data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A standard deck of 52 cards is arranged randomly. What is the probability that the top three cards alternate in color, starting with a red card, then a black card, followed by another red card? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{102}"
} | {
"split": "train",
"index": 26749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given numbers \\(a, b, c\\) satisfy \\(a b c+a+c-b\\). Then the maximum value of the algebraic expression \\(\\frac{1}{1+a^{2}}-\\frac{1}{1+b^{2}}+\\frac{1}{1+c^{2}}\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 26750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaofan checked the step count on the smartwatch app before going out and found that the step count was a two-digit number. After walking downstairs, he found that the tens digit and the units digit had swapped. When he reached the entrance of the residential area, he found tha... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 26751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. W... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive multiple of $23$ that is $4$ more than a multiple of $89$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "805"
} | {
"split": "train",
"index": 26753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\\frac{2\\sqrt{2}}{3}$, the line $y=\\frac{1}{2}$ intersects $C$ at points $A$ and $B$, where $|AB|=3\\sqrt{3}$. \n$(1)$ Find the equation of $C$; \n$(2)$ Let the left and right foci of $C$ ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles \\(P\\), \\(Q\\), and \\(R\\) are externally tangent to each other and internally tangent to circle \\(S\\). Circles \\(Q\\) and \\(R\\) are congruent. Circle \\(P\\) has radius 2 and passes through the center of \\(S\\). What is the radius of circle \\(Q\\)? Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 26755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA regular 100-sided polygon is placed on a table, with the numbers $1, 2, \\ldots, 100$ written at its vertices. These numbers are then rewritten in order of their distance from the front edge of the table. If two vertices are at an equal distance from the edge, the left numb... | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"split": "train",
"index": 26756,
"solution": ""
} |
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": "97"
} | {
"split": "train",
"index": 26757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=a\\ln x-x^{2}+1$.\n\n(I) If the tangent line of the curve $y=f(x)$ at $x=1$ is $4x-y+b=0$, find the values of the real numbers $a$ and $b$;\n\n(II) Discuss the monotonicity of the function $f(x)$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 26758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\\{1,2,\\ldots,2012\\}$ with $|A|=k$ , there exist three elements $x,y,z$ in $A$ such that $x=a+b$ , $y=b+c$ , $z=c+a$ , where $a,b,c$ are in $S$ and are distinct integers.\n\n*Proposed... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 26759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the system \n\n\\[\n\\left\\{\n\\begin{array}{l}\nx^{2} + 4y^{2} = 1 \\\\\nx + 2y = a\n\\end{array}\n\\right.\n\\]\n\nhas a unique solution. If necessary, round your answer to two decimal places. If there are no solutions, answer with 0. L... | math | {
"style": "rule",
"ground_truth": "-1.41"
} | {
"split": "train",
"index": 26760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\frac{x^5-1}3$, find $f^{-1}(-31/96)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac12"
} | {
"split": "train",
"index": 26761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the modular inverse of $27$, modulo $28$.\n\nExpress your answer as an integer from $0$ to $27$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 26762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill in each box in the equation $\\square \\square+\\square \\square=\\square \\square$ with a digit from $0, 1, 2, \\ldots, 9$ (digits in the boxes can be the same, and no number can start with a zero) such that the equation holds true. There are $\\qquad$ ways to fill in the... | math | {
"style": "rule",
"ground_truth": "4095"
} | {
"split": "train",
"index": 26763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number\n\\[\\text{cis } 75^\\circ + \\text{cis } 83^\\circ + \\text{cis } 91^\\circ + \\dots + \\text{cis } 147^\\circ\\]is expressed in the form $r \\, \\text{cis } \\theta$, where $r > 0$ and $0^\\circ \\le \\theta < 360^\\circ$. Find $\\theta$ in degrees. Let's think st... | math | {
"style": "rule",
"ground_truth": "111^\\circ"
} | {
"split": "train",
"index": 26764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a certain type of car, the purchase cost is $10$ ten thousand yuan, and the annual expenses for insurance, road maintenance, and car fees are about $0.9$ ten thousand yuan. The maintenance fee for the first year is $0.2$ ten thousand yuan, and it increases by $0.2$ ten thou... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$. What is the probability that it rains on at most 2 days in June? Express your answer as a decimal to the nearest thousandth. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "0.411"
