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": "There are 10 different points \\( P_{1}(x_{1}, y_{1}), P_{2}(x_{2}, y_{2}), \\cdots, P_{10}(x_{10}, y_{10}) \\) in the Cartesian plane. If \\( x_{i}=x_{j} \\) or \\( y_{i}=y_{j} \\), t... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "5/8",
"index": 267,
"problem": "There are 10 different points \\( P_{1}(x_{1}, y_{1}), P_{2}(x_{2}, y_{2}), \\cdots, P_{10}(x_{10}, y_{10}) \\) in the Cartesian plane. If \\( x_{i}=x_{j} \\) or \\( y_{i}=y_{j} \\), then \\( P_{i} \\) and \\( P_{j} \\) are called a \"same-coordina... | 3 | 267 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are two straight lines, each of which passes through four points of the form $(1,0,a), (b,1,0), (0,c,1),$ and $(6d,6d,-d),$ where $a,b,c,$ and $d$ are real numbers, not necessari... | math | {
"ground_truth": "\\frac{1}{3}, \\frac{1}{8}",
"style": "rule"
} | {
"answer": "\\frac{1}{3}, \\frac{1}{8}",
"difficulty": "1/8",
"index": 268,
"problem": "There are two straight lines, each of which passes through four points of the form $(1,0,a), (b,1,0), (0,c,1),$ and $(6d,6d,-d),$ where $a,b,c,$ and $d$ are real numbers, not necessarily in that order. Enter all possible v... | 4 | 268 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "To actively create a \"demonstration school for the prevention and control of myopia in children and adolescents in the city\" and cultivate students' good eye habits, a certain school... | math | {
"ground_truth": "1755",
"style": "rule"
} | {
"answer": "1755",
"difficulty": "5/8",
"index": 269,
"problem": "To actively create a \"demonstration school for the prevention and control of myopia in children and adolescents in the city\" and cultivate students' good eye habits, a certain school conducted a competition on correct eye knowledge this semest... | 5 | 269 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "As shown in the figure, each small square has a side length of $10 \\mathrm{~km}$. There are 2 gas stations in the figure. A car starts from point $A$ and travels along the edges of th... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"answer": "18",
"difficulty": "1/8",
"index": 271,
"problem": "As shown in the figure, each small square has a side length of $10 \\mathrm{~km}$. There are 2 gas stations in the figure. A car starts from point $A$ and travels along the edges of the small squares to point $B$. If the car needs to refuel every ... | 7 | 271 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "$\\left.\\frac{\\text { Auxiliary similar triangles }}{[\\quad \\text { Law of Cosines }}\\right]$\n\nIn triangle $ABC$, a point $D$ is taken on side $AC$, such that $AD=3$, $\\cos \\a... | math | {
"ground_truth": "11",
"style": "rule"
} | {
"answer": "11",
"difficulty": "2/8",
"index": 272,
"problem": "$\\left.\\frac{\\text { Auxiliary similar triangles }}{[\\quad \\text { Law of Cosines }}\\right]$\n\nIn triangle $ABC$, a point $D$ is taken on side $AC$, such that $AD=3$, $\\cos \\angle BDC=13/20$, and $\\angle B+\\angle ADB=180^{\\circ}$. Find... | 0 | 272 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in pairs to each of the $2010$ players sitting around a round table. The game consis... | math | {
"ground_truth": "2009",
"style": "rule"
} | {
"answer": "2009",
"difficulty": "3/8",
"index": 273,
"problem": "There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in pairs to each of the $2010$ players sitting around a round table. The game consists of several rounds, each of which players simultaneously ha... | 1 | 273 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In a game, \\( N \\) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds ... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"answer": "34",
"difficulty": "3/8",
"index": 275,
"problem": "In a game, \\( N \\) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be mu... | 3 | 275 |
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 cover a $3 \\times 8$ rectangle with 12 identical dominoes? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "153",
"style": "rule"
} | {
"answer": "153",
"difficulty": "7/8",
"index": 276,
"problem": "How many ways are there to cover a $3 \\times 8$ rectangle with 12 identical dominoes?"
} | 4 | 276 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How ma... | math | {
"ground_truth": "396",
"style": "rule"
} | {
"answer": "396",
"difficulty": "3/8",
"index": 277,
"problem": "Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?... | 5 | 277 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are two equal plywood triangles, one of the angles of which is $\\alpha$ (these angles are marked). Position them on the plane so that some three vertices form an angle equal to ... | math | {
"ground_truth": "\\alpha/2",
"style": "rule"
} | {
"answer": "\\alpha/2",
"difficulty": "2/8",
"index": 279,
"problem": "There are two equal plywood triangles, one of the angles of which is $\\alpha$ (these angles are marked). Position them on the plane so that some three vertices form an angle equal to $\\alpha / 2$. (No tools, not even a pencil, can be used... | 7 | 279 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "From a \\(6 \\times 6\\) square grid, gray triangles were cut out. What is the area of the remaining shape? The length of each side of the cells is 1 cm. Provide your answer in square ... | math | {
"ground_truth": "27",
"style": "rule"
} | {
"answer": "27",
"difficulty": "1/8",
"index": 280,
"problem": "From a \\(6 \\times 6\\) square grid, gray triangles were cut out. What is the area of the remaining shape? The length of each side of the cells is 1 cm. Provide your answer in square centimeters."
} | 0 | 280 |
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 set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$? Let's think step by step and o... | math | {
"ground_truth": "1011100",
"style": "rule"
} | {
"answer": "1011100",
"difficulty": "3/8",
"index": 281,
"problem": "Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$?"
} | 1 | 281 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For a pyramid S-ABCD, each vertex is colored with one color, and the two ends of the same edge are colored differently. If there are exactly 5 colors available, calculate the number of... | math | {
"ground_truth": "420",
"style": "rule"
} | {
"answer": "420",
"difficulty": "6/8",
"index": 283,
"problem": "For a pyramid S-ABCD, each vertex is colored with one color, and the two ends of the same edge are colored differently. If there are exactly 5 colors available, calculate the number of different coloring methods."
