data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Define the sequence $(b_i)$ by $b_{n+2} = \\frac{b_n + 2011}{1 + b_{n+1}}$ for $n \\geq 1$ with all terms being positive integers. Determine the minimum possible value of $b_1 + b_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"split": "train",
"index": 19749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2, so he does not use it. Additionally, he dislikes when two identical digits stand next to each other. Furthermore, he wants the first digit to match the last. How many possible combinations mus... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 19750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the shortest chord is cut by the line $y = kx + 1$ on the circle $C: x^2 + y^2 - 2x - 3 = 0$, then $k = \\boxed{\\_\\_\\_\\_\\_\\_\\_\\_}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=2021$ is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the ... | math | {
"style": "rule",
"ground_truth": "6561"
} | {
"split": "train",
"index": 19752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a\\equiv (3^{-1}+5^{-1}+7^{-1})^{-1}\\pmod{11}$. What is the remainder when $a$ is divided by $11$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 19754,
"solution": "1. **Define the form of a five-digit palindrome**: A five-digit palindrome can be represented as $\\overline{abcba}$, where $a, b, c$ are digits and $a \\neq 0$ (since it is a five-digit number).\n\n2. **Calculate the total number of five-digit palindromes**: \n ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\\mathcal{T}$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 19755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(2, 3, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 3, -8),$ with radius $r.$ Find $r.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15}"
} | {
"split": "train",
"index": 19757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system, point \\( O(0,0) \\), \\( A(0,6) \\), \\( B(-3,2) \\), \\( C(-2,9) \\), and \\( P \\) is any point on line segment \\( OA \\) (including endpoints). Find the minimum value of \\( PB + PC \\). Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{13}"
} | {
"split": "train",
"index": 19758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a table near the sea, there are $N$ glass boxes where $N<2021$ , each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the b... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 19759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The incircle of triangle \\( ABC \\) with center \\( O \\) touches the sides \\( AB \\), \\( BC \\), and \\( AC \\) at points \\( M \\), \\( N \\), and \\( K \\) respectively. It is given that angle \\( AOC \\) is four times larger than angle \\( MKN \\). Find angle \\( B \\). ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 19760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \\ldots\\}$. Find the number of posit... | math | {
"style": "rule",
"ground_truth": "958"
} | {
"split": "train",
"index": 19761,
"solution": "We see that a number $n$ is $p$-safe if and only if the residue of $n \\mod p$ is greater than $2$ and less than $p-2$; thus, there are $p-5$ residues $\\mod p$ that a $p$-safe number can have. Therefore, a number $n$ satisfying the conditions of the problem can ha... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_{n}\\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\\{b_{n}\\}$, where $\\left[x\\right)$ is the smallest integer greater than $x$ and $b_n = \\left[\\frac{n(n+1)}{a_n}\\right)$. Le... | math | {
"style": "rule",
"ground_truth": "4045"
} | {
"split": "train",
"index": 19762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{1}{q(x)}$ is shown below. If $q(x)$ is a quadratic and $q(2) = 6$, find $q(x).$\n\n[asy]\nsize(8cm);\nimport graph;\n\nLabel f; \nf.p=fontsize(6); \n\nreal f(real x) {return 1/(2*(x+1)*(x-1));}\n\nint gridsize = 5;\ndraw((-gridsize,0)-... | math | {
"style": "rule",
"ground_truth": "2x^2 - 2"
} | {
"split": "train",
"index": 19763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In this square array of 16 dots, four dots are to be chosen at random. What is the probability that the four dots will be collinear? Express your answer as a common fraction.\n\n[asy]\nsize(59);\nfor(int i = 0; i<4; ++i)\nfor(int j = 0; j<4; ++j)\ndot((i,j),linewidth(7));\n[/as... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{182}"
} | {
"split": "train",
"index": 19764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute ... | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 19765,
