data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given $(x^2+1)(x-2)^8 = a + a_1(x-1) + a_2(x-1)^2 + \\ldots + a_{10}(x-1)^{10}$, find the value of $a_1 + a_2 + \\ldots + a_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $-765^\\circ$, convert this angle into the form $2k\\pi + \\alpha$ ($0 \\leq \\alpha < 2\\pi$), where $k \\in \\mathbb{Z}$."
}
] | math | {
"style": "rule",
"ground_truth": "-6\\pi + \\frac{7\\pi}{4}"
} | {
"index": 12501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $-\\frac{\\pi}{2} < x < 0$, $\\sin x + \\cos x = \\frac{1}{5}$.\n\n(1) Find the value of $\\sin x - \\cos x$.\n\n(2) Find the value of $\\tan x$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 12502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $-π < x < 0$, $\\sin x + \\cos x = \\frac{1}{5}$,\n(1) Find the value of $\\sin x - \\cos x$;\n(2) Find the value of $\\frac{3\\sin^2 \\frac{x}{2} - 2\\sin \\frac{x}{2}\\cos \\frac{x}{2} + \\cos^2 \\frac{x}{2}}{\\tan x + \\frac{1}{\\tan x}}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{132}{125}"
} | {
"index": 12503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 < \\alpha < \\pi$, $\\tan\\alpha = -2$.\n(1) Find the value of $\\sin\\left(\\alpha + \\frac{\\pi}{6}\\right)$;\n(2) Calculate the value of $$\\frac{2\\cos\\left(\\frac{\\pi}{2} + \\alpha\\right) - \\cos(\\pi - \\alpha)}{\\sin\\left(\\frac{\\pi}{2} - \\alpha\\right) - ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{5}"
} | {
"index": 12504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 < \\beta < \\alpha < \\frac{\\pi}{2}$, point $P(1,4 \\sqrt{3})$ is a point on the terminal side of angle $\\alpha$, and $\\sin \\alpha \\sin \\left(\\frac{\\pi}{2}-\\beta \\right)+\\cos \\alpha \\cos \\left(\\frac{\\pi}{2}+\\beta \\right)= \\frac{3 \\sqrt{3}}{14}$, cal... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 12505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 < \\beta < \\frac{\\pi}{2} < \\alpha < \\pi$ and $\\cos \\left(\\alpha- \\frac{\\beta}{2}\\right)=- \\frac{1}{9}, \\sin \\left( \\frac{\\alpha}{2}-\\beta\\right)= \\frac{2}{3}$, calculate the value of $\\cos (\\alpha+\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{239}{729}"
} | {
"index": 12506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 < \\beta < \\frac{\\pi}{2} < \\alpha < \\pi$ and $\\cos\\left(\\alpha - \\frac{\\beta}{2}\\right) = -\\frac{1}{9}$, $\\sin\\left(\\frac{\\alpha}{2} - \\beta\\right) = \\frac{2}{3}$, find the value of $\\cos(\\alpha + \\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{239}{729}"
} | {
"index": 12507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 < a < 1$, and $\\left[a+\\frac{1}{2020}\\right]+\\left[a+\\frac{2}{2020}\\right]+\\cdots+\\left[a+\\frac{2019}{2020}\\right] = 2018$, calculate the value of $[1010 a]$."
}
] | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"index": 12508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 \\le x_0 < 1$, let\n\\[x_n = \\begin{cases} 2x_{n-1} & \\text{ if } 2x_{n-1} < 1 \\\\ 2x_{n-1} - 1 & \\text{ if } 2x_{n-1} \\ge 1 \\end{cases}\\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$?"
