data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given the line $x - 3y + m = 0$ ($m \\neq 0$) and the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let points $A$ and $B$ be the intersections of the line with the two asymptotes of the hyperbola. If point $P(m, 0)$ satisfies $|PA| = |PB|$, find the ecc... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 25049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with a common difference of $\\frac{{2π}}{3}$, let $S=\\{\\cos a_{n}|n\\in N^{*}\\}$. If $S=\\{a,b\\}$, find the value of $ab$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 25050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three numbers $a$, $b$, and $c$ is 99. If we increase $a$ by 6, decrease $b$ by 6 and multiply $c$ by 5, the three resulting numbers are equal. What is the value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 25051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the square root of $\\dfrac{8!}{70}$. (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is left when $\\frac{1}{8}$ is subtracted from $\\frac{3}{4}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 25053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the eve of the 2010 Guangzhou Asian Games, a 12-person tour group took a commemorative photo near a venue of the Asian Games. They initially stood in a formation with 4 people in the front row and 8 people in the back row. Now, the photographer plans to keep the order of the... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 25054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{a}{36-a}+\\frac{b}{48-b}+\\frac{c}{72-c}=9$, evaluate $\\frac{4}{36-a}+\\frac{6}{48-b}+\\frac{9}{72-c}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{3}"
} | {
"split": "train",
"index": 25055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=|\\log_{x}|$, $a > b > 0$, $f(a)=f(b)$, calculate the minimum value of $\\frac{a^2+b^2}{a-b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 25056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the coordinate plane, a point is called a $\\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$ . Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right a... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 25057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of a chord intercepted on the circle $x^2+y^2-2x+4y-20=0$ by the line $5x-12y+c=0$ is 8. Find the value(s) of $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-68"
} | {
"split": "train",
"index": 25058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle \\\\(O: x^2 + y^2 = 2\\\\) and a line \\\\(l: y = kx - 2\\\\).\n\n\\\\((1)\\\\) If line \\\\(l\\\\) intersects circle \\\\(O\\\\) at two distinct points \\\\(A\\\\) and \\\\(B\\\\), and \\\\(\\angle AOB = \\frac{\\pi}{2}\\\\), find the value of \\\\(k\\\\).\n\n\\... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 25059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large chest contains 10 smaller chests. In each of the smaller chests, either 10 even smaller chests are placed or nothing is placed. In each of those smaller chests, either 10 smaller chests are placed or none, and so on. After this, there are exactly 2006 chests with conten... | math | {
"style": "rule",
"ground_truth": "18054"
} | {
"split": "train",
"index": 25060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first n terms of a geometric sequence $\\{a_n\\}$ is $S_n$. If for any positive integer $k$, it holds that $a_k = \\lim_{n\\to\\infty}(S_n - S_k)$, then the common ratio $q = \\boxed{\\frac{1}{2}}$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 25061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the first ten positive multiples of $13$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "715"
} | {
"split": "train",
"index": 25062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a particular value of the angle $\\theta$ we can take the product of the two complex numbers $(8+i)\\sin\\theta+(7+4i)\\cos\\theta$ and $(1+8i)\\sin\\theta+(4+7i)\\cos\\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find t... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 25063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 25064,
