data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let a, b be positive integers such that $5 \\nmid a, b$ and $5^5 \\mid a^5+b^5$ . What is the minimum possible value of $a + b$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 33449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel, or perpendicular. Through each point, lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersecti... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 33450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve\n\\[\\arcsin x + \\arcsin 2x = \\frac{\\pi}{3}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{21}}{14}"
} | {
"split": "train",
"index": 33451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\in \\mathbb{N}^*$, $a_n$ be the sum of the coefficients of the expanded form of $(x+4)^n - (x+1)^n$, $c=\\frac{3}{4}t-2$, $t \\in \\mathbb{R}$, and $b_n = \\left[\\frac{a_1}{5}\\right] + \\left[\\frac{2a_2}{5^2}\\right] + ... + \\left[\\frac{na_n}{5^n}\\right]$ (where ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{25}"
} | {
"split": "train",
"index": 33452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remain... | math | {
"style": "rule",
"ground_truth": "(m, n) = (36, 6)"
} | {
"split": "train",
"index": 33453,
"solution": "\nTo solve this problem, we have to determine the total number of medals, \\( m \\), and the number of days, \\( n \\), based on the distribution given over the days.\n\nLet's denote the number of remaining medals after each day as \\( R_i \\) for day \\( i \\). In... |
deepscaler | [
{
"role": "user",
"content": "Find an ordered pair $(x,y)$ that satisfies both of the equations below: \\begin{align*} 2x - 3y &= -5,\\\\ 5x - 2y &= 4. \\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,3)"
} | {
"split": "train",
"index": 33454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is inscribed in a circle with $\\angle BAC=70^{\\circ}, \\angle ADB=40^{\\circ}, AD=4,$ and $BC=6$. What is $AC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33455,
"solution": "1. **Identify Angles Subtended by the Same Arc**: \n Since $\\angle ADB$ and $\\angle ACB$ are both subtended by the same arc $AB$ in the circle, by the Inscribed Angle Theorem, we have:\n \\[\n \\angle ACB = \\angle ADB = 40^\\circ.\n \\]\n\n2. **Calculat... |
deepscaler | [
{
"role": "user",
"content": "Find all \\( x \\in [1,2) \\) such that for any positive integer \\( n \\), the value of \\( \\left\\lfloor 2^n x \\right\\rfloor \\mod 4 \\) is either 1 or 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"split": "train",
"index": 33456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$ .\n Let's think step b... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a $5 \\times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 33458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the center of the ellipse C is at the origin, its eccentricity is equal to $\\frac{1}{2}$, and one of its minor axis endpoints is the focus of the parabola x²=$4\\sqrt{3}y$.\n1. Find the standard equation of the ellipse C.\n2. The left and right foci of the ellipse a... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}\\pi"
} | {
"split": "train",
"index": 33459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33460,
"solution": "1. **Identify the segments of Samantha's route:**\n - From her house to the southwest corner of City Park.\n - Through City Park from the southwest corner to the northeast corner.\n - From the northeast corner of City Park to her school.\n\n2. **Calculate the... |
deepscaler | [
{
"role": "user",
"content": "For some complex number $\\omega$ with $|\\omega| = 2016$ , there is some real $\\lambda>1$ such that $\\omega, \\omega^{2},$ and $\\lambda \\omega$ form an equilateral triangle in the complex plane. Then, $\\lambda$ can be written in the form $\\tfrac{a + \\sqrt{b}... | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 33461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an increasing geometric sequence $\\{a_{n}\\}$ with a common ratio greater than $1$ such that $a_{2}+a_{4}=20$, $a_{3}=8$.<br/>$(1)$ Find the general formula for $\\{a_{n}\\}$;<br/>$(2)$ Let $b_{m}$ be the number of terms of $\\{a_{n}\\}$ in the interval $\\left(0,m\\righ... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 33462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 33463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $C... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 33464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the terms in an infinite geometric series is 15, and the sum of their squares is 45. Find the first term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 33465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( f: \\mathbf{N}^{\\star} \\rightarrow \\mathbf{R} \\) satisfies \\( f(1)=1003 \\), and for any positive integer \\( n \\), it holds that\n\\[ f(1) + f(2) + \\cdots + f(n) = n^2 f(n). \\]\n\nFind the value of \\( f(2006) \\). Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2007}"