} | 3 | 283 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$. Let's think step by step and output the final answer w... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"answer": "45",
"difficulty": "5/8",
"index": 284,
"problem": "Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$."
} | 4 | 284 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Do there exist numbers \\(a\\) and \\(b\\) that satisfy the equation \\(a^2 + 3b^2 + 2 = 3ab\\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "u... | math | {
"ground_truth": "No",
"style": "rule"
} | {
"answer": "No",
"difficulty": "7/8",
"index": 285,
"problem": "Do there exist numbers \\(a\\) and \\(b\\) that satisfy the equation \\(a^2 + 3b^2 + 2 = 3ab\\)?"
} | 5 | 285 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For integers \\( a \\) and \\( b \\), the equation \\( \\sqrt{a-1} + \\sqrt{b-1} = \\sqrt{a b + k} \\) (with \\( k \\in \\mathbb{Z} \\)) has only one ordered pair of real solutions. Fi... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"answer": "0",
"difficulty": "7/8",
"index": 287,
"problem": "For integers \\( a \\) and \\( b \\), the equation \\( \\sqrt{a-1} + \\sqrt{b-1} = \\sqrt{a b + k} \\) (with \\( k \\in \\mathbb{Z} \\)) has only one ordered pair of real solutions. Find \\( k \\)."
} | 7 | 287 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\\geq3$.\n\nLet $m,n\\geq1$ be integers. Find the minimal de... | math | {
"ground_truth": "n",
"style": "rule"
} | {
"answer": "n",
"difficulty": "4/8",
"index": 288,
"problem": "The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\\geq3$.\n\nLet $m,n\\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1... | 0 | 288 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given that $a, b, c$ are not all zero, find the maximum value of $\\frac{a b + 2 b c}{a^{2} + b^{2} + c^{2}}$. Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"answer": "\\frac{\\sqrt{5}}{2}",
"difficulty": "6/8",
"index": 289,
"problem": "Given that $a, b, c$ are not all zero, find the maximum value of $\\frac{a b + 2 b c}{a^{2} + b^{2} + c^{2}}$."
} | 1 | 289 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In triangle \\(ABC\\), \\(\\angle A = 2 \\angle C\\). Suppose that \\(AC = 6\\), \\(BC = 8\\), and \\(AB = \\sqrt{a} - b\\), where \\(a\\) and \\(b\\) are positive integers. Compute \\... | math | {
"ground_truth": "7303",
"style": "rule"
} | {
"answer": "7303",
"difficulty": "6/8",
"index": 291,
"problem": "In triangle \\(ABC\\), \\(\\angle A = 2 \\angle C\\). Suppose that \\(AC = 6\\), \\(BC = 8\\), and \\(AB = \\sqrt{a} - b\\), where \\(a\\) and \\(b\\) are positive integers. Compute \\(100a + b\\)."
} | 3 | 291 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "7. The \"Hua Luogeng\" Golden Cup Junior Mathematics Invitational Competition, the first session was held in 1986, the second session in 1988, the third session in 1991, and thereafter... | math | {
"ground_truth": "629",
"style": "rule"
} | {
"answer": "629",
"difficulty": "3/8",
"index": 292,
"problem": "7. The \"Hua Luogeng\" Golden Cup Junior Mathematics Invitational Competition, the first session was held in 1986, the second session in 1988, the third session in 1991, and thereafter every 2 years. The sum of the digits of the year in which the... | 4 | 292 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In how many ways can you rearrange the letters of \"HMMTHMMT\" such that the consecutive substring \"HMMT\" does not appear? Let's think step by step and output the final answer within... | math | {
"ground_truth": "361",
"style": "rule"
} | {
"answer": "361",
"difficulty": "0/8",
"index": 293,
"problem": "In how many ways can you rearrange the letters of \"HMMTHMMT\" such that the consecutive substring \"HMMT\" does not appear?"
} | 5 | 293 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "If \\( S = \\sum_{k=1}^{99} \\frac{(-1)^{k+1}}{\\sqrt{k(k+1)}(\\sqrt{k+1}-\\sqrt{k})} \\), find the value of \\( 1000 S \\). Let's think step by step and output the final answer within... | math | {
"ground_truth": "1100",
"style": "rule"
} | {
"answer": "1100",
"difficulty": "6/8",
"index": 295,
"problem": "If \\( S = \\sum_{k=1}^{99} \\frac{(-1)^{k+1}}{\\sqrt{k(k+1)}(\\sqrt{k+1}-\\sqrt{k})} \\), find the value of \\( 1000 S \\)."
} | 7 | 295 |
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 positive integers \\( j \\leq 3^{2013} \\) such that\n\\[ j = \\sum_{k=0}^{m} \\left((-1)^k \\cdot 3^{a_k}\\right) \\]\nfor some strictly increasing sequence of nonn... | math | {
"ground_truth": "2^{2013}",
"style": "rule"
} | {
"answer": "2^{2013}",
"difficulty": "2/8",
"index": 296,
"problem": "Find the number of positive integers \\( j \\leq 3^{2013} \\) such that\n\\[ j = \\sum_{k=0}^{m} \\left((-1)^k \\cdot 3^{a_k}\\right) \\]\nfor some strictly increasing sequence of nonnegative integers \\( \\{a_k\\} \\). For example, we may w... | 0 | 296 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "5. Connecting the centers of the faces of a regular polyhedron, we get a new regular polyhedron, which we call the positive sub-polyhedron of the original polyhedron. A cube $T_{1}$ ha... | math | {
"ground_truth": "18 + 3\\sqrt{3}",
"style": "rule"
} | {
"answer": "18 + 3\\sqrt{3}",
"difficulty": "3/8",
"index": 297,
"problem": "5. Connecting the centers of the faces of a regular polyhedron, we get a new regular polyhedron, which we call the positive sub-polyhedron of the original polyhedron. A cube $T_{1}$ has a surface area of $a_{1}=16$, its positive sub-p... | 1 | 297 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins a... | math | {
"ground_truth": "1011",
"style": "rule"
} | {
"answer": "1011",
"difficulty": "6/8",
"index": 299,
"problem": "Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is a... | 3 | 299 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In an $8 \\times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same co... | math | {
"ground_truth": "234",
"style": "rule"
} | {
"answer": "234",
"difficulty": "5/8",
"index": 300,
"problem": "In an $8 \\times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What i... | 4 | 300 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triang... | math | {
"ground_truth": "\\textbf{(D)}4",
"style": "rule"
} | {
"answer": "\\textbf{(D)}4",
"difficulty": "1/8",
"index": 301,
"problem": "How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\... | 5 | 301 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Compute \\(\\tan \\left(\\frac{\\pi}{7}\\right) \\tan \\left(\\frac{2 \\pi}{7}\\right) \\tan \\left(\\frac{3 \\pi}{7}\\right)\\). Let's think step by step and output the final answer w... | math | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"answer": "\\sqrt{7}",
"difficulty": "5/8",
"index": 303,
"problem": "Compute \\(\\tan \\left(\\frac{\\pi}{7}\\right) \\tan \\left(\\frac{2 \\pi}{7}\\right) \\tan \\left(\\frac{3 \\pi}{7}\\right)\\)."
