data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) \\) is a function defined on \\(\\mathbf{R}\\), for any \\( x, y \\in \\mathbf{R} \\), it always holds that\n\\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\\]\n\nFind \\( f(x) \\) and calculate the value of \\( f(\\sqrt{2014}) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1006"
} | {
"index": 13000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x)=\\frac{2x+3}{x-1} \\), the graph of the function \\( y=g(x) \\) is symmetric with the graph of the function \\( y=f^{-1}(x+1) \\) with respect to the line \\( y=x \\). Find \\( g(3) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 13001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x)=a \\sin ((x+1) \\pi)+b \\sqrt[3]{x-1}+2 \\), where \\( a \\) and \\( b \\) are real numbers and \\( f(\\lg 5) = 5 \\), find \\( f(\\lg 20) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f_{1}(x)=-\\frac{2x+7}{x+3}, \\) and \\( f_{n+1}(x)=f_{1}(f_{n}(x)), \\) for \\( x \\neq -2, x \\neq -3 \\), find the value of \\( f_{2022}(2021) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 13003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( m = \\frac{\\sin x}{\\sin (y-z)}, n = \\frac{\\sin y}{\\sin (z-x)}, p = \\frac{\\sin z}{\\sin (x-y)} \\), find the value of \\( m n + n p + p m \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( m = n^{4} + x \\), where \\( n \\) is a natural number and \\( x \\) is a two-digit positive integer, what value of \\( x \\) will make \\( m \\) a composite number?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 13005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( m > n \\geqslant 1 \\), find the smallest value of \\( m + n \\) such that\n\\[ 1000 \\mid 1978^{m} - 1978^{n} . \\"
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 13006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( n = 1990 \\), find the value of\n\n\\[\n\\frac{1}{2^{n}}\\left(1 - 3 \\binom{n}{2} + 3^{2} \\binom{n}{4} - 3^{3} \\binom{n}{6} + \\cdots + 3^{994} \\binom{n}{1988} - 3^{995} \\binom{n}{1990}\\right).\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( n = 7^{3} \\times 11^{2} \\times 13^{4} \\), find the number of integers that are divisors of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 13008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( n \\) numbers \\( a_{1}, a_{2}, \\cdots, a_{n} \\), their root mean square is defined as \\(\\left(\\frac{a_{1}^{2}+a_{2}^{2}+\\cdots+a_{n}^{2}}{n}\\right)^{\\frac{1}{2}} \\). Let \\( M \\) be the set of values of \\( n \\) (for \\( n > 1 \\)) such that the root mean ... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 13009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( n \\in \\mathbf{N}, n > 4 \\), and the set \\( A = \\{1, 2, \\cdots, n\\} \\). Suppose there exists a positive integer \\( m \\) and sets \\( A_1, A_2, \\cdots, A_m \\) with the following properties:\n1. \\( \\bigcup_{i=1}^{m} A_i = A \\);\n2. \\( |A_i| = 4 \\) for \\... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 13010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( s = 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{10^6}} \\), what is the integer part of \\( s \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 13011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( w \\) and \\( z \\) are complex numbers such that \\( |w+z|=2 \\) and \\( |w^2+z^2|=8 \\), find the smallest possible value of \\( |w^3+z^3| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x = -2272 \\), \\( y = 10^3 + 10^2 c + 10 b + a \\), and \\( z = 1 \\), which satisfy the equation \\( a x + b y + c z = 1 \\), where \\( a \\), \\( b \\), \\( c \\) are positive integers and \\( a < b < c \\). Find \\( y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1987"
} | {