} | {
"split": "train",
"index": 33466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The median of the numbers 3, 7, x, 14, 20 is equal to the mean of those five numbers. Calculate the sum of all real numbers \\( x \\) for which this is true. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 33467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many $n=2,3,4,\\ldots,99,100$ is the base-$n$ number $235236_n$ a multiple of $7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 33468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least natural number that can be added to 71,382 to create a palindrome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.\nA) $\\frac{70}{7}$\nB) $\\frac{75}{7}$\nC) $\\frac{80}{7}$\nD) $10$\nE) $15$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{75}{7}"
} | {
"split": "train",
"index": 33470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $AB = 3,$ $AC = 6,$ $BC = 8,$ and $D$ lies on $\\overline{BC}$ such that $\\overline{AD}$ bisects $\\angle BAC.$ Find $\\cos \\angle BAD.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{34}}{12}"
} | {
"split": "train",
"index": 33471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80 - 320i$, $yz = 60$, and $zx = -96 + 24i$, where $i$ $=$ $\\sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x + y + z = a + bi$. Find $a^2 + b^2$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 33472,
"solution": "We can turn the expression $x+y+z$ into $\\sqrt{x^2+y^2+z^2+2xy+2yz+2xz}$, and this would allow us to plug in the values after some computations.\n Based off of the given products, we have\n$xy^2z=60(-80-320i)$\n$xyz^2=60(-96+24i)$\n$x^2yz=(-96+24i)(-80-320i)$.\n D... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = 2$, where $\\alpha$ is an angle in the first quadrant, find the value of $\\sin 2\\alpha + \\cos \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4 + \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 33473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The symbol $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the solution to the equation $\\left[3 x - 4 \\frac{5}{6}\\right] - 2 x - 1 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 33474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $l\\|k$. What is the number of degrees in $\\angle SRQ$? [asy]\ndraw((-.4,-.4)--(2,2)--(2,-.4));\ndraw((-.5,0)--(3,0),Arrows);\ndraw((-.5,1)--(3,1),Arrows);\ndraw((1.9,0)--(1.9,.1)--(2,.1));\nlabel(\"$S$\",(1,1),NNW);\nlabel(\"$R$\",(2,2),N);\nlabel(\"$Q$\",(2,1... | math | {
"style": "rule",
"ground_truth": "40^\\circ"
} | {
"split": "train",
"index": 33475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3 \\) are equal to the numbers \\( 1, 2, 3, \\ldots, 9 \\) in some order. Find the smallest possible value of\n\\[ x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3. \\] Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "214"
} | {
"split": "train",
"index": 33476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incr... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 33477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 33478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24, q(0)=30$, and $p(q(x))=q(p(x))$ for all real numbers $x$. Find the ordered pair $(p(3), q(6))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3,-24)"
} | {
"split": "train",
"index": 33479,
"solution": "Note that the polynomials $f(x)=a x^{3}$ and $g(x)=-a x^{3}$ commute under composition. Let $h(x)=x+b$ be a linear polynomial, and note that its inverse $h^{-1}(x)=x-b$ is also a linear polynomial. The composite polynomials $h^{-1} f h$ and $h^{-1} g h$ commute, si... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $\\overrightarrow{a} \\cdot (\\overrightarrow{a} - 2\\overrightarrow{b}) = 3$, and $|\\overrightarrow{a}| = 1$, $\\overrightarrow{b} = (1,1)$, find the angle between $\\overrightarrow{a}$ and $\\overrigh... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 33480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a geometric sequence $\\\\{a\\_n\\\\}$, $a\\_n > 0 (n \\in \\mathbb{N}^*)$, the common ratio $q \\in (0, 1)$, and $a\\_1a\\_5 + 2aa\\_5 + a\\_2a\\_8 = 25$, and the geometric mean of $a\\_3$ and $a\\_5$ is $2$.\n\n(1) Find the general term formula of the sequence $\\\\{a\\_n\... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 33481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive numbers $a$ and $b$ satisfying $a+b=1$, $c\\in R$, find the minimum value of $\\frac{3a}{b{c}^{2}+b}+\\frac{1}{ab{c}^{2}+ab}+3c^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} - 3"