} | {
"split": "train",
"index": 26766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \\sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S = \\{(x,y) | x = 1, 2, \\ldots, 1993, y = 1, 2, 3, 4\\}$. If $T \\subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "5183"
} | {
"split": "train",
"index": 26768,
"solution": "\nLet \\( S = \\{(x,y) \\mid x = 1, 2, \\ldots, 1993, y = 1, 2, 3, 4\\} \\). We aim to find the maximum possible value of \\( |T| \\) for a subset \\( T \\subset S \\) such that there are no squares in \\( T \\).\n\nTo solve this, we need to ensure that no four poi... |
deepscaler | [
{
"role": "user",
"content": "Simplify $(3-2i)^2$. (Your answer should be of the form $a+bi$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5-12i"
} | {
"split": "train",
"index": 26769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $a^3 + b^3$ given that $a+b=10$ and $ab=17$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "490"
} | {
"split": "train",
"index": 26770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon. Let's thi... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"split": "train",
"index": 26771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point outside of circle $O.$ A segment is drawn from $P$ such that it is tangent to circle $O$ at point $T.$ Meanwhile, a secant from $P$ intersects $O$ at $A$ and $B,$ such that $PA < PB.$ If $PA = 3$ and $PT = AB - PA,$ then what is $PB$? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are ther... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 26773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B each independently toss a fair coin. A tosses the coin 10 times, and B tosses the coin 11 times. What is the probability that the number of heads B gets is greater than the number of heads A gets? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 26774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of all the positive integer divisors of \\( 6^{16} \\) equals \\( 6^k \\) for some integer \\( k \\). Determine the value of \\( k \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2312"
} | {
"split": "train",
"index": 26775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A floor 9 feet by 12 feet, is to be tiled with 4-inch-by-6-inch tiles. How many tiles are needed to cover the floor? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"split": "train",
"index": 26776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a four-digit number \\( A \\). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \\( A \\), and the smallest possible number is 594 less than \\( A \\). What is \\( A \\)? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "1963"
} | {
"split": "train",
"index": 26777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Karlsson eats three jars of jam and one jar of honey in 25 minutes, while Little Brother does it in 55 minutes. Karlsson eats one jar of jam and three jars of honey in 35 minutes, while Little Brother does it in 1 hour 25 minutes. How long will it take them to eat six jars of j... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\n\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that d... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 26779,
"solution": "To find $D(96)$, we need to consider all possible ways to write $96$ as a product of integers greater than $1$, where the order of factors matters. We start by noting the prime factorization of $96$:\n\\[ 96 = 2^5 \\cdot 3 \\]\n\nWe will consider different cases ba... |
deepscaler | [
{
"role": "user",
"content": "A group of students during a sports meeting lined up for a team photo. When they lined up in rows of 5, there were two students left over. When they formed rows of 6 students, there were three extra students, and when they lined up in rows of 8, there were four students left ov... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 26780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the points $(1,1)$ and $(4,7)$. 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": "3\\sqrt{5}"
} | {
"split": "train",
"index": 26781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an $8 \\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "20736"
} | {
"split": "train",
"index": 26782,
"solution": "Number both the rows and the columns from 1 to 8, and say that black squares are the ones where the rows and columns have the same parity. We will use, e.g. 'even rows' to refer to rows 2, 4, 6,8. Choosing 8 squares all in different rows and columns is equivalent t... |
deepscaler | [
{
"role": "user",
