data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 27249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $A(-2,α)$ lies on the terminal side of angle $α$ and $\\sin α=- \\dfrac{ \\sqrt{5}}{5}$.\n\n(1) Find the values of $α$, $\\cos α$, and $\\tan α$.\n\n(2) Find the value of $\\dfrac{\\cos ( \\dfrac{π}{2}+α)\\sin (-π-α)}{\\cos ( \\dfrac{11π}{2}-α)\\sin ( \\dfrac{9... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{2}"
} | {
"split": "train",
"index": 27250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have valu... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{1024}"
} | {
"split": "train",
"index": 27251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of shots required in the game \"Battleship\" on a 7x7 board to definitely hit a four-cell battleship (which consists of four consecutive cells in a single row)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=5$, $| \\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ on vector $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 27253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\( 105 \\times 95 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9975"
} | {
"split": "train",
"index": 27254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?\n$\\text{(A) } 2\\quad \\text{(B) } 3\\quad \\text{(C) } 4\\quad \\text{(D) } 5\\quad \\text{(E) } 6$\n Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \\geq t_0$ by the following properties: \\begin{enumerate} \\item[(a)] $f(t)$ is continuous for $t \\geq t_... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 27256,
"solution": "The minimum value of $T$ is 29. Write $t_{n+1} = t_0+T$ and define $s_k = t_k-t_{k-1}$ for $1\\leq k\\leq n+1$. On $[t_{k-1},t_k]$, we have $f'(t) = k(t-t_{k-1})$ and so $f(t_k)-f(t_{k-1}) = \\frac{k}{2} s_k^2$. Thus if we define \\[ g(s_1,\\ldots,s_{n+1}) = \\sum_... |
deepscaler | [
{
"role": "user",
"content": "Given $(1+x) + (1+x)^2 + \\ldots + (1+x)^5 = a_0 + a_1(1-x) + a_2(1-x)^2 + \\ldots + a_5(1-x)^5$, calculate the value of $a_1 + a_2 + a_3 + a_4 + a_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-57"
} | {
"split": "train",
"index": 27257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total distance on a Cartesian coordinate plane starting at $(2, -3)$, going to $(8, 9)$, and then moving to $(3, 2)?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{5} + \\sqrt{74}"
} | {
"split": "train",
"index": 27258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln x-\\frac{a}{{x+1}}$. \n$(1)$ Discuss the monotonicity of the function $f(x)$. \n$(2)$ If the function $f(x)$ has two extreme points $x_{1}$ and $x_{2}$, and $k{e^{f({{x_1}})+f({{x_2}})-4}}+\\ln\\frac{k}{{{x_1}+{x_2}-2}}≥0$ always holds, find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 27259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitude of an equilateral triangle is $\\sqrt6$ units. What is the area of the triangle, in square units? 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": "2\\sqrt{3}"
} | {
"split": "train",
"index": 27260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overset{→}{a\\_n}=\\left(\\cos \\frac{nπ}{6},\\sin \\frac{nπ}{6}\\right)$, $n∈ℕ^∗$, $\\overset{→}{b}=\\left( \\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$, calculate the value of $y={\\left| \\overset{→}{{a\\_1}}+ \\overset{→}{b}\\right|}^{2}+{\\left| \\overset{→}{{... | math | {
"style": "rule",
"ground_truth": "4029"
} | {
"split": "train",
"index": 27261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a fifty-cent piece. What is the probability that at least 40 cents worth of coins land on heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $g$ from the positive integers to the positive integers with the following properties:\n(i) $g$ is increasing.\n(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.\n(iii) If $m \\neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.\n\nCompute all ... | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"split": "train",
"index": 27263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if all three triplets are in the starting lineup? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 27264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The five-digit number $246\\underline{\\hspace{5pt}}8$ is divisible by 9. What is the missing digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 27265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the digits $1, 3, 7, 8, 9$, find the smallest difference that can be achieved in the subtraction problem\n\\[\\begin{tabular}[t]{cccc} & \\boxed{} & \\boxed{} & \\boxed{} \\\\ - & & \\boxed{} & \\boxed{} \\\\ \\hline \\end{tabular}\\] Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 27266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Encrypt integers using the following method: each digit of the number becomes the units digit of its product with 7, then replace each digit $a$ with $10-a$. If a number is encrypted using the above method and the result is 473392, then the original number is ______. Let's thin... | math | {
"style": "rule",
"ground_truth": "891134"
} | {
"split": "train",
