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": "Rice grains were placed on the cells of a chessboard. The number of grains on any two adjacent cells differed by exactly\n\n1. At the same time, there were three grains on one of the c... | math | {
"ground_truth": "80",
"style": "rule"
} | {
"answer": "80",
"difficulty": "2/8",
"index": 533,
"problem": "Rice grains were placed on the cells of a chessboard. The number of grains on any two adjacent cells differed by exactly\n\n1. At the same time, there were three grains on one of the cells of the board, and 17 grains on another. A rooster pecked a... | 5 | 533 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, ... | math | {
"ground_truth": "108",
"style": "rule"
} | {
"answer": "108",
"difficulty": "2/8",
"index": 535,
"problem": "Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$... | 7 | 535 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "8th Putnam 1948 Problem A5 Let α 1 , α 2 , ... , α n be the nth roots of unity. Find ∏ i<j (α i - α j ) 2 . Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "(-1)^{\\frac{(n-1)(n-2)}{2}} n^{n}",
"style": "rule"
} | {
"answer": "(-1)^{\\frac{(n-1)(n-2)}{2}} n^{n}",
"difficulty": "0/8",
"index": 536,
"problem": "8th Putnam 1948 Problem A5 Let α 1 , α 2 , ... , α n be the nth roots of unity. Find ∏ i<j (α i - α j ) 2 ."
} | 0 | 536 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "1/8",
"index": 537,
"problem": "Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie o... | 1 | 537 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability ... | math | {
"ground_truth": "\\textbf{(B)}\\frac{1}{14}",
"style": "rule"
} | {
"answer": "\\textbf{(B)}\\frac{1}{14}",
"difficulty": "1/8",
"index": 539,
"problem": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row... | 3 | 539 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 6 people standing in a line numbered 1 through 6 in a left to right order.\nEach person has the following attributes: Job, Hobby, Movie-Genre, Nationality, Transport.\nThe a... | math | {
"ground_truth": "manager",
"style": "rule"
} | {
"answer": "manager",
"difficulty": "1/8",
"index": 540,
"problem": "There are 6 people standing in a line numbered 1 through 6 in a left to right order.\nEach person has the following attributes: Job, Hobby, Movie-Genre, Nationality, Transport.\nThe attributes have the following possible values:\n- Job: entr... | 4 | 540 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\\sqrt{n... | math | {
"ground_truth": "682",
"style": "rule"
} | {
"answer": "682",
"difficulty": "5/8",
"index": 541,
"problem": "Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not di... | 5 | 541 |
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": "Holly",
"style": "rule"
} | {
"answer": "Holly",
"difficulty": "0/8",
"index": 543,
"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 | 543 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(A\\) and \\(B\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the dis... | math | {
"ground_truth": "2\\sqrt{14}",
"style": "rule"
} | {
"answer": "2\\sqrt{14}",
"difficulty": "4/8",
"index": 544,
"problem": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(A\\) and \\(B\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \\(\\sqrt{305}\\). The ... | 0 | 544 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( xyzt = 1 \\). If \\( R = \\frac{1}{1+x+xy+xyz} + \\frac{1}{1+y+yz+yzt} + \\frac{1}{1+z+zt+ztx} + \\frac{1}{1+t+tx+txy} \\), find the value of \\( R \\). Let's think step by ste... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "4/8",
"index": 545,
"problem": "Let \\( xyzt = 1 \\). If \\( R = \\frac{1}{1+x+xy+xyz} + \\frac{1}{1+y+yz+yzt} + \\frac{1}{1+z+zt+ztx} + \\frac{1}{1+t+tx+txy} \\), find the value of \\( R \\)."
} | 1 | 545 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "14. (6 points) Two cars, A and B, start from points $A$ and $B$ respectively and travel towards each other, meeting at a point 108 kilometers from $B$. After reaching each other's star... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"answer": "48",
"difficulty": "5/8",
"index": 547,
"problem": "14. (6 points) Two cars, A and B, start from points $A$ and $B$ respectively and travel towards each other, meeting at a point 108 kilometers from $B$. After reaching each other's starting point, they immediately return to their original locations... | 3 | 547 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the derivative.\n\n$$\ny = \\operatorname{arctg} \\frac{2 \\sin x}{\\sqrt{9 \\cos ^{2} x - 4}}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"ro... | math | {
"ground_truth": "\\frac{2}{\\cosx\\sqrt{9\\cos^2x-4}}",
"style": "rule"
} | {
"answer": "\\frac{2}{\\cosx\\sqrt{9\\cos^2x-4}}",
"difficulty": "7/8",
"index": 548,
"problem": "Find the derivative.\n\n$$\ny = \\operatorname{arctg} \\frac{2 \\sin x}{\\sqrt{9 \\cos ^{2} x - 4}}\n$$"
} | 4 | 548 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: Let's think step by step and output the final answer wi... | math | {
"ground_truth": "\\frac {\\sqrt {8d^2 - p^2}}{2}",
"style": "rule"
} | {
"answer": "\\frac {\\sqrt {8d^2 - p^2}}{2}",
"difficulty": "5/8",
"index": 549,
"problem": "If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is:"
} | 5 | 549 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_n =\\sum_{d|n} \\frac{1}{2^{d+ \\frac{n}{d}}}$ . In other words, $a_n$ is the sum of $\\frac{1}{2^{d+ \\frac{n}{d}}}$ over all divisors $d$ of $n$ . \nFind $$ \\frac{\\... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "7/8",
"index": 551,
"problem": "Let $a_n =\\sum_{d|n} \\frac{1}{2^{d+ \\frac{n}{d}}}$ . In other words, $a_n$ is the sum of $\\frac{1}{2^{d+ \\frac{n}{d}}}$ over all divisors $d$ of $n$ . \nFind $$ \\frac{\\sum_{k=1} ^{\\infty}ka_k}{\\sum_{k=1}^{\\infty} a_k} =\\frac{a_1... | 7 | 551 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "13. Given $f(x)=\\frac{x^{2}}{x^{2}-100 x+5000}$, find the value of $f(1)+f(2)+\\cdots+f(100)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "us... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"answer": "101",
"difficulty": "4/8",
"index": 552,
"problem": "13. Given $f(x)=\\frac{x^{2}}{x^{2}-100 x+5000}$, find the value of $f(1)+f(2)+\\cdots+f(100)$."