"content": "Club Truncator is now in a soccer league with four other teams, each of which it plays once. In any of its 4 matches, the probabilities that Club Truncator will win, lose, or tie are $\\frac{1}{3}$, $\\frac{1}{3}$, and $\\frac{1}{3}$ respectively. The probability that Club Trun... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 26783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive two-digit integers are multiples of 5 and of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Esther and Frida are supposed to fill a rectangular array of 16 columns and 10 rows with the numbers 1 to 160. Esther fills it row-wise so that the first row is numbered 1, 2, ..., 16 and the second row is 17, 18, ..., 32 and so on. Frida fills it column-wise, so that her first... | math | {
"style": "rule",
"ground_truth": "322"
} | {
"split": "train",
"index": 26785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertices of a $3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these circles?\n\n[asy]unitsize(1cm);\ndraw(Circle((1.8,2.4),1),linewidth(0.7));\ndraw(Circle((0,0),2),linewidth(0.7));\ndraw(C... | math | {
"style": "rule",
"ground_truth": "14\\pi"
} | {
"split": "train",
"index": 26786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using Green's theorem, evaluate the line integral \\( I = \\oint_{L} 2(x^{2}+y^{2}) \\, dx + (x+y)^{2} \\, dy \\) along the contour \\( L \\) of triangle \\( ABC \\) with vertices \\( A(1,1) \\), \\( B(2,2) \\), \\( C(1,3) \\). Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 26787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\cdots, 10\\} \\). If a subset \\( T \\) of \\( S \\) has at least 2 elements and the absolute difference between any two elements in \\( T \\) is greater than 1, then \\( T \\) is said to have property \\( P \\). Find the number of different subsets of \\... | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 26788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $a \\nabla b$ is defined by $a \\nabla b=\\frac{a+b}{a-b}$ for all integers $a$ and $b$ with $a \\neq b$. If $3 \\nabla b=-4$, 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": "5"
} | {
"split": "train",
"index": 26789,
"solution": "Using the definition, $3 \\nabla b=\\frac{3+b}{3-b}$. Assuming $b \\neq 3$, the following equations are equivalent: $3 \\nabla b =-4$, $\\frac{3+b}{3-b} =-4$, $3+b =-4(3-b)$, $3+b =-12+4 b$, $15 =3 b$ and so $b=5$."
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a}$ and $\\mathbf{b}$ are vectors such that $\\|\\mathbf{a}\\| = 7$ and $\\|\\mathbf{b}\\| = 11$, then find all possible values of $\\mathbf{a} \\cdot \\mathbf{b}$.\n\nSubmit your answer in interval notation. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "[-77,77]"
} | {
"split": "train",
"index": 26790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\\circ}$ and $85^{\\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\\circ}$ more than twice $C$. Find the measure of the largest angle. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "205^\\circ"
} | {
"split": "train",
"index": 26791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=3$, and $\\left(a+b\\right)\\sin B=\\left(\\sin A+\\sin C\\right)\\left(a+b-c\\right)$.<br/>$(1)$ Find angle $A$;<br/>$(2)$ If $acosB+bcosA=\\sqrt{3}$, find th... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 26792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the ellipse $C: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1\\; (a > b > 0)$, a line with slope $k(k > 0)$ intersects the ellipse at the left vertex $A$ and another point $B$. The projection of point $B$ on the $x$-axis is exactly the right focus $F$. If the eccentricity of ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 26793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a\\_n\\}$ satisfies $a\\_1=1$, $a\\_2=3$, and $a_{n+2}=|a_{n+1}|-a_{n}$, where $n∈N^{*}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence. Find $S_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 26794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 26795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r$, and $s$ be positive real numbers such that\n\\[\n\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}\np^2+q^2&=&r^2+s^2&=&2500,\\\\\npr&=&qs&=&1200.\n\\end{array}\n\\]\nCompute the value of $\\lfloor T \\rfloor$, where $T=p+q+r+s... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 26796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, wh... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 26797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\frac{\\frac{1}{3} + \\frac{1}{4}}{ \\frac{2}{5} - \\frac{1}{6}}$. Express your answer as a fraction in simplest form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 26798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\\frac {x^{2}}{16}+ \\frac {y^{2}}{9}=1$, and a line passing through point $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|AB|=5$, calculate $|AF_{1}|+|BF_{1}|$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 26799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain company implements an annual salary system, where an employee's annual salary consists of three components: basic salary, housing allowance, and medical expenses, as specified below:\n\n| Item | Salary in the First Year (in ten thousand yuan) | Calculation Me... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"split": "train",
