data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive 3-digit numbers are divisible by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 31050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\\leqslant x\\leqslant n\\left(m \\lt n\\right)$, and satisfies $M-N=2$, then the function is called the \"range function\" on $m\\leqslant x\\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 31051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\frac{1}{10} + \\frac{2}{10} + \\frac{3}{10} + \\frac{4}{10} + \\frac{5}{10} + \\frac{6}{10} + \\frac{7}{10} + \\frac{8}{10} + \\frac{9}{10} + \\frac{55}{10}= Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 31052,
"solution": "1. **Convert fractions to a common denominator**: All the fractions given in the problem have the same denominator, which is 10. Therefore, we can focus on adding the numerators directly.\n\n2. **Add the numerators**: The numerators are $1, 2, 3, 4, 5, 6, 7, 8, 9, ... |
deepscaler | [
{
"role": "user",
"content": "A frog is at the point $(0,0)$. Every second, he can jump one unit either up or right. He can only move to points $(x, y)$ where $x$ and $y$ are not both odd. How many ways can he get to the point $(8,14)$? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "330"
} | {
"split": "train",
"index": 31053,
"solution": "When the frog is at a point $(x, y)$ where $x$ and $y$ are both even, then if that frog chooses to move right, his next move will also have to be a step right; similarly, if he moves up, his next move will have to be up. If we 'collapse' each double step into one s... |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(1+x){(x-\\frac{2}{x})}^{3}$, calculate the coefficient of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 31054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the determinant of the matrix:\n\\[\n\\begin{vmatrix} 2 & 4 & -2 \\\\ 0 & 3 & -1 \\\\ 5 & -1 & 2 \\end{vmatrix}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C. \\Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{9}"
} | {
"split": "train",
"index": 31056,
"solution": "First, note that a homothety $h$ centered at $A$ takes $\\Gamma$ to the circumcircle of $A B C, D$ to $B$ and $E$ to $C$, since the two circles are tangent. As a result, we have $D E \\| B C$. Now, let $P$ be the center of $\\Gamma$ and $O$ be the circumcenter of $... |
deepscaler | [
{
"role": "user",
"content": "Let $Q$ be the product of the first $50$ positive even integers. Find the largest integer $j$ such that $Q$ is divisible by $2^j$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 31057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man walked a certain distance at a constant rate. If he had gone $\\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\\frac{1}{2}$ mile per hour slower, he would have been $2\\frac{1}{2}$ hours longer on the road.... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 31058,
"solution": "1. **Set up the equations based on the problem statement:**\n Let $x$ be the man's usual speed in miles per hour, and $t$ be the usual time in hours it takes him to walk the distance. The distance he walks can be represented as $d = xt$.\n\n2. **Equation for incr... |
deepscaler | [
{
"role": "user",
"content": "Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicir... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 31059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $-\\left(14\\div 2\\cdot 9-60+3\\cdot 9\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 31060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine $\\sqrt[4]{105413504}$ without a calculator. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 31061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Approximate the increase in the volume of a cylinder with a height of \\( H = 40 \\) cm and a base radius of \\( R = 30 \\) cm when the radius is increased by \\( 0.5 \\) cm. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200\\pi"
} | {
"split": "train",
"index": 31062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the four vertices of the quadrilateral $MNPQ$ are on the graph of the function $f(x)=\\log_{\\frac{1}{2}} \\frac{ax+1}{x+b}$, and it satisfies $\\overrightarrow{MN}= \\overrightarrow{QP}$, where $M(3,-1)$, $N\\left( \\frac{5}{3},-2\\right)$, then the area of the quad... | math | {
"style": "rule",
"ground_truth": "\\frac{26}{3}"
} | {
"split": "train",
"index": 31063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 31064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=2\\sin \\left(x+ \\frac {\\pi}{6}\\right)\\cos \\left(x+ \\frac {\\pi}{6}\\right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\\sin 2x$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 31065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the cells of a $100 \\times 100$ square, the numbers $1, 2, \\ldots, 10000$ were placed, each exactly once, such that numbers differing by 1 are recorded in adjacent cells along the side. After that, the distances between the centers of each two cells, where the numbers in t... | math | {