} | 0 | 552 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABC$ be a triangle and $d$ a line parallel to the side $[BC]$ intersecting the segment $[AB]$ at point $E$ and the segment $[AC]$ at point $F$. Show that the intersection point of... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "2/8",
"index": 553,
"problem": "Let $ABC$ be a triangle and $d$ a line parallel to the side $[BC]$ intersecting the segment $[AB]$ at point $E$ and the segment $[AC]$ at point $F$. Show that the intersection point of the lines $(EC)$ and $(BF)$ lies on the median from vertex $A$.... | 1 | 553 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "\\(ABC\\) is a triangle such that \\(BC = 10\\), \\(CA = 12\\). Let \\(M\\) be the midpoint of side \\(AC\\). Given that \\(BM\\) is parallel to the external bisector of \\(\\angle A\\... | math | {
"ground_truth": "8\\sqrt{14}",
"style": "rule"
} | {
"answer": "8\\sqrt{14}",
"difficulty": "7/8",
"index": 555,
"problem": "\\(ABC\\) is a triangle such that \\(BC = 10\\), \\(CA = 12\\). Let \\(M\\) be the midpoint of side \\(AC\\). Given that \\(BM\\) is parallel to the external bisector of \\(\\angle A\\), find the area of triangle \\(ABC\\). (Lines \\(AB\\... | 3 | 555 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "## Problem 2.\n\nDetermine the largest natural number $n$ for which the following statement is true:\n\nThere exist $n$ distinct non-zero natural numbers $x_{1}, x_{2}, \\ldots, x_{n}$... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"answer": "9",
"difficulty": "4/8",
"index": 556,
"problem": "## Problem 2.\n\nDetermine the largest natural number $n$ for which the following statement is true:\n\nThere exist $n$ distinct non-zero natural numbers $x_{1}, x_{2}, \\ldots, x_{n}$ with the property that for any numbers $a_{1}, a_{2}, \\ldots, ... | 4 | 556 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the maximum number of self-intersection points that a closed polyline with 7 segments can have? Let's think step by step and output the final answer within \\boxed{}.",
"ro... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"answer": "14",
"difficulty": "2/8",
"index": 557,
"problem": "What is the maximum number of self-intersection points that a closed polyline with 7 segments can have?"
} | 5 | 557 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger th... | math | {
"ground_truth": "4060",
"style": "rule"
} | {
"answer": "4060",
"difficulty": "6/8",
"index": 559,
"problem": "Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. Determine the num... | 7 | 559 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The areas of triangles \\(ABC\\) and \\(A_1B_1C_1\\) are \\(S\\) and \\(S_1\\) respectively, where triangle \\(ABC\\) is not obtuse. The largest of the ratios \\(a_1 / a, b_1 / b,\\) a... | math | {
"ground_truth": "S_{1}\\lek^{2}S",
"style": "rule"
} | {
"answer": "S_{1}\\lek^{2}S",
"difficulty": "4/8",
"index": 560,
"problem": "The areas of triangles \\(ABC\\) and \\(A_1B_1C_1\\) are \\(S\\) and \\(S_1\\) respectively, where triangle \\(ABC\\) is not obtuse. The largest of the ratios \\(a_1 / a, b_1 / b,\\) and \\(c_1 / c\\) equals \\(k\\). Prove that \\(S_1... | 0 | 560 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "9.6. Waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing the watermelon on one scale pan with one or two w... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "3/8",
"index": 561,
"problem": "9.6. Waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing the watermelon on one scale pan with one or two weights on the other pan (possibly identical). In the process, th... | 1 | 561 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the ... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"answer": "12",
"difficulty": "4/8",
"index": 563,
"problem": "Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to ."
} | 3 | 563 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the function\n\\[ f(x) = ax - bx^2 \\quad \\text{(where } a, b \\in \\mathbb{R}^+ \\text{)} \\]\n(1) If for any \\( x \\in \\mathbb{R} \\), \\( f(x) \\leq 1 \\), prove that \\( a... | math | {
"ground_truth": "1\\le\\le2\\sqrt{b}",
"style": "rule"
} | {
"answer": "1\\le\\le2\\sqrt{b}",
"difficulty": "6/8",
"index": 564,
"problem": "Given the function\n\\[ f(x) = ax - bx^2 \\quad \\text{(where } a, b \\in \\mathbb{R}^+ \\text{)} \\]\n(1) If for any \\( x \\in \\mathbb{R} \\), \\( f(x) \\leq 1 \\), prove that \\( a \\leq 2\\sqrt{b} \\).\n\n(2) When \\( b > 1 \... | 4 | 564 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Shapovalov A.V.\n\nEach face of a cube $6 \\times 6 \\times 6$ is divided into $1 \\times 1$ cells. The cube is covered with $2 \\times 2$ squares such that each square covers exactly ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "2/8",
"index": 565,
"problem": "Shapovalov A.V.\n\nEach face of a cube $6 \\times 6 \\times 6$ is divided into $1 \\times 1$ cells. The cube is covered with $2 \\times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by the ... | 5 | 565 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Count the number of functions $f: \\mathbb{Z} \\rightarrow\\{$ 'green', 'blue' $\\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(... | math | {
"ground_truth": "39601",
"style": "rule"
} | {
"answer": "39601",
"difficulty": "4/8",
"index": 567,
"problem": "Count the number of functions $f: \\mathbb{Z} \\rightarrow\\{$ 'green', 'blue' $\\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'."