"index": 27267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram \\(ABCD\\), the angle at vertex \\(A\\) is \\(60^{\\circ}\\), \\(AB = 73\\) and \\(BC = 88\\). The angle bisector of \\(\\angle ABC\\) intersects segment \\(AD\\) at point \\(E\\) and ray \\(CD\\) at point \\(F\\). Find the length of segment \\(EF\\). Let's thi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2 - \\sin^{2}(x + 2y - 1) = \\frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\\qquad$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/9"
} | {
"split": "train",
"index": 27269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property? Let's think s... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different numbers are randomly selected from the set $\\{ - 2, -1, 0, 3, 4, 5\\}$ and multiplied together. What is the probability that the product is $0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 27271,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Identify the Total Number of Ways to Choose Two Numbers:**\n The set given is $\\{-2, -1, 0, 3, 4, 5\\}$, which contains 6 elements. The number of ways to choose 2 different numbers from this set is given by the combination f... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be real numbers satisfying $3 \\leqslant xy^2 \\leqslant 8$ and $4 \\leqslant \\frac{x^2}{y} \\leqslant 9$. Find the maximum value of $\\frac{x^3}{y^4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 27272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = (m^2+2m)x^{m^2+m-1}$, for what value(s) of $m$ does: \n(Ⅰ) $f(x)$ become a power function; \n(Ⅱ) $f(x)$ become a direct proportionality function; \n(Ⅲ) $f(x)$ become an inverse proportionality function. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 27273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 3 zeros and 2 ones are randomly arranged in a row, calculate the probability that the 2 ones are not adjacent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 27274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle such that $A B=13, B C=14, C A=15$ and let $E, F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $A E F$ be $\\omega$. We draw three lines, tangent to the circumcircle of triangle $A E F$ at $A, E$, and $F... | math | {
"style": "rule",
"ground_truth": "\\frac{462}{5}"
} | {
"split": "train",
"index": 27275,
"solution": "Note that $A E F \\sim A B C$. Let the vertices of the triangle whose area we wish to compute be $P, Q, R$, opposite $A, E, F$ respectively. Since $H, O$ are isogonal conjugates, line $A H$ passes through the circumcenter of $A E F$, so $Q R \\| B C$. Let $M$ be th... |
deepscaler | [
{
"role": "user",
"content": "A certain store sells a product, and because the purchase price decreased by 6.4% compared to the original purchase price, the profit margin increased by 8 percentage points. What was the original profit margin for selling this product? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "17\\%"
} | {
"split": "train",
"index": 27276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All subscripts in this problem are to be considered modulo 6 , that means for example that $\\omega_{7}$ is the same as $\\omega_{1}$. Let $\\omega_{1}, \\ldots \\omega_{6}$ be circles of radius $r$, whose centers lie on a regular hexagon of side length 1 . Let $P_{i}$ be the i... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 27277,
"solution": "Consider two consecutive circles $\\omega_{i}$ and $\\omega_{i+1}$. Let $Q_{i}, Q_{i}^{\\prime}$ be two points on $\\omega_{i}$ and $Q_{i+1}, Q_{i+1}^{\\prime}$ on $\\omega_{i+1}$ such that $Q_{i}, P_{i}$ and $Q_{i+1}$ are colinear and also $Q_{i}^{\\prime}, P_{i}$... |
deepscaler | [
{
"role": "user",
"content": "Find the probability that, when five different numbers are randomly chosen from the set $\\{1, 2, \\ldots, 20\\}$, at least two of them are consecutive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{232}{323}"
} | {
"split": "train",
"index": 27278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x+\\frac1x = -5$, what is $x^5+\\frac1{x^5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2525"
} | {
"split": "train",
"index": 27279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of coefficients $A$, $B$, $C$, and $D$ for the simplified polynomial form of the function \n\\[ y = \\frac{x^3 - 4x^2 - 9x + 36}{x - 3} \\]\nwhich is defined everywhere except at $x = D$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 27280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, one of the sides is equal to 6, the radius of the inscribed circle is 2, and the radius of the circumscribed circle is 5. Find the perimeter. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 27281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\frac{2 \\times 3 + 4}{2 + 3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27282,
"solution": "Evaluating, $\\frac{2 \\times 3+4}{2+3}=\\frac{6+4}{5}=\\frac{10}{5}=2$."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $A$ has $n$ elements, where $n \\geq 2$, and $C$ is a 2-configuration of $A$ that is not $m$-separable for any $m<n$. What is (in terms of $n$) the smallest number of elements that $C$ can have? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\binom{n}{2}"
