)=f(x)+f(y)\\text{ and } f(P(x))=f(x)\\text{ for all }x,y\\in\\math... | math | {
"ground_truth": "f(x)=0",
"style": "rule"
} | {
"answer": "f(x)=0",
"difficulty": "0/8",
"index": 415,
"problem": "Let $P$ be a given quadratic polynomial. Find all functions $f : \\mathbb{R}\\to\\mathbb{R}$ such that $$ f(x+y)=f(x)+f(y)\\text{ and } f(P(x))=f(x)\\text{ for all }x,y\\in\\mathbb{R}. $$"
} | 7 | 415 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the rati... | math | {
"ground_truth": "r=\\frac{1}{2}",
"style": "rule"
} | {
"answer": "r=\\frac{1}{2}",
"difficulty": "7/8",
"index": 416,
"problem": "Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of triangle $BPD$ to that of triangle... | 0 | 416 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 420 miles, bought 10 gallons of ... | math | {
"ground_truth": "420",
"style": "rule"
} | {
"answer": "420",
"difficulty": "5/8",
"index": 417,
"problem": "Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 420 miles, bought 10 gallons of gas, and continued driving until his gas tank was three-quarte... | 1 | 417 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years. Let's think step by step and output the final answer w... | math | {
"ground_truth": "0.8",
"style": "rule"
} | {
"answer": "0.8",
"difficulty": "6/8",
"index": 419,
"problem": "The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years."
} | 3 | 419 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Divide 2 teachers and 6 students into 2 groups, each group consisting of 1 teacher and 3 students. Calculate the total number of different arrangements to assign them to locations A an... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"answer": "40",
"difficulty": "6/8",
"index": 420,
"problem": "Divide 2 teachers and 6 students into 2 groups, each group consisting of 1 teacher and 3 students. Calculate the total number of different arrangements to assign them to locations A and B to participate in social practice activities."
} | 4 | 420 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's... | math | {
"ground_truth": "k = 2n",
"style": "rule"
} | {
"answer": "k = 2n",
"difficulty": "3/8",
"index": 421,
"problem": "$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$."
} | 5 | 421 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If the integer part of $\\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}."... | math | {
"ground_truth": "\\sqrt{10} - 3",
"style": "rule"
} | {
"answer": "\\sqrt{10} - 3",
"difficulty": "2/8",
"index": 423,
"problem": "If the integer part of $\\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\\_\\_\\_\\_\\_\\_$."
} | 7 | 423 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as \"division exponentiation,\" denot... | math | {
"ground_truth": "-73",
"style": "rule"
} | {
"answer": "-73",
"difficulty": "2/8",
"index": 424,
"problem": "Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as \"division exponentiation,\" denoted as $a^{ⓝ}$, read as \"$a$ circle $n$ times.\" For example, ... | 0 | 424 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On an island of knights and liars (liars always lie, knights always tell the truth), each person supports exactly one football team. All island inhabitants participated in a survey. To... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"answer": "30",
"difficulty": "4/8",
"index": 425,
"problem": "On an island of knights and liars (liars always lie, knights always tell the truth), each person supports exactly one football team. All island inhabitants participated in a survey. To the question \"Do you support 'Rostov'?\", 40% of the inhabita... | 1 | 425 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine... | math | {
"ground_truth": "49",
"style": "rule"
} | {
"answer": "49",
"difficulty": "6/8",
"index": 427,
"problem": "Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine the minimum number of shots, n, she must get as bullseyes to g... | 3 | 427 |
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": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "0/8",
"index": 428,
"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: `P... | 4 | 428 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "4. Given that the center of an ellipse is at the origin, and its foci are on the coordinate axes, the line $y=x+1$ intersects the ellipse at points $P, Q$, and $O P \\perp O Q,|P Q|=\\... | math | {
"ground_truth": "\\dfrac{x^2}{2} + \\dfrac{y^2}{\\dfrac{2}{3}} = 1",
"style": "rule"
} | {
"answer": "\\dfrac{x^2}{2} + \\dfrac{y^2}{\\dfrac{2}{3}} = 1",
"difficulty": "1/8",
"index": 429,
"problem": "4. Given that the center of an ellipse is at the origin, and its foci are on the coordinate axes, the line $y=x+1$ intersects the ellipse at points $P, Q$, and $O P \\perp O Q,|P Q|=\\frac{\\sqrt{10}}... | 5 | 429 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A line passing through the center of the circumscribed circle and the intersection point of the altitudes of a scalene triangle \\(ABC\\) divides its perimeter and area in the same rat... | math | {
"ground_truth": "1:1",
"style": "rule"
} | {
"answer": "1:1",
"difficulty": "1/8",
"index": 431,
"problem": "A line passing through the center of the circumscribed circle and the intersection point of the altitudes of a scalene triangle \\(ABC\\) divides its perimeter and area in the same ratio. Find this ratio."
} | 7 | 431 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Arrange the numbers in the set \\(\\left\\{2^{x}+2^{y} \\mid x, y\\ \\text{are non-negative integers,}\\ x < y\\right\\}\\) in ascending order. What is the 60th number? (Answer in digi... | math | {
"ground_truth": "2064",
"style": "rule"
} | {
"answer": "2064",
"difficulty": "6/8",
"index": 432,
"problem": "Arrange the numbers in the set \\(\\left\\{2^{x}+2^{y} \\mid x, y\\ \\text{are non-negative integers,}\\ x < y\\right\\}\\) in ascending order. What is the 60th number? (Answer in digits)."