} | 7 | 303 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says \"One of the first forty fruit is the banana!'' Th... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"answer": "21",
"difficulty": "4/8",
"index": 304,
"problem": "Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says \"One of the first forty fruit is the banana!'' The last fruit responds \"No, one of the $\\emph{last}$ forty fru... | 0 | 304 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( m \\) and \\( n \\) be nonnegative integers. Prove that \\( m!n!(m+n)! \\) divides \\( (2m)!(2n)! \\). Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "m!n!(m+n)! \\text{ divides } (2m)!(2n)!",
"style": "rule"
} | {
"answer": "m!n!(m+n)! \\text{ divides } (2m)!(2n)!",
"difficulty": "5/8",
"index": 305,
"problem": "Let \\( m \\) and \\( n \\) be nonnegative integers. Prove that \\( m!n!(m+n)! \\) divides \\( (2m)!(2n)! \\)."
} | 1 | 305 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again... | math | {
"ground_truth": "99",
"style": "rule"
} | {
"answer": "99",
"difficulty": "2/8",
"index": 307,
"problem": "Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Gris... | 3 | 307 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Points \\( K \\) and \\( N \\) are located on the sides \\( AB \\) and \\( AC \\) of triangle \\( ABC \\) respectively, such that \\( AK = BK \\) and \\( AN = 2NC \\).\n\nIn what ratio... | math | {
"ground_truth": "4:3",
"style": "rule"
} | {
"answer": "4:3",
"difficulty": "4/8",
"index": 308,
"problem": "Points \\( K \\) and \\( N \\) are located on the sides \\( AB \\) and \\( AC \\) of triangle \\( ABC \\) respectively, such that \\( AK = BK \\) and \\( AN = 2NC \\).\n\nIn what ratio does the segment \\( KN \\) divide the median \\( AM \\) of t... | 4 | 308 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Nine hundred forty-three minus eighty-seven equals Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "856",
"style": "rule"
} | {
"answer": "856",
"difficulty": "7/8",
"index": 309,
"problem": "Nine hundred forty-three minus eighty-seven equals"
} | 5 | 309 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\\circ.$ \n$B_{1}$ is the foot of perpendicular from $B$ to $... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\frac{1}{2}",
"difficulty": "3/8",
"index": 311,
"problem": "An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\\circ.$ \n$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}... | 7 | 311 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Determine how many zeros precede the last digit of \\(2009^{2011}\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "0/8",
"index": 312,
"problem": "Determine how many zeros precede the last digit of \\(2009^{2011}\\)."
} | 0 | 312 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$? Let's thin... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"answer": "23",
"difficulty": "4/8",
"index": 313,
"problem": "Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?"
} | 1 | 313 |
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 derived from \\( y = f(x) \\) by reflecting it across the y-axis and then shifting it 3 units to the right. Provide the expression for \\( y = h(x) \\)... | math | {
"ground_truth": "h(x) = f(3 - x)",
"style": "rule"
} | {
"answer": "h(x) = f(3 - x)",
"difficulty": "0/8",
"index": 315,
"problem": "The graph of \\( y = h(x) \\) is derived from \\( y = f(x) \\) by reflecting it across the y-axis and then shifting it 3 units to the right. Provide the expression for \\( y = h(x) \\) in terms of \\( f(x) \\)."
} | 3 | 315 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "[ Ratio of areas of similar triangles ] [ Inscribed and circumscribed circles ]\n\nIn an isosceles triangle $A B C$, the angle at the base $A C$ is $\\alpha$. The circle inscribed in ... | math | {
"ground_truth": "\\cos \\alpha (1 - \\cos \\alpha)",
"style": "rule"
} | {
"answer": "\\cos \\alpha (1 - \\cos \\alpha)",
"difficulty": "4/8",
"index": 316,
"problem": "[ Ratio of areas of similar triangles ] [ Inscribed and circumscribed circles ]\n\nIn an isosceles triangle $A B C$, the angle at the base $A C$ is $\\alpha$. The circle inscribed in this triangle touches the sides ... | 4 | 316 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A fishing vessel illegally fishes in a foreign country's territorial waters, resulting in an identical loss of value for the foreign country with each cast of the net. The probability ... | math | {
"ground_truth": "k-1",
"style": "rule"
} | {
"answer": "k-1",
"difficulty": "3/8",
"index": 317,
"problem": "A fishing vessel illegally fishes in a foreign country's territorial waters, resulting in an identical loss of value for the foreign country with each cast of the net. The probability that the vessel will be detained by the foreign coast guard du... | 5 | 317 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. F... | math | {
"ground_truth": "210",
"style": "rule"
} | {
"answer": "210",
"difficulty": "1/8",
"index": 319,
"problem": "Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$."