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 12509,
"solution": "1. **Understanding the Sequence**: The sequence defined by $x_n$ is a binary sequence where each term is generated by doubling the previous term and subtracting 1 if the result is at least 1. This can be interpreted as a shift and truncate operation in binary representation.\n\n2. **B... |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 \\leq a_k \\leq 1$ for $k=1,2,\\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\\sum_{k=1}^{2020}\\left(a_{k}-a_{k+1} a_{k+2}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 12510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 \\leq x \\leq 2$, find the maximum and minimum values of the function $y = 4^{x- \\frac {1}{2}} - 3 \\times 2^{x} + 5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 12511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0 \\leq x_0 < 1$, for all integers $n > 0$, let\n$$\nx_n = \\begin{cases}\n2x_{n-1}, & \\text{if } 2x_{n-1} < 1,\\\\\n2x_{n-1} - 1, & \\text{if } 2x_{n-1} \\geq 1.\n\\end{cases}\n$$\nFind the number of initial values of $x_0$ such that $x_0 = x_6$."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 12512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $0\\leqslant x\\_0 < 1$, for all integers $n > 0$, let $x\\_n= \\begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\\\ 2x_{n-1}-1, & 2x_{n-1} \\geqslant 1 \\end{cases}$. Find the number of $x\\_0$ that makes $x\\_0=x\\_6$ true."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 12513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $2$ red and $2$ white balls, a total of $4$ balls are randomly arranged in a row. The probability that balls of the same color are adjacent to each other is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $2\\cos(2\\alpha)=\\sin\\left(\\alpha-\\frac{\\pi}{4}\\right)$ and $\\alpha\\in\\left(\\frac{\\pi}{2},\\pi\\right)$, calculate the value of $\\cos 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{8}"
} | {
"index": 12515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $2^x = 8^{y+1}$ and $9^y = 3^{x-9}$, find the value of $x+y$"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 12516,
"solution": "1. **Rewrite the equations with the same base:** \n Given $2^x = 8^{y+1}$, we know $8 = 2^3$, so:\n \\[\n 2^x = (2^3)^{y+1} = 2^{3(y+1)}\n \\]\n Similarly, given $9^y = 3^{x-9}$, and knowing $9 = 3^2$, we have:\n \\[\n 9^y = (3^2)^y = 3^{2y}\n \\]\n Therefore, the e... |
DeepScaleR | [
{
"role": "user",
"content": "Given $2^{3x} = 128$, calculate the value of $2^{-x}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2^{\\frac{7}{3}}}"
} | {
"index": 12517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 12518,
"solution": "\nGiven 30 students such that each student has at most 5 friends and for every 5 students there is a pair of students that are not friends, we need to determine the maximum \\( k \\) such that for all such possible configurations, there exists \\( k \\) students who are all not friend... |
DeepScaleR | [
{
"role": "user",
"content": "Given $31.25 : x : 13.75 = 100 : 9.6 : y$, find the value of $x + y$."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 12519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $3\\sin \\left(-3\\pi +\\theta \\right)+\\cos \\left(\\pi -\\theta \\right)=0$, then the value of $\\frac{sinθcosθ}{cos2θ}$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"index": 12520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $3\\vec{a} + 4\\vec{b} + 5\\vec{c} = 0$ and $|\\vec{a}| = |\\vec{b}| = |\\vec{c}| = 1$, calculate $\\vec{b} \\cdot (\\vec{a} + \\vec{c})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{4}{5}"
} | {
"index": 12521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values?"