} | 0 | 432 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "How many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? Let's t... | math | {
"ground_truth": "432",
"style": "rule"
} | {
"answer": "432",
"difficulty": "3/8",
"index": 433,
"problem": "How many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors?"
} | 1 | 433 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The fraction halfway between $\\frac{1}{5}$ and $\\frac{1}{3}$ (on the number line) is\n\n$\\text{(A)}\\ \\frac{1}{4} \\qquad \\text{(B)}\\ \\frac{2}{15} \\qquad \\text{(C)}\\ \\frac{4... | math | {
"ground_truth": "C",
"style": "rule"
} | {
"answer": "C",
"difficulty": "0/8",
"index": 435,
"problem": "The fraction halfway between $\\frac{1}{5}$ and $\\frac{1}{3}$ (on the number line) is\n\n$\\text{(A)}\\ \\frac{1}{4} \\qquad \\text{(B)}\\ \\frac{2}{15} \\qquad \\text{(C)}\\ \\frac{4}{15} \\qquad \\text{(D)}\\ \\frac{53}{200} \\qquad \\text{(E)}\... | 3 | 435 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, “My number is probably greater than yours.” Bob repudiates, saying, “No, my numbe... | math | {
"ground_truth": "\\frac{11}{12}",
"style": "rule"
} | {
"answer": "\\frac{11}{12}",
"difficulty": "1/8",
"index": 436,
"problem": "Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, “My number is probably greater than yours.” Bob repudiates, saying, “No, my number is probably greater than yours!” Alice concedes, ... | 4 | 436 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given \\( 0 \\leq x \\leq \\pi \\) and \\( 0 \\leq y \\leq 1 \\), find the minimum value of the function \\( f(x, y) = (2y - 1) \\sin x + (1 - y) \\sin ((1 - y) x) \\). Let's think ste... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "2/8",
"index": 437,
"problem": "Given \\( 0 \\leq x \\leq \\pi \\) and \\( 0 \\leq y \\leq 1 \\), find the minimum value of the function \\( f(x, y) = (2y - 1) \\sin x + (1 - y) \\sin ((1 - y) x) \\)."
} | 5 | 437 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Suppose the quadratic function \\( f(x)=a x^{2}+b x+c \\) (where \\( a, b, c \\in \\mathbf{R} \\) and \\( a \\neq 0 \\)) satisfies the following conditions:\n(1) For \\( x \\in \\mathb... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"answer": "9",
"difficulty": "6/8",
"index": 439,
"problem": "Suppose the quadratic function \\( f(x)=a x^{2}+b x+c \\) (where \\( a, b, c \\in \\mathbf{R} \\) and \\( a \\neq 0 \\)) satisfies the following conditions:\n(1) For \\( x \\in \\mathbf{R} \\), \\( f(x-4)=f(2-x) \\), and \\( f(x) \\geqslant x \\);\... | 7 | 439 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals o... | math | {
"ground_truth": "192",
"style": "rule"
} | {
"answer": "192",
"difficulty": "7/8",
"index": 440,
"problem": "The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals of the base and is 6 cm. Determine the lateral surface area of ... | 0 | 440 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Fox Alice thought of a two-digit number and told Pinocchio that this number is divisible by $2, 3, 4, 5,$ and $6$. However, Pinocchio found out that exactly two of these five statement... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"answer": "8",
"difficulty": "4/8",
"index": 441,
"problem": "Fox Alice thought of a two-digit number and told Pinocchio that this number is divisible by $2, 3, 4, 5,$ and $6$. However, Pinocchio found out that exactly two of these five statements are actually false. What numbers could Fox Alice have thought ... | 1 | 441 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that the perimeter of each of the nine small equilateral triangles is $6 \\mathrm{cm}$, calculate the perimeter of $\\triangle A B C$. Let's think step by step and output the fin... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"answer": "18",
"difficulty": "1/8",
"index": 443,
"problem": "Given that the perimeter of each of the nine small equilateral triangles is $6 \\mathrm{cm}$, calculate the perimeter of $\\triangle A B C$."
} | 3 | 443 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "ABC is an equilateral triangle. D is on the side AB and E is on the side AC such that DE touches the incircle. Show that AD/DB + AE/EC = 1. Let's think step by step and output the fina... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "5/8",
"index": 444,
"problem": "ABC is an equilateral triangle. D is on the side AB and E is on the side AC such that DE touches the incircle. Show that AD/DB + AE/EC = 1."