} | 7 | 319 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $M=\\{1,2,\\dots,49\\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that the... | math | {
"ground_truth": "495",
"style": "rule"
} | {
"answer": "495",
"difficulty": "0/8",
"index": 320,
"problem": "Let $M=\\{1,2,\\dots,49\\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this v... | 0 | 320 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find all functions \\( g: \\mathbb{N} \\rightarrow \\mathbb{N} \\) such that \\( (g(n)+m)(g(m)+n) \\) is a perfect square for all \\( n, m \\). Let's think step by step and output the ... | math | {
"ground_truth": "(n)=n+",
"style": "rule"
} | {
"answer": "(n)=n+",
"difficulty": "6/8",
"index": 321,
"problem": "Find all functions \\( g: \\mathbb{N} \\rightarrow \\mathbb{N} \\) such that \\( (g(n)+m)(g(m)+n) \\) is a perfect square for all \\( n, m \\)."
} | 1 | 321 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $ \\overline{AB}$ be a diameter of circle $ \\omega$ . Extend $ \\overline{AB}$ through $ A$ to $ C$ . Point $ T$ lies on $ \\omega$ so that line $ CT$ is tangent to ... | math | {
"ground_truth": "432",
"style": "rule"
} | {
"answer": "432",
"difficulty": "5/8",
"index": 323,
"problem": "Let $ \\overline{AB}$ be a diameter of circle $ \\omega$ . Extend $ \\overline{AB}$ through $ A$ to $ C$ . Point $ T$ lies on $ \\omega$ so that line $ CT$ is tangent to $ \\omega$ . Point $ P$ is the foot of the perpendicular fr... | 3 | 323 |
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 the sides of triangle $\\triangle ABC$, and let $S$ be its area. Prove that\n\\[a^{2} + b^{2} + c^{2} \\geq 4 \\sqrt{3} S + (a - b)^{2} + (b - c)^{2} + (c - a)^{2},\\]... | math | {
"ground_truth": "^2+b^2+^2\\ge4\\sqrt{3}S+(b)^2+()^2+()^2",
"style": "rule"
} | {
"answer": "^2+b^2+^2\\ge4\\sqrt{3}S+(b)^2+()^2+()^2",
"difficulty": "3/8",
"index": 324,
"problem": "Let $a, b, c$ be the sides of triangle $\\triangle ABC$, and let $S$ be its area. Prove that\n\\[a^{2} + b^{2} + c^{2} \\geq 4 \\sqrt{3} S + (a - b)^{2} + (b - c)^{2} + (c - a)^{2},\\]\nwith equality if and on... | 4 | 324 |
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 $xOy$, the parametric equation of line $l$ is\n$$\n\\begin{cases}\nx = -1 + \\frac {\\sqrt {2}}{2}t \\\\\ny = 1 + \\frac {\\sqrt {2}}{2}t\n\\end{case... | math | {
"ground_truth": "\\frac{3\\sqrt{10}}{10}",
"style": "rule"
} | {
"answer": "\\frac{3\\sqrt{10}}{10}",
"difficulty": "7/8",
"index": 325,
"problem": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is\n$$\n\\begin{cases}\nx = -1 + \\frac {\\sqrt {2}}{2}t \\\\\ny = 1 + \\frac {\\sqrt {2}}{2}t\n\\end{cases}\n(t \\text{ is the parameter}),\n$$\nan... | 5 | 325 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In circle $\\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_{1}$ and $M_{2}$ respectively, such that $P M_{1}=15$ and $P M_{2}=20$. Line $M... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"answer": "7",
"difficulty": "1/8",
"index": 327,
"problem": "In circle $\\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_{1}$ and $M_{2}$ respectively, such that $P M_{1}=15$ and $P M_{2}=20$. Line $M_{1} M_{2}$ intersects $\\omega$ at points $A$ and $B$, with $M_... | 7 | 327 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given \\( x, y, z \\in \\mathbf{R} \\) with \\( 0 < z < y < x < \\frac{\\pi}{2} \\), prove that:\n\\[ \\frac{\\pi}{2} + 2 \\sin x \\cdot \\cos y + 2 \\sin y \\cdot \\cos z > \\sin 2x +... | math | {
"ground_truth": "\\frac{\\pi}{2}+2\\sinx\\cdot\\cosy+2\\siny\\cdot\\cosz>\\sin2x+\\sin2y+\\sin2z",
"style": "rule"
} | {
"answer": "\\frac{\\pi}{2}+2\\sinx\\cdot\\cosy+2\\siny\\cdot\\cosz>\\sin2x+\\sin2y+\\sin2z",
"difficulty": "3/8",
"index": 328,
"problem": "Given \\( x, y, z \\in \\mathbf{R} \\) with \\( 0 < z < y < x < \\frac{\\pi}{2} \\), prove that:\n\\[ \\frac{\\pi}{2} + 2 \\sin x \\cdot \\cos y + 2 \\sin y \\cdot \\cos ... | 0 | 328 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( x, y, z \\) be positive numbers satisfying the system of equations:\n\n\\[ \\left\\{\\begin{array}{l}\nx^{2}+xy+y^{2}=75 \\\\\ny^{2}+yz+z^{2}=4 \\\\\nz^{2}+xz+x^{2}=79\n\\end{a... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"answer": "20",
"difficulty": "5/8",
"index": 329,
"problem": "Let \\( x, y, z \\) be positive numbers satisfying the system of equations:\n\n\\[ \\left\\{\\begin{array}{l}\nx^{2}+xy+y^{2}=75 \\\\\ny^{2}+yz+z^{2}=4 \\\\\nz^{2}+xz+x^{2}=79\n\\end{array}\\right. \\]\n\nFind the value of the expression \\( xy + ... | 1 | 329 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The p... | math | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"answer": "2\\pi",
"difficulty": "2/8",
"index": 331,
"problem": "A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the tabl... | 3 | 331 |
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": "toyota camry",
"style": "rule"
} | {
"answer": "toyota camry",
"difficulty": "5/8",
"index": 332,
"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 uniq... | 4 | 332 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Four numbers form a geometric progression. If the second number is increased by 4 and the fourth number is decreased by 28, an arithmetic progression is obtained. Determine these four ... | math | {
"ground_truth": "2,6,18,54",
"style": "rule"
} | {
"answer": "2,6,18,54",
"difficulty": "0/8",
"index": 333,
"problem": "Four numbers form a geometric progression. If the second number is increased by 4 and the fourth number is decreased by 28, an arithmetic progression is obtained. Determine these four numbers."