}
] | math | {
"style": "rule",
"ground_truth": "231"
} | {
"index": 12522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $6$ cards labeled $1$, $2$, $3$, $4$, $5$, and $6$ are drawn without replacement, calculate the probability that the product of the numbers of the $2$ cards is a multiple of $4$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 12523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $60\\%$ of students like dancing and the rest dislike it, $80\\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students w... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"index": 12524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $995 + 997 + 999 + 1001 + 1003 = 5100 - N$, determine $N$."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 12525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A = 30^\\circ$ and $B = 60^\\circ$, calculate the value of $(1+\\tan A)(1+\\tan B)$."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{4\\sqrt{3}}{3}"
} | {
"index": 12526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A(-1,\\cos \\theta)$, $B(\\sin \\theta,1)$, if $|\\overrightarrow{OA}+\\overrightarrow{OB}|=|\\overrightarrow{OA}-\\overrightarrow{OB}|$, then find the value of the acute angle $\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 12527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A(a, 1-a^{2})$, $B(b, 1-b^{2})$, where $ab \\lt 0$, tangents are drawn at points $A$ and $B$ to the parabola $y=1-x^{2}$. The minimum area enclosed by the two tangents and the $x$-axis is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{3}}{9}"
} | {
"index": 12528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=3x^{2}-x+2y-4xy$ and $B=2x^{2}-3x-y+xy$.\n\n$(1)$ Simplify $2A-3B$.\n\n$(2)$ When $x+y=\\frac{6}{7}$ and $xy=-1$, find the value of $2A-3B$.\n\n$(3)$ If the value of $2A-3B$ is independent of the value of $y$, find the value of $2A-3B$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{11}"
} | {
"index": 12529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=3x^{2}-x+2y-4xy$, $B=x^{2}-2x-y+xy-5$.\n\n$(1)$ Find $A-3B$.\n\n$(2)$ If $(x+y-\\frac{4}{5})^{2}+|xy+1|=0$, find the value of $A-3B$.\n\n$(3)$ If the value of $A-3B$ is independent of $y$, find the value of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 12530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=\\{-3,-2,-1,0,1,2,3\\}$, $a$ and $b$ are elements of $A$. How many cases are there where $|a| \\lt |b|$?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 12531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=\\{4, a^2\\}$, $B=\\{a-6, a+1, 9\\}$, if $A \\cap B = \\{9\\}$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=\\{a, b, c\\}$ and $B=\\{0, 1, 2\\}$, determine the number of mappings $f: A \\to B$ that satisfy the condition $f(a) + f(b) > f(c)$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 12533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=\\{a^{2},a+1,-3\\}$ and $B=\\{a-3,3a-1,a^{2}+1\\}$, if $A∩B=\\{-3\\}$, find the value of the real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {2}{3}"
} | {
"index": 12534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=\\{x|ax^{2}+bx+c\\leqslant 0\\left(a \\lt b\\right)\\}$ has one and only one element, then the minimum value of $M=\\frac{{a+3b+4c}}{{b-a}}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5} + 5"
} | {
"index": 12535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=\\{x|x^{3}+3x^{2}+2x > 0\\}$, $B=\\{x|x^{2}+ax+b\\leqslant 0\\}$ and $A\\cap B=\\{x|0 < x\\leqslant 2\\}$, $A\\cup B=\\{x|x > -2\\}$, then $a+b=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A=a^{2}-2ab+b^{2}$, $B=a^{2}+2ab+b^{2}$, where $a\\neq b$. <br/>$(1)$ Determine the sign of $A+B$ and explain the reason; <br/>$(2)$ If $ab$ are reciprocals of each other, find the value of $A-B$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 12537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $A_n^2 = 132$, calculate the value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $DC = 7$, $CB = 8$, $AB = \\frac{1}{4}AD$, and $ED = \\frac{4}{5}AD$, find $FC$. Express your answer as a decimal. [asy]\ndraw((0,0)--(-20,0)--(-20,16)--cycle);\ndraw((-13,0)--(-13,10.4));\ndraw((-5,0)--(-5,4));\ndraw((-5,0.5)--(-5+0.5,0.5)--(-5+0.5,0));\ndraw((-13,0.5)... | math | {