"solution": "1. **Calculate the initial lap time**: \n When Elisa started swimming, she completed 10 laps in 25 minutes. To find the time it took for one lap, we divide the total time by the number of laps:\n \\[\n \\text{Initial lap time} = \\frac{25 \\text{ minutes}}{10... |
deepscaler | [
{
"role": "user",
"content": "Given a fair 8-sided die is rolled until an odd number appears, determine the probability that the even numbers 2, 4, and 6 appear in strictly ascending order at least once before the first occurrence of any odd number. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{512}"
} | {
"split": "train",
"index": 25065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 25066,
"solution": "1. **Define Variables:**\n Let $x$ be the number of cows and $y$ be the number of chickens.\n\n2. **Set Up Equations:**\n - Each cow has 4 legs and each chicken has 2 legs.\n - The total number of legs is given by $4x + 2y$.\n - The total number of heads (s... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 7$, $BC = 24$, and the area of triangle $ABC$ is 84 square units. Given that the length of median $AM$ from $A$ to $BC$ is 12.5, find $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 25067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C\\_1$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the hyperbola $C\\_2$: $x^{2}- \\frac{y^{2}}{4}=1$ share a common focus. One of the asymptotes of $C\\_2$ intersects with the circle having the major axis of $C\\_1$ as its diameter at poi... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 25068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A taxi has a starting fare of 10 yuan. After exceeding 10 kilometers, for every additional kilometer, the fare increases by 1.50 yuan (if the increase is less than 1 kilometer, it is rounded up to 1 kilometer; if the increase is more than 1 kilometer but less than 2 kilometers,... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 25069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(-3,1)$, $\\overrightarrow{b}=(1,-2)$, and $\\overrightarrow{n}=\\overrightarrow{a}+k\\overrightarrow{b}$ ($k\\in\\mathbb{R}$).\n\n$(1)$ If $\\overrightarrow{n}$ is perpendicular to the vector $2\\overrightarrow{a}-\\overrightarrow{b}$, find t... | math | {
"style": "rule",
"ground_truth": "-\\frac {1}{3}"
} | {
"split": "train",
"index": 25070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following, when rounded to the nearest hundredth, does not round to 65.14?\nA) 65.141\nB) 65.138\nC) 65.1339999\nD) 65.1401\nE) 65.14444\n\nYour answer should be a letter: A, B, C, D, or E. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "C"
} | {
"split": "train",
"index": 25071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is 0.3 more than 29.8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30.1"
} | {
"split": "train",
"index": 25072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ that satisfy the conditions\n\\[f(1+xy)-f(x+y)=f(x)f(y) \\quad \\text{for all } x,y \\in \\mathbb{R},\\]\nand $f(-1) \\neq 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x - 1"
} | {
"split": "train",
"index": 25073,
"solution": "\nTo find all functions \\( f:\\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the functional equation\n\n\\[\nf(1+xy) - f(x+y) = f(x)f(y)\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\), and also given that \\( f(-1) \\neq 0 \\), we proceed as follows:\n\n### Ste... |
deepscaler | [
{
"role": "user",
"content": "Given a deck of cards consisting of four red cards numbered 1, 2, 3, 4; four green cards numbered 1, 2, 3, 4; and four yellow cards numbered 1, 2, 3, 4, calculate the probability of drawing a winning pair, where a winning pair consists of two cards of the same color or two card... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"split": "train",
"index": 25074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ratio of the area of $\\triangle BCX$ to the area of $\\triangle ACX$ in the diagram if $CX$ bisects $\\angle ACB$. In the diagram, $AB = 34$, $BC = 35$, and $AC = 39$. Let the coordinates be:\n- $A = (-17,0)$,\n- $B = (17,0)$,\n- $C = (0,30)$.\nExpress your answer as ... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{39}"
} | {
"split": "train",