} | 5 | 333 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Determine the range of the function\n\n\\[ f(x)=\\frac{4-3 \\sin ^{6} x-3 \\cos ^{6} x}{\\sin x \\cos x} \\]\n\nin the interval \\(\\left(0, \\frac{\\pi}{2}\\right)\\). Let's think ste... | math | {
"ground_truth": "[6,\\infty)",
"style": "rule"
} | {
"answer": "[6,\\infty)",
"difficulty": "7/8",
"index": 335,
"problem": "Determine the range of the function\n\n\\[ f(x)=\\frac{4-3 \\sin ^{6} x-3 \\cos ^{6} x}{\\sin x \\cos x} \\]\n\nin the interval \\(\\left(0, \\frac{\\pi}{2}\\right)\\)."
} | 7 | 335 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A pedestrian left point $A$ for point $B$, and with some delay, a second pedestrian left. When the first pedestrian had covered half the distance, the second pedestrian had walked 15 k... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"answer": "40",
"difficulty": "6/8",
"index": 336,
"problem": "A pedestrian left point $A$ for point $B$, and with some delay, a second pedestrian left. When the first pedestrian had covered half the distance, the second pedestrian had walked 15 km, and when the second pedestrian had covered half the distance... | 0 | 336 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Captain Avery is a stranded sailor navigating through treacherous waters. He has a small lifeboat that can hold up to 8 people, and currently, there are 3 people aboard, including hims... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"answer": "5",
"difficulty": "3/8",
"index": 337,
"problem": "Captain Avery is a stranded sailor navigating through treacherous waters. He has a small lifeboat that can hold up to 8 people, and currently, there are 3 people aboard, including himself. Captain Avery sees two other lifeboats nearby, one with 5 p... | 1 | 337 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that if any two nodes fail, it still remains possible to transmit informati... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"answer": "15",
"difficulty": "6/8",
"index": 339,
"problem": "What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that if any two nodes fail, it still remains possible to transmit information between any two remaining nodes (at least through a chain vi... | 3 | 339 |
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_{200}\\) be natural numbers greater than 2 (not necessarily distinct). In a \\(200 \\times 200\\) table, the numbers are arranged as follows: at the int... | math | {
"ground_truth": "-40000",
"style": "rule"
} | {
"answer": "-40000",
"difficulty": "4/8",
"index": 340,
"problem": "Let \\(x_{1}, x_{2}, \\ldots, x_{200}\\) be natural numbers greater than 2 (not necessarily distinct). In a \\(200 \\times 200\\) table, the numbers are arranged as follows: at the intersection of the \\(i\\)-th row and the \\(k\\)-th column, ... | 4 | 340 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the ... | math | {
"ground_truth": "Peter",
"style": "rule"
} | {
"answer": "Peter",
"difficulty": "0/8",
"index": 341,
"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... | 5 | 341 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "12.169. In the square \\(ABCD\\), an isosceles triangle \\(AEF\\) is inscribed; point \\(E\\) lies on side \\(BC\\), point \\(F\\) lies on side \\(CD\\), and \\(AE = EF\\). The tangent... | math | {
"ground_truth": "3 - \\sqrt{5}",
"style": "rule"
} | {
"answer": "3 - \\sqrt{5}",
"difficulty": "5/8",
"index": 343,
"problem": "12.169. In the square \\(ABCD\\), an isosceles triangle \\(AEF\\) is inscribed; point \\(E\\) lies on side \\(BC\\), point \\(F\\) lies on side \\(CD\\), and \\(AE = EF\\). The tangent of angle \\(AEF\\) is 2. Find the tangent of angle ... | 7 | 343 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "## Task 3 - 030823\n\nIn an auditorium, 300 chairs are arranged in several equally long rows. If 3 chairs are removed from each row to form a central aisle and these chairs are used to... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"answer": "15",
"difficulty": "5/8",
"index": 344,
"problem": "## Task 3 - 030823\n\nIn an auditorium, 300 chairs are arranged in several equally long rows. If 3 chairs are removed from each row to form a central aisle and these chairs are used to create 5 new rows (with a central aisle), the total number of ... | 0 | 344 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Quadrilateral $ABCD$ is inscribed in a circle with $\\angle BAC=70^{\\circ}, \\angle ADB=40^{\\circ}, AD=4,$ and $BC=6$. What is $AC$?\n$\\textbf{(A)}\\; 3+\\sqrt{5} \\qquad\\textbf{(B... | math | {
"ground_truth": "\\textbf{(B)}\\;6",
"style": "rule"
} | {
"answer": "\\textbf{(B)}\\;6",
"difficulty": "0/8",
"index": 345,
"problem": "Quadrilateral $ABCD$ is inscribed in a circle with $\\angle BAC=70^{\\circ}, \\angle ADB=40^{\\circ}, AD=4,$ and $BC=6$. What is $AC$?\n$\\textbf{(A)}\\; 3+\\sqrt{5} \\qquad\\textbf{(B)}\\; 6 \\qquad\\textbf{(C)}\\; \\dfrac{9}{2}\\s... | 1 | 345 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Compute\n\\[\\cos^6 0^\\circ + \\cos^6 1^\\circ + \\cos^6 2^\\circ + \\dots + \\cos^6 90^\\circ.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role":... | math | {
"ground_truth": "\\frac{229}{8}",
"style": "rule"
} | {
"answer": "\\frac{229}{8}",
"difficulty": "5/8",
"index": 347,
"problem": "Compute\n\\[\\cos^6 0^\\circ + \\cos^6 1^\\circ + \\cos^6 2^\\circ + \\dots + \\cos^6 90^\\circ.\\]"
} | 3 | 347 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selec... | math | {
"ground_truth": "76",
"style": "rule"
} | {
"answer": "76",
"difficulty": "7/8",
"index": 348,
"problem": "Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none o... | 4 | 348 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Prove that the necessary and sufficient condition for the median from $B$ in the triangle $ABC$ to be divided into three equal parts by the incircle of the triangle is\n\n$$\n\\frac{a}... | math | {
"ground_truth": "\\frac{}{5}=\\frac{b}{10}=\\frac{}{13}",
"style": "rule"
} | {
"answer": "\\frac{}{5}=\\frac{b}{10}=\\frac{}{13}",
"difficulty": "2/8",
"index": 349,
"problem": "Prove that the necessary and sufficient condition for the median from $B$ in the triangle $ABC$ to be divided into three equal parts by the incircle of the triangle is\n\n$$\n\\frac{a}{5}=\\frac{b}{10}=\\frac{c}... | 5 | 349 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given unit vectors $a$ and $b$ satisfying $|a+3b|=\\sqrt{13}$, find the angle between $a$ and $b$. Let's think step by step and output the final answer within \\boxed{}.",
"role": ... | math | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"answer": "\\frac{\\pi}{3}",
"difficulty": "6/8",
"index": 351,
"problem": "Given unit vectors $a$ and $b$ satisfying $|a+3b|=\\sqrt{13}$, find the angle between $a$ and $b$."