} | 4 | 444 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switch... | math | {
"ground_truth": "499",
"style": "rule"
} | {
"answer": "499",
"difficulty": "6/8",
"index": 445,
"problem": "There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switches numbered 2, 3, and 5 are flipped, how many lamps remain on?... | 5 | 445 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Which fraction \\( \\frac{p}{q} \\), where \\( p \\) and \\( q \\) are positive integers less than 100, is closest to \\( \\sqrt{2} \\)? Find all digits after the decimal point in the ... | math | {
"ground_truth": "\\frac{99}{70}",
"style": "rule"
} | {
"answer": "\\frac{99}{70}",
"difficulty": "2/8",
"index": 447,
"problem": "Which fraction \\( \\frac{p}{q} \\), where \\( p \\) and \\( q \\) are positive integers less than 100, is closest to \\( \\sqrt{2} \\)? Find all digits after the decimal point in the decimal representation of this fraction that coinci... | 7 | 447 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a cyclic quadrilateral \\(A B C D\\), the rays \\(A B\\) and \\(D C\\) intersect at point \\(K\\). It is found that the points \\(B\\), \\(D\\), and the midpoints of segments \\(... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"answer": "90",
"difficulty": "0/8",
"index": 448,
"problem": "Given a cyclic quadrilateral \\(A B C D\\), the rays \\(A B\\) and \\(D C\\) intersect at point \\(K\\). It is found that the points \\(B\\), \\(D\\), and the midpoints of segments \\(A C\\) and \\(K C\\) lie on the same circle. What values can th... | 0 | 448 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the ci... | math | {
"ground_truth": "95",
"style": "rule"
} | {
"answer": "95",
"difficulty": "4/8",
"index": 449,
"problem": "Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$,... | 1 | 449 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minute... | math | {
"ground_truth": "22",
"style": "rule"
} | {
"answer": "22",
"difficulty": "7/8",
"index": 451,
"problem": "This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours... | 3 | 451 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the kite \\(ABCD\\), \\(AB = AD\\) and \\(BC = CD\\). Let \\(O\\) be the intersection point of \\(AC\\) and \\(BD\\). Draw two lines through \\(O\\), which intersect \\(AD\\) at \\(... | math | {
"ground_truth": "IO=OJ",
"style": "rule"
} | {
"answer": "IO=OJ",
"difficulty": "2/8",
"index": 452,
"problem": "In the kite \\(ABCD\\), \\(AB = AD\\) and \\(BC = CD\\). Let \\(O\\) be the intersection point of \\(AC\\) and \\(BD\\). Draw two lines through \\(O\\), which intersect \\(AD\\) at \\(E\\) and \\(BC\\) at \\(F\\), \\(AB\\) at \\(G\\) and \\(CD\... | 4 | 452 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On the side \\( BC \\) of the isosceles triangle \\( ABC \\), a circle is constructed with \\( BC \\) as its diameter, intersecting the base of this triangle at point \\( D \\). Find t... | math | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"answer": "\\sqrt{7}",
"difficulty": "4/8",
"index": 453,
"problem": "On the side \\( BC \\) of the isosceles triangle \\( ABC \\), a circle is constructed with \\( BC \\) as its diameter, intersecting the base of this triangle at point \\( D \\). Find the distance from vertex \\( A \\) to the center of the c... | 5 | 453 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and \\[5^n<2^m<2^{m+2}<5^{n+1}?\\]\n$\\textbf{(A... | math | {
"ground_truth": "\\textbf{(B)}279",
"style": "rule"
} | {
"answer": "\\textbf{(B)}279",
"difficulty": "1/8",
"index": 455,
"problem": "The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and \\[5^n<2^m<2^{m+2}<5^{n+1}?\\]\n$\\textbf{(A) }278\\qquad \\textbf{(B) }279\\qquad \\textbf{(... | 7 | 455 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and ... | math | {
"ground_truth": "p\\equiv1\\pmod{4}",
"style": "rule"
} | {
"answer": "p\\equiv1\\pmod{4}",
"difficulty": "7/8",
"index": 456,
"problem": "Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\\mid f(i)f(-i)$ , where $i$ is the imag... | 0 | 456 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a natural number \\( n \\), we write out all fractions of the form \\( \\frac{1}{pq} \\), where \\( p \\) and \\( q \\) are coprime, \\( 0 < p < q \\leq n \\), and \\( p + q > n ... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\frac{1}{2}",
"difficulty": "0/8",
"index": 457,
"problem": "Given a natural number \\( n \\), we write out all fractions of the form \\( \\frac{1}{pq} \\), where \\( p \\) and \\( q \\) are coprime, \\( 0 < p < q \\leq n \\), and \\( p + q > n \\). Prove that the sum of all such fractions is \\( ... | 1 | 457 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If $m$ is a root of the equation $4^{x+ \\frac {1}{2}}-9\\cdot2^{x}+4=0$, then the eccentricity of the conic section $x^{2}+ \\frac {y^{2}}{m}=1$ is \\_\\_\\_\\_\\_\\_. Let's think ste... | math | {
"ground_truth": "\\sqrt {2}",
"style": "rule"
} | {
"answer": "\\sqrt {2}",
"difficulty": "2/8",
"index": 459,
"problem": "If $m$ is a root of the equation $4^{x+ \\frac {1}{2}}-9\\cdot2^{x}+4=0$, then the eccentricity of the conic section $x^{2}+ \\frac {y^{2}}{m}=1$ is \\_\\_\\_\\_\\_\\_."