} | {
"split": "train",
"index": 27283,
"solution": "We claim that every pair of elements of \\( A \\) must belong to \\( C \\), so that the answer is \\( \\binom{n}{2} \\). Indeed, if \\( a, b \\in A \\) and \\( \\{a, b\\} \\) is not in the 2-configuration, then we can assign the other elements of \\( A \\) the numb... |
deepscaler | [
{
"role": "user",
"content": "Harry and Sandy wish to meet at a point in Hogwarts that is at the midpoint of their coordinates. Harry is standing at $(9,-2)$, and Sandy is standing at $(1,6)$. At what coordinate will they meet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(5,2)"
} | {
"split": "train",
"index": 27284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \\ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\\left(a_{n}-1\\right)\\left(a_{n}+1\\right)$ for all positive integers $n$. Find the sum of all good numbers. Let's thin... | math | {
"style": "rule",
"ground_truth": "296"
} | {
"split": "train",
"index": 27285,
"solution": "By the condition of the problem statement, we have $a_{n}^{2}-a_{n-1} a_{n+1}=1=a_{n-1}^{2}-a_{n-2} a_{n}$. This is equivalent to $\\frac{a_{n-2}+a_{n}}{a_{n-1}}=\\frac{a_{n-1}+a_{n+1}}{a_{n}}$. Let $k=\\frac{a_{0}+a_{2}}{a_{1}}$. Then we have $\\frac{a_{n-1}+a_{n+... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$ = (4, -7) and $\\overrightarrow {b}$ = (3, -4), find the projection of $\\overrightarrow {a}$ - $2\\overrightarrow {b}$ in the direction of $\\overrightarrow {b}$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 27286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin (2x+ \\frac {\\pi}{4})$, let $f_1(x)$ denote the function after translating and transforming $f(x)$ to the right by $φ$ units and compressing every point's abscissa to half its original length, then determine the minimum value of $φ$ for which $f... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{8}"
} | {
"split": "train",
"index": 27287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the operation $\\#$ be defined as $\\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$, and $c$. Define a new operation $\\oplus$ by $\\oplus(a, b, c, d) = \\#(a, b + d, c) - \\#(a, b, c)$. What is the value of $\\oplus(2, 4, 1, 3)$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 27288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what values of $z$ is $z^2-40z+340\\le 4$? Express your answer in interval notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[12,28]"
} | {
"split": "train",
"index": 27289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$ as measured from the vertex of its acute angle. What fraction of the rhombus's area is the area of the circle inscribed in it? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi \\sqrt{15}}{16}"
} | {
"split": "train",
"index": 27290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using systematic sampling, \\\\(32\\\\) people are selected from \\\\(960\\\\) for a questionnaire survey. They are randomly numbered from \\\\(1\\\\) to \\\\(960\\\\), and then grouped. The number drawn by simple random sampling in the first group is \\\\(9\\\\). Among the \\\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that ship $A$ is located at $80^{\\circ}$ north by east from lighthouse $C$, and the distance from $A$ to $C$ is $2km$. Ship $B$ is located at $40^{\\circ}$ north by west from lighthouse $C$, and the distance between ships $A$ and $B$ is $3km$. Find the distance fro... | math | {
"style": "rule",
"ground_truth": "\\sqrt {6}-1"
} | {
"split": "train",
"index": 27292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many sets of two or more consecutive positive integers have a sum of $15$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 27293,
"solution": "1. **Identify the nature of the problem**: We need to find sets of two or more consecutive positive integers that sum to 15.\n\n2. **Formulate the sum of an arithmetic progression (AP)**: The sum of the first $n$ terms of an AP where the first term is $a$ and the c... |
deepscaler | [
{
"role": "user",
"content": "In a sequence, every (intermediate) term is half of the arithmetic mean of its neighboring terms. What relationship exists between any term, and the terms that are 2 positions before and 2 positions after it? The first term of the sequence is 1, and the 9th term is 40545. What ... | math | {
"style": "rule",
"ground_truth": "57424611447841"
} | {
"split": "train",
"index": 27294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x=0.4$ using Horner's method, and then determine the value of $v_1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.2"
} | {
"split": "train",
"index": 27295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first two terms of a given sequence are 1 and 1 respectively, and each successive term is the sum of the two preceding terms. What is the value of the first term which exceeds 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1597"
} | {
"split": "train",