} | 7 | 351 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given positive real numbers \\( a \\) and \\( b \\) such that the function\n\\[ f(x) = x^3 + ax^2 + 2bx - 1 \\]\nhas three distinct zeros, and the function\n\\[ g(x) = 2x^2 + 2bx + a \... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "5/8",
"index": 352,
"problem": "Given positive real numbers \\( a \\) and \\( b \\) such that the function\n\\[ f(x) = x^3 + ax^2 + 2bx - 1 \\]\nhas three distinct zeros, and the function\n\\[ g(x) = 2x^2 + 2bx + a \\]\nhas no zeros. Prove that \\( a - b > 1 \\)."
} | 0 | 352 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let a convex n-gon \\( A_1 A_2 \\cdots A_n \\) be inscribed in a unit circle. Find the maximum value of the sum of the squares of all its sides and diagonals, and determine when this m... | math | {
"ground_truth": "n^2",
"style": "rule"
} | {
"answer": "n^2",
"difficulty": "6/8",
"index": 353,
"problem": "Let a convex n-gon \\( A_1 A_2 \\cdots A_n \\) be inscribed in a unit circle. Find the maximum value of the sum of the squares of all its sides and diagonals, and determine when this maximum value is achieved."
} | 1 | 353 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: Let's think step by step and outp... | math | {
"ground_truth": "79",
"style": "rule"
} | {
"answer": "79",
"difficulty": "3/8",
"index": 355,
"problem": "Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is:"
} | 3 | 355 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For a given increasing function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\), define the function \\( g(x, y) \\) as follows:\n\n\\[ g(x, y) = \\frac{f(x+y) - f(x)}{f(x) - f(x-y)}, ... | math | {
"ground_truth": "\\frac{1}{14}<(x,y)<14",
"style": "rule"
} | {
"answer": "\\frac{1}{14}<(x,y)<14",
"difficulty": "1/8",
"index": 356,
"problem": "For a given increasing function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\), define the function \\( g(x, y) \\) as follows:\n\n\\[ g(x, y) = \\frac{f(x+y) - f(x)}{f(x) - f(x-y)}, \\quad x \\in \\mathbb{R}, \\; y > 0. \\]\n... | 4 | 356 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blan... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\frac{1}{2}",
"difficulty": "2/8",
"index": 357,
"problem": "The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solvi... | 5 | 357 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The number of solutions in positive integers of $2x+3y=763$ is:\n$\\textbf{(A)}\\ 255 \\qquad \\textbf{(B)}\\ 254\\qquad \\textbf{(C)}\\ 128 \\qquad \\textbf{(D)}\\ 127 \\qquad \\textb... | math | {
"ground_truth": "\\textbf{(D)}\\127",
"style": "rule"
} | {
"answer": "\\textbf{(D)}\\127",
"difficulty": "1/8",
"index": 359,
"problem": "The number of solutions in positive integers of $2x+3y=763$ is:\n$\\textbf{(A)}\\ 255 \\qquad \\textbf{(B)}\\ 254\\qquad \\textbf{(C)}\\ 128 \\qquad \\textbf{(D)}\\ 127 \\qquad \\textbf{(E)}\\ 0$"
} | 7 | 359 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive integer $n$ such th... | math | {
"ground_truth": "150",
"style": "rule"
} | {
"answer": "150",
"difficulty": "7/8",
"index": 360,
"problem": "Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive integer $n$ such that $2^n$ divides $K$."
} | 0 | 360 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probabili... | math | {
"ground_truth": "\\dfrac{1001}{2003}",
"style": "rule"
} | {
"answer": "\\dfrac{1001}{2003}",
"difficulty": "7/8",
"index": 361,
"problem": "A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shade... | 1 | 361 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a\\cdot (a-b)=3$, find the angle between $a$ and $b$. Let's think step by step and output the final answer within \\boxed{... | math | {
"ground_truth": "\\frac{\\pi }{3}",
"style": "rule"
} | {
"answer": "\\frac{\\pi }{3}",
"difficulty": "4/8",
"index": 363,
"problem": "Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a\\cdot (a-b)=3$, find the angle between $a$ and $b$."
} | 3 | 363 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An exchange point conducts two types of operations:\n\n1) Give 2 euros - receive 3 dollars and a candy as a gift;\n2) Give 5 dollars - receive 3 euros and a candy as a gift.\n\nWhen th... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"answer": "10",
"difficulty": "4/8",
"index": 364,
"problem": "An exchange point conducts two types of operations:\n\n1) Give 2 euros - receive 3 dollars and a candy as a gift;\n2) Give 5 dollars - receive 3 euros and a candy as a gift.\n\nWhen the wealthy Buratino came to the exchange point, he had only doll... | 4 | 364 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "For which values of the constant \\( c \\) does the equation\n\\[ \nx^{2} - 2\\left|x + \\frac{1}{4}\\right| + c = 0 \n\\]\nhave exactly three distinct real roots? Let's think step by ... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"answer": "\\frac{1}{2}",
"difficulty": "0/8",
"index": 365,
"problem": "For which values of the constant \\( c \\) does the equation\n\\[ \nx^{2} - 2\\left|x + \\frac{1}{4}\\right| + c = 0 \n\\]\nhave exactly three distinct real roots?"