} | 7 | 567 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \\frac{x^{2}}{4}-\\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, an... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "1/8",
"index": 568,
"problem": "6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \\frac{x^{2}}{4}-\\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\\triangle P F_{1} F_{2}$ opposite to $\\angl... | 0 | 568 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What percent of the square $EFGH$ is shaded? All angles in the diagram are right angles, and the side length of the square is 8 units. In this square:\n- A smaller square in one corner... | math | {
"ground_truth": "17.1875\\%",
"style": "rule"
} | {
"answer": "17.1875\\%",
"difficulty": "7/8",
"index": 569,
"problem": "What percent of the square $EFGH$ is shaded? All angles in the diagram are right angles, and the side length of the square is 8 units. In this square:\n- A smaller square in one corner measuring 2 units per side is shaded.\n- A larger squa... | 1 | 569 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:\nThe last digit is deleted from the number on the board and then t... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"answer": "39",
"difficulty": "2/8",
"index": 571,
"problem": "Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:\nThe last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) ... | 3 | 571 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a given regular $n$-gon with $n \\geq 10$, select $k$ points from the $n$ vertices such that there exist 4 points among the $k$ points which form the vertices of a quadrilateral. Th... | math | {
"ground_truth": "\\lfloor\\frac{3n}{4}\\rfloor+1",
"style": "rule"
} | {
"answer": "\\lfloor\\frac{3n}{4}\\rfloor+1",
"difficulty": "0/8",
"index": 572,
"problem": "In a given regular $n$-gon with $n \\geq 10$, select $k$ points from the $n$ vertices such that there exist 4 points among the $k$ points which form the vertices of a quadrilateral. This quadrilateral must have three o... | 4 | 572 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the convex quadrilateral \\(ABCD\\), the angles at vertices \\(B\\), \\(C\\), and \\(D\\) are \\(30^\\circ\\), \\(90^\\circ\\), and \\(120^\\circ\\) respectively. Find the length of... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "4/8",
"index": 573,
"problem": "In the convex quadrilateral \\(ABCD\\), the angles at vertices \\(B\\), \\(C\\), and \\(D\\) are \\(30^\\circ\\), \\(90^\\circ\\), and \\(120^\\circ\\) respectively. Find the length of segment \\(AB\\), given that \\(AD = CD = 2\\)."
} | 5 | 573 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ bei... | math | {
"ground_truth": "\\frac{16}{5}",
"style": "rule"
} | {
"answer": "\\frac{16}{5}",
"difficulty": "3/8",
"index": 575,
"problem": "Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.\n\n$(1)$ Find the ... | 7 | 575 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In the figure, segment \\(DE\\) divides square sheet \\(ABCD\\) into \\(\\triangle ADE\\) and quadrilateral \\(EDCB\\). Given that the area ratio \\(S_{\\triangle ADE} : S_{EDCB} = 5 :... | math | {
"ground_truth": "\\frac{15}{22}",
"style": "rule"
} | {
"answer": "\\frac{15}{22}",
"difficulty": "6/8",
"index": 576,
"problem": "In the figure, segment \\(DE\\) divides square sheet \\(ABCD\\) into \\(\\triangle ADE\\) and quadrilateral \\(EDCB\\). Given that the area ratio \\(S_{\\triangle ADE} : S_{EDCB} = 5 : 19\\), determine the ratio of the perimeter of \\(... | 0 | 576 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Alex, a seasoned budget traveler, is planning a trip to three different cities. He has found the following deals for flights: \n\n- A flight to City A costs $120, but he has a 15% disc... | math | {
"ground_truth": "452",
"style": "rule"
} | {
"answer": "452",
"difficulty": "3/8",
"index": 577,
"problem": "Alex, a seasoned budget traveler, is planning a trip to three different cities. He has found the following deals for flights: \n\n- A flight to City A costs $120, but he has a 15% discount code he can use.\n- A flight to City B is advertised as \... | 1 | 577 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Xiaoming saw a tractor pulling a rope slowly on the road. Xiaoming decided to measure the length of the rope. If Xiaoming walks in the same direction as the tractor, it takes 140 steps... | math | {
"ground_truth": "35",
"style": "rule"
} | {
"answer": "35",
"difficulty": "7/8",
"index": 579,
"problem": "Xiaoming saw a tractor pulling a rope slowly on the road. Xiaoming decided to measure the length of the rope. If Xiaoming walks in the same direction as the tractor, it takes 140 steps to walk from one end of the rope to the other end; if Xiaoming... | 3 | 579 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\(a, b, c\\) be positive integers. All the roots of each of the quadratics\n\\[ a x^{2}+b x+c, \\quad a x^{2}+b x-c, \\quad a x^{2}-b x+c, \\quad a x^{2}-b x-c \\]\nare integers. ... | math | {
"ground_truth": "(1,10,24)",
"style": "rule"
} | {
"answer": "(1,10,24)",
"difficulty": "1/8",
"index": 580,
"problem": "Let \\(a, b, c\\) be positive integers. All the roots of each of the quadratics\n\\[ a x^{2}+b x+c, \\quad a x^{2}+b x-c, \\quad a x^{2}-b x+c, \\quad a x^{2}-b x-c \\]\nare integers. Over all triples \\((a, b, c)\\), find the triple with t... | 4 | 580 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a linear function \\( f(x)=ax+b \\) that satisfies \\( |f(x)+f(y)-xy| \\leq \\frac{1}{4} \\) for any \\( x, y \\in [0,1] \\), determine the function \\( f(x) \\). Let's think ste... | math | {
"ground_truth": "f(x)=\\frac{1}{2}x-\\frac{1}{8}",
"style": "rule"
} | {
"answer": "f(x)=\\frac{1}{2}x-\\frac{1}{8}",
"difficulty": "0/8",
"index": 581,
"problem": "Given a linear function \\( f(x)=ax+b \\) that satisfies \\( |f(x)+f(y)-xy| \\leq \\frac{1}{4} \\) for any \\( x, y \\in [0,1] \\), determine the function \\( f(x) \\)."