"solution": "The main claim is that if the light pulse reflects vertically (on the left/right edges) $a$ times and horizontally $b$ times, then $\\operatorname{gcd}(a+1, b+1)=1$, and the number of regions is $\\frac{(a+2)(b+2)}{2}$. This claim can be conjectured by looking at s... |
deepscaler | [
{
"role": "user",
"content": "Imagine you own 8 shirts, 5 pairs of pants, 4 ties, and 3 different jackets. If an outfit consists of a shirt, a pair of pants, and optionally a tie and/or a jacket, how many different outfits can you create? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 19766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the quinary (base-5) number $234_{(5)}$ to a decimal (base-10) number, then convert it to a binary (base-2) number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000101_{(2)}"
} | {
"split": "train",
"index": 19767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function defined by $g(x) = \\frac{px + q}{rx + s}$, where $p$, $q$, $r$ and $s$ are nonzero real numbers, and the function has the properties $g(31)=31$, $g(41)=41$, and $g(g(x))=x$ for all values except $\\frac{-s}{r}$. Determine the unique number that is not in ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 19768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(x + 2)(3x^2 - x + 5) = Ax^3 + Bx^2 + Cx + D$, what is the value of $A + B + C + D$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 19769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $e^{i \\theta} = \\frac{2 + i \\sqrt{5}}{3},$ then find $\\sin 4 \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{8 \\sqrt{5}}{81}"
} | {
"split": "train",
"index": 19770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the refreshment point, four athletes drank all the stock of Coke-Loco lemonade. If only athlete Bystrov had drunk half as much, one-tenth of the lemonade would have been left. If, additionally, athlete Shustrov had also drunk half as much, one-eighth of the lemonade would ha... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 19771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ that satisfies\n$$\na_{n-1} = a_{n} + a_{n-2} \\quad (n \\geqslant 3),\n$$\n\nlet $S_{n}$ be the sum of the first $n$ terms. If $S_{2018} = 2017$ and $S_{2019} = 2018$, then find $S_{20200}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 19772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the foci of the ellipse \n\\[\\frac{x^2}{36} + \\frac{y^2}{16} = 8.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{10}"
} | {
"split": "train",
"index": 19773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{ax^{2}+bx+c}{e^{x}} (a > 0)$ whose derivative $y=f′(x)$ has two zeros at $-3$ and $0$.\n\n1. Find the monotonic intervals of $f(x)$;\n2. If the minimum value of $f(x)$ is $-e^{3}$, find the maximum value of $f(x)$ on the interval $[-5,+\\infty)$.... | math | {
"style": "rule",
"ground_truth": "5e^{5}"
} | {
"split": "train",
"index": 19774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When purchasing goods with a total amount of at least 1000 rubles, the store provides a 50% discount on subsequent purchases. With 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of 300 ruble... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 19775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elon Musk's Starlink project belongs to his company SpaceX. He plans to use tens of thousands of satellites to provide internet services to every corner of the Earth. A domestic company also plans to increase its investment in the development of space satellite networks to deve... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 19776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = 12x + 5 \\). Find the sum of all \\( x \\) that satisfy the equation \\( f^{-1}(x) = f((3x)^{-1}) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 19777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the triangle \\(ABC\\), a point \\(M\\) is taken such that \\(\\angle MBA = 30^\\circ\\) and \\(\\angle MAB = 10^\\circ\\). Find \\(\\angle AMC\\) if \\(\\angle ACB = 80^\\circ\\) and \\(AC = BC\\). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 19778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 19779,
"solution": "Let the three numbers be $a$, $b$, and $c$ such that $a < b < c$. We are given the sums of these numbers taken in pairs:\n1. $a + b = 12$\n2. $a + c = 17$\n3. $b + c = 19$\n\nWe aim to find the value of $b$, the middle number. To do this, we can manipulate the give... |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor of natural numbers \\( m \\) and \\( n \\) is 1. What is the greatest possible value of \\(\\text{GCD}(m + 2000n, n + 2000m) ?\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3999999"
} | {
"split": "train",
