data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given a moving circle $M$ that passes through the fixed point $F(0,-1)$ and is tangent to the line $y=1$. The trajectory of the circle's center $M$ forms a curve $C$. Let $P$ be a point on the line $l$: $x-y+2=0$. Draw two tangent lines $PA$ and $PB$ from point $P$ to the curve... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 29649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a given integer $n\\ge 2$, let $a_0,a_1,\\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\\{ a_i+a_j \\mid 0\\le i \\le j \\le n \\}$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "3n"
} | {
"split": "train",
"index": 29650,
"solution": "\nFor a given integer \\( n \\ge 2 \\), let \\( a_0, a_1, \\ldots, a_n \\) be integers satisfying \\( 0 = a_0 < a_1 < \\ldots < a_n = 2n-1 \\). We aim to find the smallest possible number of elements in the set \\( \\{ a_i + a_j \\mid 0 \\le i \\le j \\le n \\} \\)... |
deepscaler | [
{
"role": "user",
"content": "Given the inequality $\\frac{x-2}{ax-1} > 0$ with the solution set $(-1,2)$, determine the constant term in the expansion of the binomial $(ax-\\frac{1}{x^{2}})^{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $A_n^2 = 132$, calculate the value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory produces two types of products, A and B, with profits P and Q (in ten thousand yuan), respectively. The relationship between the profits and the invested capital m (in ten thousand yuan) follows the empirical formulas P = (1/3)m + 65 and Q = 76 + 4√m. Now, 150 ten tho... | math | {
"style": "rule",
"ground_truth": "203"
} | {
"split": "train",
"index": 29653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $sin(\\alpha-\\beta)=\\frac{1}{3}$ and $cos\\alpha sin\\beta=\\frac{1}{6}$, find $\\cos \\left(2\\alpha +2\\beta \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 29654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $DEF$ below, where $DF = 8$ and $\\angle D = 45^\\circ$.\n\n[asy]\nunitsize(1inch);\npair D,E,F;\nD = (0,0);\nE= (sqrt(2),0);\nF = (0,sqrt(2));\ndraw (D--E--F--D,linewidth(0.9));\ndraw(rightanglemark(E,D,F,3));\nlabel(\"$D$\",D,S);\nlabel(\"$E$\",E,S);... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 29655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $121_b$, written in the integral base $b$, is the square of an integer, for Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "b > 2"
} | {
"split": "train",
"index": 29656,
"solution": "1. **Convert the base-b number to base-10**: The number $121_b$ in base $b$ can be expressed in base 10 as:\n \\[\n 1 \\cdot b^2 + 2 \\cdot b^1 + 1 \\cdot b^0 = b^2 + 2b + 1\n \\]\n\n2. **Factorize the expression**: The expression $b^2 + 2b + 1$ can be rewrit... |
deepscaler | [
{
"role": "user",
"content": "A math teacher randomly selects 3 questions for analysis from a test paper consisting of 12 multiple-choice questions, 4 fill-in-the-blank questions, and 6 open-ended questions. The number of different ways to select questions such that at least one multiple-choice question and... | math | {
"style": "rule",
"ground_truth": "864"
} | {
"split": "train",
"index": 29657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\dfrac{3 \\times 4}{6}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 29658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the perimeter of a triangle with points $A(1,2)$, $B(1,8)$, and $C(5,5)$ on a Cartesian coordinate plane. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The denominator of a fraction is 7 less than 3 times the numerator. If the fraction is equivalent to $2/5$, what is the numerator of the fraction? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 29660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 29661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1660"
} | {
"split": "train",
"index": 29662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59049"
} | {
"split": "train",
"index": 29663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a class of 30 students is seated in a movie theater, then in any case at least two classmates will be in the same row. If the same is done with a class of 26 students, then at least three rows will be empty. How many rows are in the theater?\n\n Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 29664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{28}/R_8$ when $R_{28}$ is divided by $R_8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 29665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x < a < 0$ means that $x$ and $a$ are numbers such that $x$ is less than $a$ and $a$ is less than zero, then: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 > ax > a^2"