} | 3 | 459 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( S = \\{1, 2, \\ldots, n\\} \\), and let \\( T \\) be the set consisting of all nonempty subsets of \\( S \\). The function \\( f: T \\rightarrow S \\) is \"garish\" if there do... | math | {
"ground_truth": "n!",
"style": "rule"
} | {
"answer": "n!",
"difficulty": "2/8",
"index": 460,
"problem": "Let \\( S = \\{1, 2, \\ldots, n\\} \\), and let \\( T \\) be the set consisting of all nonempty subsets of \\( S \\). The function \\( f: T \\rightarrow S \\) is \"garish\" if there do not exist sets \\( A, B \\in T \\) such that \\( A \\) is a pr... | 4 | 460 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "62nd Putnam 2001 Problem B3 Let @n denote the closest integer to √n. Find ∑ (2 @n + 2 -@n )/2 n , where the sum is taken from n = 1 to n = infinity. Solution Let's think step by step a... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "0/8",
"index": 461,
"problem": "62nd Putnam 2001 Problem B3 Let @n denote the closest integer to √n. Find ∑ (2 @n + 2 -@n )/2 n , where the sum is taken from n = 1 to n = infinity. Solution"
} | 5 | 461 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \\fra... | math | {
"ground_truth": "\\frac{3 \\sqrt{2}}{2}",
"style": "rule"
} | {
"answer": "\\frac{3 \\sqrt{2}}{2}",
"difficulty": "6/8",
"index": 463,
"problem": "In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \\frac{\\sqrt{2}}{2}$. The point $P(2, 1)$ lies ... | 7 | 463 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many ... | math | {
"ground_truth": "250500",
"style": "rule"
} | {
"answer": "250500",
"difficulty": "4/8",
"index": 464,
"problem": "The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible?"
} | 0 | 464 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A plane intersects each side of the quadrilateral \\(ABCD\\) at an internal point. Following a traversing direction, we write down the ratio in which each intersection point divides it... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "4/8",
"index": 465,
"problem": "A plane intersects each side of the quadrilateral \\(ABCD\\) at an internal point. Following a traversing direction, we write down the ratio in which each intersection point divides its corresponding side. What will be the product of the four ratio... | 1 | 465 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "[ Examples and counterexamples. Constructions ] [ Partitions into pairs and groups; bijections ] [ Decimal number system ]\n\nA subset $X$ of the set of \"two-digit\" numbers $00,01,... | math | {
"ground_truth": "55",
"style": "rule"
} | {
"answer": "55",
"difficulty": "4/8",
"index": 467,
"problem": "[ Examples and counterexamples. Constructions ] [ Partitions into pairs and groups; bijections ] [ Decimal number system ]\n\nA subset $X$ of the set of \"two-digit\" numbers $00,01, \\ldots, 98,99$ is such that in any infinite sequence of digit... | 3 | 467 |
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 the expression \\(\\cos ^{4} \\frac{7 \\pi}{24}+\\sin ^{4} \\frac{11 \\pi}{24}+\\sin ^{4} \\frac{17 \\pi}{24}+\\cos ^{4} \\frac{13 \\pi}{24}\\). Let's think step by s... | math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"answer": "\\frac{3}{2}",
"difficulty": "5/8",
"index": 468,
"problem": "Find the value of the expression \\(\\cos ^{4} \\frac{7 \\pi}{24}+\\sin ^{4} \\frac{11 \\pi}{24}+\\sin ^{4} \\frac{17 \\pi}{24}+\\cos ^{4} \\frac{13 \\pi}{24}\\)."
} | 4 | 468 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On a straight line $\\ell$, there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing ... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"answer": "30",
"difficulty": "3/8",
"index": 469,
"problem": "On a straight line $\\ell$, there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is fo... | 5 | 469 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be an integer number greater than $2$ , let $x_{1},x_{2},\\ldots ,x_{n}$ be $n$ positive real numbers such that\n\\[\\sum_{i=1}^{n}\\frac{1}{x_{i}+1}=1\\]\nand let $k$ ... | math | {
"ground_truth": "\\frac{n}{(n-1)^k+1}",
"style": "rule"
} | {
"answer": "\\frac{n}{(n-1)^k+1}",
"difficulty": "6/8",
"index": 471,
"problem": "Let $n$ be an integer number greater than $2$ , let $x_{1},x_{2},\\ldots ,x_{n}$ be $n$ positive real numbers such that\n\\[\\sum_{i=1}^{n}\\frac{1}{x_{i}+1}=1\\]\nand let $k$ be a real number greater than $1$ . Show th... | 7 | 471 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the functions\n$$\nf(x)=a \\sin x(a \\in \\mathbf{R}), g(x)=\\mathrm{e}^{x}.\n$$\n(1) If $0<a \\leqslant 1$, determine the monotonicity of\n$$\nF(x)=f(1-x)+\\ln x\n$$\nin the int... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "6/8",
"index": 472,
"problem": "Given the functions\n$$\nf(x)=a \\sin x(a \\in \\mathbf{R}), g(x)=\\mathrm{e}^{x}.\n$$\n(1) If $0<a \\leqslant 1$, determine the monotonicity of\n$$\nF(x)=f(1-x)+\\ln x\n$$\nin the interval $(0,1)$;\n\n(2) Prove that:\n$$\n\\sin \\frac{1}{2^{2}}+\\... | 0 | 472 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the plane quadrilateral $\\mathrm{ABCD}$, given $\\mathrm{AB}=1, \\mathrm{BC}=4, \\mathrm{CD}=2, \\mathrm{DA}=3$, find the value of $\\overrightarrow{\\mathrm{AC}} \\cdot \\overrigh... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"answer": "10",
"difficulty": "6/8",
"index": 473,
"problem": "In the plane quadrilateral $\\mathrm{ABCD}$, given $\\mathrm{AB}=1, \\mathrm{BC}=4, \\mathrm{CD}=2, \\mathrm{DA}=3$, find the value of $\\overrightarrow{\\mathrm{AC}} \\cdot \\overrightarrow{\\mathrm{BD}}$."
} | 1 | 473 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the least upper bound for the set of values \\((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\\), where \\(x_i\\) are real numbers, not all zero. Let's think ste... | math | {
"ground_truth": "\\frac{\\sqrt{2}+1}{2}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{2}+1}{2}",
"difficulty": "0/8",
"index": 475,
"problem": "Find the least upper bound for the set of values \\((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\\), where \\(x_i\\) are real numbers, not all zero."