"index": 19780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the ellipse \\[9(x-1)^2 + y^2 = 36.\\]Let $A$ be one of the endpoints of its major axis, and let $B$ be one of the endpoints of its minor axis. Find the distance $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 19781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a sequence of complex numbers by $z_1 = 0$ and\n\\[z_{n + 1} = z_n^2 + i\\]for all $n \\ge 1.$ In the complex plane, how far from the origin is $z_{111}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 19782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be nonzero vectors such that\n\\[\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = \\|\\mathbf{a} + \\mathbf{b}\\|.\\]Find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 19783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 36 students in a club. If any 33 of them attend a session, girls will always be in the majority. However, if 31 students attend, it might happen that boys are in the majority. How many girls are in the club? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\left(a > b > 0\\right)$ that passes through the point $(0,1)$, and its eccentricity is $\\frac{\\sqrt{3}}{2}$.\n\n$(1)$ Find the standard equation of the ellipse $E$;\n\n$(2)$ Suppose a line $l: y = \\frac{1}{2}x + ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"split": "train",
"index": 19785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a series of squares where each square has a border of dots around it, each border containing dots on all four sides. The smallest square (first in the series) has a single dot at the center. Each successive square surrounds the previous one with an additional border of... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 19786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number \\( N \\) greater than 20 is a palindrome in both base 14 and base 20 (a palindrome is a number that reads the same forward and backward, such as \\( 12321 \\) and \\( 3443 \\), but \\( 12331 \\) is not a palindrome). What is the smallest value of \\( N \\) (ex... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 19787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 7,999,999,999 has at most two prime factors, find its largest prime factor. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4,002,001"
} | {
"split": "train",
"index": 19788,
"solution": "7,999,999,999=8 \\cdot 10^{9}-1=2000^{3}-1=(2000-1)\\left(2000^{2}+2000+1\\right)$, so \\left(2000^{2}+2000+1\\right)=4,002,001$ is its largest prime factor."
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle XYZ$, we have $\\angle X = 90^\\circ$ and $\\tan Z = 7$. If $YZ = 100$, then what is $XY$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70\\sqrt{2}"
} | {
"split": "train",
"index": 19789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Students from three different schools worked on a summer project. Six students from Atlas school worked for $4$ days, five students from Beacon school worked for $6$ days, and three students from Cedar school worked for $10$ days. The total payment for the students' work was $7... | math | {
"style": "rule",
"ground_truth": "276.43"
} | {
"split": "train",
"index": 19790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \\geq v \\geq f(x) - x$ , then $f(z) = v + z$ , for some number $z$ between $x$ and $y$ . $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions o... | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"split": "train",
"index": 19791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \\ldots + a_{11}(x+2)^{11}$, then the value of $a_0 + a_1 + \\ldots + a_{11}$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 19792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On her previous five attempts Sarah had achieved times, in seconds, of 86, 94, 97, 88 and 96, for swimming 50 meters. After her sixth try she brought her median time down to 92 seconds. What was her time, in seconds, for her sixth attempt? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 19793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be positive real numbers such that $ab^2c^3 = 256$. Find the minimum value of\n\\[a^2 + 8ab + 16b^2 + 2c^5.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 19794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1936"
} | {
"split": "train",
"index": 19795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((0,-3)-... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 19796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $(\\sqrt{100}+\\sqrt{9}) \\times(\\sqrt{100}-\\sqrt{9})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 19797,
"solution": "Simplifying, $(\\sqrt{100}+\\sqrt{9}) \\times(\\sqrt{100}-\\sqrt{9})=(10+3) \\times(10-3)=13 \\times 7=91$."