} | {
"split": "train",
"index": 29666,
"solution": "Given the conditions $x < a < 0$, we know that both $x$ and $a$ are negative numbers. Let's analyze each part of the options provided:\n\n1. **$x^2$ and $a^2$ are positive**: Since squaring a negative number results in a positive number, both $x^2$ and $a^2$ are po... |
deepscaler | [
{
"role": "user",
"content": "Each valve $A$, $B$, and $C$, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. The number of hours ... | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"split": "train",
"index": 29667,
"solution": "1. **Define the rates and volume**: Let the rate of water flow through valve $A$ be $a$ liters per hour, through valve $B$ be $b$ liters per hour, and through valve $C$ be $c$ liters per hour. Assume the volume of the tank is 1 liter.\n\n2. **Set up the equations b... |
deepscaler | [
{
"role": "user",
"content": "Each pair of vertices of a regular $67$ -gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 29668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with side length 2 has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and for $1\\le i\\le 4,$ vertices $Q_i$ and $Q_i'$ are opposite to each other. A regular octahedron has one vertex in each of the s... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 29669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\Gamma$ denote the circumcircle of triangle $A B C$. Point $D$ is on $\\overline{A B}$ such that $\\overline{C D}$ bisects $\\angle A C B$. Points $P$ and $Q$ are on $\\Gamma$ such that $\\overline{P Q}$ passes through $D$ and is perpendicular to $\\overline{C D}$. Comput... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{745}"
} | {
"split": "train",
"index": 29670,
"solution": "Suppose that $P$ lies between $A$ and $B$ and $Q$ lies between $A$ and $C$, and let line $P Q$ intersect lines $A C$ and $B C$ at $E$ and $F$ respectively. As usual, we write $a, b, c$ for the lengths of $B C, C A, A B$. By the angle bisector theorem, $A D / D B=A ... |
deepscaler | [
{
"role": "user",
"content": "A meal at a diner includes a burger weighing 150 grams, of which 40 grams are filler. What percent of the burger is not filler? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "73.33\\%"
} | {
"split": "train",
"index": 29671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 20-step path is to go from $(-5,-5)$ to $(5,5)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the rectangle $-3 \\le x \\le 3$, $-1 \\le y \\le 1$ at each step? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "2126"
} | {
"split": "train",
"index": 29672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n\\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\\left\\{1;\\;2;\\;...;\\;r\\right\\}$ in a way such that the following two conditions are fulfilled:\n\n[b]1.[/b] Each number from the set $\\left\\{1;\\... | math | {
"style": "rule",
"ground_truth": "{\\frac{n!(n-1)!}{2^{n-1}}}"
} | {
"split": "train",
"index": 29673,
"solution": "\nTo solve the given problem, we consider a regular \\( n \\)-gon with sides and diagonals labeled from a set \\(\\{1, 2, \\ldots, r\\}\\). The goal is to find the maximal \\( r \\) such that the labeling satisfies the provided conditions. \n\n### Part (a): Finding... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac13\\cdot\\frac92\\cdot\\frac1{27}\\cdot\\frac{54}{1}\\cdot\\frac{1}{81}\\cdot\\frac{162}{1}\\cdot\\frac{1}{243}\\cdot\\frac{486}{1}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One writes, initially, the numbers $1,2,3,\\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\\frac{ab}{f(a,b)}$ , where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$ , one will make this until remain... | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"split": "train",
"index": 29675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the graph of the equation $(x - 1)^2 + (y - 1)^2 = |x - 1| + |y - 1|$?\nA) $\\frac{\\pi}{4}$\nB) $\\frac{\\pi}{2}$\nC) $\\frac{\\pi}{3}$\nD) $\\pi$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 29676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $(-1)^{47} + 2^{(3^3+4^2-6^2)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 29677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming arranges chess pieces in a two-layer hollow square array (the diagram shows the top left part of the array). After arranging the inner layer, 60 chess pieces are left. After arranging the outer layer, 32 chess pieces are left. How many chess pieces does Xiao Ming have... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 29678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set $\\{[x]+[2x]+[3x] \\mid x \\in \\mathbf{R}\\} \\bigcap \\{1, 2, \\cdots, 100\\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 29679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Solve the trigonometric inequality: $\\cos x \\geq \\frac{1}{2}$\n2. In $\\triangle ABC$, if $\\sin A + \\cos A = \\frac{\\sqrt{2}}{2}$, find the value of $\\tan A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2 - \\sqrt{3}"