"style": "rule",
"ground_truth": "10.4"
} | {
"index": 12539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $F$ is a point on diagonal $BC$ of the unit square $ABCD$ such that $\\triangle{ABF}$ is isosceles right triangle with $AB$ as the hypotenuse, consider a strip inside $ABCD$ parallel to $AD$ ranging from $y=\\frac{1}{4}$ to $y=\\frac{3}{4}$ of the unit square, calculate t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $F(x) = \\int_{0}^{x} (t^{2} + 2t - 8) \\, dt$, where $x > 0$.\n1. Determine the intervals of monotonicity for $F(x)$.\n2. Find the maximum and minimum values of the function $F(x)$ on the interval $[1, 3]$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{28}{3}"
} | {
"index": 12541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\\angle F_{1}PF_{2}=60^{\\circ}$, determine the eccentricity of the ellipse in this pair of related curves."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{3}"
} | {
"index": 12542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $F_{1}$ and $F_{2}$ are the foci of the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, a regular triangle $M F_{1} F_{2}$ is constructed with $F_{1} F_{2}$ as one side. If the midpoint of the side $M F_{1}$ lies on the hyperbola, what is the eccentrici... | math | {
"style": "rule",
"ground_truth": "$\\sqrt{3}+1$"
} | {
"index": 12543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $M$ be the second smallest positive integer that is divisible by every positive integer less than 9, find the sum of the digits of $M$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $M=\\{1,2,x\\}$, we call the set $M$, where $1$, $2$, $x$ are elements of set $M$. The elements in the set have definiteness (such as $x$ must exist), distinctiveness (such as $x\\neq 1, x\\neq 2$), and unorderedness (i.e., changing the order of elements does not change t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $O$ as the circumcenter of $\\triangle ABC$ and $D$ as the midpoint of $BC$. If $\\overrightarrow{AO} \\cdot \\overrightarrow{AD}=4$ and $BC=2 \\sqrt{6}$, then find the length of $AD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 12546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $S = \\{1, 2, 3, 4\\}$. Let $a_{1}, a_{2}, \\cdots, a_{k}$ be a sequence composed of numbers from $S$, which includes all permutations of $(1, 2, 3, 4)$ that do not end with 1. That is, if $\\left(b_{1}, b_{2}, b_{3}, b_{4}\\right)$ is a permutation of $(1, 2, 3, 4)$ and ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$."
}
] | math | {
"style": "rule",
"ground_truth": "111"
} | {
"index": 12548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $S$, $P$ (not the origin) are two different points on the parabola $y=x^{2}$, the tangent line at point $P$ intersects the $x$ and $y$ axes at $Q$ and $R$, respectively.\n\n(Ⅰ) If $\\overrightarrow{PQ}=\\lambda \\overrightarrow{PR}$, find the value of $\\lambda$;\n\n(Ⅱ) I... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{9}"
} | {
"index": 12549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $X \\sim N(5, 4)$, find $P(1 < X \\leq 7)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.9759"
} | {
"index": 12550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $X \\sim N(\\mu, \\sigma^2)$, $P(\\mu-\\sigma < X \\leq \\mu+\\sigma) = 0.68$, $P(\\mu-2\\sigma < X \\leq \\mu+2\\sigma) = 0.95$. In a city-wide exam with 20,000 participants, the math scores approximately follow a normal distribution $N(100, 100)$. How many students scor... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 12551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\alpha $, $\\beta \\in (0, \\frac{π}{2})$, $\\sin \\alpha = \\frac{{\\sqrt{5}}}{5}$, $\\cos \\beta = \\frac{1}{{\\sqrt{10}}}$, find the value of $\\alpha - \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{4}"
} | {
"index": 12552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\alpha \\in (0, \\pi)$, if $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{3}}{3}$, calculate the value of $\\cos^2 \\alpha - \\sin^2 \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{3}"