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)\\frac{\\sqrt{8}-\\sqrt{24}}{\\sqrt{2}}+|1-\\sqrt{3}|$;<br/>$(2)(\\frac{1}{2})^{-1}-2cos30°+|2-\\sqrt{3}|-(2\\sqrt{2}+1)^{0}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{3}"
} | {
"split": "train",
"index": 19798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $15\\%$ of $N$ is $45\\%$ of 2003, what is the value of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6009"
} | {
"split": "train",
"index": 19799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2? 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": 19800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .\n\n(A *common external tangent* i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 19801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some positive integers are initially written on a board, where each $2$ of them are different.\nEach time we can do the following moves:\n(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 number... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 19802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points $A$ and $B$ lie on the curves $C_{1}: x^{2}-y+1=0$ and $C_{2}: y^{2}-x+1=0$ respectively, what is the minimum value of the distance $|AB|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 19803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be distinct real numbers such that\n\\[\n\\begin{vmatrix} 1 & 4 & 9 \\\\ 3 & x & y \\\\ 3 & y & x \\end{vmatrix}\n= 0.\\]Find $x + y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 19804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A graph shows the number of books read in June by the top readers in a school library. The data points given are:\n- 4 readers read 3 books each\n- 5 readers read 5 books each\n- 2 readers read 7 books each\n- 1 reader read 10 books\nDetermine the mean (average) number of books... | math | {
"style": "rule",
"ground_truth": "5.0833"
} | {
"split": "train",
"index": 19805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a s... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{13}"
} | {
"split": "train",
"index": 19806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\\angle AMD = \\angle CMD$. What is the degree measure of $\\angle AMD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 19807,
"solution": "1. **Identify the given information and the goal:** We are given a rectangle $ABCD$ with $AB = 6$ and $BC = 3$. We need to find the degree measure of $\\angle AMD$ given that $\\angle AMD = \\angle CMD$.\n\n2. **Assign variables and use the Pythagorean Theorem:** L... |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer $n$ such that $7350$ is a factor of $n!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The famous skater Tony Hawk rides a skateboard (segment \\( AB \\)) on a ramp, which is a semicircle with diameter \\( PQ \\). Point \\( M \\) is the midpoint of the skateboard, and \\( C \\) is the foot of the perpendicular dropped from point \\( A \\) to the diameter \\( PQ \... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the perimeter of the figure shown? [asy]\n\ndraw((0,0)--(0,2)--(1,2+sqrt(3))--(2,2)--(4,2)--(4,0)--(3,-sqrt(3))--(2,0)--cycle);\n\ndraw((-.2,1)--(.2,1));\n\ndraw((3.8,1)--(4.2,1));\n\ndraw((1,-.2)--(1,.2));\n\ndraw((3,1.8)--(3,2.2));\n\ndraw((0.15,0)--(0.15,0.15)--(0,0.... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 19810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet squares of one kind have a side of \\(a\\) units, another kind have a side of \\(b\\) units, and the original square have a side of \\(c\\) units. Then the area of the original square is given by \\(c^{2}=n a^{2}+n b^{2}\\).\n\nNumbers satisfying this equation can be obta... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 19811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=x^{2}-ax$, $g(x)=\\ln x$, $h(x)=f(x)+g(x)$,\n(1) Find the range of values for the real number $a$ such that $f(x) \\geq g(x)$ holds true for any $x$ within their common domain;\n(2) Suppose $h(x)$ has two critical points $x_{1}$, $x_{2}$, with $x_{1} \\in (0, \\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4} - \\ln 2"
} | {
"split": "train",
"index": 19812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a line $y=-x+1$ and an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B.\n(1) If the eccentricity of the ellipse is $\\frac{\\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB.\n(2) (For Liberal Ar... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 19813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a supermarket with 100 customers, the table shows the number of customers in each age group and payment method category. Determine the number of customers aged between 40 and 60 years old who do not use mobile payment. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 19814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the data from a 2×2 contingency table calculates $k=4.073$, there is a \\_\\_\\_\\_\\_\\_ confidence that the two variables are related, knowing that $P(k^2 \\geq 3.841) \\approx 0.05$, $P(k^2 \\geq 5.024) \\approx 0.025$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "95\\%"
} | {
"split": "train",