} | {
"split": "train",
"index": 29680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following equations: \n(1) $x^{2}-3x=4$; \n(2) $x(x-2)+x-2=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive solution to\n\\[\\sqrt{x + 2 + \\sqrt{x + 2 + \\dotsb}} = \\sqrt{x \\sqrt{x \\dotsm}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3}"
} | {
"split": "train",
"index": 29682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The truncated right circular cone has a large base radius 8 cm and a small base radius of 4 cm. The height of the truncated cone is 6 cm. How many $\\text{cm}^3$ are in the volume of this solid? [asy]\n\nimport olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;\n\nd... | math | {
"style": "rule",
"ground_truth": "224\\pi"
} | {
"split": "train",
"index": 29683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\begin{cases} x-5, & x\\geq 2000 \\\\ f[f(x+8)], & x<2000 \\end{cases}$, calculate $f(1996)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"split": "train",
"index": 29684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amelia has a coin that lands heads with a probability of $\\frac{3}{7}$, and Blaine has a coin that lands on heads with a probability of $\\frac{1}{4}$. Initially, they simultaneously toss their coins once. If both get heads, they stop; otherwise, if either gets a head and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{14}"
} | {
"split": "train",
"index": 29685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 29686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 121 is a palindrome, because it reads the same backwards as forward. How many integer palindromes are between 100 and 500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 29687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My co-worker Larry only likes numbers that are divisible by 4, such as 20, or 4,004. How many different ones digits are possible in numbers that Larry likes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with all positive terms and $\\lg=6$, calculate the value of $a_1 \\cdot a_{15}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10^4"
} | {
"split": "train",
"index": 29689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\\] Find $|f(0)|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 29690,
"solution": "Because a cubic must come in a \"wave form\" with two points of inflection, we can see that $f(1)=f(5)=f(6)$, and $f(2)=f(3)=f(7)$. By symmetry, $f(4)=0$. Now, WLOG let $f(1)=12$, and $f(2)=f(3)=-12$. Then, we can use finite differences to get that the third (const... |
deepscaler | [
{
"role": "user",
"content": "Vitya Perestukin always incorrectly calculates percentages during surveys: he divides the number of respondents who answered a certain way by the number of all remaining respondents. For instance, in the survey \"What is your name?\" conducted among 7 Annas, 9 Olgas, 8 Julias, ... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 29691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $3\\cdot\\frac{11}{4}\\cdot \\frac{16}{-55}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{5}"
} | {
"split": "train",
"index": 29692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the graph of $\\frac{x^2+3x+2}{x^3+x^2-2x}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for a total of 8 hours per day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand to coincide with the hour hand once. Calculate the am... | math | {
"style": "rule",
"ground_truth": "2.60"
} | {
"split": "train",
"index": 29694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\\frac ab,$ wh... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 29695,
"solution": "1. **Understanding the Problem:**\n The problem asks us to find the possible values of the slope $m$ of a line such that exactly $300$ lattice points from the set $S$ (where $S$ consists of points $(x,y)$ with $1 \\leq x, y \\leq 30$) lie on or below the line $y ... |
deepscaler | [
{
"role": "user",
"content": "The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8192"
} | {
"split": "train",