} | 5 | 365 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given the polynomial \n$$\n\\begin{aligned}\nf(x)= & a_{2020} x^{2020}+a_{2019} x^{2019}+\\cdots+ \\\\\n& a_{2018} x^{2018}+a_{1} x+a_{0},\n\\end{aligned}\n$$\n\nwhere $a_{i} \\in \\ma... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"answer": "8",
"difficulty": "1/8",
"index": 367,
"problem": "Given the polynomial \n$$\n\\begin{aligned}\nf(x)= & a_{2020} x^{2020}+a_{2019} x^{2019}+\\cdots+ \\\\\n& a_{2018} x^{2018}+a_{1} x+a_{0},\n\\end{aligned}\n$$\n\nwhere $a_{i} \\in \\mathbf{Z}$ for $i=0,1, \\cdots, 2020$, determine the number of ord... | 7 | 367 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "9.1. In a five-digit number, one of the digits was crossed out, and this four-digit number was subtracted from the original number. As a result, the number 54321 was obtained. Find the... | math | {
"ground_truth": "60356",
"style": "rule"
} | {
"answer": "60356",
"difficulty": "6/8",
"index": 368,
"problem": "9.1. In a five-digit number, one of the digits was crossed out, and this four-digit number was subtracted from the original number. As a result, the number 54321 was obtained. Find the original number."
} | 0 | 368 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "At the vertices of a regular $n$-gon, the numbers are arranged such that there are $n-1$ zeros and one one. It is allowed to increase by 1 all numbers at the vertices of any regular $k... | math | {
"ground_truth": "No",
"style": "rule"
} | {
"answer": "No",
"difficulty": "2/8",
"index": 369,
"problem": "At the vertices of a regular $n$-gon, the numbers are arranged such that there are $n-1$ zeros and one one. It is allowed to increase by 1 all numbers at the vertices of any regular $k$-gon inscribed in this polygon. Is it possible to make all the... | 1 | 369 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pant... | math | {
"ground_truth": "216",
"style": "rule"
} | {
"answer": "216",
"difficulty": "7/8",
"index": 371,
"problem": "The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pants, a shirt, and shoes, and wore a jacket for some lessons. On ... | 3 | 371 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a quadratic function \\( f(x) = ax^2 + bx + 2 \\) (where \\( a \\neq 0 \\)), and for any real number \\( x \\), it holds that \\( |f(x)| \\geq 2 \\). Find the coordinates of the ... | math | {
"ground_truth": "\\left(0, 2 + \\dfrac{1}{4a}\\right)",
"style": "rule"
} | {
"answer": "\\left(0, 2 + \\dfrac{1}{4a}\\right)",
"difficulty": "4/8",
"index": 372,
"problem": "Given a quadratic function \\( f(x) = ax^2 + bx + 2 \\) (where \\( a \\neq 0 \\)), and for any real number \\( x \\), it holds that \\( |f(x)| \\geq 2 \\). Find the coordinates of the focus of this parabolic curve... | 4 | 372 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers. Let's think step by step and output the final answe... | math | {
"ground_truth": "1111111110000",
"style": "rule"
} | {
"answer": "1111111110000",
"difficulty": "2/8",
"index": 373,
"problem": "Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers."
} | 5 | 373 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "3.52 A passenger on a train knows that the speed of this train on this section of the track is 40 km/h. As soon as a passing train started to go by the window, the passenger started a ... | math | {
"ground_truth": "50",
"style": "rule"
} | {
"answer": "50",
"difficulty": "5/8",
"index": 375,
"problem": "3.52 A passenger on a train knows that the speed of this train on this section of the track is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noticed that the passing train went by the window... | 7 | 375 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( x \\) be a complex number such that \\( x + x^{-1} \\) is a root of the polynomial \\( p(t) = t^3 + t^2 - 2t - 1 \\). Find all possible values of \\( x^7 + x^{-7} \\). Let's th... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"answer": "2",
"difficulty": "7/8",
"index": 376,
"problem": "Let \\( x \\) be a complex number such that \\( x + x^{-1} \\) is a root of the polynomial \\( p(t) = t^3 + t^2 - 2t - 1 \\). Find all possible values of \\( x^7 + x^{-7} \\)."
} | 0 | 376 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Compare the number \\(a\\) with one, if\n$$\na = 0.99999^{1.00001} \\cdot 1.00001^{0.99999}.\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "u... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"answer": "1",
"difficulty": "6/8",
"index": 377,
"problem": "Compare the number \\(a\\) with one, if\n$$\na = 0.99999^{1.00001} \\cdot 1.00001^{0.99999}.\n$$"
} | 1 | 377 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such a... | math | {
"ground_truth": "\\frac{6}{4165}",
"style": "rule"
} | {
"answer": "\\frac{6}{4165}",
"difficulty": "6/8",
"index": 379,
"problem": "We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that f... | 3 | 379 |
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": "ford f150",
"style": "rule"
} | {
"answer": "ford f150",
"difficulty": "0/8",
"index": 380,
"problem": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique ... | 4 | 380 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the ... | math | {
"ground_truth": "mountain",
"style": "rule"
} | {
"answer": "mountain",
"difficulty": "3/8",
"index": 381,
"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 n... | 5 | 381 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Given a positive integer $n (n \\geqslant 2)$. From the numbers $1, 2, \\cdots, 3n$, select $m$ numbers such that there must be four distinct numbers $a, b, c, d$ among them, which sat... | math | {
"ground_truth": "2n+2",
"style": "rule"
} | {
"answer": "2n+2",
"difficulty": "0/8",
"index": 383,
"problem": "Given a positive integer $n (n \\geqslant 2)$. From the numbers $1, 2, \\cdots, 3n$, select $m$ numbers such that there must be four distinct numbers $a, b, c, d$ among them, which satisfy $a = b + c + d$. Find the minimum value of $m$."