"index": 27296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $P(-2,0)$ and the parabola $C$: $y^{2}=4x$, the line passing through $P$ intersects $C$ at points $A$ and $B$, where $|PA|= \\frac {1}{2}|AB|$. Determine the distance from point $A$ to the focus of parabola $C$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 27297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a right triangle with $\\angle ABC = 90^\\circ$, and let $AB = 10\\sqrt{21}$ be the hypotenuse. Point $E$ lies on $AB$ such that $AE = 10\\sqrt{7}$ and $EB = 20\\sqrt{7}$. Let $F$ be the foot of the altitude from $C$ to $AB$. Find the distance $EF$. Expr... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 27298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$, and $c$ are real numbers, and the roots of the equation \\[x^4 - 10x^3 + ax^2 + bx + c = 0\\] are four distinct positive integers. Compute $a + b + c.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 27299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 40 identical looking coins, among which 3 are counterfeit - they weigh the same and are lighter than the genuine coins (the genuine coins also weigh the same). How can you use three weighings on a balance scale without weights to select 16 genuine coins? Let's think s... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 27300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is inside a square $A B C D$ such that $\\angle A P B=135^{\\circ}, P C=12$, and $P D=15$. Compute the area of this square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "123+6\\sqrt{119}"
} | {
"split": "train",
"index": 27301,
"solution": "Let $x=A P$ and $y=B P$. Rotate $\\triangle B A P$ by $90^{\\circ}$ around $B$ to get $\\triangle B C Q$. Then, $\\triangle B P Q$ is rightisosceles, and from $\\angle B Q C=135^{\\circ}$, we get $\\angle P Q C=90^{\\circ}$. Therefore, by Pythagorean's theorem, $P ... |
deepscaler | [
{
"role": "user",
"content": "The average age of five children is 6 years old. Each child is a different age and there is a difference of two years in the ages of any two consecutive children. In years, how old is the oldest child? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 27303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 27304,
"solution": "Let the positive number be $x$. According to the problem, the number $x$ should have been multiplied by $6$, which would give the correct result as $6x$. However, it was mistakenly divided by $6$, giving the erroneous result $\\frac{x}{6}$.\n\nTo find the error com... |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle shares a common side with a square as shown. What is the number of degrees in $m\\angle CDB$? [asy] pair A,E,C,D,B; D = dir(60); C = dir(0); E = (0,-1); B = C+E; draw(B--D--C--B--E--A--C--D--A); label(\"D\",D,N); label(\"C\",C,dir(0)); label(\"B\",B,dir(... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola with its left and right foci being $F_1$ and $F_2$ respectively, and the length of chord $AB$ on the left branch passing through $F_1$ is 5. If $2a=8$, calculate the perimeter of $\\triangle ABF_2$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 27306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(A\\) and \\(B\\) are digits such that:\n\n\\[ \\begin{array}{r}\nAAA \\\\\nAAB \\\\\nABB \\\\\n+\\ BBB \\\\\n\\hline\n1503 \\\\\n\\end{array} \\]\n\nWhat is the value of \\(A^3 + B^2\\)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 27307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 15 are between 25 and 225? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ordered pair of integers $(x,y)$ for which $12x + 21y = 15$ and $21x + 12y = 51$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3,-1)"
} | {
"split": "train",
"index": 27309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fractional part of a positive number, its integer part, and the number itself form an increasing geometric progression. Find all such numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"split": "train",
"index": 27310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vectors $\\overrightarrow{a} \\cdot (\\overrightarrow{a}+2\\overrightarrow{b})=0$ and the magnitudes $|\\overrightarrow{a}|=|\\overrightarrow{b}|=2$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 27311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98752"
} | {
"split": "train",
"index": 27312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon has one side along the diameter of a semicircle, and the two opposite vertices on the semicircle. Find the area of the hexagon if the diameter of the semicircle is 1. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} / 26"
} | {
"split": "train",
"index": 27313,
"solution": "The midpoint of the side of the hexagon on the diameter is the center of the circle. Draw the segment from this center to a vertex of the hexagon on the circle. This segment, whose length is $1 / 2$, is the hypotenuse of a right triangle whose legs have lengths $a ... |
deepscaler | [
{
"role": "user",