"index": 29696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $\\cos C+(\\cos A- \\sqrt {3}\\sin A)\\cos B=0$.\n(1) Find the measure of angle $B$;\n(2) If $\\sin (A- \\frac {π}{3})= \\frac {3}{5}$, find $\\sin 2C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {24+7 \\sqrt {3}}{50}"
} | {
"split": "train",
"index": 29697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain exam, students' math scores follow a normal distribution $N(100,100)$. It is known that there are 1000 students taking this exam. Then the number of students whose math scores are between 70 and 110 points is approximately ____.\n\n(Reference data: $P(\\mu -\\sigma... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 29698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point inside regular pentagon $A B C D E$ such that $\\angle P A B=48^{\\circ}$ and $\\angle P D C=42^{\\circ}$. Find $\\angle B P C$, in degrees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84^{\\circ}"
} | {
"split": "train",
"index": 29699,
"solution": "Since a regular pentagon has interior angles $108^{\\circ}$, we can compute $\\angle P D E=66^{\\circ}, \\angle P A E=60^{\\circ}$, and $\\angle A P D=360^{\\circ}-\\angle A E D-\\angle P D E-\\angle P A E=126^{\\circ}$. Now observe that drawing $P E$ divides quadr... |
deepscaler | [
{
"role": "user",
"content": "Given a right quadrilateral pyramid \\( V-ABCD \\) with the height equal to half the length of \\( AB \\), \\( M \\) is the midpoint of the lateral edge \\( VB \\), and \\( N \\) is a point on the lateral edge \\( VD \\) such that \\( DN=2VN \\). Find the cosine of the angle be... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{11}}{11}"
} | {
"split": "train",
"index": 29700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the tetrahedron \\( P-ABC \\), \\( \\triangle ABC \\) is an equilateral triangle with a side length of \\( 2\\sqrt{3} \\), \\( PB = PC = \\sqrt{5} \\), and the dihedral angle between \\( P-BC \\) and \\( BC-A \\) is \\( 45^\\circ \\). Find the surface area of the circumscrib... | math | {
"style": "rule",
"ground_truth": "25\\pi"
} | {
"split": "train",
"index": 29701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $s(n)$ be the number of 1's in the binary representation of $n$ . Find the number of ordered pairs of integers $(a,b)$ with $0 \\leq a < 64, 0 \\leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$ .\n\n*Author:Anderson Wang* Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "1458"
} | {
"split": "train",
"index": 29702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rachel has two identical basil plants and an aloe plant. She also has two identical white lamps and two identical red lamps she can put each plant under (she can put more than one plant under a lamp, but each plant is under exactly one lamp). How many ways are there for Rachel ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 29703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\\%$ longer than the minor axis, the length of the major axis is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"split": "train",
"index": 29704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ms. Carr expands her reading list to 12 books and asks each student to choose any 6 books. Harold and Betty each randomly select 6 books from this list. Calculate the probability that there are exactly 3 books that they both select. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{405}{2223}"
} | {
"split": "train",
"index": 29705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the end of the school year, teachers of the third grade met with the parents of some of their students; exactly 31 people were present at this meeting. The Latin teacher was asked questions by 16 parents, the French teacher by 17 parents, the English teacher by 18 parents, a... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 29706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. Calculate the time Xiaoming ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a rectangular garden is 60 feet. If the length of the field is twice the width, what is the area of the field, in square feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 29708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In space, the following four propositions are given: \n(1) Through a point, there is exactly one plane perpendicular to a given line; \n(2) If the distances from two points outside a plane to the plane are equal, then the line passing through these two points must be parallel... | math | {
"style": "rule",
"ground_truth": "(1)(4)"
} | {
"split": "train",