"index": 13013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x = 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{10^{6}}} \\), calculate the value of \\([x]\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 13014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x \\) and \\( y \\) are in \\( (0, +\\infty) \\), and \\(\\frac{19}{x} + \\frac{98}{y} = 1\\). What is the minimum value of \\( x + y \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "117 + 14\\sqrt{38}"
} | {
"index": 13015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x \\) satisfies \\(\\log _{5 x} 2 x = \\log _{625 x} 8 x\\), find the value of \\(\\log _{2} x\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln 5}{2 \\ln 2 - 3 \\ln 5}"
} | {
"index": 13016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x \\in (0,1) \\) and \\( \\frac{1}{x} \\notin \\mathbf{Z} \\), define\n\\[ \na_{n}=\\frac{n x}{(1-x)(1-2 x) \\cdots(1-n x)} \\quad \\text{for} \\quad n=1,2, \\cdots \n\\]\n\nA number \\( x \\) is called a \"good number\" if and only if it makes the sequence \\(\\{a_{n... | math | {
"style": "rule",
"ground_truth": "61/210"
} | {
"index": 13017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x \\in [0, 2\\pi] \\), determine the maximum value of the function \n\\[ \nf(x) = \\sqrt{4 \\cos^2 x + 4 \\sqrt{6} \\cos x + 6} + \\sqrt{4 \\cos^2 x - 8 \\sqrt{6} \\cos x + 4 \\sqrt{2} \\sin x + 22}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2(\\sqrt{6} + \\sqrt{2})"
} | {
"index": 13018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x \\in \\mathbb{R} \\), find the maximum value of \\(\\frac{\\sin x(2-\\cos x)}{5-4 \\cos x}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 13019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x \\in \\mathbf{R} \\), the sum of the maximum and minimum values of the function \\( f(x)=\\max \\left\\{\\sin x, \\cos x, \\frac{\\sin x+\\cos x}{\\sqrt{2}}\\right\\} \\) is equal to?"
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}"
} | {
"index": 13020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x, y, z > 0 \\) and \\( x + y + z = 1 \\), find the maximum value of \n$$\nf(x, y, z) = \\sum \\frac{x(2y - z)}{1 + x + 3y}.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "1/7"
} | {
"index": 13021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x, y, z \\in (0, +\\infty) \\) and \\(\\frac{x^2}{1+x^2} + \\frac{y^2}{1+y^2} + \\frac{z^2}{1+z^2} = 2 \\), find the maximum value of \\(\\frac{x}{1+x^2} + \\frac{y}{1+y^2} + \\frac{z}{1+z^2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x, y, z \\in [0, 1] \\), find the maximum value of \\( M = \\sqrt{|x-y|} + \\sqrt{|y-z|} + \\sqrt{|z-x|} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 13023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x, y, z \\in \\mathbb{R}^{+} \\) and \\( x + 2y + 3z = 1 \\), find the minimum value of \\( \\frac{16}{x^{3}}+\\frac{81}{8y^{3}}+\\frac{1}{27z^{3}} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1296"
} | {
"index": 13024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x, y, z \\in \\mathbb{Z}_{+} \\) with \\( x \\leq y \\leq z \\), how many sets of solutions satisfy the equation \\(\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{1}{2}\\) ?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x, y, z \\in \\mathbf{R} \\) such that \\( x^2 + y^2 + xy = 1 \\), \\( y^2 + z^2 + yz = 2 \\), \\( x^2 + z^2 + xz = 3 \\), find \\( x + y + z \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3 + \\sqrt{6}}"
} | {