"style": "rule",
"ground_truth": "50\\sqrt{2}"
} | {
"split": "train",
"index": 31066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the total number of occurrences of the digits $0,1 \\ldots, 9$ in the entire guts round. If your answer is $X$ and the actual value is $Y$, your score will be $\\max \\left(0,20-\\frac{|X-Y|}{2}\\right)$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "559"
} | {
"split": "train",
"index": 31067,
"solution": "To compute the answer, I extracted the flat text from the PDF file and ran word-count against the list of digit matches. ``` evan@ArchMega ~ /Downloads/November $ pdftotext HMMTNovember2016GutsTest.pdf guts-test-text.txt evan@ArchMega ~ /Downloads/November $ cat gu... |
deepscaler | [
{
"role": "user",
"content": "If $x^2+x+4 = y - 4$ and $x = -7$, what is the value of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 31068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 31069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{28}"
} | {
"split": "train",
"index": 31070,
"solution": "To solve this problem, we need to calculate the probability that there are at least two adjacent empty spaces among the 16 spaces after 12 cars have parked. Each car occupies one space, and they choose their spaces randomly.\n\n#### Step 1: Total ways to park 12 ca... |
deepscaler | [
{
"role": "user",
"content": "How many square units are in the area of the triangle whose vertices are the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 31071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of the coordinates of the midpoint of the segment? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 31072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x=1-2a$, $y=3a-4$. \n$(1)$ Given that the arithmetic square root of $x$ is $3$, find the value of $a$; \n$(2)$ If $x$ and $y$ are both square roots of the same number, find this number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 31073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fictional country of Isoland uses an alphabet with ten unique letters: A, B, D, E, I, L, N, O, R, U. License plates in Isoland are structured with five letters. How many different license plates are possible if they must begin with A or I, end with R, cannot include the let... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 31074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$ . A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$ . Points $C$ , $A$ , and $B$ are collinear. The line seg... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{19}"
} | {
"split": "train",
"index": 31075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways are distinguished by the sizes of the square and the triangle). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer $ n$, for all positive integers $ a_1, a_2, \\cdots, a_n$ that satisfy $ a_1 \\equal{} 1$, $ a_{i \\plus{} 1} \\leq a_i \\plus{} 1$, find $ \\displaystyle \\sum_{i \\equal{} 1}^{n} a_1a_2 \\cdots a_i$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "(2n-1)!!"
} | {
"split": "train",
"index": 31077,
"solution": "\nGiven a positive integer \\( n \\), for all positive integers \\( a_1, a_2, \\cdots, a_n \\) that satisfy \\( a_1 = 1 \\) and \\( a_{i+1} \\leq a_i + 1 \\), we aim to find the sum \\( \\displaystyle \\sum_{i=1}^{n} a_1 a_2 \\cdots a_i \\).\n\nTo solve this proble... |
deepscaler | [
{
"role": "user",
"content": "The number of trees in a park must be more than 80 and fewer than 150. The number of trees is 2 more than a multiple of 4, 3 more than a multiple of 5, and 4 more than a multiple of 6. How many trees are in the park? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 31078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Given $x \\gt 0$, $y \\gt 0$, and $2x+3y=6$, find the maximum value of $xy$;<br/>$(2)$ Given $x＜\\frac{1}{2}$, find the maximum value of $y=2x+\\frac{4}{2x-1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 31079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many roots does the equation \n\n$$\n\\overbrace{f(f(\\ldots f}^{10 \\text{ times }}(x) \\ldots))+\\frac{1}{2}=0\n$$\n\nwhere $f(x)=|x|-1$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 2015 can be represented as a sum of consecutive integers in several ways, for example, $2015 = 1007 + 1008$ or $2015 = 401 + 402 + 403 + 404 + 405$. How many ways can this be done? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\\angle A D C=60^{\\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.\n\na) Find the area of the circle obtained in the cross-section of t... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 31082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{C} \\to \\mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \\text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "399"