"index": 25075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x) = \\log_2 x$ is shifted 1 unit to the left, and then the part below the $x$-axis is reflected across the $x$-axis to obtain the graph of function $g(x)$. Suppose real numbers $m$ and $n$ ($m < n$) satisfy $g(m) = g\\left(-\\frac{n+1}{n+2}\\right)... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{15}"
} | {
"split": "train",
"index": 25076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all odd positive integers $n>1$ such that there is a permutation $a_{1}, a_{2}, \\ldots, a_{n}$ of the numbers $1,2, \\ldots, n$, where $n$ divides one of the numbers $a_{k}^{2}-a_{k+1}-1$ and $a_{k}^{2}-a_{k+1}+1$ for each $k, 1 \\leq k \\leq n$ (we assume $a_{n+1}=a_{1}$... | math | {
"style": "rule",
"ground_truth": "n=3"
} | {
"split": "train",
"index": 25077,
"solution": "Since $\\{a_{1}, a_{2}, \\ldots, a_{n}\\}=\\{1,2, \\ldots, n\\}$ we conclude that $a_{i}-a_{j}$ : $n$ only if $i=j$. From the problem conditions it follows that $$a_{k+1}=a_{k}^{2}+\\varepsilon_{k}-n b_{k}$$ where $b_{k} \\in \\mathbb{Z}$ and $\\varepsilon_{k}= \\p... |
deepscaler | [
{
"role": "user",
"content": "Given a moving circle $C$ that passes through points $A(4,0)$ and $B(0,-2)$, and intersects with the line passing through point $M(1,-2)$ at points $E$ and $F$. Find the minimum value of $|EF|$ when the area of circle $C$ is at its minimum. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 25078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many distinct four-digit positive integers exist such that the product of their digits equals 18. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $a_1, a_2, \\ldots, a_n, \\ldots$, insert 3 numbers between each pair of consecutive terms to form a new sequence. Determine the term number of the 69th term of the new sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 25080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, find the value of $g(-1)$, where $g(x) = f(x) + 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 25081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( AB \\) and \\( AC \\) are tangents to a circle with center \\( O \\), \\( M \\) is the point of intersection of the line \\( AO \\) with the circle; \\( DE \\) is the segment of the tangent drawn through the point \\( M \\) between \\( AB \\) and \\( AC \\). Find \\( DE \\)... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r, s, t, u$ be positive real numbers such that $p+q+r+s+t+u = 11$. Find the minimum value of\n\\[\n\\frac{1}{p} + \\frac{9}{q} + \\frac{25}{r} + \\frac{49}{s} + \\frac{81}{t} + \\frac{121}{u}.\n\\] Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{1296}{11}"
} | {
"split": "train",
"index": 25083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \\begin{tabular}{|c|c|c|}\\hline 1 & &\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular} Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 25084,
"solution": "1. **Fill in the first row**: We start with the given number in the top left corner, which is $1$. Since each row and column must contain each of the numbers $1, 2, 3$, the middle cell in the top row cannot be $1$. It also cannot be $2$ because there is already a $... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$, and $c$. If $P(1,2,3)=1$, compute $P(2,4,8)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 25085,
"solution": "Since $P(a, a, b)=0,(x-y)$ is a factor of $P$, which means $(y-z)$ and $(z-x)$ are also factors by the symmetry of the polynomial. So, $$\\frac{P(x, y, z)}{(x-y)(y-z)(z-x)}$$ is a symmetric homogeneous degree 1 polynomial, so it must be $k(x+y+z)$ for some real $k$... |
deepscaler | [
{
"role": "user",
"content": "Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. If all three coins show heads, roll an additional fourth die. Determine the probability that the sum of the die rolls is odd. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 25086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 25087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 25088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a group of boys and girls, 15 girls leave. There are then left two boys for each girl. After this 45 boys leave. There are then 5 girls for each boy. The number of girls in the beginning was: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 25089,