} | 5 | 581 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "21. Prime Minister Turtle ordered the shrimp soldiers and crab soldiers to stand in a solid square formation facing the Dragon Palace. Little shrimp Nini has 4 people directly in front... | math | {
"ground_truth": "256",
"style": "rule"
} | {
"answer": "256",
"difficulty": "4/8",
"index": 583,
"problem": "21. Prime Minister Turtle ordered the shrimp soldiers and crab soldiers to stand in a solid square formation facing the Dragon Palace. Little shrimp Nini has 4 people directly in front of her, and 3 people directly to her left; Little crab Momo h... | 7 | 583 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows ... | math | {
"ground_truth": "\\textbf{(B)}",
"style": "rule"
} | {
"answer": "\\textbf{(B)}",
"difficulty": "1/8",
"index": 584,
"problem": "Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?\n$\\textbf{(A)}\\ \\text{If Lewis did not... | 0 | 584 |
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 integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leadin... | math | {
"ground_truth": "99",
"style": "rule"
} | {
"answer": "99",
"difficulty": "3/8",
"index": 585,
"problem": "Find the number of integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leading 0 s (i.e. no 0 s at the very left), its rightmost digit is st... | 1 | 585 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "21. The volume of tetrahedron $ZABC$ is 5. A plane is drawn through the midpoint $K$ of edge $ZA$ and the midpoint $P$ of edge $BC$, intersecting edge $ZC$ at point $M$, such that $ZM:... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "1/8",
"index": 587,
"problem": "21. The volume of tetrahedron $ZABC$ is 5. A plane is drawn through the midpoint $K$ of edge $ZA$ and the midpoint $P$ of edge $BC$, intersecting edge $ZC$ at point $M$, such that $ZM: MC = 2: 3$. If the distance from vertex $A$ to the plane is 1, ... | 3 | 587 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let the functions $f_{0}(x)=|x|$, $f_{1}(x)=\\left|f_{0}(x)-1\\right|$, and $f_{2}(x)=\\left|f_{1}(x)-2\\right|$. Determine the area of the closed region formed by the graph of $y=f_{2... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"answer": "7",
"difficulty": "7/8",
"index": 588,
"problem": "Let the functions $f_{0}(x)=|x|$, $f_{1}(x)=\\left|f_{0}(x)-1\\right|$, and $f_{2}(x)=\\left|f_{1}(x)-2\\right|$. Determine the area of the closed region formed by the graph of $y=f_{2}(x)$ and the $x$-axis."
} | 4 | 588 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that \\( x^{3}+y^{3}+z^{3}-3 x y z-3\\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\\right)=0 \\) where \\( x, y, z \\) are not all equal positive real numbers.\n\nProve:\n(1) \\( x+y+z=3 \... | math | {
"ground_truth": "x^2(1+y)+y^2(1+z)+z^2(1+x)>6",
"style": "rule"
} | {
"answer": "x^2(1+y)+y^2(1+z)+z^2(1+x)>6",
"difficulty": "2/8",
"index": 589,
"problem": "Given that \\( x^{3}+y^{3}+z^{3}-3 x y z-3\\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\\right)=0 \\) where \\( x, y, z \\) are not all equal positive real numbers.\n\nProve:\n(1) \\( x+y+z=3 \\);\n(2) \\( x^{2}(1+y)+y^{2}(1+z)+z^... | 5 | 589 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For positive integer $n$ , let $s(n)$ denote the sum of the digits of $n$ . Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$ . Let's think step by step and o... | math | {
"ground_truth": "695",
"style": "rule"
} | {
"answer": "695",
"difficulty": "2/8",
"index": 591,
"problem": "For positive integer $n$ , let $s(n)$ denote the sum of the digits of $n$ . Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$ ."
} | 7 | 591 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condit... | math | {
"ground_truth": "62",
"style": "rule"
} | {
"answer": "62",
"difficulty": "7/8",
"index": 592,
"problem": "Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condition?"
} | 0 | 592 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( x \\) and \\( y \\) be positive integers. Can \\( x^{2} + 2y \\) and \\( y^{2} + 2x \\) both be squares? Let's think step by step and output the final answer within \\boxed{}."... | math | {
"ground_truth": "No",
"style": "rule"
} | {
"answer": "No",
"difficulty": "3/8",
"index": 593,
"problem": "Let \\( x \\) and \\( y \\) be positive integers. Can \\( x^{2} + 2y \\) and \\( y^{2} + 2x \\) both be squares?"
} | 1 | 593 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c ... | math | {
"ground_truth": "\\dfrac {3}{4}",
"style": "rule"
} | {
"answer": "\\dfrac {3}{4}",
"difficulty": "7/8",
"index": 595,
"problem": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\\cos B$ in terms of $a$ and $r... | 3 | 595 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "3. Ivan Semenovich leaves for work at the same time every day, drives at the same speed, and arrives exactly at 9:00. One day he overslept and left 40 minutes later than usual. To avoi... | math | {
"ground_truth": "30\\%",
"style": "rule"
} | {
"answer": "30\\%",
"difficulty": "5/8",
"index": 596,
"problem": "3. Ivan Semenovich leaves for work at the same time every day, drives at the same speed, and arrives exactly at 9:00. One day he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich drove at a speed 60% faster th... | 4 | 596 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 6 people standing in a line numbered 1 through 6 in a left to right order.\nEach person has the following attributes: Nationality, Movie-Genre, Beverage, Pet, Sport, Transpo... | math | {
"ground_truth": "lemonade",
"style": "rule"
} | {
"answer": "lemonade",
"difficulty": "0/8",
"index": 597,
"problem": "There are 6 people standing in a line numbered 1 through 6 in a left to right order.\nEach person has the following attributes: Nationality, Movie-Genre, Beverage, Pet, Sport, Transport.\nThe attributes have the following possible values:\n... | 5 | 597 |
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": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "3/8",
"index": 599,
"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: `E... | 7 | 599 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find all non-zero polynomials with real coefficients \\( P(x) \\) that satisfy the equation \\( P(x) \\cdot P\\left(2 x^{2}\\right) \\equiv P\\left(2 x^{3}+x\\right) \\) for all \\( x ... | math | {
"ground_truth": "(x^2+1)^k",
"style": "rule"
} | {
"answer": "(x^2+1)^k",
"difficulty": "2/8",
"index": 600,
"problem": "Find all non-zero polynomials with real coefficients \\( P(x) \\) that satisfy the equation \\( P(x) \\cdot P\\left(2 x^{2}\\right) \\equiv P\\left(2 x^{3}+x\\right) \\) for all \\( x \\in \\mathbb{R} \\)."