"index": 29709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A floor decoration is a circle with eight rays pointing from the center. The rays form eight congruent central angles. One of the rays points due north. What is the measure in degrees of the smaller angle formed between the ray pointing East and the ray pointing Southwest?\n... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 29710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( \\frac{1}{3} \\leqslant a \\leqslant 1 \\), if \\( f(x)=a x^{2}-2 x+1 \\) attains its maximum value \\( M(a) \\) and minimum value \\( N(a) \\) on the interval \\([1,3]\\), and let \\( g(a)=M(a)-N(a) \\), then the minimum value of \\( g(a) \\) is \\(\\quad\\) . Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 29711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe wizard revealed the secret of wisdom. For those who wanted to know precisely where the secret is hidden, he left a clue in his magic book:\n\n\\[ 5 \\cdot \\text{BANK} = 6 \\cdot \\text{SAD} \\]\n\nEach letter in this clue represents a certain digit. Find these digits and... | math | {
"style": "rule",
"ground_truth": "55.543065317"
} | {
"split": "train",
"index": 29712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lily and Sarah are playing a game. They each choose a real number at random between -1 and 1. They then add the squares of their numbers together. If the result is greater than or equal to 1, Lily wins, and if the result is less than 1, Sarah wins. What is the probability that ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 29713,
"solution": "If we let $x$ denote Lily's choice of number and $y$ denote Sarah's, then all possible outcomes are represented by the square with vertices $(-1,-1),(-1,1),(1,-1)$, and $(1,1)$. Sarah wins if $x^{2}+y^{2} \\leq 1$, which is the area inside the unit circle. Since th... |
deepscaler | [
{
"role": "user",
"content": "A line $l$ passes through two points $P(-1,2)$ and $Q(2,-2)$, and intersects the hyperbola $(y-2)^{2}-x^{2}=1$ at two points $A$ and $B$. \n$(1)$ Write the parametric equation of $l$ as required by the question; \n$(2)$ Find the distance between the midpoint $M$ of $AB$ and p... | math | {
"style": "rule",
"ground_truth": "5 \\sqrt {65}"
} | {
"split": "train",
"index": 29714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( g(2022) \\) if for any real numbers \\( x, y \\) the following equality holds:\n\n\\[ g(x-y) = g(x) + g(y) - 2021(x+y) \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4086462"
} | {
"split": "train",
"index": 29715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible? Let... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29716,
"solution": "1. **Identify the Problem Context**: We are given a regular tetrahedron with each face painted either red, white, or blue. We need to find the number of distinguishable colorings, considering that colorings are indistinguishable if one tetrahedron can be rotated to... |
deepscaler | [
{
"role": "user",
"content": "The length of the interval of solutions of the inequality $a \\le 2x + 3 \\le b$ is $10$. What is $b - a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 29717,
"solution": "1. **Rewrite the inequality in terms of $x$:** \n Given the inequality $a \\le 2x + 3 \\le b$, we first isolate $x$. Subtract $3$ from all parts of the inequality:\n \\[\n a - 3 \\le 2x \\le b - 3\n \\]\n \n2. **Divide by $2$ to solve for $x$:** \n Ne... |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\ldots, 98\\} \\). Find the smallest natural number \\( n \\) such that from any subset of \\( S \\) containing \\( n \\) elements, we can always select 10 numbers. No matter how we divide these 10 numbers into two groups, there will always be one group co... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 29718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 29719,
"solution": "The ant goes in the opposite direction every $3$ moves, going $(1/2)^3=1/8$ the distance backwards. Using geometric series, he travels $1-1/8+1/64-1/512...=(7/8)(1+1/64+1/4096...)=(7/8)(64/63)=8/9$ the distance of the first three moves over infinity moves. Now, we ... |
deepscaler | [
{
"role": "user",
"content": "Two families of three, with a total of $4$ adults and $2$ children, agree to go on a countryside trip together on Sunday using two cars, an \"Audi\" and a \"Jetta\". Each car can accommodate a maximum of $4$ people, and the two children cannot be left alone in one car. Determin... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 29720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 29721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\\{1,2,...,n\\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $3... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29722,