} | {
"split": "train",
"index": 31083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all four-digit numbers, in which only the digits $1, 2, 3, 4, 5$ are used, and each digit occurs no more than once. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "399960"
} | {
"split": "train",
"index": 31084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 31085,
"solution": "1. **Organize the Data**: First, we arrange the given numbers in ascending order:\n \\[\n 0, 0, 1, 2, 3, 3, 3, 4\n \\]\n\n2. **Finding the Mode**: The mode is the number that appears most frequently in the data set. From the ordered list, the number $3$ appea... |
deepscaler | [
{
"role": "user",
"content": "On Tony's map, the distance from Saint John, NB to St. John's, NL is $21 \\mathrm{~cm}$. The actual distance between these two cities is $1050 \\mathrm{~km}$. What is the scale of Tony's map? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:5 000 000"
} | {
"split": "train",
"index": 31086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incom... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 31087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any sequence of real numbers $A=(a_1,a_2,a_3,\\ldots)$, define $\\Delta A^{}_{}$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\\ldots)$, whose $n^{\\mbox{th}}_{}$ term is $a_{n+1}-a_n^{}$. Suppose that all of the terms of the sequence $\\Delta(\\Delta A^{}_{})$ are $1^{}_{}... | math | {
"style": "rule",
"ground_truth": "819"
} | {
"split": "train",
"index": 31088,
"solution": "Note that the $\\Delta$s are reminiscent of differentiation; from the condition $\\Delta(\\Delta{A}) = 1$, we are led to consider the differential equation \\[\\frac{d^2 A}{dn^2} = 1\\] This inspires us to guess a quadratic with leading coefficient 1/2 as the solut... |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a}$, $\\mathbf{b}$, $\\mathbf{c}$, and $\\mathbf{d}$ are unit vectors, then 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 | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.2, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability that the phone call is answered within the first four rings. Let's thi... | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 31090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x \\sim N(4, 1)$ and $f(x < 3) = 0.0187$, then $f(x < 5) = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.9813"
} | {
"split": "train",
"index": 31091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is the power of the triple. Compute the sum of powers of all triples $(x,y,z)$ where $x,y,z \\leq 9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7290"
} | {
"split": "train",
"index": 31092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\\triangle ABC$. What is $BI$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 31093,
"solution": "1. **Identify the triangle and its sides**: We are given a triangle $ABC$ with sides $AB = 27$, $AC = 26$, and $BC = 25$.\n\n2. **Incircle and angle bisectors**: Let $I$ be the incenter of $\\triangle ABC$, which is the point of intersection of the internal angle b... |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer \\( n \\) such that \\( n \\leq 2016 \\) and \\(\\left\\{\\frac{n}{2}\\right\\}+\\left\\{\\frac{n}{4}\\right\\}+\\left\\{\\frac{n}{6}\\right\\}+\\left\\{\\frac{n}{12}\\right\\}=3\\), where \\(\\{x\\} = x - \\lfloor x \\rfloor\\) and \\(\\lfloor x \\rflo... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 31094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A finite set $S$ of points in the coordinate plane is called [i]overdetermined[/i] if $|S|\\ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S|-2$, satisfying $P(x)=y$ for every point $(x,y)\\in S$. \n \nFor each integer $n\\ge ... | math | {
"style": "rule",
"ground_truth": "2^{n-1} - n"
} | {
"split": "train",
"index": 31095,
"solution": "\nGiven a finite set \\( S \\) of points in the coordinate plane, a set \\( S \\) is called \\textit{overdetermined} if \\( |S| \\ge 2 \\) and there exists a nonzero polynomial \\( P(t) \\) with real coefficients of degree at most \\( |S| - 2 \\), such that \\( P(x... |
deepscaler | [
{
"role": "user",
"content": "Given that $x > 0$, $y > 0$ and $x + y = 4$, find the minimum value of $$\\frac{x^2}{x + 1} + \\frac{y^2}{y + 2}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{7}"
} | {
"split": "train",
"index": 31096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\\circ}$ to the base, is $9 \\mathrm{~cm}^{3}$. Find the total surface area of the pyramid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\sqrt{3} (1 + \\sqrt{2})"
} | {
"split": "train",