"solution": "1. **Define Variables:**\n Let $b$ represent the number of boys and $g$ represent the number of girls initially.\n\n2. **Translate the Problem into Equations:**\n - After 15 girls leave, the number of girls becomes $g - 15$. According to the problem, there are ... |
deepscaler | [
{
"role": "user",
"content": "What is the largest $2$-digit prime factor of the integer $n = {300 \\choose 150}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 25090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon is inscribed in a circle of radius 2 units. In square units, what is the area of the hexagon? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"split": "train",
"index": 25091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a}$, $\\mathbf{b}$, $\\mathbf{c}$, and $\\mathbf{d}$ are unit vectors, find the largest possible value of\n\\[\n\\|\\mathbf{a} - \\mathbf{b}\\|^2 + \\|\\mathbf{a} - \\mathbf{c}\\|^2 + \\|\\mathbf{a} - \\mathbf{d}\\|^2 + \\|\\mathbf{b} - \\mathbf{c}\\|^2 + \\|\\math... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f^{-1}(g(x))=x^3-1$ and $g$ has an inverse, find $g^{-1}(f(7))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 25093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ be an arbitrary positive real number. Determine for this number $\\alpha$ the greatest real number $C$ such that the inequality $$ \\left(1+\\frac{\\alpha}{x^2}\\right)\\left(1+\\frac{\\alpha}{y^2}\\right)\\left(1+\\frac{\\alpha}{z^2}\\right)\\geq C\\left(\... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) =\n\\begin{cases}\n3x^2 + 2&\\text{if } x\\le 3, \\\\\nax - 1 &\\text{if } x>3.\n\\end{cases}\n\\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 25095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the acute-angled triangle \\(ABC\\), it is known that \\(\\sin (A+B)=\\frac{3}{5}\\), \\(\\sin (A-B)=\\frac{1}{5}\\), and \\(AB=3\\). Find the area of \\(\\triangle ABC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6 + 3\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 25096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the ellipse given by $x^2 + 6x + 4y^2 - 8y + 9 = 0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\pi"
} | {
"split": "train",
"index": 25097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( a_{n} = 4^{2n - 1} + 3^{n - 2} \\) (for \\( n = 1, 2, 3, \\cdots \\)), where \\( p \\) is the smallest prime number dividing infinitely many terms of the sequence \\( a_{1}, a_{2}, a_{3}, \\cdots \\), and \\( q \\) is the smallest prime number dividing every term of t... | math | {
"style": "rule",
"ground_truth": "5 \\times 13"
} | {
"split": "train",
"index": 25098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, \\ldots$ be an arithmetic sequence and $b_{1}, b_{2}, \\ldots$ be a geometric sequence. Suppose that $a_{1} b_{1}=20$, $a_{2} b_{2}=19$, and $a_{3} b_{3}=14$. Find the greatest possible value of $a_{4} b_{4}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{4}"
} | {
"split": "train",
"index": 25099,
"solution": "Solution 1. Let $\\{a_{n}\\}$ have common difference $d$ and $\\{b_{n}\\}$ have common ratio $r$; for brevity, let $a_{1}=a$ and $b_{1}=b$. Then we have the equations $a b=20,(a+d) b r=19$, and $(a+2 d) b r^{2}=14$, and we want to maximize $(a+3 d) b r^{3}$. The eq... |
deepscaler | [
{
"role": "user",
"content": "A nut has the shape of a regular hexagonal prism. Each lateral face of the nut is painted in one of three colors: white, red, or blue, with adjacent faces painted in different colors. How many different nut paintings are possible? (It is not necessary to use all three colors in... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 25100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer that cannot be expressed in the form $\\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 25101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ be a positive integer such that $2a$ has units digit 4. What is the sum of the possible units digits of $3a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 25102,
"solution": "If $2a$ has last digit 4, then the last digit of $a$ is either 2 or 7. In the former case, $3a$ has last digit 6, and in the latter case, $3a$ has last digit 1. This gives a final answer of $6+1=7$."