"index": 19815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest sum of distances from an arbitrary point on the plane to the vertices of a unit square.\n\nA problem of the shortest connection for four points. Four points: \\(A, B, C,\\) and \\(D\\) are the vertices of a square with side length 1. How should these points be... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 19816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $c$ denote the number of positive integers $n \\leq 1000$ with $S(n)$ odd, and let $d$ den... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 19817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The orchestra has more than 150 members but fewer than 300 members. When they line up in rows of 6 there are two extra people; when they line up in rows of 8 there are four extra people; and when they line up in rows of 9 there are six extra people. How many members are in the ... | math | {
"style": "rule",
"ground_truth": "212"
} | {
"split": "train",
"index": 19818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ is constructed along diameter $AB$ of a semicircle, as shown. The semicircle and square $ABCD$ are coplanar. Line segment $AB$ has a length of 6 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MC$? Express your answer in simpl... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{10}"
} | {
"split": "train",
"index": 19819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $x$ such that the product $(x + 2i)((x + 1) + 2i)((x + 2) + 2i)((x + 3) + 2i)$ is purely imaginary. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 19820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY=25$ and $XZ=14$. The angle bisector of $\\angle X$ intersects $YZ$ at point $E$, and point $N$ is the midpoint of $XE$. Let $Q$ be the point of the intersection of $XZ$ and $YN$. The ratio of $ZQ$ to $QX$ can be expressed in the form $\\dfrac{p}{q}$, wher... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 19821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man bought a number of ping-pong balls where a 16% sales tax is added. If he did not have to pay tax, he could have bought 3 more balls for the same amount of money. If \\( B \\) is the total number of balls that he bought, find \\( B \\). Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "18.75"
} | {
"split": "train",
"index": 19822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let positive integers \\( a \\) and \\( b \\) be such that \\( 15a + 16b \\) and \\( 16a - 15b \\) are both perfect squares. Find the smallest possible value of the smaller of these two squares.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "481^2"
} | {
"split": "train",
"index": 19823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school has a total of 1,000 students, including 380 first-year students and 180 male second-year students. It is known that the probability of drawing a female second-year student from all the students is 0.19. If stratified sampling (by grade) is used to draw 100 stu... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 19824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the square root of $\\dfrac{9!}{210}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216\\sqrt{3}"
} | {
"split": "train",
"index": 19825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student has five different physics questions numbered 1, 2, 3, 4, and 5, and four different chemistry questions numbered 6, 7, 8, and 9. The student randomly selects two questions, each with an equal probability of being chosen. Let the event `(x, y)` represent \"the two ques... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 19826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "962"
} | {
"split": "train",
"index": 19827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum and the average of the prime numbers between 20 and 40? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 19828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 19829,
"solution": "WLOG let E be be between C & D (as in solution 1). Assume $AD = 3m$. We use power of a point to get that $AG = DE = \\sqrt{2}m$ and $AB = AG + GB = AG + BE = 10+2\\sqrt{2} m$\n Since now we have $AC = 10$, $BC = 20, AB = 10+2\\sqrt{2} m$ in triangle $\\triangle ABC... |
deepscaler | [
{
"role": "user",
"content": "In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly selected from the pot, the probability that at least one dumpling ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{48}{91}"
} | {
"split": "train",
"index": 19830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. For example, since $2^3 - 1 = 7$, and 3 is a prime number, 7 is a Mersenne prime. What is the largest Mersenne prime less than 200? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 19831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that the divisors of $n$ can be partitioned into three sets with equal sums. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 19832,
"solution": "I claim the answer is 120. First, note that $120=2^{3} \\cdot 3 \\cdot 5$, so the sum of divisors is $(1+2+4+8)(1+3)(1+5)=15 \\cdot 4 \\cdot 6=360$. Thus, we need to split the divisors into groups summing to 120 . But then we can just take $\\{120\\},\\{20,40,60\\}... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( PQR \\), the median \\( PA \\) and the angle bisector \\( QB \\) (where \\( A \\) and \\( B \\) are the points of their intersection with the corresponding sides of the triangle) intersect at point \\( O \\). It is known that \\( 3PQ = 5QR \\). Find the ratio of... | math | {