} | 3 | 475 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Mientka Publishing Company prices its bestseller Where's Walter? as follows:\n$C(n) = \\begin{cases} 12n, & \\text{if } 1 \\le n \\le 24 \\\\ 11n, & \\text{if } 25 \\le n \\le 48 \\\\ ... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "7/8",
"index": 476,
"problem": "Mientka Publishing Company prices its bestseller Where's Walter? as follows:\n$C(n) = \\begin{cases} 12n, & \\text{if } 1 \\le n \\le 24 \\\\ 11n, & \\text{if } 25 \\le n \\le 48 \\\\ 10n, & \\text{if } 49 \\le n \\end{cases}$\nwhere $n$ is the num... | 4 | 476 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( f(x) \\) be a function defined on \\( \\mathbf{R} \\) such that \\( f(0) = 0 \\), \\( f(x) + f(1-x) = 1 \\), \\( f\\left(\\frac{x}{3}\\right) = \\frac{1}{2} f(x) \\), and for \... | math | {
"ground_truth": "\\dfrac{1}{128}",
"style": "rule"
} | {
"answer": "\\dfrac{1}{128}",
"difficulty": "5/8",
"index": 477,
"problem": "Let \\( f(x) \\) be a function defined on \\( \\mathbf{R} \\) such that \\( f(0) = 0 \\), \\( f(x) + f(1-x) = 1 \\), \\( f\\left(\\frac{x}{3}\\right) = \\frac{1}{2} f(x) \\), and for \\( 0 \\leqslant x_{1} < x_{2} \\leqslant 1 \\), we... | 5 | 477 |
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 \\), let \\( N=\\varphi(1)+\\varphi(2)+\\ldots+\\varphi(n) \\). Show that there exists a sequence \n\\[ \na_{1}, a_{2}, \\ldots, a_{N}\n\\]\ncontaining e... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "0/8",
"index": 479,
"problem": "For any positive integer \\( n \\), let \\( N=\\varphi(1)+\\varphi(2)+\\ldots+\\varphi(n) \\). Show that there exists a sequence \n\\[ \na_{1}, a_{2}, \\ldots, a_{N}\n\\]\ncontaining exactly \\( \\varphi(k) \\) instances of \\( k \\) for all positi... | 7 | 479 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints. Let's think step ... | math | {
"ground_truth": "9\\pi",
"style": "rule"
} | {
"answer": "9\\pi",
"difficulty": "4/8",
"index": 480,
"problem": "Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints."
} | 0 | 480 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume t... | math | {
"ground_truth": "503",
"style": "rule"
} | {
"answer": "503",
"difficulty": "1/8",
"index": 481,
"problem": "Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at a... | 1 | 481 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "4. In $\\triangle A B C$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are denoted as $a$, $b$, and $c$ respectively. If $b^{2}=a^{2}+c^{2}-a c$, and $c-a$ equals th... | math | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule"
} | {
"answer": "\\dfrac{1}{2}",
"difficulty": "3/8",
"index": 483,
"problem": "4. In $\\triangle A B C$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are denoted as $a$, $b$, and $c$ respectively. If $b^{2}=a^{2}+c^{2}-a c$, and $c-a$ equals the height $h$ from vertex $A$ to side $A C$, then $\\... | 3 | 483 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For given integer $n \\geq 3$ , set $S =\\{p_1, p_2, \\cdots, p_m\\}$ consists of permutations $p_i$ of $(1, 2, \\cdots, n)$ . Suppose that among every three distinct numbers in ... | math | {
"ground_truth": "2^{n-1}",
"style": "rule"
} | {
"answer": "2^{n-1}",
"difficulty": "2/8",
"index": 484,
"problem": "For given integer $n \\geq 3$ , set $S =\\{p_1, p_2, \\cdots, p_m\\}$ consists of permutations $p_i$ of $(1, 2, \\cdots, n)$ . Suppose that among every three distinct numbers in $\\{1, 2, \\cdots, n\\}$ , one of these number does not l... | 4 | 484 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are ne... | math | {
"ground_truth": "351",
"style": "rule"
} | {
"answer": "351",
"difficulty": "3/8",
"index": 485,
"problem": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same... | 5 | 485 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Problem 3. A square with side length 5 is divided into unit squares by parallel to its sides lines. Let $A$ be the set of the vertexes of the unit squares which are not on the sides of... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "4/8",
"index": 487,
"problem": "Problem 3. A square with side length 5 is divided into unit squares by parallel to its sides lines. Let $A$ be the set of the vertexes of the unit squares which are not on the sides of the given square. How many points from $A$ can be chosen at mos... | 7 | 487 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Determine the number of 8-digit positive integers such that after deleting any one digit, the remaining 7-digit number is divisible by 7. Let's think step by step and output the final ... | math | {
"ground_truth": "64",
"style": "rule"
} | {
"answer": "64",
"difficulty": "0/8",
"index": 488,
"problem": "Determine the number of 8-digit positive integers such that after deleting any one digit, the remaining 7-digit number is divisible by 7."