"index": 26800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the li... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {3} - 3}{4}"
} | {
"split": "train",
"index": 26801,
"solution": "To solve this problem, we need to determine the probability that the distance between the centers of circles $A$ and $B$ is less than or equal to 2 (since each circle has a radius of 1, and two circles intersect if the distance between their centers is less than or... |
deepscaler | [
{
"role": "user",
"content": "If a $5\\times 5$ chess board exists, in how many ways can five distinct pawns be placed on the board such that each column and row contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 26802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\frac{5}{8}$ times the radius of cir... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 26803,
"solution": "1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The dis... |
deepscaler | [
{
"role": "user",
"content": "The distance between cities $A$ and $B$ is 435 km. A train left city $A$ at a speed of 45 km/h. After 40 minutes, another train left city $B$ heading towards the first train at a speed of 55 km/h. What distance will be between them one hour before they meet? Let's think step by... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of 9 standard dice rolled has the same probability of occurring as the sum of 20. Find the value of the sum of 9 standard dice that shares this same probability. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 26805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the odd function $f(x)$ is a monotonically increasing function defined on $\\mathbb{R}$ and the sequence $\\{x_n\\}$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, find the value of $x_{2012}$. Let's... | math | {
"style": "rule",
"ground_truth": "4005"
} | {
"split": "train",
"index": 26806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 26807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circle centered at $(3,-2)$ and with radius $5$ intersects the circle centered at $(3,4)$ and with radius $3$ at two points $C$ and $D$. Find $(CD)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{224}{9}"
} | {
"split": "train",
"index": 26808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits). Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"split": "train",
"index": 26809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) In an arithmetic sequence $\\{ a_n \\}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.\n\n(2) Determine the interval of decrease for the function $y = \\log_{\\frac{1}{2}}(x^2 + 2x - 3)$.\n\n(3) In triangle $ABC$, given $a = \\sqrt{3}$, $b = 1$, and $\\an... | math | {
"style": "rule",
"ground_truth": "10\\pi"
} | {
"split": "train",
"index": 26810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Originally, there were 5 books on the bookshelf. If 2 more books are added, but the relative order of the original books must remain unchanged, then there are $\\boxed{\\text{different ways}}$ to place the books. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 26811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number $x$ is equal to $6 \\cdot 18 \\cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "441"
} | {
"split": "train",
"index": 26812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we roll a standard fair 6-sided die. What is the probability that a perfect square is rolled? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac13"
} | {
"split": "train",
"index": 26813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the square is $s^2$ and the area of the rectangle is $3s \\times \\frac{9s}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.41\\%"
} | {
"split": "train",
"index": 26814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $73^{-1} \\pmod{74}$, as a residue modulo 74. (Give an answer between 0 and 73, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 26815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression \\((x+y+z)^{2024} + (x-y-z)^{2024}\\), the parentheses are expanded and like terms are combined. How many monomials \\(x^{a} y^{b} z^{c}\\) have a non-zero coefficient? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1026169"
} | {
"split": "train",
"index": 26816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\{b_n\\}$ is an arithmetic sequence with $$\nb_1+b_2+ \\cdots +b_{150}=150 \\quad \\text{and} \\quad\nb_{151}+b_{152}+ \\cdots + b_{300}=450.\n$$What is the value of $b_2 - b_1$? Express your answer as a common fraction. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{75}"