} |
deepscaler | [
{
"role": "user",
"content": "Adam filled a $3 \\times 3$ table with the numbers from 1 to 9 as follows:\n\n| 7 | 6 | 4 |\n| :--- | :--- | :--- |\n| 1 | 2 | 8 |\n| 9 | 3 | 5 |\n\nFor this arrangement, the sum of the numbers along every side of the table remains the same. Adam found that the numbers can be a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the distance between the two (non-intersecting) face diagonals on adjacent faces of a unit cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 25104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest prime factor of 1739? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1739"
} | {
"split": "train",
"index": 25105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\n$3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3)))))))))$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88572"
} | {
"split": "train",
"index": 25106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the vertices of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ lie on the surface of a sphere $O$, the lateral area of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ is $6$, and the base area is $\\sqrt{3}$, calculate the surface area of the sphere $O$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{19\\pi}{3}"
} | {
"split": "train",
"index": 25107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, given that point $P(3,4)$ is a point on the terminal side of angle $\\alpha$, if $\\cos(\\alpha+\\beta)=\\frac{1}{3}$, where $\\beta \\in (0,\\pi)$, then $\\cos \\beta =\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 8\\sqrt{2}}{15}"
} | {
"split": "train",
"index": 25108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 45? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "945"
} | {
"split": "train",
"index": 25109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The top section of an 8 cm by 6 cm rectangular sheet of paper is folded along a straight line so that when the top section lies flat on the bottom section, corner $P$ lies on top of corner $R$. What is the length of the crease? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 25110,
"solution": "Suppose that the crease intersects $PS$ at $X$, $QR$ at $Y$, and the line $PR$ at $Z$. We want to determine the length of $XY$. Since $P$ folds on top of $R$, then line segment $PZ$ folds on top of line segment $RZ$, since after the fold $Z$ corresponds with itself... |
deepscaler | [
{
"role": "user",
"content": "One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the s... | math | {
"style": "rule",
"ground_truth": "82"
} | {
"split": "train",
"index": 25111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a subset with four elements chosen from \\{1,2, \\ldots, 10\\}$. Michael notes that there is a way to label the vertices of a square with elements from $S$ such that no two vertices have the same label, and the labels adjacent to any side of the square differ by at l... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25112,
"solution": "Let the four numbers be $a, b, c, d$ around the square. Assume without loss of generality that $a$ is the largest number, so that $a>b$ and $a>d$. Note that $c$ cannot be simultaneously smaller than one of $b, d$ and larger than the other because, e.g. if $b>c>d$, ... |
deepscaler | [
{
"role": "user",
"content": "Let $X$ be the number of sequences of integers $a_{1}, a_{2}, \\ldots, a_{2047}$ that satisfy all of the following properties: - Each $a_{i}$ is either 0 or a power of 2 . - $a_{i}=a_{2 i}+a_{2 i+1}$ for $1 \\leq i \\leq 1023$ - $a_{1}=1024$. Find the remainder when $X$ is divi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 25113,
"solution": "This problem can be visualized as a complete binary tree with 16 leaves, such that each node contains the sum of its two children. Let $f(p)$ be the number of ways to fill in a binary tree with $2^{p}$ leaves and the root having value $2^{p}$. We want $f(10)$. Sinc... |
deepscaler | [
{
"role": "user",
"content": "Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "871"
} | {
"split": "train",
"index": 25114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible value of $n$ if a solid cube is made of white plastic and has dimensions $n \\times n \\times n$, the six faces of the cube are completely covered with gold paint, the cube is then cut into $n^{3}$ cubes, each of which has dimensions $1 \\times 1 \... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 25115,
"solution": "We call the $n \\times n \\times n$ cube the \"large cube\", and we call the $1 \\times 1 \\times 1$ cubes \"unit cubes\". The unit cubes that have exactly 0 gold faces are those unit cubes that are on the \"inside\" of the large cube. \n In other words, these are ... |
deepscaler | [
{
"role": "user",
"content": "The set $\\{[x] + [2x] + [3x] \\mid x \\in \\mathbb{R}\\} \\mid \\{x \\mid 1 \\leq x \\leq 100, x \\in \\mathbb{Z}\\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 25116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Name the smallest four-digit number in which all digits are different and the second digit is 6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1602"