} | 0 | 488 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(P(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \\ldots + a_{1} x + a_{0}\\), where \\(n\\) is a natural number. It is known that the numbers \\(a_{0}, a_{1}, \\ldots, a_{n}\\) are integ... | math | {
"ground_truth": "2021^2",
"style": "rule"
} | {
"answer": "2021^2",
"difficulty": "4/8",
"index": 489,
"problem": "Let \\(P(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \\ldots + a_{1} x + a_{0}\\), where \\(n\\) is a natural number. It is known that the numbers \\(a_{0}, a_{1}, \\ldots, a_{n}\\) are integers, with \\(a_{n} \\neq 0\\), \\(a_{n-k} = a_{k}\\) for al... | 1 | 489 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a country, some cities are connected by direct bus routes. The residents of this country consider an odd natural number \\( n \\) to be unlucky, so there does not exist a cyclic rou... | math | {
"ground_truth": "n = 101",
"style": "rule"
} | {
"answer": "n = 101",
"difficulty": "3/8",
"index": 491,
"problem": "In a country, some cities are connected by direct bus routes. The residents of this country consider an odd natural number \\( n \\) to be unlucky, so there does not exist a cyclic route consisting of exactly \\( n \\) trips (and not entering... | 3 | 491 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On side AC of triangle ABC with a 120-degree angle at vertex B, points D and E are marked such that AD = AB and CE = CB. A perpendicular DF is dropped from point D to line BE. Find the... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "5/8",
"index": 492,
"problem": "On side AC of triangle ABC with a 120-degree angle at vertex B, points D and E are marked such that AD = AB and CE = CB. A perpendicular DF is dropped from point D to line BE. Find the ratio BD / DF."
} | 4 | 492 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the second-order derivative \\( y_{xx}^{\\prime \\prime} \\) of the function defined parametrically by:\n\n\\[\n\\begin{cases}\nx = \\cos t \\\\\ny = \\ln (\\sin t)\n\\end{cases}\... | math | {
"ground_truth": "-\\frac{1+\\cos^2}{\\sin^4}",
"style": "rule"
} | {
"answer": "-\\frac{1+\\cos^2}{\\sin^4}",
"difficulty": "3/8",
"index": 493,
"problem": "Find the second-order derivative \\( y_{xx}^{\\prime \\prime} \\) of the function defined parametrically by:\n\n\\[\n\\begin{cases}\nx = \\cos t \\\\\ny = \\ln (\\sin t)\n\\end{cases}\n\\]"
} | 5 | 493 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Compute \\((1+i^{-100}) + (2+i^{-99}) + (3+i^{-98}) + \\cdots + (101+i^0) + (102+i^1) + \\cdots + (201+i^{100})\\). Let's think step by step and output the final answer within \\boxed{... | math | {
"ground_truth": "20302",
"style": "rule"
} | {
"answer": "20302",
"difficulty": "6/8",
"index": 495,
"problem": "Compute \\((1+i^{-100}) + (2+i^{-99}) + (3+i^{-98}) + \\cdots + (101+i^0) + (102+i^1) + \\cdots + (201+i^{100})\\)."
} | 7 | 495 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For the quadrilateral \\(ABCD\\), it is known that \\(AB = BD\\), \\(\\angle ABD = \\angle DBC\\), and \\(\\angle BCD = 90^\\circ\\). On segment \\(BC\\), point \\(E\\) is marked such ... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"answer": "17",
"difficulty": "3/8",
"index": 496,
"problem": "For the quadrilateral \\(ABCD\\), it is known that \\(AB = BD\\), \\(\\angle ABD = \\angle DBC\\), and \\(\\angle BCD = 90^\\circ\\). On segment \\(BC\\), point \\(E\\) is marked such that \\(AD = DE\\). What is the length of segment \\(BD\\) if i... | 0 | 496 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. Let's think step by step and output th... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"answer": "21",
"difficulty": "1/8",
"index": 497,
"problem": "On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted."
} | 1 | 497 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Pentagon $J A M E S$ is such that $A M=S J$ and the internal angles satisfy $\\angle J=\\angle A=\\angle E=90^{\\circ}$, and $\\angle M=\\angle S$. Given that there exists a diagonal o... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"answer": "\\frac{1}{4}",
"difficulty": "0/8",
"index": 499,
"problem": "Pentagon $J A M E S$ is such that $A M=S J$ and the internal angles satisfy $\\angle J=\\angle A=\\angle E=90^{\\circ}$, and $\\angle M=\\angle S$. Given that there exists a diagonal of $J A M E S$ that bisects its area, find the ratio o... | 3 | 499 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Observe the following equations:\n\n1. $\\cos 2\\alpha = 2\\cos^2\\alpha - 1$;\n2. $\\cos 4\\alpha = 8\\cos^4\\alpha - 8\\cos^2\\alpha + 1$;\n3. $\\cos 6\\alpha = 32\\cos^6\\alpha - 48... | math | {
"ground_truth": "962",
"style": "rule"
} | {
"answer": "962",
"difficulty": "7/8",
"index": 500,
"problem": "Observe the following equations:\n\n1. $\\cos 2\\alpha = 2\\cos^2\\alpha - 1$;\n2. $\\cos 4\\alpha = 8\\cos^4\\alpha - 8\\cos^2\\alpha + 1$;\n3. $\\cos 6\\alpha = 32\\cos^6\\alpha - 48\\cos^4\\alpha + 18\\cos^2\\alpha - 1$;\n4. $\\cos 8\\alpha = ... | 4 | 500 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Amon... | math | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"answer": "\\frac{5}{16}",
"difficulty": "4/8",
"index": 501,
"problem": "A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there... | 5 | 501 |
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": "german",
"style": "rule"
} | {
"answer": "german",
"difficulty": "0/8",
"index": 503,
"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 nam... | 7 | 503 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\\overline{AC}$ with $V$ on $\\overline{AW}$, points $X$ and $Y$ are on $\\overline{BC}$ with $X$ on ... | math | {