"index": 31097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the $x O y$ coordinate plane, there is a Chinese chess \"knight\" at the origin $(0,0)$. The \"knight\" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. Calculate the minimum number of moves required. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "1328"
} | {
"split": "train",
"index": 31098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, \\(FGHI\\) is a trapezium with side \\(GF\\) parallel to \\(HI\\). The lengths of \\(FG\\) and \\(HI\\) are 50 and 20 respectively. The point \\(J\\) is on the side \\(FG\\) such that the segment \\(IJ\\) divides the trapezium into two parts of equal ar... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 31099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The clock runs 3 minutes fast every 24 hours. The clock was set accurately. After what minimum time will the clock show the correct time again? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 31100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different five-letter words can be formed such that they start and end with the same letter, and the middle letter is always 'A'? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17576"
} | {
"split": "train",
"index": 31101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The simple quadrilateral \\(ABCD\\) has sides \\(AB\\), \\(BC\\), and \\(CD\\) with lengths 4, 5, and 20, respectively. If the angles \\(B\\) and \\(C\\) are obtuse, and \\(\\sin C = -\\cos B = \\frac{3}{5}\\), calculate the length of the side \\(AD\\). Let's think step by step... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 31102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{1}{1+\\sqrt{2}}\\cdot\\frac{1}{1-\\sqrt{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 31103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The dollar is now worth $\\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\\frac{1}{2{}^2{}^{n-1}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\\frac{1}{b... | math | {
"style": "rule",
"ground_truth": "506"
} | {
"split": "train",
"index": 31104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the length of the major axis of the ellipse is 4, the left vertex is on the parabola \\( y^2 = x - 1 \\), and the left directrix is the y-axis, find the maximum value of the eccentricity of such an ellipse. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 31105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer $x$ for which $x<2x-7$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 31106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, $BE$ is the height from vertex $B$ to side $AD$, and segment $ED$ is extended from $D$ such that $ED = 8$. The base $BC$ of the parallelogram is $14$. The entire parallelogram has an area of $126$. Determine the area of the shaded region $BEDC$. Let's t... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 31107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 31108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all primes $p$ such that $p^2-p+1$ is a perfect cube. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 31109,
"solution": "\nTo solve the problem of finding all primes \\( p \\) such that \\( p^2 - p + 1 \\) is a perfect cube, we want \\( p^2 - p + 1 = n^3 \\) for some integer \\( n \\).\n\n1. **Case Analysis: Small Values of \\( p \\):**\n\n Start with small values of \\( p \\):\n ... |
deepscaler | [
{
"role": "user",
"content": "A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 31110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31111,
"solution": "1. **Formulate the given condition**: The problem states that four times the reciprocal of the circumference of a circle equals the diameter of the circle. Let the radius of the circle be $r$. The circumference of the circle is $2\\pi r$, and the diameter is $2r$. ... |
deepscaler | [
{
"role": "user",
"content": "In the figure, $\\angle EAB$ and $\\angle ABC$ are right angles, $AB=4,$ $BC=6,$ $AE=8,$ and $\\overline{AC}$ and $\\overline{BE}$ intersect at $D.$ What is the difference between the areas of $\\triangle ADE$ and $\\triangle BDC?$ [asy]\npair A,B,C,D,I;\nI=(0,8);\nA=(0,0);\nB=... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\\left[0, \\frac{\\pi}{4}\\right]$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 31113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The smallest positive integer \\( n \\) that satisfies \\( \\sqrt{n} - \\sqrt{n-1} < 0.01 \\) is:\n(29th Annual American High School Mathematics Examination, 1978) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2501"
} | {
"split": "train",
"index": 31114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( A B C \\), the base of the height \\( C D \\) lies on side \\( A B \\), and the median \\( A E \\) is equal to 5. The height \\( C D \\) is equal to 6.\n\nFind the area of triangle \\( A B C \\), given that the area of triangle \\( A D C \\) is three times the a... | math | {
"style": "rule",