} | {
"split": "train",
"index": 33482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=(x-a)^{2}+(2\\ln x-2a)^{2}$, where $x > 0, a \\in \\mathbb{R}$, find the value of the real number $a$ such that there exists $x_{0}$ such that $f(x_{0}) \\leqslant \\frac{4}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 33483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}} \\right\\}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{101}"
} | {
"split": "train",
"index": 33484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 33485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"split": "train",
"index": 33486,
"solution": "To solve this problem, we need to determine the number of four-digit numbers that meet the given conditions:\n1. The leftmost digit is odd.\n2. The second digit is even.\n3. All four digits are different.\n\n#### Step 1: Determine the choices for the first digit\nT... |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x)=\\sin 2x+\\cos 2x$ is translated to the left by $\\varphi (\\varphi > 0)$ units, and the resulting graph is symmetric about the $y$-axis, then find the minimum value of $\\varphi$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"split": "train",
"index": 33487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joe has written 5 questions of different difficulties for a test with problems numbered 1 though 5. He wants to make sure that problem $i$ is harder than problem $j$ whenever $i-j \\geq 3$. In how many ways can he order the problems for his test? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 33488,
"solution": "We will write $p_{i}>p_{j}$ for integers $i, j$ when the $i$ th problem is harder than the $j$ th problem. For the problem conditions to be true, we must have $p_{4}>p_{1}, p_{5}>p_{2}$, and $p_{5}>p_{1}$. Then, out of $5!=120$ total orderings, we see that in half ... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\\angle{BAC}$ intersects $\\overline{BC}$ in point $D$, and intersects the circumcircle of $\\triangle{ABC}$ in point $E \\ne A$. The circumcircle of $\\triangle{BED}$ intersects the line $AB$ in ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 33489,
"solution": "1. **Identify the triangle and given lengths**: We are given a triangle $ABC$ with side lengths $AB = 11$, $BC = 24$, and $CA = 20$. \n\n2. **Angle bisector and point definitions**: The bisector of $\\angle BAC$ intersects $BC$ at point $D$ and the circumcircle of ... |
deepscaler | [
{
"role": "user",
"content": "Given that the domain of the function $f(x)$ is $D$, if for any $x\\_1$, $x\\_2 \\in D$, when $x\\_1 < x\\_2$, we always have $f(x\\_1) \\leqslant f(x\\_2)$, then the function $f(x)$ is called a non-decreasing function on $D$. Suppose $f(x)$ is a non-decreasing function on $[0,... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"split": "train",
"index": 33490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $3x^2$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 9. When $x^2$ was subtracted from it, its minimum value decreased by 9. How will the minimum value of $f(x)$ change if $x^2$ is added to it? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 33491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f : \\mathbb{C} \\to \\mathbb{C} \\) be defined by \\( f(z) = z^2 - 2iz + 2 \\). Determine the number of complex numbers \\( z \\) such that \\( \\text{Im}(z) > 0 \\) and both the real and the imaginary parts of \\( f(z) \\) are integers within the range \\(-5\\) to \\(... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 33492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1=24$ and form the sequence $a_n$ , $n\\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 33493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 33494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express as a fraction in lowest terms: $0.\\overline{1} + 0.\\overline{01}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{33}"
} | {
"split": "train",
"index": 33496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are integers, and $a-2b=4$, $ab+c^2-1=0$, find the value of $a+b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 33497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ b... | math | {
"style": "rule",
"ground_truth": "162"
} | {
"split": "train",
"index": 33498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Problem\nSteve is piling $m\\geq 1$ indistinguishable stones on the squares of an $n\\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any fo... | math | {
"style": "rule",
"ground_truth": "\\binom{n+m-1}{m}^{2}"
} | {
"split": "train",
"index": 33499,