"ground_truth": "318",
"style": "rule"
} | {
"answer": "318",
"difficulty": "1/8",
"index": 504,
"problem": "In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\\overline{AC}$ with $V$ on $\\overline{AW}$, points $X$ and $Y$ are on $\\overline{BC}$ with $X$ on $\\overline{CY}$, and points $Z$ and $U$ are on $\\overline{AB... | 0 | 504 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "# Task 14. (1 point)\n\nCalculate the amount of personal income tax (NDFL) paid by Sergey for the past year, if he is a Russian resident and during this period had a stable income of 3... | math | {
"ground_truth": "10400",
"style": "rule"
} | {
"answer": "10400",
"difficulty": "0/8",
"index": 505,
"problem": "# Task 14. (1 point)\n\nCalculate the amount of personal income tax (NDFL) paid by Sergey for the past year, if he is a Russian resident and during this period had a stable income of 30000 rub./month and a one-time vacation bonus of 20000 rub. ... | 1 | 505 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given in the figure:\n- The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$.\n- The area of the triangle $\\triangle FEC$ is $40$ cm$^2$.\n- The sum of the... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"answer": "60",
"difficulty": "3/8",
"index": 507,
"problem": "Given in the figure:\n- The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$.\n- The area of the triangle $\\triangle FEC$ is $40$ cm$^2$.\n- The sum of the areas of the triangles $\\triangle ADF$ and $\\triangle DBE$ i... | 3 | 507 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the sum of all positive integers $n$ such that $\\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3... | math | {
"ground_truth": "2345",
"style": "rule"
} | {
"answer": "2345",
"difficulty": "7/8",
"index": 508,
"problem": "Let $S$ be the sum of all positive integers $n$ such that $\\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3$ . Compute $\\frac{S}{36}$ ."
} | 4 | 508 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The graph of \\(y = h(x)\\) is defined in parts as follows:\n1. For \\(x \\geq -4\\) and \\(x \\leq 1\\), \\(y = 1 - x\\).\n2. For \\(x \\geq 1\\) and \\(x \\leq 3\\), \\(y = \\sqrt{4 ... | math | {
"ground_truth": "j(x) = h(6 - x)",
"style": "rule"
} | {
"answer": "j(x) = h(6 - x)",
"difficulty": "5/8",
"index": 509,
"problem": "The graph of \\(y = h(x)\\) is defined in parts as follows:\n1. For \\(x \\geq -4\\) and \\(x \\leq 1\\), \\(y = 1 - x\\).\n2. For \\(x \\geq 1\\) and \\(x \\leq 3\\), \\(y = \\sqrt{4 - (x - 3)^2} + 1\\).\n3. For \\(x \\geq 3\\) and \... | 5 | 509 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The legs of a compass are positioned at the nodes of an infinite sheet of ruled paper, where the cells are squares with side length 1. It is allowed, without changing the compass openi... | math | {
"ground_truth": "\\text{No}",
"style": "rule"
} | {
"answer": "\\text{No}",
"difficulty": "3/8",
"index": 511,
"problem": "The legs of a compass are positioned at the nodes of an infinite sheet of ruled paper, where the cells are squares with side length 1. It is allowed, without changing the compass opening, to move one leg to another node on the sheet by rot... | 7 | 511 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The residents of the village of Razumeyevo, which is 3 km away from the river, like to visit the village of Vkusnoteevo, located 3.25 km downstream on the opposite bank of the river an... | math | {
"ground_truth": "1.5",
"style": "rule"
} | {
"answer": "1.5",
"difficulty": "0/8",
"index": 512,
"problem": "The residents of the village of Razumeyevo, which is 3 km away from the river, like to visit the village of Vkusnoteevo, located 3.25 km downstream on the opposite bank of the river and 1 km away from the riverbank. The river width is 500 meters,... | 0 | 512 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A professional scriptwriter named Alex is working on a new holiday movie screenplay. To make the movie unique, Alex decides to include 5 different holiday traditions from various cultu... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"answer": "-3",
"difficulty": "0/8",
"index": 513,
"problem": "A professional scriptwriter named Alex is working on a new holiday movie screenplay. To make the movie unique, Alex decides to include 5 different holiday traditions from various cultures. Each tradition is to be featured in a 10-minute scene. Ale... | 1 | 513 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\\ 45$ represents $3$ minutes, $45$ seconds, wh... | math | {
"ground_truth": "147.5",
"style": "rule"
} | {
"answer": "147.5",
"difficulty": "7/8",
"index": 515,
"problem": "The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median lengt... | 3 | 515 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the figure, there is a square $A B C D$, in which four identical isosceles triangles $A B E, B C F, C D G$, and $D A H$ are placed, all shaded gray. The sides of the square $A B C D... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"answer": "24",
"difficulty": "2/8",
"index": 516,
"problem": "In the figure, there is a square $A B C D$, in which four identical isosceles triangles $A B E, B C F, C D G$, and $D A H$ are placed, all shaded gray. The sides of the square $A B C D$ are the bases of these isosceles triangles. We know that the ... | 4 | 516 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500. Let's think step by step and output the final... | math | {
"ground_truth": "1587",
"style": "rule"
} | {
"answer": "1587",
"difficulty": "4/8",
"index": 517,
"problem": "Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500."