"ground_truth": "96/7"
} | {
"split": "train",
"index": 31115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of $8!$ and $10!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"split": "train",
"index": 31116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ ABC$ , $ 3\\sin A \\plus{} 4\\cos B \\equal{} 6$ and $ 4\\sin B \\plus{} 3\\cos A \\equal{} 1$ . Then $ \\angle C$ in degrees is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$ . An estimate $E$ earns $\\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer.\n\n*2022 CCA Math Bonanza Lightning Round 5.2* Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 31118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To investigate the height of high school students, a stratified sampling method is used to draw a sample of 100 students from three grades. 24 students are sampled from grade 10, 26 from grade 11. If there are 600 students in grade 12, then the total number of students in the s... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 31119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\\frac {2a+b}{c}= \\frac {\\cos (A+C)}{\\cos C}$. \n(I) Find the magnitude of angle $C$, \n(II) If $c=2$, find the maximum area of $\\triangle ABC$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 31120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quantity $\\sqrt{45} - 2\\sqrt{5} + \\frac{\\sqrt{360}}{\\sqrt{2}}$ can be expressed as $\\sqrt{N}$, where $N$ is an integer. Find $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "245"
} | {
"split": "train",
"index": 31121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find distinct digits to replace the letters \\(A, B, C, D\\) such that the following division in the decimal system holds:\n\n$$\n\\frac{ABC}{BBBB} = 0,\\overline{BCDB \\, BCDB \\, \\ldots}\n$$\n\n(in other words, the quotient should be a repeating decimal). Let's think step by... | math | {
"style": "rule",
"ground_truth": "219"
} | {
"split": "train",
"index": 31122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 31123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\displaystyle\\frac{21}{\\sqrt{21}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\!\\sqrt{21}"
} | {
"split": "train",
"index": 31124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5,1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hamme... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{5}"
} | {
"split": "train",
"index": 31125,
"solution": "After they go to $y=x$, we reflect the remainder of their path in $y=x$, along with the second pipe and their headquarters. Now, they must go from $(5,1)$ to $y=7$ crossing $y=x$, and then go to $(1,5)$. When they reach $y=7$, we reflect the remainder of their path... |
deepscaler | [
{
"role": "user",
"content": "Karlanna places 600 marbles into $m$ total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of $m$ can this be done? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 31126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(\\alpha) = \\cos\\alpha \\sqrt{\\frac{\\cot\\alpha - \\cos\\alpha}{\\cot\\alpha + \\cos\\alpha}} + \\sin\\alpha \\sqrt{\\frac{\\tan\\alpha - \\sin\\alpha}{\\tan\\alpha + \\sin\\alpha}}$, and $\\alpha$ is an angle in the second quadrant. \n(1) Simplify $f(\\alpha)... | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{12}"
} | {
"split": "train",
"index": 31127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive number such as 3.27, 3 is called the integer part of the number, and 0.27 is called the decimal part of the number. Find a positive number whose decimal part, integer part, and the number itself form three consecutive terms of a geometric sequence.\n(The 7th Cana... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 31128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=(2-a)(x-1)-2\\ln x$ $(a\\in \\mathbb{R})$.\n\n(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the decreasing interval of the function $g(x)$;\n\n(Ⅱ) If the function $y=f(x)$ has no zeros ... | math | {
"style": "rule",
"ground_truth": "2-4\\ln 2"
} | {
"split": "train",
"index": 31129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a cone with a height of 4 and a slant height of 8, three spheres of radius \\( r \\) are placed. They touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \\( r \\). Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{5+2\\sqrt{3}}"
} | {
"split": "train",
"index": 31130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ such that\n\\[a^2 + b^2 + c^2 + d^2 \\ge ab + \\lambda bc + cd\\]for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 31131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four-digit positive integer is called [i]virtual[/i] if it has the form $\\overline{abab}$, where $a$ and $b$ are digits and $a \\neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some p... | math | {
"style": "rule",
"ground_truth": "8282"
} | {