} | 0 | 600 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "On a drying rack, there are $n$ socks hanging in a random order (as they were taken out of the washing machine). Among them are two favorite socks of the Absent-minded Scientist. The s... | math | {
"ground_truth": "\\frac{2(n+1)}{3}",
"style": "rule"
} | {
"answer": "\\frac{2(n+1)}{3}",
"difficulty": "5/8",
"index": 601,
"problem": "On a drying rack, there are $n$ socks hanging in a random order (as they were taken out of the washing machine). Among them are two favorite socks of the Absent-minded Scientist. The socks are blocked by a drying sheet, so the Scien... | 1 | 601 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Define the sequence \\(a_1 = 1\\), \\(a_2, a_3, \\ldots \\) by \\(a_{n+1} = a_1^2 + a_2^2 + a_3^2 + \\ldots + a_n^2 + n\\). Show that 1 is the only square in the sequence. Let's think ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "7/8",
"index": 603,
"problem": "Define the sequence \\(a_1 = 1\\), \\(a_2, a_3, \\ldots \\) by \\(a_{n+1} = a_1^2 + a_2^2 + a_3^2 + \\ldots + a_n^2 + n\\). Show that 1 is the only square in the sequence."
} | 3 | 603 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"answer": "14",
"difficulty": "2/8",
"index": 604,
"problem": "Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs ... | 4 | 604 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA = 5$ and $CD = 2$. Point $E$ lies on side $CA$ such that angle $DEF =... | math | {
"ground_truth": "989",
"style": "rule"
} | {
"answer": "989",
"difficulty": "0/8",
"index": 605,
"problem": "Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA = 5$ and $CD = 2$. Point $E$ lies on side $CA$ such that angle $DEF = 60^{\\circ}$. The area of triangle $DEF$ is $14\\sqrt{3}$. Th... | 5 | 605 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squa... | math | {
"ground_truth": "72",
"style": "rule"
} | {
"answer": "72",
"difficulty": "1/8",
"index": 607,
"problem": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, an... | 7 | 607 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An event critic is attending a series of upscale events to review unique locations and services. On the first day, they visit a luxurious art gallery where each ticket costs $45. The g... | math | {
"ground_truth": "280",
"style": "rule"
} | {
"answer": "280",
"difficulty": "3/8",
"index": 608,
"problem": "An event critic is attending a series of upscale events to review unique locations and services. On the first day, they visit a luxurious art gallery where each ticket costs $45. The gallery hosts a special event, and the critic receives two comp... | 0 | 608 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A domino is a $2 \\times 1$ or $1 \\times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \\times 2 n$ chessboard so that every... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "2/8",
"index": 609,
"problem": "A domino is a $2 \\times 1$ or $1 \\times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \\times 2 n$ chessboard so that every $2 \\times 2$ square contains at least two uncovered unit squar... | 1 | 609 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Prove that if \\( a \\) and \\( b \\) are positive numbers, then\n\n$$\n\\left(1+\\frac{1}{a}\\right)\\left(1+\\frac{1}{b}\\right) \\geq \\frac{8}{1+ab}\n$$ Let's think step by step an... | math | {
"ground_truth": "(1+\\frac{1}{})(1+\\frac{1}{b})\\ge\\frac{8}{1+}",
"style": "rule"
} | {
"answer": "(1+\\frac{1}{})(1+\\frac{1}{b})\\ge\\frac{8}{1+}",
"difficulty": "6/8",
"index": 611,
"problem": "Prove that if \\( a \\) and \\( b \\) are positive numbers, then\n\n$$\n\\left(1+\\frac{1}{a}\\right)\\left(1+\\frac{1}{b}\\right) \\geq \\frac{8}{1+ab}\n$$"
} | 3 | 611 |
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": "prince",
"style": "rule"
} | {
"answer": "prince",
"difficulty": "1/8",
"index": 612,
"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 nam... | 4 | 612 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "10.042. One of the angles of the trapezoid is $30^{\\circ}$, and the lines containing the lateral sides of the trapezoid intersect at a right angle (Fig. 10.42). Find the length of the... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "5/8",
"index": 613,
"problem": "10.042. One of the angles of the trapezoid is $30^{\\circ}$, and the lines containing the lateral sides of the trapezoid intersect at a right angle (Fig. 10.42). Find the length of the shorter lateral side of the trapezoid if its midline is 10 cm, ... | 5 | 613 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in te... | math | {
"ground_truth": "\\frac{8k-168}{k-12}",
"style": "rule"
} | {
"answer": "\\frac{8k-168}{k-12}",
"difficulty": "5/8",
"index": 615,
"problem": "The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$?"
} | 7 | 615 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "【Question 6】\nA bottle contains 1001 blue balls, 1000 red balls, and 1000 green balls. At the same time, Xiaoming has enough balls of these three colors in his hand. Next, Xiaoming tak... | math | {
"ground_truth": "red",
"style": "rule"
} | {
"answer": "red",
"difficulty": "1/8",
"index": 616,
"problem": "【Question 6】\nA bottle contains 1001 blue balls, 1000 red balls, and 1000 green balls. At the same time, Xiaoming has enough balls of these three colors in his hand. Next, Xiaoming takes out two balls from the bottle each time, and then puts one ... | 0 | 616 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The inclination angle of the line $x-y+1=0$ can be calculated. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"answer": "\\frac{\\pi}{4}",
"difficulty": "2/8",
"index": 617,
"problem": "The inclination angle of the line $x-y+1=0$ can be calculated."