"style": "rule",
"ground_truth": "2.6"
} | {
"split": "train",
"index": 19833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, suppose the terminal side of the obtuse angle $\\alpha$ intersects the circle $O: x^{2}+y^{2}=4$ at point $P(x_{1},y_{1})$. If point $P$ moves clockwise along the circle for a unit arc length of $\\frac{2\\pi}{3}$ to reach point $Q(x_{2... | math | {
"style": "rule",
"ground_truth": "\\frac{1-3\\sqrt{5}}{4}"
} | {
"split": "train",
"index": 19834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\\cdots \\pm 1$ ! is prime. Be prepared to justify your answer for $\\left\\{\\begin{array}{c}n, \\\\ {\\left[\\frac{n+225}{10}\\right],}\\end{array} n \\leq 25\\right.$ points, where $[N]$ is the greatest integer less than $N$. Le... | math | {
"style": "rule",
"ground_truth": "3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160"
} | {
"split": "train",
"index": 19835,
"solution": "$3,4,5,6,7,8,10,15,19,41$ (26 points), 59, 61 (28 points), 105 (33 points), 160 (38 points) are the only ones less than or equal to 335. If anyone produces an answer larger than 335, then we ask for justification to call their bluff. It is not known whether or not ... |
deepscaler | [
{
"role": "user",
"content": "Determine how many divisors of \\(9!\\) are multiples of 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 19836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set \\( A = \\{0, 1, 2, 3, 4, 5, 6, 7\\} \\), how many mappings \\( f \\) from \\( A \\) to \\( A \\) satisfy the following conditions?\n1. For all \\( i, j \\in A \\) with \\( i \\neq j \\), \\( f(i) \\neq f(j) \\).\n2. For all \\( i, j \\in A \\) with \\( i + j = 7 ... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 19837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 8 people be seated in a row of chairs if two of the people, Alice and Bob, must not sit next to each other, and Charlie has to sit at one end of the row? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7200"
} | {
"split": "train",
"index": 19838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cooperative farm can purchase two types of feed mixtures from a neighboring farm to feed its animals. The Type I feed costs $30 per sack and contains 10 kg of component A and 10 kg of component B. The Type II feed costs $50 per sack and contains 10 kg of component A, 20 kg of... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 19839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ whose general term formula is $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, then for any $n > m$ (where $m, n \\in \\mathbb{N}^*$), the maximum value of $S_n - S_m$ is __. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum number of real roots to a polynomial of the form\n\\[x^n + x^{n - 1} + \\dots + x + 1 = 0,\\]where $n$ is a positive integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 19841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance between Ivan's house and his grandmother's house is 12 km. Exactly at 12:00, Ivan left his house and walked along the straight road to his grandmother's house at a speed of 1 m/s. At 12:30, Ivan's parents called his grandmother, informed her that Ivan was coming to... | math | {
"style": "rule",
"ground_truth": "12:47"
} | {
"split": "train",
"index": 19842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{19}"
} | {
"split": "train",
"index": 19843,
"solution": "1. **Prime Factorization of $21!$**:\n To find the prime factorization of $21!$, we consider all prime numbers less than or equal to 21. We then count how many times each prime divides $21!$. This is done by summing the integer parts of $21$ divided by each prime... |
deepscaler | [
{
"role": "user",
"content": "Car X is traveling at a constant speed of 90 km/h and has a length of 5 meters, while Car Y is traveling at a constant speed of 91 km/h and has a length of 6 meters. Given that Car Y starts behind Car X and eventually passes Car X, calculate the length of time between the insta... | math | {
"style": "rule",
"ground_truth": "39.6"
} | {
"split": "train",
"index": 19844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 19845,
"solution": "1. **Using the Remainder Theorem**: Given that $x+2$ and $x-1$ are factors of $2x^3 - hx + k$, we can use the Remainder Theorem to set up equations. The Remainder Theorem states that if $x - c$ is a factor of a polynomial $p(x)$, then $p(c) = 0$.\n\n2. **Setting up... |
deepscaler | [
{
"role": "user",
"content": "Given that the sides of triangle $\\triangle ABC$ opposite to angles $A$, $B$, and $C$ are in arithmetic sequence and $C=2\\left(A+B\\right)$, calculate the ratio $\\frac{b}{a}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 19846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the following as a common fraction: $\\sqrt[3]{5\\div 15.75}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt[3]{20}}{\\sqrt[3]{63}}"
} | {
"split": "train",
"index": 19847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively. Le... | math | {
"style": "rule",
"ground_truth": "20/29"
} | {
"split": "train",
"index": 19848,
"solution": ""
} |
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