"split": "train",
"index": 31132,
"solution": "\nTo solve the problem of finding all virtual numbers of the form \\( n^2 + 1 \\), we need to express a virtual number in the required form and establish conditions for \\( n \\).\n\nA virtual number \\(\\overline{abab}\\) can be expressed mathematically as:\n\\[\n... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\lfloor-5.77\\rfloor+\\lceil-3.26\\rceil+\\lfloor15.93\\rfloor+\\lceil32.10\\rceil$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 31133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\Delta ABC$ is isosceles with $AC = BC$. If $m\\angle C = 40^{\\circ}$, what is the number of degrees in $m\\angle CBD$? [asy] pair A,B,C,D,E;\nC = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);\ndraw(B--C--A--E); dot(D);\nlabel(\"$A$\",A,S); label(\"$B$\",B,S); label(\"$... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 31134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular pentagon $ABCDE$ and regular hexagon $AEFGHI$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$? [asy]\ndraw((0,2.5)--(0,7.5)--(4,10)--(8,7.5)--(8,2.5)--(4,0)--cycle,linewidth(1));\ndraw((8,2... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 31135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To reach the Solovyov family's dacha from the station, one must first travel 3 km on the highway and then 2 km on a path. Upon arriving at the station, the mother called her son Vasya at the dacha and asked him to meet her on his bicycle. They started moving towards each other ... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 31136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways is it possible to arrange the digits of 1150 to get a four-digit multiple of 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming and Xiaojun start simultaneously from locations A and B, heading towards each other. If both proceed at their original speeds, they meet after 5 hours. If both increase their speeds by 2 km/h, they meet after 3 hours. The distance between locations A and B is 30 km. Le... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company that sells widgets has to pay $\\$500$ in maintenance fees each day and then it pays each worker $\\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\\$3.10$ each. What is the least number of workers the company has to hire in order to make a pr... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 31139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_{k} \\) be the coefficient of \\( x^{k} \\) in the expansion of \\( (1+2x)^{100} \\), where \\( 0 \\leq k \\leq 100 \\). Find the number of integers \\( r \\) such that \\( 0 \\leq r \\leq 99 \\) and \\( a_{r} < a_{r+1} \\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 31140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit integers are divisible by both 7 and 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "257"
} | {
"split": "train",
"index": 31141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of $x$ so that $\\arccos x > \\arcsin x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ -1, \\frac{1}{\\sqrt{2}} \\right)"
} | {
"split": "train",
"index": 31142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "217"
} | {
"split": "train",
"index": 31143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain type of beverage with a prize promotion has bottle caps printed with either \"reward one bottle\" or \"thank you for your purchase\". If a bottle is purchased and the cap is printed with \"reward one bottle\", it is considered a winning bottle, with a winning probabil... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{27}"
} | {
"split": "train",
"index": 31144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We toss a die 30 times. Determine, with an accuracy of 8 decimal places, the probability that there will be at least one six among the numbers rolled, and that no five or four appears before the first six. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "0.33333333"
} | {
"split": "train",
"index": 31145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1997,0)"
} | {
"split": "train",
"index": 31146,
"solution": "We are given that $m$ and $b$ are real numbers and $mb > 0$. This implies that both $m$ and $b$ are either both positive or both negative.\n\n1. **Plugging in $(0, 1997)$ into $y = mx + b$:**\n \\[\n 1997 = 0 \\cdot m + b \\implies b = 1997\n \\]\n Since $b... |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 31147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many $5$ -digit numbers $N$ (in base $10$ ) contain no digits greater than $3$ and satisfy the equality $\\gcd(N,15)=\\gcd(N,20)=1$ ? (The leading digit of $N$ cannot be zero.)\n\n*Based on a proposal by Yannick Yao* Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 31148,
"solution": ""
} |
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