} | 1 | 617 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be positive integer. Define a sequence $\\{a_k\\}$ by \n\\[a_1=\\frac{1}{n(n+1)},\\ a_{k+1}=-\\frac{1}{k+n+1}+\\frac{n}{k}\\sum_{i=1}^k a_i\\ \\ (k=1,\\ 2,\\ 3,\\ \\cdots).... | math | {
"ground_truth": "\\ln2",
"style": "rule"
} | {
"answer": "\\ln2",
"difficulty": "4/8",
"index": 619,
"problem": "Let $n$ be positive integer. Define a sequence $\\{a_k\\}$ by \n\\[a_1=\\frac{1}{n(n+1)},\\ a_{k+1}=-\\frac{1}{k+n+1}+\\frac{n}{k}\\sum_{i=1}^k a_i\\ \\ (k=1,\\ 2,\\ 3,\\ \\cdots).\\]\n\n(1) Find $a_2$ and $a_3$ .\n\n(2) Find the general... | 3 | 619 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that \\( P \\) is any point on the arc \\( \\overparen{AD} \\) of the circumscribed circle of square \\( ABCD \\), find the value of \\( \\frac{PA + PC}{PB} \\). Let's think step... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"answer": "\\sqrt{2}",
"difficulty": "2/8",
"index": 620,
"problem": "Given that \\( P \\) is any point on the arc \\( \\overparen{AD} \\) of the circumscribed circle of square \\( ABCD \\), find the value of \\( \\frac{PA + PC}{PB} \\)."
} | 4 | 620 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $a_1, a_2, a_3, \\dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \\[ a_{a_1} + a_{a_2} + ... | math | {
"ground_truth": "365",
"style": "rule"
} | {
"answer": "365",
"difficulty": "7/8",
"index": 621,
"problem": "Suppose $a_1, a_2, a_3, \\dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \\]*Proposed by Evan Chen*"
} | 5 | 621 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "10.276. A right triangle $ABC$ is divided by the altitude $CD$, drawn to the hypotenuse, into two triangles $BCD$ and $ACD$. The radii of the circles inscribed in triangles $BCD$ and $... | math | {
"ground_truth": "5\\sqrt{2}",
"style": "rule"
} | {
"answer": "5\\sqrt{2}",
"difficulty": "5/8",
"index": 623,
"problem": "10.276. A right triangle $ABC$ is divided by the altitude $CD$, drawn to the hypotenuse, into two triangles $BCD$ and $ACD$. The radii of the circles inscribed in triangles $BCD$ and $ACD$ are 4 and $3 \\text{ cm}$, respectively. Find the ... | 7 | 623 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Three circles are tangent to each other, as shown in the figure. The area of the region of the outer circle that is not covered by the two inner circles is $2 \\pi$. Determine the leng... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "3/8",
"index": 624,
"problem": "Three circles are tangent to each other, as shown in the figure. The area of the region of the outer circle that is not covered by the two inner circles is $2 \\pi$. Determine the length of the segment $PQ$."
} | 0 | 624 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Between 1000 and 9999, how many four-digit integers with all different digits have an absolute difference of 2 between the first and last digits? Let's think step by step and output th... | math | {
"ground_truth": "840",
"style": "rule"
} | {
"answer": "840",
"difficulty": "7/8",
"index": 625,
"problem": "Between 1000 and 9999, how many four-digit integers with all different digits have an absolute difference of 2 between the first and last digits?"
} | 1 | 625 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A palindrome is a positive integer which is unchanged if you reverse the order of its digits. For example, 23432. If all palindromes are written in increasing order, what possible prim... | math | {
"ground_truth": "11",
"style": "rule"
} | {
"answer": "11",
"difficulty": "5/8",
"index": 627,
"problem": "A palindrome is a positive integer which is unchanged if you reverse the order of its digits. For example, 23432. If all palindromes are written in increasing order, what possible prime values can the difference between successive palindromes take... | 3 | 627 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $n\\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of the... | math | {
"ground_truth": "2^n - 1",
"style": "rule"
} | {
"answer": "2^n - 1",
"difficulty": "4/8",
"index": 628,
"problem": "Let $n\\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than $1$ that is a perfect $n^{\\m... | 4 | 628 |
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. Find the largest integer \\( k \\) such that it is possible to form \\( k \\) subsets from a set with \\( n \\) elements, where the intersection of... | math | {
"ground_truth": "2^{n-1}",
"style": "rule"
} | {
"answer": "2^{n-1}",
"difficulty": "6/8",
"index": 629,
"problem": "Let \\( n \\) be a positive integer. Find the largest integer \\( k \\) such that it is possible to form \\( k \\) subsets from a set with \\( n \\) elements, where the intersection of any two subsets is non-empty."
} | 5 | 629 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive number $\\lambda$, such that for any $12$ points on the plane $P_1,P_2,\\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed ... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"answer": "48",
"difficulty": "0/8",
"index": 631,
"problem": "Find the smallest positive number $\\lambda$, such that for any $12$ points on the plane $P_1,P_2,\\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\\sum_{1\\le i<j\\le 12} |P_iP_j|^2\\le \\lambda$."
... | 7 | 631 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A week-old set's three-element subsets need to be colored such that if the intersection of two subsets is empty, then their colors differ. How many colors do we need at least? Let's th... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "5/8",
"index": 632,
"problem": "A week-old set's three-element subsets need to be colored such that if the intersection of two subsets is empty, then their colors differ. How many colors do we need at least?"