"index": 13026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x^{2} + y^{2} - 2x - 2y + 1 = 0 \\) where \\( x, y \\in \\mathbb{R} \\), find the minimum value of \\( F(x, y) = \\frac{x + 1}{y} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"index": 13027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_1, x_2, \\cdots, x_{1993} \\) that satisfy\n\\[\n\\begin{array}{l}\n\\left|x_1 - x_2\\right| + \\left|x_2 - x_3\\right| + \\cdots + \\left|x_{1992} - x_{1993}\\right| = 1993, \\\\\ny_k = \\frac{x_1 + x_2 + \\cdots + x_k}{k}, \\quad (k=1, 2, \\cdots, 1993)\n\\end{arr... | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"index": 13028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{0} > 0 \\), \\( x_{0} \\neq \\sqrt{3} \\), a point \\( Q\\left( x_{0}, 0 \\right) \\), and a point \\( P(0, 4) \\), the line \\( PQ \\) intersects the hyperbola \\( x^{2} - \\frac{y^{2}}{3} = 1 \\) at points \\( A \\) and \\( B \\). If \\( \\overrightarrow{PQ} = t ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{1}, x_{2}, \\cdots, x_{1993} \\) satisfy:\n\\[\n\\left|x_{1}-x_{2}\\right|+\\left|x_{2}-x_{3}\\right|+\\cdots+\\left|x_{1992}-x_{1993}\\right|=1993,\n\\]\nand\n\\[\ny_{k}=\\frac{x_{1}+x_{2}+\\cdots+x_{k}}{k} \\quad (k=1,2,\\cdots,1993),\n\\]\nwhat is the maximum po... | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"index": 13030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{1}, x_{2}, x_{3} \\in [0, 12] \\),\n\\[ x_{1} x_{2} x_{3} = \\left(\\left(12 - x_{1}\\right)\\left(12 - x_{2}\\right)\\left(12 - x_{3}\\right)\\right)^{2}. \\]\n\nFind the maximum value of \\( f = x_{1} x_{2} x_{3} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 13031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{i} \\geq 0 \\) for \\( i = 1, 2, \\cdots, n \\) and \\( \\sum_{i=1}^{n} x_{i} = 1 \\) with \\( n \\geq 2 \\), find the maximum value of \\( \\sum_{1 \\leq i \\leq j \\leq n} x_{i} x_{j} (x_{i} + x_{j}) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{i} \\in \\mathbf{R}, x_{i} \\geq 0 \\) for \\( i=1,2,3,4,5 \\), and \\( \\sum_{i=1}^{5} x_{i} = 1 \\), find the minimum value of \\(\\max \\left\\{ x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, x_{4} + x_{5} \\right\\} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{i} \\in \\{-1,1\\}, i=1,2,\\cdots,2021 \\), and \\( x_{1}+x_{2}+\\cdots+x_{k} \\geq 0 \\) for \\( k=1,2,\\cdots,2020 \\), with \\( x_{1}+x_{2}+\\cdots+x_{2021}=-1 \\). How many ordered arrays \\( (x_{1}, x_{2}, \\cdots, x_{2021}) \\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1011} \\binom{2020}{1010}"
} | {
"index": 13034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( x_{i}=\\frac{i}{101} \\), find the value of \\( S=\\sum_{i=1}^{101} \\frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 13035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( z \\in \\mathbf{C} \\) and \\( z^{7} = 1 \\) (where \\( z \\neq 1 \\)), find the value of \\( \\cos \\alpha + \\cos 2 \\alpha + \\cos 4 \\alpha \\), where \\(\\alpha\\) is the argument of \\(z\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( |z|=2 \\) and \\( u=\\left|z^{2}-z+1\\right| \\), find the minimum value of \\( u \\) where \\( z \\in \\mathbf{C} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} \\sqrt{3}"
} | {
"index": 13037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \\cdots + a_{20} x^{20}\\), find \\( a_0 + a_1 + 2a_2 + 3a_3 + \\cdots + 20a_{20} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 13038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\((a + b i)^2 = 3 + 4i\\), where \\(a, b \\in \\mathbf{R}\\), and \\(i\\) is the imaginary unit, find the value of \\(a^2 + b^2\\)."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 13039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\((n+1)^{\\alpha+1}-n^{\\alpha+1} < n^{\\alpha}(\\alpha+1) < n^{\\alpha+1}-(n-1)^{\\alpha+1}, -1 < \\alpha < 0\\). Let \\(x = \\sum_{k=4}^{10^{6}} \\frac{1}{\\sqrt[3]{k}}\\), find the integer part of \\(x\\)."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 13040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(0 < a < \\sqrt{3} \\cos \\theta\\), \\(\\theta \\in \\left[-\\frac{\\pi}{4}, \\frac{\\pi}{3}\\right]\\), find the minimum value of \\(f(a, \\theta) = \\cos^3 \\theta + \\frac{4}{3a \\cos^2 \\theta - a^3}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17 \\sqrt{2}}{4}"
} | {