} | {
"split": "train",
"index": 26817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 6 is drawn centered at the origin. How many squares of side length 1 and integer coordinate vertices intersect the interior of this circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 26818,
"solution": "By symmetry, the answer is four times the number of squares in the first quadrant. Let's identify each square by its coordinates at the bottom-left corner, $(x, y)$. When $x=0$, we can have $y=0 \\ldots 5$, so there are 6 squares. (Letting $y=6$ is not allowed beca... |
deepscaler | [
{
"role": "user",
"content": "What is the number of ways in which one can choose $60$ unit squares from a $11 \\times 11$ chessboard such that no two chosen squares have a side in common? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 26819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the even integers between $200$ and $400$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30300"
} | {
"split": "train",
"index": 26820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vijay chooses three distinct integers \\(a, b, c\\) from the set \\(\\{1,2,3,4,5,6,7,8,9,10,11\\}\\). If \\(k\\) is the minimum value taken on by the polynomial \\(a(x-b)(x-c)\\) over all real numbers \\(x\\), and \\(l\\) is the minimum value taken on by the polynomial \\(a(x-b... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 26821,
"solution": "Quadratics are minimized at the average of their roots, so \\(k=a\\left(\\frac{b+c}{2}-b\\right)\\left(\\frac{b+c}{2}-c\\right)=-\\frac{a(b-c)^{2}}{4}\\), and \\(l=a\\left(\\frac{b-c}{2}-b\\right)\\left(\\frac{b-c}{2}+c\\right)=-\\frac{a(b+c)^{2}}{4}\\). Therefore,... |
deepscaler | [
{
"role": "user",
"content": "Compute $$100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\\ldots+4^{2}+3^{2}-2^{2}-1^{2}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"split": "train",
"index": 26822,
"solution": "Note that $(n+3)^{2}-(n+2)^{2}-(n+1)^{2}+n^{2}=4$ for every $n$. Therefore, adding $0^{2}$ to the end of the given sum and applying this identity for every four consecutive terms after $100^{2}$, we see that the given sum is equivalent to $100^{2}+25 \\cdot 4=10100... |
deepscaler | [
{
"role": "user",
"content": "Minimize \\(\\boldsymbol{F}=\\boldsymbol{x}_{2}-\\boldsymbol{x}_{1}\\) for non-negative \\(x_{1}\\) and \\(x_{2}\\), subject to the system of constraints:\n\n$$\n\\left\\{\\begin{aligned}\n-2 x_{1}+x_{2}+x_{3} &=2 \\\\\nx_{1}-2 x_{2}+x_{4} &=2 \\\\\nx_{1}+x_{2}+x_{5} &=5\n\\end... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 26823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the following expression:\n\\[\n\\frac{(1 + 15) \\left( 1 + \\dfrac{15}{2} \\right) \\left( 1 + \\dfrac{15}{3} \\right) \\dotsm \\left( 1 + \\dfrac{15}{21} \\right)}{(1 + 17) \\left( 1 + \\dfrac{17}{2} \\right) \\left( 1 + \\dfrac{17}{3} \\right) \\dotsm \\left( 1 + \\d... | math | {
"style": "rule",
"ground_truth": "272"
} | {
"split": "train",
"index": 26824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(ABCD\\), the sides \\(AB\\) and \\(CD\\) are parallel and \\(CD = 2AB\\). Points \\(P\\) and \\(Q\\) are chosen on sides \\(AD\\) and \\(BC\\), respectively, such that \\(DP : PA = 2\\) and \\(BQ : QC = 3 : 4\\). Find the ratio of the areas of quadrilaterals \\(... | math | {
"style": "rule",
"ground_truth": "19/44"
} | {
"split": "train",
"index": 26825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the sm... | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"split": "train",
"index": 26826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Leticia has a $9\\times 9$ board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of... | math | {
"style": "rule",
"ground_truth": "486"
} | {
"split": "train",
"index": 26827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ has these properties: (1) The maximum value of $y = ax^2 + bx + c$ is 4, which occurs at $x = 2$. (2) The graph passes through the point $(0,-16)$. If the graph also passes through the point $(5,n)$, what is the value of $n$? Let's... | math | {
"style": "rule",
"ground_truth": "-41"
} | {
"split": "train",
"index": 26829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations\n\\begin{align*}\n 13x + by + cz &= 0 \\\\\n ax + 23y + cz &= 0 \\\\\n ax + by + 42z &= 0.\n\\end{align*}Suppose that $ a \\ne 13$ and $ x \\ne 0$. What is the value of\n\\[ \\... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of six-digit palindromes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"split": "train",
"index": 26831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\\ge 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26832,
"solution": "Let the largest odd number below the sequence be the $q$th positive odd number, and the largest odd number in the sequence be the $p$th positive odd number. Therefore, the sum is $p^2-q^2=(p+q)(p-q)$ by sum of consecutive odd numbers. Note that $p+q$ and $p-q$ have... |