"solution": "1. **Define the problem and constraints**: We need to find a 6-digit integer $abcdef$ such that:\n - The digits are a permutation of $\\{1, 2, 3, 4, 5, 6\\}$.\n - For each $k = 1, 2, \\ldots, 6$, the number formed by the first $k$ digits is divisible by $k$.\n\... |
deepscaler | [
{
"role": "user",
"content": "The $8 \\times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 29723,
"solution": "1. **Understanding the problem**: We are given an $8 \\times 18$ rectangle that is cut into two congruent hexagons. These hexagons are rearranged to form a square. We need to find the value of $y$, which is a dimension in the hexagon.\n\n2. **Area of the rectangle*... |
deepscaler | [
{
"role": "user",
"content": "(1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 2146 and 1813. \n(2) Use the Horner's method to calculate the value of $v_4$ for the function $f(x) = 2x^5 + 3x^4 + 2x^3 - 4x + 5$ when $x = 2$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 29724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\Gamma$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus at $F(3,0)$, and its upper and lower vertices at $A$ and $B$ respectively. The line $AF$ intersects $\\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 29725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 3)$. All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{\\pi}{2}"
} | {
"split": "train",
"index": 29726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$-inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its bas... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 29727,
"solution": "The scale factor is uniform in all dimensions, so the volume of the liquid is $\\left(\\frac{3}{4}\\right)^{3}$ of the container. The remaining section of the volume is $\\frac{1-\\left(\\frac{3}{4}\\right)^{3}}{1}$ of the volume, and therefore $\\frac{\\left(1-\\l... |
deepscaler | [
{
"role": "user",
"content": "A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\\overline{AB}$ and $\\overline{CD}$ are parallel edges of the cube, and $\\overline{AC}$ and $\\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $... | math | {
"style": "rule",
"ground_truth": "751"
} | {
"split": "train",
"index": 29728,
"solution": "Denote $h(X)$ the distance from point $X$ to $\\mathcal{P}, h(A) = 0, h(B) = 2,$ $h(C) = 8, h(D) = 10, h(G) = h(I) = h(H) = 7, AB = a, AC = a \\sqrt{2}.$\n Let slope $AB$ to $\\mathcal{P}$ be $\\alpha.$ Notation is shown in the diagram. \\[\\tan \\alpha = \\frac {\... |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r$ be primes such that $2 p+3 q=6 r$. Find $p+q+r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 29729,
"solution": "First, it is known that $3 q=6 r-2 p=2(3 r-p)$, thus $q$ is even. The only even prime is 2 so $q=2$. Further, $2 p=6 r-3 q=3(2 r-q)$, which means that $p$ is a multiple of 3 and thus $p=3$. This means that $2 \\cdot 3+3 \\cdot 2=6 r \\Longrightarrow r=2$. Therefore... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence \\( A = (a_1, a_2, \\cdots, a_{10}) \\) that satisfies the following four conditions:\n1. \\( a_1, a_2, \\cdots, a_{10} \\) is a permutation of \\{1, 2, \\cdots, 10\\};\n2. \\( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \\);\n3. \\( a_2 > a_3, a_4... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 29730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 29731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair six-sided die is rolled twice, and the resulting numbers are denoted as $a$ and $b$.\n\n(1) Find the probability that $a^2 + b^2 = 25$.\n\n(2) Given three line segments with lengths $a$, $b$, and $5$, find the probability that they can form an isosceles triangle (includi... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"split": "train",
"index": 29732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $A=\\{x|ax^{2}+bx+c\\leqslant 0\\left(a \\lt b\\right)\\}$ has one and only one element, then the minimum value of $M=\\frac{{a+3b+4c}}{{b-a}}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5} + 5"
} | {
"split": "train",
"index": 29733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = l \\sqrt{2}$, find $l$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{60}{17}"
} | {
"split": "train",
"index": 29734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory conducted a survey on the defective parts produced by a team. The number of defective parts produced by the team each day in 7 days is as follows (unit: pieces): 3, 3, 0, 2, 3, 0, 3. What is the value of the variance of the number of defective parts produced by the te... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"split": "train",