} | {
"split": "train",
"index": 25117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"split": "train",
"index": 25118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vector $\\begin{pmatrix} 1 \\\\ 2 \\\\ 2 \\end{pmatrix}$ is rotated $90^\\circ$ about the origin. During the rotation, it passes through the $x$-axis. Find the resulting vector. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 \\sqrt{2} \\\\ -\\frac{1}{\\sqrt{2}} \\\\ -\\frac{1}{\\sqrt{2}} \\end{pmatrix}"
} | {
"split": "train",
"index": 25119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 25120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The function $f(x)=2\\cos x\\sin (x-A)+\\sin A (x\\in R)$ reaches its maximum value at $x=\\frac{5\\pi}{12}$.\n(1) Find the range of the function $f(x)$ when $x\\in(0,\\frac{... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"split": "train",
"index": 25121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle PQR is a right triangle with PQ = 6, QR = 8, and PR = 10. Point S is on PR, and QS bisects the right angle at Q. The inscribed circles of triangles PQS and QRS have radii rp and rq, respectively. Find rp/rq. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{28}\\left(10-\\sqrt{2}\\right)"
} | {
"split": "train",
"index": 25122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest three-digit integer whose digits are distinct and form a geometric sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "964"
} | {
"split": "train",
"index": 25123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, and angles $A, B, C$ form an arithmetic sequence.\n\n(1) Find the measure of angle $B$.\n\n(2) If $a=4$ and the area of $\\triangle ABC$ is $S=5\\sqrt{3}$, find the value of $b$. Let's think... | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 25124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If an item is sold for $x$ dollars, there is a loss of $15\\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\\%$ based on the cost. The ratio of $y:x$ is: Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "23:17"
} | {
"split": "train",
"index": 25125,
"solution": "1. **Define the cost price**: Let the cost price of the item be denoted as $c$.\n\n2. **Calculate the selling price for a loss of 15%**: \n - When the item is sold for $x$ dollars, there is a loss of 15%. This means that the selling price $x$ is 85% of the cost p... |
deepscaler | [
{
"role": "user",
"content": "Find the total number of sets of positive integers \\((x, y, z)\\), where \\(x, y\\) and \\(z\\) are positive integers, with \\(x < y < z\\) such that\n$$\nx + y + z = 203.\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3333"
} | {
"split": "train",
"index": 25126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first three terms of a geometric progression are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. Find the fourth term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 25127,
"solution": "1. **Identify the sequence and rewrite terms with fractional exponents:**\n The given terms are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. We can rewrite these terms using fractional exponents:\n \\[\n \\sqrt{2} = 2^{1/2}, \\quad \\sqrt[3]{2} = 2^{1/3}, \\quad ... |
deepscaler | [
{
"role": "user",
"content": "By how much did the dollar exchange rate change over the course of 2014 (from January 1, 2014, to December 31, 2014)? Provide the answer in rubles, rounded to the nearest whole number (answer - whole number). Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the price of a stamp is 50 cents, what is the maximum number of stamps that could be purchased with $50? Furthermore, if a customer buys more than 80 stamps, they receive a discount of 5 cents per stamp. How many stamps would then be purchased at maximum? Let's think step by... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 25129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 25130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 25131,
"solution": "To find the median number of children in the families of the students in Ms. Jordan's English class, we follow these steps:\n\n1. **Identify the Total Number of Data Points**: The total number of families (data points) is given as thirteen.\n\n2. **Determine the Po... |
deepscaler | [
{
"role": "user",
"content": "Points \\( M \\) and \\( N \\) are taken on the diagonals \\( AB_1 \\) and \\( BC_1 \\) of the faces of the parallelepiped \\( ABCD A_1 B_1 C_1 D_1 \\), and the segments \\( MN \\) and \\( A_1 C \\) are parallel. Find the ratio of these segments. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"split": "train",
"index": 25132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2 \\times 2 \\times 3 \\times 3 \\times 5 \\times 6=5 \\times 6 \\times n \\times n$, what is a possible value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 25133,
"solution": "We rewrite the left side of the given equation as $5 \\times 6 \\times(2 \\times 3) \\times(2 \\times 3)$. Since $5 \\times 6 \\times(2 \\times 3) \\times(2 \\times 3)=5 \\times 6 \\times n \\times n$, then a possible value of $n$ is $2 \\times 3$ or 6."