"content": "Let set $A=\\{(x,y) \\,|\\, |x|+|y| \\leq 2\\}$, and $B=\\{(x,y) \\in A \\,|\\, y \\leq x^2\\}$. Calculate the probability that a randomly selected element $P(x,y)$ from set $A$ belongs to set $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {17}{24}"
} | {
"split": "train",
"index": 27314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $y=\\frac{x+2}{5x-7}$, at what $x$-value is there a vertical asymptote? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"split": "train",
"index": 27315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate or simplify:\n\n1. $\\frac{\\sqrt{1 - 2\\sin 15^{\\circ}\\cos 15^{\\circ}}}{\\cos 15^{\\circ} - \\sqrt{1 - \\cos^{2} 165^{\\circ}}}$\n2. Given $|\\vec{a}| = 4$, $|\\vec{b}| = 2$, and the angle between $\\vec{a}$ and $\\vec{b}$ is $\\frac{2\\pi}{3}$, find the value of $... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 27316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My school's Physics Club has 22 members. It needs to select 3 officers: chairman, vice-chairman, and sergeant-at-arms. Each person can hold at most one office. Two of the members, Penelope and Quentin, will only be officers if the other one is also an officer. (In other wor... | math | {
"style": "rule",
"ground_truth": "6960"
} | {
"split": "train",
"index": 27317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be an interior point of triangle $ABC$ . Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \\min\\left(\\frac{PA}{p-a},\\frac{PB}{p-b},\\frac{PC}{p-c}\\right) $$ taking into consideratio... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{3}}"
} | {
"split": "train",
"index": 27318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A team of fishermen planned to catch 1800 centners of fish within a certain timeframe. During one-third of this period, there was a storm, causing them to fall short of their daily target by 20 centners each day. However, on the remaining days, the team managed to catch 20 cent... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 27319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 2023 are congruent to 7 modulo 13? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 27320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The science club has 25 members: 10 boys and 15 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{475}{506}"
} | {
"split": "train",
"index": 27321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the coefficient of the $x^3$ term in the expansion of $\\left(x+a\\right)\\left(x-2\\right)^5$ is $-60$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary wants to buy one large pizza, one medium pizza, and three drinks. The drinks cost $p$ dollars each, the medium pizza costs two times as much as one drink, and the large pizza costs three times as much as one drink. If Mary started with $30$ dollars, how much money would sh... | math | {
"style": "rule",
"ground_truth": "30-8p"
} | {
"split": "train",
"index": 27323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The asymptotes of a hyperbola are \\(y = 2x + 3\\) and \\(y = -2x + 1\\). The hyperbola also passes through the point \\((2, 1)\\). Find the distance between the foci of the hyperbola. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{30}"
} | {
"split": "train",
"index": 27324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $i^{11} + i^{16} + i^{21} + i^{26} + i^{31}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"split": "train",
"index": 27325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \\ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \\ldots + a_{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "364"
} | {
"split": "train",
"index": 27326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are 6 balls of each of the four colors: red, blue, yellow, and green, each numbered from 1 to 6, calculate the number of ways to select 3 balls with distinct numbers, such that no two balls have the same color or consecutive numbers. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 27327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 27328,
"solution": "\n#### Step 1: Determine the percentage of birds that are not swans.\nGiven that 25% of the birds are swans, the percentage of birds that are not swans is:\n\\[ 100\\% - 25\\% = 75\\% \\]\n\n#### Step 2: Calculate the percentage of geese among the non-swan birds.\n... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with left and right foci $F_1$ and $F_2$, respectively. Point A is the upper vertex of the ellipse, $|F_{1}A|= \\sqrt {2}$, and the area of △$F_{1}AF_{2}$ is 1.\n(1) Find the standard equation of the ... | math | {
"style": "rule",
"ground_truth": "y=- \\frac {1}{3}"
} | {
"split": "train",
"index": 27329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle centered at $O$ is circumscribed about $\\triangle ABC$ as follows: [asy]\npair pA, pB, pC, pO;\npO = (0, 0);\npA = pO + dir(-20);\npB = pO + dir(90);\npC = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pO--pA);\ndraw(pO--pB);\ndraw(pO--pC);\nlabel(\"$O$\", pO, S);\nlabe... | math | {
"style": "rule",
"ground_truth": "50^\\circ"
} | {
"split": "train",