deepscaler | [
{
"role": "user",
"content": "Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3:30 PM"
} | {
"split": "train",
"index": 26833,
"solution": "1. **Calculate the total time for the first two tasks**: Marie starts the first task at 1:00 PM and finishes the second task at 2:40 PM. The total time taken for these two tasks is calculated by subtracting the start time from the end time:\n \\[\n \\text{Total... |
deepscaler | [
{
"role": "user",
"content": "What is the probability that \\(2^{n}\\), where \\(n\\) is a randomly chosen positive integer, ends with the digit 2? What is the probability that \\(2^{n}\\) ends with the digits 12? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.05"
} | {
"split": "train",
"index": 26834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four points \\( K, L, M, N \\) that are not coplanar. A sphere touches the planes \\( K L M \\) and \\( K L N \\) at points \\( M \\) and \\( N \\) respectively. Find the surface area of the sphere, knowing that \\( M L = 1 \\), \\( K M = 2 \\), \\( \\angle M N L = 60^\\c... | math | {
"style": "rule",
"ground_truth": "\\frac{64\\pi}{11}"
} | {
"split": "train",
"index": 26835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set \\( A = \\{x \\mid (x-2)(x-6) \\geqslant 3, x \\in \\mathbf{Z}, 0 \\leqslant x \\leq 7\\} \\), find the number of non-empty subsets of \\( A \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 26836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the roots and the sum of the roots of the equation $24x^2 + 60x - 600 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2.5"
} | {
"split": "train",
"index": 26837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0$. Suppose that $a + \\frac{1}{b}$ and $b + \\frac{1}{a}$ are the roots of the equation $x^2-px+q=0$. What is $q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 26838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper $A B C D$ is folded and flattened such that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ landing on side $A B$. Given that $\\angle 1 = 22^{\\circ}$, find $\\angle 2$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 26839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y=x^2$ and the moving line $y=(2t-1)x-c$ have common points $(x_1, y_1)$, $(x_2, y_2)$, and $x_1^2+x_2^2=t^2+2t-3$.\n\n(1) Find the range of the real number $t$; \n(2) When does $t$ take the minimum value of $c$, and what is the minimum value of $c$? Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{11-6\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 26840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of e... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 26841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an arithmetic sequence $\\{a_n\\}$ with a non-zero common difference, $a_1$, $a_2$, and $a_5$ form a geometric sequence, and the sum of the first $10$ terms of this sequence is $100$. The sum of the first $n$ terms of the sequence $\\{b_n\\}$ is $S_n$, and it satisfies $S_n=... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{49}"
} | {
"split": "train",
"index": 26842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club consists of five leaders and some regular members. Each year, all leaders leave the club and each regular member recruits three new people to join as regular members. Subsequently, five new leaders are elected from outside the club to join. Initially, there are eighteen ... | math | {
"style": "rule",
"ground_truth": "3164"
} | {
"split": "train",
"index": 26843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many six-digit numbers with the penultimate digit being 1 are divisible by 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18000"
} | {
"split": "train",
"index": 26844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be distinct integers, and let $\\omega$ be a complex number such that $\\omega^3 = 1$ and $\\omega \\neq 1.$ Find the smallest possible value of\n\\[|a + b \\omega + c \\omega^2|.\\] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 26845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be the smallest positive integer that satisfies the following conditions: (1) \\( n \\) is a multiple of 75; (2) \\( n \\) has exactly 75 positive integer factors (including 1 and itself). Find \\(\\frac{n}{75}\\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 26846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 137$, what is the minimal possible value of $|a - b|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the infinite product $(2^{1/3})(4^{1/9})(8^{1/27})(16^{1/81}) \\dotsm.$ Enter your answer in the form \"\\sqrt[a]{b}\", which stands for $\\sqrt[a]{b}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{8}"
} | {
"split": "train",
"index": 26848,
"solution": ""
} |
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