} | 7 | 383 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "## 9. Rhombi\n\nRhombus $A_{0} B_{0} C_{0} D_{0}$ is located inside rhombus $A B C D$ such that their sides are parallel, and their centers are at the same point. The segments $\\overl... | math | {
"ground_truth": "864",
"style": "rule"
} | {
"answer": "864",
"difficulty": "1/8",
"index": 384,
"problem": "## 9. Rhombi\n\nRhombus $A_{0} B_{0} C_{0} D_{0}$ is located inside rhombus $A B C D$ such that their sides are parallel, and their centers are at the same point. The segments $\\overline{A A_{0}}, \\overline{B B_{0}}, \\overline{C C_{0}}$, and $... | 0 | 384 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "A positive integer $N$ greater than $1$ is described as special if in its base- $8$ and base- $9$ representations, both the leading and ending digit of $N$ are equal to $1$ . ... | math | {
"ground_truth": "793",
"style": "rule"
} | {
"answer": "793",
"difficulty": "4/8",
"index": 385,
"problem": "A positive integer $N$ greater than $1$ is described as special if in its base- $8$ and base- $9$ representations, both the leading and ending digit of $N$ are equal to $1$ . What is the smallest special integer in decimal representation... | 1 | 385 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "2. Each of the islanders is either a knight, who always tells the truth, or a liar, who always lies (both types are present on the island). Every resident knows whether each other is a... | math | {
"ground_truth": "No",
"style": "rule"
} | {
"answer": "No",
"difficulty": "5/8",
"index": 387,
"problem": "2. Each of the islanders is either a knight, who always tells the truth, or a liar, who always lies (both types are present on the island). Every resident knows whether each other is a knight or a liar. Some of the islanders stated that there is a... | 3 | 387 |
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 non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\\sigma$ be a permutation of the set $\\{1,2,...,n\\}$ . Determine ... | math | {
"ground_truth": "n",
"style": "rule"
} | {
"answer": "n",
"difficulty": "4/8",
"index": 388,
"problem": "Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\\sigma$ be a permutation of the set $\\{1,2,...,n\\}$ . Determine the least value the sum\n\\[\\sum_{i=1}^{n}\\frac{P(x_i^2)}{P(x_... | 4 | 388 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "## Task 5 - 261245\n\nDetermine all natural numbers $n \\geq 3$ for which the following statement holds:\n\nEvery plane convex $n$-gon $A_{1} A_{2} \\ldots A_{n}$ is completely covered... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"answer": "4",
"difficulty": "2/8",
"index": 389,
"problem": "## Task 5 - 261245\n\nDetermine all natural numbers $n \\geq 3$ for which the following statement holds:\n\nEvery plane convex $n$-gon $A_{1} A_{2} \\ldots A_{n}$ is completely covered by the areas of the $n$ circles that have the segments $A_{i} A... | 5 | 389 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The factors of $x^4+64$ are: Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x^2-4x+8)(x^2+4x+8)",
"style": "rule"
} | {
"answer": "(x^2-4x+8)(x^2+4x+8)",
"difficulty": "2/8",
"index": 391,
"problem": "The factors of $x^4+64$ are:"
} | 7 | 391 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "3. A $4 \\mathrm{~cm} \\times 4 \\mathrm{~cm}$ square is split into four rectangular regions using two line segments parallel to the sides.\nHow many ways are there to do this so that ... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"answer": "17",
"difficulty": "1/8",
"index": 392,
"problem": "3. A $4 \\mathrm{~cm} \\times 4 \\mathrm{~cm}$ square is split into four rectangular regions using two line segments parallel to the sides.\nHow many ways are there to do this so that each region has an area equal to an integer number of square ce... | 0 | 392 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $A=30^{\\circ}$, $AB=\\sqrt {3}$, $BC=1$, find the area of $\\triangle ABC$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "... | math | {
"ground_truth": "\\frac {\\sqrt {3}}{4}",
"style": "rule"
} | {
"answer": "\\frac {\\sqrt {3}}{4}",
"difficulty": "0/8",
"index": 393,
"problem": "In $\\triangle ABC$, $A=30^{\\circ}$, $AB=\\sqrt {3}$, $BC=1$, find the area of $\\triangle ABC$."
} | 1 | 393 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are the... | math | {
"ground_truth": "615",
"style": "rule"
} | {
"answer": "615",
"difficulty": "7/8",
"index": 395,
"problem": "An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?"
} | 3 | 395 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "# 3. Clone 1\n\nA rope was divided into 19 equal parts and arranged in a snake-like pattern. After that, a cut was made along the dotted line. The rope split into 20 pieces: the longes... | math | {
"ground_truth": "152",
"style": "rule"
} | {
"answer": "152",
"difficulty": "2/8",
"index": 396,
"problem": "# 3. Clone 1\n\nA rope was divided into 19 equal parts and arranged in a snake-like pattern. After that, a cut was made along the dotted line. The rope split into 20 pieces: the longest of them is 8 meters, and the shortest is 2 meters. What was ... | 4 | 396 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "Consider the sequence $$ 1,7,8,49,50,56,57,343\\ldots $$ which consists of sums of distinct powers of $7$ , that is, $7^0$ , $7^1$ , $7^0+7^1$ , $7^2$ , $\\ldots$ in increasing ... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"answer": "36",
"difficulty": "5/8",
"index": 397,
"problem": "Consider the sequence $$ 1,7,8,49,50,56,57,343\\ldots $$ which consists of sums of distinct powers of $7$ , that is, $7^0$ , $7^1$ , $7^0+7^1$ , $7^2$ , $\\ldots$ in increasing order. At what position will $16856$ occur in this sequence?"... | 5 | 397 |
POLARIS-Project/Polaris-Dataset-53K | [
{
"content": "Please reason step by step and put the final answer in \\boxed{}.",
"role": "system"
},
{
"content": "The measures of the interior angles of a triangle are in a 3:4:9 ratio. If the largest angle is decreased by $x$ degrees, and it still maintains a triangle with the sum of angles being... | math | {
"ground_truth": "33.75",
"style": "rule"
} | {
"answer": "33.75",
"difficulty": "5/8",
"index": 399,
"problem": "The measures of the interior angles of a triangle are in a 3:4:9 ratio. If the largest angle is decreased by $x$ degrees, and it still maintains a triangle with the sum of angles being 180 degrees, find $x$ such that the smallest angle doubles ... | 7 | 399 |
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