} | {
"index": 12553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\alpha \\in \\left(0, \\frac{\\pi}{2}\\right)$, $\\beta \\in \\left(\\frac{\\pi}{2}, \\pi\\right)$, $\\cos\\beta = -\\frac{1}{3}$, $\\sin(\\alpha + \\beta) = \\frac{7}{9}$. \n(1) Find the value of $\\tan \\frac{\\beta}{2}$. \n(2) Find the value of $\\sin\\alpha$."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\alpha \\in \\left(0,\\pi \\right)$, $sin\\alpha+cos\\alpha=\\frac{\\sqrt{3}}{3}$, find $\\cos 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{3}"
} | {
"index": 12555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\alpha \\in \\left(0,\\pi \\right)$, $tan2\\alpha=\\frac{sin\\alpha}{2+cos\\alpha}$, find the value of $\\ tan \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{15}"
} | {
"index": 12556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\alpha$ and $\\beta \\in \\left(0,\\pi \\right)$, where $\\tan \\alpha$ and $\\tan \\beta$ are two roots of the equation ${x^2}+3\\sqrt{3}x+4=0$, find the value of $\\alpha +\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"index": 12557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\angle1+\\angle2=180^\\circ$ and $\\angle3=\\angle4,$ find $\\angle4.$ Express your answer in degrees. [asy]\n/* AMC8 1997 #12 Problem */\npair A=(0,0), B=(24,0), C=(48,0), D=(18,24), E=(12,48);\npen p=1mm+black;\ndraw(A--C);\ndraw(A--E);\ndraw(B--E);\ndraw(D--C);\nlabe... | math | {
"style": "rule",
"ground_truth": "35^\\circ"
} | {
"index": 12558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = 5,$ find $\\begin{vmatrix} 2a & 2b \\\\ 2c & 2d \\end{vmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\begin{vmatrix} p & q \\\\ r & s \\end{vmatrix} = 6,$ find\n\\[\\begin{vmatrix} p & 9p + 4q \\\\ r & 9r + 4s \\end{vmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\begin{vmatrix} x & y \\\\ z & w \\end{vmatrix} = 3,$ find $\\begin{vmatrix} 3x & 3y \\\\ 3z & 3w \\end{vmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 12561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\binom{18}{11}=31824$, $\\binom{18}{12}=18564$, and $\\binom{20}{13}=77520$, find the value of $\\binom{19}{13}$."
}
] | math | {
"style": "rule",
"ground_truth": "27132"
} | {
"index": 12562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\boldsymbol{a} = (\\cos \\alpha, \\sin \\alpha)$ and $\\boldsymbol{b} = (\\cos \\beta, \\sin \\beta)$, the relationship between $\\boldsymbol{a}$ and $\\boldsymbol{b}$ is given by $|k \\boldsymbol{a} + \\boldsymbol{b}| - \\sqrt{3}|\\boldsymbol{a} - k \\boldsymbol{b}|$, ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos (\\alpha - \\pi) = -\\frac{5}{13}$, and $\\alpha$ is an angle in the fourth quadrant, find the value of $\\sin (-2\\pi + \\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{13}"
} | {
"index": 12564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos (\\frac{\\pi}{4}-\\frac{\\theta}{2})=\\frac{2}{3}$, find the value of $\\sin \\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"index": 12565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos (\\pi+\\alpha)=- \\frac { \\sqrt {10}}{5}$, and $\\alpha\\in(- \\frac {\\pi}{2},0)$, determine the value of $\\tan \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{\\sqrt{6}}{2}"
} | {
"index": 12566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\alpha = \\frac{1}{7}$ and $\\cos (\\alpha-\\beta) = \\frac{13}{14}$, with $0 < \\beta < \\alpha < \\frac{\\pi}{2}$, find $\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 12567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\alpha= \\frac {1}{3}$, $\\cos (\\alpha+\\beta)=- \\frac {1}{3}$, and $\\alpha$, $\\beta\\in(0, \\frac {\\pi}{2})$, find the value of $\\cos (\\alpha-\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{27}"
} | {
"index": 12568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\alpha= \\frac {1}{7}$, $\\cos (\\alpha+\\beta)=- \\frac {11}{14}$, and $\\alpha\\in(0, \\frac {\\pi}{2})$, $\\alpha+\\beta\\in( \\frac {\\pi}{2},\\pi)$, find the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 12569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\left( \\frac {π}{4}-α \\right) = \\frac {3}{5}$, and $\\sin \\left( \\frac {5π}{4}+β \\right) = - \\frac {12}{13}$, with $α \\in \\left( \\frac {π}{4}, \\frac {3π}{4} \\right)$ and $β \\in (0, \\frac {π}{4})$, find the value of $\\sin (α+β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {56}{65}"