} | 5 | 517 |
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": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "4/8",
"index": 519,
"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: `P... | 7 | 519 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\\omega$ at $B$ meets line $AC... | math | {
"ground_truth": "2 \\sqrt{42}",
"style": "rule"
} | {
"answer": "2 \\sqrt{42}",
"difficulty": "5/8",
"index": 520,
"problem": "Let $\\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F... | 0 | 520 |
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}, \\ldots, x_{1970} \\) be positive integers satisfying \\( x_{1} + x_{2} + \\cdots + x_{1970} = 2007 \\). Determine the largest possible value of \\( x_{1}^{3} + x... | math | {
"ground_truth": "56841",
"style": "rule"
} | {
"answer": "56841",
"difficulty": "5/8",
"index": 521,
"problem": "Let \\( x_{1}, x_{2}, \\ldots, x_{1970} \\) be positive integers satisfying \\( x_{1} + x_{2} + \\cdots + x_{1970} = 2007 \\). Determine the largest possible value of \\( x_{1}^{3} + x_{2}^{3} + \\cdots + x_{1970}^{3} \\)."
} | 1 | 521 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "87. Bobo has 640 points, Guagua has 1000 points. Now both of them exchange their points for the same watch. As a result, the points Guagua has left are 4 times the points Bobo has left... | math | {
"ground_truth": "520",
"style": "rule"
} | {
"answer": "520",
"difficulty": "2/8",
"index": 523,
"problem": "87. Bobo has 640 points, Guagua has 1000 points. Now both of them exchange their points for the same watch. As a result, the points Guagua has left are 4 times the points Bobo has left. Therefore, exchanging for such a watch requires $\\qquad$ po... | 3 | 523 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "4. The random variable $a$ is uniformly distributed on the interval $[-2 ; 1]$. Find the probability that all roots of the quadratic equation $a x^{2}-x-4 a+1=0$ in absolute value exce... | math | {
"ground_truth": "\\dfrac{7}{9}",
"style": "rule"
} | {
"answer": "\\dfrac{7}{9}",
"difficulty": "2/8",
"index": 524,
"problem": "4. The random variable $a$ is uniformly distributed on the interval $[-2 ; 1]$. Find the probability that all roots of the quadratic equation $a x^{2}-x-4 a+1=0$ in absolute value exceed 1."
} | 4 | 524 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \\ldots + k^l$ . Let's think step by step and output the final ans... | math | {
"ground_truth": "(2,1,1,1)",
"style": "rule"
} | {
"answer": "(2,1,1,1)",
"difficulty": "4/8",
"index": 525,
"problem": "Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \\ldots + k^l$ ."
} | 5 | 525 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that the central angle of a sector is $\\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \\_\\_\\_\\_\\_\\_ cm, and the area of the sector is \\... | math | {
"ground_truth": "27",
"style": "rule"
} | {
"answer": "27",
"difficulty": "6/8",
"index": 527,
"problem": "Given that the central angle of a sector is $\\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \\_\\_\\_\\_\\_\\_ cm, and the area of the sector is \\_\\_\\_\\_\\_\\_ cm<sup>2</sup>."
} | 7 | 527 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A certain orange orchard has a total of 120 acres, consisting of both flat and hilly land. To estimate the average yield per acre, a stratified sampling method is used to survey a tota... | math | {
"ground_truth": "84",
"style": "rule"
} | {
"answer": "84",
"difficulty": "2/8",
"index": 528,
"problem": "A certain orange orchard has a total of 120 acres, consisting of both flat and hilly land. To estimate the average yield per acre, a stratified sampling method is used to survey a total of 10 acres. If the number of hilly acres sampled is 2 times ... | 0 | 528 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The function \\( f \\) is defined on the set of real numbers and satisfies the equations \\( f(2 - x) = f(2 + x) \\) and \\( f(7 - x) = f(7 + x) \\) for all real numbers \\( x \\). Giv... | math | {
"ground_truth": "401",
"style": "rule"
} | {
"answer": "401",
"difficulty": "5/8",
"index": 529,
"problem": "The function \\( f \\) is defined on the set of real numbers and satisfies the equations \\( f(2 - x) = f(2 + x) \\) and \\( f(7 - x) = f(7 + x) \\) for all real numbers \\( x \\). Given that \\( x = 0 \\) is a root of \\( f(x) = 0 \\), let \\( N... | 1 | 529 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A herd of elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, and a herd of 37 elephants could do so in 5 days. How many d... | math | {
"ground_truth": "365",
"style": "rule"
} | {
"answer": "365",
"difficulty": "7/8",
"index": 531,
"problem": "A herd of elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, and a herd of 37 elephants could do so in 5 days. How many days will it take for 1 elephant to drink the lake dry?"
} | 3 | 531 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A rectangular tank holds 216 cubic feet of water. Determine the volume of this tank in cubic meters, knowing that 1 cubic meter is approximately 35.315 cubic feet. Let's think step by ... | math | {
"ground_truth": "6.12",
"style": "rule"
} | {
"answer": "6.12",
"difficulty": "6/8",
"index": 532,
"problem": "A rectangular tank holds 216 cubic feet of water. Determine the volume of this tank in cubic meters, knowing that 1 cubic meter is approximately 35.315 cubic feet."
} | 4 | 532 |
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