"index": 27330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\frac{\\tan^2 20^\\circ - \\sin^2 20^\\circ}{\\tan^2 20^\\circ \\sin^2 20^\\circ}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 27331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways ... | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 27332,
"solution": "1. **Understanding the Problem:**\n - We have 6 individuals: Adam, Benin, Chiang, Deshawn, Esther, and Fiona.\n - Each individual has the same number of internet friends within this group.\n - We need to determine the number of ways this can happen, consideri... |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "86.4"
} | {
"split": "train",
"index": 27333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\\circ}$. Find the ratio of the radii of the circles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 27334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define mutually externally tangent circles $\\omega_1$ , $\\omega_2$ , and $\\omega_3$ . Let $\\omega_1$ and $\\omega_2$ be tangent at $P$ . The common external tangents of $\\omega_1$ and $\\omega_2$ meet at $Q$ . Let $O$ be the center of $\\omega_3$ . If $QP ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 27335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f_1(x)=x^2-1$ , and for each positive integer $n \\geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial $f_{2004}$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"split": "train",
"index": 27336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In equilateral triangle $ABC,$ let points $D$ and $E$ trisect $\\overline{BC}$. Find $\\sin \\angle DAE.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{14}"
} | {
"split": "train",
"index": 27337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\Omega$ be a sphere of radius 4 and $\\Gamma$ be a sphere of radius 2 . Suppose that the center of $\\Gamma$ lies on the surface of $\\Omega$. The intersection of the surfaces of $\\Omega$ and $\\Gamma$ is a circle. Compute this circle's circumference. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\pi \\sqrt{15}"
} | {
"split": "train",
"index": 27338,
"solution": "Take a cross-section of a plane through the centers of $\\Omega$ and $\\Gamma$, call them $O_{1}$ and $O_{2}$, respectively. The resulting figure is two circles, one of radius 4 and center $O_{1}$, and the other with radius 2 and center $O_{2}$ on the circle of rad... |
deepscaler | [
{
"role": "user",
"content": "Santa Claus has 36 identical gifts divided into 8 bags. The number of gifts in each of the 8 bags is at least 1 and all are different. You need to select some of these bags to evenly distribute all their gifts to 8 children, such that all gifts are distributed completely (each ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 27339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andrey found the product of all the numbers from 1 to 11 inclusive and wrote the result on the board. During a break, someone accidentally erased three digits, leaving the number $399 * 68 * *$. Help restore the missing digits without recalculating the product. Let's think step... | math | {
"style": "rule",
"ground_truth": "39916800"
} | {
"split": "train",
"index": 27340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix that corresponds to a dilation centered at the origin with scale factor $-3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -3 & 0 \\\\ 0 & -3 \\end{pmatrix}"
} | {
"split": "train",
"index": 27341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A notebook with 75 pages numbered from 1 to 75 is renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the old and new numbering systems. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$, $y$, $z$ be the three sides of a triangle, and let $\\xi$, $\\eta$, $\\zeta$ be the angles opposite them, respectively. If $x^2 + y^2 = 2023z^2$, find the value of\n\\[\\frac{\\cot \\zeta}{\\cot \\xi + \\cot \\eta}.\\] Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"split": "train",
"index": 27343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Segment $s_1$ has endpoints at $(4,1)$ and $(-8,5)$. Segment $s_2$ is obtained by translating $s_1$ by $2$ units to the right and $3$ units up. Find the midpoint of segment $s_2$. Express your answer as $(a,b)$ with $a$ and $b$ integers. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "(0,6)"
} | {
"split": "train",
"index": 27344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a = \\log 9$ and $b = \\log 16,$ compute\n\\[4^{a/b} + 3^{b/a}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 27345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer $n$ such that $6375$ is a factor of $n!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 27346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest product obtainable from two integers whose sum is 2016? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1016064"
} | {
"split": "train",
"index": 27347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem? Let's think st... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 27348,
"solution": ""
} |
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