} | {
"index": 12570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\left(40^{\\circ}-\\theta \\right)+\\cos \\left(40^{\\circ}+\\theta \\right)+\\cos \\left(80^{\\circ}-\\theta \\right)=0$, calculate the value of $\\tan \\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"index": 12571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\left(\\alpha- \\frac {\\beta}{2}\\right)=- \\frac {1}{9}$ and $\\sin \\left( \\frac {\\alpha}{2}-\\beta\\right)= \\frac {2}{3}$, with $0 < \\beta < \\frac {\\pi}{2} < \\alpha < \\pi$, find $\\sin \\frac {\\alpha+\\beta}{2}=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {22}{27}"
} | {
"index": 12572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\left(\\alpha- \\frac{\\pi}{6}\\right) + \\sin \\alpha = \\frac{4}{5} \\sqrt{3}$, find the value of $\\sin \\left(\\alpha+ \\frac{7\\pi}{6}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{5}"
} | {
"index": 12573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos \\left(a- \\frac{\\pi}{6}\\right) + \\sin a = \\frac{4 \\sqrt{3}}{5}$, find the value of $\\sin \\left(a+ \\frac{7\\pi}{6}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{5}"
} | {
"index": 12574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos x + \\cos y = \\frac{1}{2}$ and $\\sin x + \\sin y = \\frac{1}{3}$, find the value of $\\cos (x - y)$."
}
] | math | {
"style": "rule",
"ground_truth": "$-\\frac{59}{72}$"
} | {
"index": 12575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos(\\alpha + \\frac{\\pi}{4}) = \\frac{\\sqrt{2}}{4}$, find the value of $\\sin(2\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 12576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos(x+y) \\cdot \\sin x - \\sin(x+y) \\cdot \\cos x = \\frac{12}{13}$, and $y$ is an angle in the fourth quadrant, express $\\tan \\frac{y}{2}$ in terms of a rational number."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"index": 12577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos(α-β)= \\frac{3}{5}$, $\\sin β= -\\frac{5}{13}$, and $α∈(0, \\frac{π}{2})$, $β∈(-\\frac{π}{2},0)$, find $\\sin α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"index": 12578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos\\alpha + \\cos\\beta = \\frac{1}{2}$, $\\sin\\alpha + \\sin\\beta = \\frac{\\sqrt{3}}{2}$, then $\\cos(\\alpha - \\beta) =$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 12579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos\\alpha = \\frac{5}{13}$ and $\\cos(\\alpha - \\beta) = \\frac{4}{5}$, with $0 < \\beta < \\alpha < \\frac{\\pi}{2}$,\n$(1)$ Find the value of $\\tan 2\\alpha$;\n$(2)$ Find the value of $\\cos\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 12580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos\\alpha =\\frac{\\sqrt{5}}{5}$ and $\\sin\\beta =\\frac{3\\sqrt{10}}{10}$, with $0 < \\alpha$, $\\beta < \\frac{\\pi}{2}$, determine the value of $\\alpha +\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 12581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos\\alpha= \\frac {4}{5}$, $\\cos\\beta= \\frac {3}{5}$, $\\beta\\in\\left(\\frac {3\\pi}{2}, 2\\pi\\right)$, and $0<\\alpha<\\beta$, calculate the value of $\\sin(\\alpha+\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{25}"
} | {
"index": 12582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos\\left(\\alpha + \\frac{\\pi}{6}\\right) = \\frac{1}{3}$, where $\\alpha$ is in the interval $\\left(0, \\frac{\\pi}{2}\\right)$, find the values of $\\sin\\alpha$ and $\\sin\\left(2\\alpha + \\frac{5\\pi}{6}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"index": 12583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\cos\\left(\\alpha- \\frac{\\beta}{2}\\right) = -\\frac{1}{9}$ and $\\sin\\left(\\frac{\\alpha}{2} - \\beta\\right) = \\frac{2}{3}$, with $\\frac{\\pi}{2} < \\alpha < \\pi$ and $0 < \\beta < \\frac{\\pi}{2}$, find $\\cos(\\alpha + \\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{239}{729}"
} | {