} | 0 | 632 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the ... | math | {
"ground_truth": "fish",
"style": "rule"
} | {
"answer": "fish",
"difficulty": "4/8",
"index": 633,
"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:... | 1 | 633 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Cara is sitting at a circular table with her seven friends. One of her friends, named Emma, must always sit next to her. How many different possible pairs of people could Cara be sitti... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "3/8",
"index": 635,
"problem": "Cara is sitting at a circular table with her seven friends. One of her friends, named Emma, must always sit next to her. How many different possible pairs of people could Cara be sitting between, given that one of them must always be Emma?"
} | 3 | 635 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "[asy]\ndraw((-7,0)--(7,0),black+linewidth(.75));\ndraw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));\ndraw((-2*s... | math | {
"ground_truth": "12\\sqrt{3}",
"style": "rule"
} | {
"answer": "12\\sqrt{3}",
"difficulty": "0/8",
"index": 636,
"problem": "[asy]\ndraw((-7,0)--(7,0),black+linewidth(.75));\ndraw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));\ndraw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt... | 4 | 636 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "To find the smallest value of \\( n \\) such that the system of equations\n$$\n\\left\\{\\begin{array}{l}\n\\sin x_{1}+\\sin x_{2}+\\cdots+\\sin x_{n}=0, \\\\\n\\sin x_{1}+2 \\sin x_{2... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"answer": "20",
"difficulty": "6/8",
"index": 637,
"problem": "To find the smallest value of \\( n \\) such that the system of equations\n$$\n\\left\\{\\begin{array}{l}\n\\sin x_{1}+\\sin x_{2}+\\cdots+\\sin x_{n}=0, \\\\\n\\sin x_{1}+2 \\sin x_{2}+\\cdots+n \\sin x_{n}=100 .\n\\end{array}\\right.\n$$\nhas a ... | 5 | 637 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probabil... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"answer": "23",
"difficulty": "3/8",
"index": 639,
"problem": "Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probability of Carina winning the first game is \\(\\frac{1}{2}\\). Aft... | 7 | 639 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? Let's think step by step and output the final answer within \\boxed{}.... | math | {
"ground_truth": "$-\\frac{2\\sqrt{3}}{3}$",
"style": "rule"
} | {
"answer": "$-\\frac{2\\sqrt{3}}{3}$",
"difficulty": "3/8",
"index": 640,
"problem": "The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?"
} | 0 | 640 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \\( PQ \\) of the larger circle is a tangent to both unshaded circles. The length of... | math | {
"ground_truth": "\\frac{9\\pi}{2}",
"style": "rule"
} | {
"answer": "\\frac{9\\pi}{2}",
"difficulty": "5/8",
"index": 641,
"problem": "The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \\( PQ \\) of the larger circle is a tangent to both unshaded circles. The length of \\( PQ \\) is 6 units. What is the area, in squa... | 1 | 641 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The ratio of two numbers is $5: a$. If 12 is added to each of them, the ratio becomes $3: 4$. If $b$ is the difference of the original numbers and $b>0$, find the value of $b$. Let's t... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"answer": "9",
"difficulty": "3/8",
"index": 643,
"problem": "The ratio of two numbers is $5: a$. If 12 is added to each of them, the ratio becomes $3: 4$. If $b$ is the difference of the original numbers and $b>0$, find the value of $b$."
} | 3 | 643 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases:... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"answer": "10",
"difficulty": "3/8",
"index": 644,
"problem": "Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: \"My neighbor on the left is a liar\" or \"My neighbor on the ... | 4 | 644 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The trisectors of angles $B$ and $C$ of scalene triangle $ABC$ meet at points $P$ and $Q$ as shown. Angle $A$ measures 39 degrees and angle $QBP$ measures 14 degrees. What is the measu... | math | {
"ground_truth": "133",
"style": "rule"
} | {
"answer": "133",
"difficulty": "4/8",
"index": 645,
"problem": "The trisectors of angles $B$ and $C$ of scalene triangle $ABC$ meet at points $P$ and $Q$ as shown. Angle $A$ measures 39 degrees and angle $QBP$ measures 14 degrees. What is the measure of angle $BPC$? [asy]\nimport olympiad; import geometry; si... | 5 | 645 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "[ Arithmetic. Mental calculation, etc.]\n\nFrom the ten-digit number 2946835107, five digits were erased. What is the largest number that could result from this?\n\n# Let's think step ... | math | {
"ground_truth": "98517",
"style": "rule"
} | {
"answer": "98517",
"difficulty": "6/8",
"index": 647,
"problem": "[ Arithmetic. Mental calculation, etc.]\n\nFrom the ten-digit number 2946835107, five digits were erased. What is the largest number that could result from this?\n\n#"
} | 7 | 647 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, ... | math | {
"ground_truth": "\\frac{159}{323}",
"style": "rule"
} | {
"answer": "\\frac{159}{323}",
"difficulty": "6/8",
"index": 648,
"problem": "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\... | 0 | 648 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers \\( k \\) such that the Diophantine equation \\( x^2 + y^2 = kxy - 1 \\) has a solution in positive integers \\( x \\) and \\( y \\). Let's think step by ste... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": "2/8",
"index": 649,
"problem": "Find all positive integers \\( k \\) such that the Diophantine equation \\( x^2 + y^2 = kxy - 1 \\) has a solution in positive integers \\( x \\) and \\( y \\)."
} | 1 | 649 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find [the decimal form of] the largest prime divisor of $100111011_6$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "181",
"style": "rule"
} | {
"answer": "181",
"difficulty": "6/8",
"index": 651,
"problem": "Find [the decimal form of] the largest prime divisor of $100111011_6$."
} | 3 | 651 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The radii of two circles are 2 and 4. Their common internal tangents are perpendicular to each other. Find the length of each of them. Let's think step by step and output the final ans... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": "4/8",
"index": 652,
"problem": "The radii of two circles are 2 and 4. Their common internal tangents are perpendicular to each other. Find the length of each of them."