} |
deepscaler | [
{
"role": "user",
"content": "Cylinder $C$'s height is equal to the diameter of cylinder $D$ and cylinder $C$'s diameter is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \\pi h^3$ cubic units.... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 25134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a number is tripled and then decreased by 5, the result is 16. What is the original number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 25135,
"solution": "To get back to the original number, we undo the given operations. We add 5 to 16 to obtain 21 and then divide by 3 to obtain 7. These are the 'inverse' operations of decreasing by 5 and multiplying by 3."
} |
deepscaler | [
{
"role": "user",
"content": "Find the repetend in the decimal representation of $\\frac{5}{17}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "294117647058823529"
} | {
"split": "train",
"index": 25136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(\\omega x+\\varphi)$ is monotonically increasing on the interval ($\\frac{π}{6}$,$\\frac{{2π}}{3}$), and the lines $x=\\frac{π}{6}$ and $x=\\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, calculate the value of $f... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 25137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, when $x > 0$, $f(x)=2^{x}+ \\ln \\frac{x}{4}$. Let $a_{n}=f(n-5)$, then the sum of the first $8$ terms of the sequence $\\{a_{n}\\}$ is $\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 25138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide a square into 25 smaller squares, where 24 of these smaller squares are unit squares, and the remaining piece can also be divided into squares with a side length of 1. Find the area of the original square. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 25139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume that the probability of a certain athlete hitting the bullseye with a dart is $40\\%$. Now, the probability that the athlete hits the bullseye exactly once in two dart throws is estimated using a random simulation method: first, a random integer value between $0$ and $9$... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 25140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\mathbb{N}\\) be the set of all positive integers. A function \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) satisfies \\( f(m + n) = f(f(m) + n) \\) for all \\( m, n \\in \\mathbb{N} \\), and \\( f(6) = 2 \\). Also, no two of the values \\( f(6), f(9), f(12) \\), and ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 25141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 25142,
"solution": "1. **Calculate the fraction of the pizza Peter ate alone:**\n Peter ate one whole slice out of a 12-slice pizza. Therefore, the fraction of the pizza he ate alone is:\n \\[\n \\frac{1}{12}\n \\]\n\n2. **Calculate the fraction of the pizza Peter shared with ... |
deepscaler | [
{
"role": "user",
"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 put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi }{3}"
} | {
"split": "train",
"index": 25143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 25144,
"solution": "Notice that all odd numbers can be obtained by using $(a+1)^2-a^2=2a+1,$ where $a$ is a nonnegative integer. All multiples of $4$ can be obtained by using $(b+1)^2-(b-1)^2 = 4b$, where $b$ is a positive integer. Numbers congruent to $2 \\pmod 4$ cannot be obtained ... |
deepscaler | [
{
"role": "user",
"content": "Two circular poles, with diameters of 8 inches and 24 inches, touch each other at a single point. A wire is wrapped around them such that it goes around the entire configuration including a straight section tangential to both poles. Find the length of the shortest wire that suf... | math | {
"style": "rule",
"ground_truth": "16\\sqrt{3} + 32\\pi"
} | {
"split": "train",
"index": 25145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 25146,
"solution": "1. **Prime Factorization of $12!$:**\n The prime factorization of $12!$ (12 factorial) is necessary to determine the number of divisors and the condition for these divisors to be perfect squares. We start by finding the prime factorization of $12!$:\n \\[\n 1... |
deepscaler | [
{
"role": "user",
"content": "A digital 12-hour clock has a malfunction such that every time it should display a \"2\", it instead shows a \"5\". For example, when it is 2:27 PM, the clock incorrectly shows 5:57 PM. What fraction of the day will the clock show the correct time? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 25147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that $\\frac{n}{n+150}$ is equal to a terminating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 25148,
"solution": ""
} |
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