"index": 29735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bernardo randomly picks 3 distinct numbers from the set $\\{1,2,3,4,5,6,7,8,9\\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\\{1,2,3,4,5,6,7,8\\}$ and also arranges them in descending order to form a 3... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{56}"
} | {
"split": "train",
"index": 29736,
"solution": "We will solve this problem by considering two cases based on whether Bernardo picks the number 9 or not.\n\n#### Case 1: Bernardo picks 9.\nIf Bernardo picks a 9, then any three-digit number he forms will be larger than any three-digit number Silvia can form, since... |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 29737,
"solution": "Let's consider a quadrilateral $ABCD$ inscribed in a circle. We need to find the sum of the angles inscribed in the four segments outside the quadrilateral.\n\n1. **Identify the Segments and Angles**:\n - The four segments outside the quadrilateral are the region... |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an integer, and let $\\triangle ABC$ be a right-angles triangle with right angle at $C$ . It is given that $\\sin A$ and $\\sin B$ are the roots of the quadratic equation \\[(5n+8)x^2-(7n-20)x+120=0.\\] Find the value of $n$ Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 29738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the positive divisors of \\( 2014^2 \\) be \\( d_{1}, d_{2}, \\cdots, d_{k} \\). Then\n$$\n\\frac{1}{d_{1}+2014}+\\frac{1}{d_{2}+2014}+\\cdots+\\frac{1}{d_{k}+2014} = \n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{4028}"
} | {
"split": "train",
"index": 29739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $g(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $g(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $g(x) \\le g(y).$\n(iii) $g(1 - x) = 1 - g(x)$ for all $0 \\le x \\le 1.$\n(iv) $g\\left( \\frac{2x}{5} \\right) = \\frac{g(x)}{3}$ for $0 \... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 29740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3^{2x}+9=10\\left(3^{x}\\right)$, then the value of $(x^2+1)$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 or 5"
} | {
"split": "train",
"index": 29741,
"solution": "1. **Rewrite the equation**: Start by rewriting the given equation:\n \\[\n 3^{2x} + 9 = 10 \\cdot 3^x\n \\]\n Rearrange the terms to form a quadratic equation in terms of $3^x$:\n \\[\n 3^{2x} - 10 \\cdot 3^x + 9 = 0\n \\]\n\n2. **Substitute $u$ for ... |
deepscaler | [
{
"role": "user",
"content": "Given the variance of a sample is $$s^{2}= \\frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 29742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \\geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two t... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{2}"
} | {
"split": "train",
"index": 29743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$? \n[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29744,
"solution": "1. **Determine the interior angle of the decagon**: \n The formula for the interior angle of a regular polygon with $m$ sides is given by:\n \\[\n \\text{Interior angle} = \\frac{(m-2) \\times 180^\\circ}{m}\n \\]\n For a decagon ($m=10$), the interior a... |
deepscaler | [
{
"role": "user",
"content": "The number of students in Teresa's graduating class is more than 50 and fewer than 100 and is 1 less than a multiple of 3, 2 less than a multiple of 4, and 3 less than a multiple of 5. How many students are in Teresa's graduating class? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 29745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\) and \\( B \\) be the endpoints of a semicircular arc of radius \\( 3 \\). The arc is divided into five congruent arcs by four equally spaced points \\( C_1, C_2, C_3, C_4 \\). All chords of the form \\( \\overline{AC_i} \\) or \\( \\overline{BC_i} \\) are drawn. F... | math | {
"style": "rule",
"ground_truth": "32805"
} | {
"split": "train",
"index": 29746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are la... | math | {
"style": "rule",
"ground_truth": "650"
} | {
"split": "train",
"index": 29747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the word 'ARROW', find the probability that a random arrangement of its letters will have both R's next to each other. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 29748,
"solution": ""
} |
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