} | 4 | 652 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The sequence $\\left\\{a_{n}\\right\\}$ is defined as follows: $a_{1}=\\frac{1}{2}$, and $a_{n+1}=\\frac{a_{n}^{2}}{a_{n}^{2}-a_{n}+1}$ for $n=1,2,\\ldots$. Prove that for every positi... | math | {
"ground_truth": "a_{1}+a_{2}+\\cdots+a_{n}<1",
"style": "rule"
} | {
"answer": "a_{1}+a_{2}+\\cdots+a_{n}<1",
"difficulty": "4/8",
"index": 653,
"problem": "The sequence $\\left\\{a_{n}\\right\\}$ is defined as follows: $a_{1}=\\frac{1}{2}$, and $a_{n+1}=\\frac{a_{n}^{2}}{a_{n}^{2}-a_{n}+1}$ for $n=1,2,\\ldots$. Prove that for every positive integer $n$, the inequality $a_{1}+... | 5 | 653 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let there be $k$ red balls and $N$ green balls arranged in a line, where $k$ is a positive integer and $N$ is a positive multiple of 10. Determine the least value of $N$ such that the ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "6/8",
"index": 655,
"problem": "Let there be $k$ red balls and $N$ green balls arranged in a line, where $k$ is a positive integer and $N$ is a positive multiple of 10. Determine the least value of $N$ such that the probability that at least $\\frac{3}{5}$ of the green balls are ... | 7 | 655 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Cara is sitting at a circular table with her five friends as shown below. How many different possible pairs of people could Cara be sitting between?\n\n[asy]\ndraw(circle((0,0),1));\nl... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"answer": "10",
"difficulty": "7/8",
"index": 656,
"problem": "Cara is sitting at a circular table with her five friends as shown below. How many different possible pairs of people could Cara be sitting between?\n\n[asy]\ndraw(circle((0,0),1));\nlabel(\"$\\_$\",1.5dir(0));\nlabel(\"$\\_$\",1.5dir(60));\nlabel... | 0 | 656 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "4. In a regular tetrahedron $P-ABC$, it is known that the angle between the lateral edge and the base is $45^{\\circ}$. Then the cosine value of the dihedral angle between two adjacent... | math | {
"ground_truth": "\\dfrac{1}{5}",
"style": "rule"
} | {
"answer": "\\dfrac{1}{5}",
"difficulty": "3/8",
"index": 657,
"problem": "4. In a regular tetrahedron $P-ABC$, it is known that the angle between the lateral edge and the base is $45^{\\circ}$. Then the cosine value of the dihedral angle between two adjacent lateral faces is $\\qquad$"
} | 1 | 657 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Rectangle $ABCD$ has sides $\\overline{AB}$ of length 5 and $\\overline{CB}$ of length 4. Divide $\\overline{AB}$ into 200 congruent segments with points $A=P_0, P_1, \\ldots, P_{200}=... | math | {
"ground_truth": "200\\sqrt{41}",
"style": "rule"
} | {
"answer": "200\\sqrt{41}",
"difficulty": "4/8",
"index": 659,
"problem": "Rectangle $ABCD$ has sides $\\overline{AB}$ of length 5 and $\\overline{CB}$ of length 4. Divide $\\overline{AB}$ into 200 congruent segments with points $A=P_0, P_1, \\ldots, P_{200}=B$, and divide $\\overline{CB}$ into 200 congruent s... | 3 | 659 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 6 people standing in a line numbered 1 through 6 in a left to right order.\nEach person has the following attributes: Nationality, Hobby, Beverage, Pet, Sport, Transport.\nT... | math | {
"ground_truth": "german",
"style": "rule"
} | {
"answer": "german",
"difficulty": "0/8",
"index": 660,
"problem": "There are 6 people standing in a line numbered 1 through 6 in a left to right order.\nEach person has the following attributes: Nationality, Hobby, Beverage, Pet, Sport, Transport.\nThe attributes have the following possible values:\n- Nation... | 4 | 660 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"answer": "-3",
"difficulty": "7/8",
"index": 661,
"problem": "Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$."
} | 5 | 661 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Solve the system of equations:\n\n\\[\n\\left\\{\n\\begin{array}{l}\nx^{2} y + xy^{2} - 2x - 2y + 10 = 0, \\\\\nx^{3} y - xy^{3} - 2x^{2} + 2y^{2} - 30 = 0.\n\\end{array}\n\\right.\n\\... | math | {
"ground_truth": "(-4,-1)",
"style": "rule"
} | {
"answer": "(-4,-1)",
"difficulty": "5/8",
"index": 663,
"problem": "Solve the system of equations:\n\n\\[\n\\left\\{\n\\begin{array}{l}\nx^{2} y + xy^{2} - 2x - 2y + 10 = 0, \\\\\nx^{3} y - xy^{3} - 2x^{2} + 2y^{2} - 30 = 0.\n\\end{array}\n\\right.\n\\]"
} | 7 | 663 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the pairs of positive integers \\((a, b)\\) such that \\(\\frac{a b^{2}}{a+b}\\) is a prime number, where \\(a \\neq b\\). Let's think step by step and output the final answer wit... | math | {
"ground_truth": "(6,2)",
"style": "rule"
} | {
"answer": "(6,2)",
"difficulty": "2/8",
"index": 664,
"problem": "Find the pairs of positive integers \\((a, b)\\) such that \\(\\frac{a b^{2}}{a+b}\\) is a prime number, where \\(a \\neq b\\)."
} | 0 | 664 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the maximum and minimum values of \\( \\cos x + \\cos y + \\cos z \\), where \\( x, y, z \\) are non-negative reals with sum \\( \\frac{4\\pi}{3} \\). Let's think step by step and... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "1/8",
"index": 665,
"problem": "Find the maximum and minimum values of \\( \\cos x + \\cos y + \\cos z \\), where \\( x, y, z \\) are non-negative reals with sum \\( \\frac{4\\pi}{3} \\)."
} | 1 | 665 |
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