"index": 12584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\delta(x) = 3x + 8$ and $\\phi(x) = 8x + 7$, what is $x$ if $\\delta(\\phi(x)) = 7$?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{11}{12}"
} | {
"index": 12585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\frac{\\sin \\alpha \\cos \\alpha }{1-\\cos 2\\alpha }=\\frac{1}{4}$, $\\tan (\\alpha-\\beta)=2$, calculate $\\tan (\\beta-2\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 12586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\frac{cos2α}{sin(α-\\frac{π}{4})}=-\\frac{\\sqrt{6}}{2}$, express $cos(α-\\frac{π}{4})$ in terms of radicals."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"index": 12587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\frac{e}{f}=\\frac{3}{4}$ and $\\sqrt{e^{2}+f^{2}}=15$, find $ef$."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 12588,
"solution": "We know that $\\frac{e}{f}=\\frac{3}{4}$ and $\\sqrt{e^{2}+f^{2}}=15$. Solving for $e$ and $f$, we find that $e^{2}+f^{2}=225$, so $16 e^{2}+16 f^{2}=3600$, so $(4 e)^{2}+(4 f)^{2}=3600$, so $(3 f)^{2}+(4 f)^{2}=3600$, so $f^{2}\\left(3^{2}+4^{2}\\right)=3600$, so $25 f^{2}=3600$, so ... |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\frac{x}{y}=\\frac{10}{4}.$ If $y = 18$, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 12589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\left(a + \\frac{1}{a}\\right)^2 = 5$, find the value of $a^3 + \\frac{1}{a^3}$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 12590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\left(x+y\\right)^{2}=1$ and $\\left(x-y\\right)^{2}=49$, find the values of $x^{2}+y^{2}$ and $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 12591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\lg 2=0.3010$ and $\\lg 3=0.4771$, at which decimal place does the first non-zero digit of $\\left(\\frac{6}{25}\\right)^{100}$ occur?\n(Shanghai Middle School Mathematics Competition, 1984)"
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 12592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\log_9 \\Big(\\log_4 (\\log_3 x) \\Big) = 1$, calculate the value of $x^{-2/3}$."
}
] | math | {
"style": "rule",
"ground_truth": "3^{-174762.6667}"
} | {
"index": 12593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\mathbf{a} = \\begin{pmatrix} -7 \\\\ 0 \\\\ 1 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 4 \\\\ 2 \\\\ -1 \\end{pmatrix},$ find $\\mathbf{a} - 3 \\mathbf{b}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -19 \\\\ -6 \\\\ 4 \\end{pmatrix}"
} | {
"index": 12594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=( \\sqrt {2},m)(m > 0)$, $\\overrightarrow{b}=(\\sin x,\\cos x)$, and the maximum value of the function $f(x)= \\overrightarrow{a} \\cdot \\overrightarrow{b}$ is $2$.\n1. Find $m$ and the smallest positive period of the function $f(x)$;\n2. In $\\tria... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{4}"
} | {
"index": 12595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(-3,4)$, $\\overrightarrow{b}=(5,2)$, find $|\\overrightarrow{a}|$, $|\\overrightarrow{b}|$, and $\\overrightarrow{a}\\cdot \\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 12596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(1+\\cos \\omega x,-1)$, $\\overrightarrow{b}=( \\sqrt {3},\\sin \\omega x)$ ($\\omega > 0$), and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\\pi$. \n(1) Find the expression... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}+4}{10}"
} | {
"index": 12597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(1,-1)$ and $\\overrightarrow{b}=(1,2)$, calculate the projection of $\\overrightarrow{b}$ onto $\\overrightarrow{a}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"index": 12598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(1,2)$ and $\\overrightarrow{b}=(-3,2)$, for what value of $k$ does\n(1) $k \\overrightarrow{a}+ \\overrightarrow{b}$ and $\\overrightarrow{a}-3 \\overrightarrow{b}$ are perpendicular?\n(2) $k \\overrightarrow{a}+ \\overrightarrow{b}$ and $\\overright... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 12599,
"solution": ""
} |
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