"solution": "Let the number of stones in row $i$ be $r_i$ and let the number of stones in column $i$ be $c_i$ . Since there are $m$ stones, we must have $\\sum_{i=1}^n r_i=\\sum_{i=1}^n c_i=m$ \nLemma 1: If any $2$ pilings are equivalent, then $r_i$ and $c_i$ are the same in bo... |
deepscaler | [
{
"role": "user",
"content": "Define the sequence $a_1, a_2, a_3, \\ldots$ by $a_n = \\sum\\limits_{k=1}^n \\sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "628"
} | {
"split": "train",
"index": 33500,
"solution": "If $n = 1$, $a_n = \\sin(1) > 0$. Then if $n$ satisfies $a_n < 0$, $n \\ge 2$, and \\[a_n = \\sum_{k=1}^n \\sin(k) = \\cfrac{1}{\\sin{1}} \\sum_{k=1}^n\\sin(1)\\sin(k) = \\cfrac{1}{2\\sin{1}} \\sum_{k=1}^n\\cos(k - 1) - \\cos(k + 1) = \\cfrac{1}{2\\sin(1)} [\\cos(0... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the 2023 fractions of the form $\\frac{2}{n(n+3)}$ for $n$ as the positive integers from 1 through 2023? Express your answer as a decimal to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.222"
} | {
"split": "train",
"index": 33501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-6-3+\\left(-7\\right)-\\left(-2\\right)$;<br/>$(2)\\left(-1\\right)^{2023}+5\\times \\left(-2\\right)-12\\div \\left(-4\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 33502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games, meaning into unordered pairs of unordered pairs of people? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 33503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( \\mathrm{n} \\) is a positive integer not greater than 100 and not less than 10, and \\( \\mathrm{n} \\) is a multiple of the sum of its digits. How many such \\( \\mathrm{n} \\) are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set $\\{1,2,3, \\cdots, 10\\}$, six distinct integers are chosen at random. What is the probability that the second smallest number among the chosen integers is 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 33505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square, points $R$ and $S$ are midpoints of two adjacent sides. A line segment is drawn from the bottom left vertex to point $S$, and another from the top right vertex to point $R$. What fraction of the interior of the square is shaded?\n\n[asy]\nfilldraw((0,0)--(2,0)--(2,... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 33506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the four equations:\n$\\textbf{(1)}\\ 3y-2x=12 \\qquad\\textbf{(2)}\\ -2x-3y=10 \\qquad\\textbf{(3)}\\ 3y+2x=12 \\qquad\\textbf{(4)}\\ 2y+3x=10$\nThe pair representing the perpendicular lines is: Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\text{(1) and (4)}"
} | {
"split": "train",
"index": 33507,
"solution": "To determine which pair of lines is perpendicular, we need to find the slopes of each line and check if the product of the slopes of any two lines is $-1$.\n\n1. **Convert each equation to slope-intercept form ($y = mx + b$), where $m$ is the slope:**\n - **Equat... |
deepscaler | [
{
"role": "user",
"content": "Points $E, F, G, H$ are chosen on segments $A B, B C, C D, D A$, respectively, of square $A B C D$. Given that segment $E G$ has length 7 , segment $F H$ has length 8 , and that $E G$ and $F H$ intersect inside $A B C D$ at an acute angle of $30^{\\circ}$, then compute the area... | math | {
"style": "rule",
"ground_truth": "\\frac{784}{19}"
} | {
"split": "train",
"index": 33508,
"solution": "Rotate $E G$ by $90^{\\circ}$ about the center of the square to $E^{\\prime} G^{\\prime}$ with $E^{\\prime} \\in A D$ and $G^{\\prime} \\in B C$. Now $E^{\\prime} G^{\\prime}$ and $F H$ intersect at an angle of $60^{\\circ}$. Then consider the translation which tak... |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all real numbers $x$ satisfying\n\\[(x^2-4x+2)^{x^2-5x+2} = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 33509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.\n\n[asy]\ndraw((-2,0)--(7,0),linewidth(1),Arrows);\ndraw((0,-1)--(0,4),linewidth(1),Arrows);\n\ndr... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 33510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One mole of an ideal monatomic gas is first heated isobarically, performing 10 Joules of work. Then it is heated isothermally, receiving the same amount of heat as in the first case. How much work does the gas perform (in Joules) in the second case? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 33511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is a periodic function defined on $\\mathbb{R}$ with a period of $2$, when $x \\in (-1, 1]$, $f(x)=\\begin{cases} -4x^{2}+ \\frac{9}{8},-1 < x\\leqslant 0, \\\\ \\log _{2}x,0 < x\\leqslant 1, \\end{cases}$, find the value of $f(f( \\frac{7}{2}))=\\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 33512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the positive integer \\( a \\) has 15 factors and the positive integer \\( b \\) has 20 factors, and \\( a + b \\) is a perfect square, find the smallest possible value of \\( a + b \\) that meets these conditions. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 33513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of digits is $2015!$ . Your score will be given by $\\max\\{\\lfloor125(\\min\\{\\tfrac{A}{C},\\tfrac{C}{A}\\}-\\tfrac{1}{5})\\rfloor,0\\}$ , where $A$ is your answer and $C$ is the actual answer. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "5787"
} | {
"split": "train",
"index": 33514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 33515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive sequence $\\{a_n\\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \\frac{5}{2}a_{n+1}a_n$, where $n \\in \\mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\\{a_n\\}$.\n1. If $a_2 = \\frac{3}{2}$, $a_3 = x$, and $a_4 ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit numbers are multiples of neither 5 nor 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "618"
} | {
"split": "train",
"index": 33517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two dice are tossed. What is the probability that the sum is greater than three? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{12}"
} | {
"split": "train",
"index": 33518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest positive integer \\( n \\) with \\( n < 500 \\) for which \\( 6048 \\cdot 28^n \\) is a perfect cube (that is, it is equal to \\( m^3 \\) for some positive integer \\( m \\) ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "497"
} | {
"split": "train",
"index": 33519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of\n\\[2023 + \\frac{1}{2} \\left( 2022 + \\frac{1}{2} \\left( 2021 + \\dots + \\frac{1}{2} \\left( 4 + \\frac{1}{2} \\cdot (3 + 1) \\right) \\right) \\dotsb \\right).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4044"
} | {
"split": "train",
"index": 33520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth select... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 33522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a beam of light starting from point $A$. It propagates within a plane and reflects between the lines $AD$ and $CD$. After several reflections, it perpendicularly hits point $B$ (which may lie on either $AD$ or $CD$), then reflects back to point $A$ along the original p... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given (1+i)x=1+yi, find |x+yi|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 33524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If an integer $a$ ($a \\neq 1$) makes the solution of the linear equation in one variable $ax-3=a^2+2a+x$ an integer, then the sum of all integer roots of this equation is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boy presses his thumb along a vertical rod that rests on a rough horizontal surface. Then he gradually tilts the rod, keeping the component of the force along the rod constant, which is applied to its end. When the tilt angle of the rod to the horizontal is $\\alpha=80^{\\cir... | math | {
"style": "rule",
"ground_truth": "0.17"
} | {
"split": "train",
"index": 33527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For \\( x \\in \\mathbb{R} \\), the function \\( f(x) \\) satisfies \\( f(x+4) + f(x-4) = f(x) \\). Thus, it is a periodic function. The common minimum period of such functions is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify and write the result as a common fraction: $$\\sqrt[4]{\\sqrt[3]{\\sqrt{\\frac{1}{65536}}}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt[3]{4}}"
} | {
"split": "train",
"index": 33529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x) = x^3 + bx^2 + cx + d$ is decreasing on the interval $[-2,2]$. The maximum value of $b+c$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 33530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that we have an equation $y=ax^2+bx+c$ whose graph is a parabola with vertex $(3,2)$, vertical axis of symmetry, and contains the point $(1,0)$.\n\nWhat is $(a, b, c)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(-\\frac{1}{2}, 3, -\\frac{5}{2}\\right)"
} | {
"split": "train",
"index": 33531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (The grid lines are spaced one unit apart.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vector... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 33532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? Let's think... | math | {
"style": "rule",
"ground_truth": "1:36"
} | {
"split": "train",
"index": 33533,
"solution": "1. **Identify the rate of burning for each candle**: \n - The first candle burns out in $3$ hours, i.e., $180$ minutes.\n - The second candle burns out in $4$ hours, i.e., $240$ minutes.\n\n2. **Set up the function for the length of each candle's stub**:\n - ... |
deepscaler | [
{
"role": "user",
"content": "A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the sur... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{54}"
} | {
"split": "train",
"index": 33534,
"solution": "To solve this problem, we need to minimize the white surface area on the larger $3$-inch cube constructed from $27$ smaller $1$-inch cubes. We have $21$ red cubes and $6$ white cubes.\n\n1. **Understanding the structure of the cube:**\n - The larger cube has dime... |
deepscaler | [
{
"role": "user",
"content": "Nathan and his two younger twin sisters' ages multiply to 72. Find the sum of their three ages. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 33535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 33536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $D$ is a point on $BC$ such that $\\frac{BD}{DC}=\\frac{1}{3}$. $E$ is the midpoint of $AC$. $AD$ and $BE$ intersect at $O$, and $CO$ intersects $AB$ at $F$. Find the ratio of the area of quadrilateral $BDOF$ to the area of $\\triangle ABC$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "1/10"
} | {
"split": "train",
"index": 33537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coordinates of the 3 vertices of triangle are \\( P(-8, 5) \\), \\( Q(-15, -19) \\), and \\( R(1, -7) \\). Find the equation of the angle bisector of \\(\\angle P\\) and express it in the form \\(ax + by + c = 0\\), where \\(a, b, c \\in \\mathbf{Z}^{+}\\) and \\((a, b, c)=... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 33538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each pair of distinct natural numbers \\(a\\) and \\(b\\), not exceeding 20, Petya drew the line \\( y = ax + b \\) on the board. That is, he drew the lines \\( y = x + 2, y = x + 3, \\ldots, y = x + 20, y = 2x + 1, y = 2x + 3, \\ldots, y = 2x + 20, \\ldots, y = 3x + 1, y =... | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 33539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \\ldots, n_{70}$ such that $k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}... | math | {
"style": "rule",
"ground_truth": "553"
} | {
"split": "train",
"index": 33540,
"solution": "Obviously $k$ is positive. Then, we can let $n_1$ equal $k^3$ and similarly let $n_i$ equal $k^3 + (i - 1)k$.\n The wording of this problem (which uses \"exactly\") tells us that $k^3+69k<(k+1)^3 = k^3 + 3k^2 + 3k+1 \\leq k^3+70k$. Taking away $k^3$ from our inequa... |
deepscaler | [
{
"role": "user",
"content": "Two couples each bring one child to visit the zoo. After purchasing tickets, they line up to enter the zoo one by one. For safety reasons, the two fathers must be positioned at the beginning and the end of the line. Moreover, the two children must be positioned together. Determ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_1 = 1$, $a_2 = 2$, $a_{n+2}$ is equal to the remainder of $a_n + a_{n+1}$ divided by 3. Find the sum of the first 89 terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 33542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = -\\dfrac{1}{x},$ what is $f(f(f(f(f(6)))))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{1}{6}"
} | {
"split": "train",
"index": 33543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The teacher fills some numbers into the circles in the diagram below (each circle can and must only contain one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point P on the hyperbola $C_1: \\frac{x^2}{16} - \\frac{y^2}{9} = 1$, a point Q on the circle $C_2: (x - 5)^2 + y^2 = 1$, and a point R on the circle $C_3: (x + 5)^2 + y^2 = 1$, find the maximum value of $|PQ| - |PR|$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an odd function $f(x)$ on $\\mathbb{R}$, for any $x \\in \\mathbb{R}$, $f(x+1) = -f(x)$, and when $x \\in (-1, 1)$, $f(x) = x$. Find the value of $f(3) + f(-7.5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 33546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A laboratory has $10$ experimental mice, among which $3$ have been infected with a certain virus, and the remaining $7$ are healthy. Random medical examinations are conducted one by one until all $3$ infected mice are identified. The number of different scenarios where the last... | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"split": "train",
"index": 33547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 33548,
"solution": "1. **Define the circles**: Consider a circle with radius $2$ (outer circle) and a concentric circle with radius $1$ (inner circle). The center of both circles is the same point.\n\n2. **Determine the significance of the inner circle**: The inner circle of radius $1... |
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