"index": 13041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(0 \\le x_0 < 1\\), let \n\\[x_n = \\left\\{ \\begin{array}{ll}\n3x_{n-1} & \\text{if } 3x_{n-1} < 1 \\\\\n3x_{n-1} - 1 & \\text{if } 1 \\le 3x_{n-1} < 2 \\\\\n3x_{n-1} - 2 & \\text{if } 3x_{n-1} \\ge 2 \n\\end{array}\\right.\\]\nfor all integers \\(n > 0\\), determine ... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 13042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(0 \\leq x_0 < 1\\), let \n\\[\nx_n = \\left\\{ \\begin{array}{ll}\n2x_{n-1} & \\text{if } 2x_{n-1} < 1 \\\\\n2x_{n-1} - 1 & \\text{if } 2x_{n-1} \\geq 1\n\\end{array} \\right.\n\\]\nfor all integers \\(n > 0\\). Determine the number of initial values of \\(x_0\\) that ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 13043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(0 \\leqslant x \\leqslant 2\\), the function \\(y=4^{x-\\frac{1}{2}}-3 \\cdot 2^{x}+5\\) reaches its minimum value at?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(0<\\theta<\\pi\\), a complex number \\(z_{1}=1-\\cos \\theta+i \\sin \\theta\\) and \\(z_{2}=a^{2}+a i\\), where \\(a \\in \\mathbb{R}\\), it is known that \\(z_{1} z_{2}\\) is a pure imaginary number, and \\(\\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\\). Determine the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 13045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(1 \\leq x^{2}+y^{2} \\leq 4\\), find the sum of the maximum and minimum values of \\(x^{2}-xy+y^{2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 13046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(1990 = 2^{\\alpha_{1}} + 2^{\\alpha_{2}} + \\cdots + 2^{\\alpha_{n}}\\), where \\(\\alpha_{1}, \\alpha_{2}, \\cdots, \\alpha_{n}\\) are distinct non-negative integers. Find \\(\\alpha_{1} + \\alpha_{2} + \\cdots + \\alpha_{n}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 13047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(2x^2 + 3xy + 2y^2 = 1\\), find the minimum value of \\(f(x, y) = x + y + xy\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{8}"
} | {
"index": 13048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(3 \\sin^{2} \\alpha + 2 \\sin^{2} \\beta = 1\\) and \\(3 (\\sin \\alpha + \\cos \\alpha)^{2} - 2 (\\sin \\beta + \\cos \\beta)^{2} = 1\\), find \\(\\cos 2 (\\alpha + \\beta) = \\quad \\) ."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 13049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(5^p + 5^3 = 140\\), \\(3^r + 21 = 48\\), and \\(4^s + 4^3 = 280\\), find the product of \\(p\\), \\(r\\), and \\(s\\)."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 13050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\alpha \\in (0, \\pi)\\), if \\(\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{3}}{3}\\), then \\(\\cos^2 \\alpha - \\sin^2 \\alpha = \\)"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{3}"
} | {
"index": 13051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\geq 0\\) and \\(\\alpha + \\beta \\leq 2\\pi\\), find the minimum value of \\(\\sin \\alpha + 2 \\cos \\beta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{5}"
} | {
"index": 13052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\in [0, \\pi]\\), find the maximum value of \\((\\sin \\alpha + \\sin (\\alpha + \\beta)) \\cdot \\sin \\beta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{3}}{9}"
} | {
"index": 13053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)\\), \\(\\cos \\alpha = \\frac{4}{5}\\), \\(\\tan (\\alpha - \\beta) = -\\frac{1}{3}\\), find \\(\\cos \\beta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\sqrt{10}}{50}"
} | {
"index": 13054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)\\), \\(\\sin \\beta = 2 \\cos (\\alpha + \\beta) \\cdot \\sin \\alpha \\left(\\alpha + \\beta \\neq \\frac{\\pi}{2}\\right)\\), find the maximum value of \\(\\tan \\beta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\in \\left[0, \\frac{\\pi}{4}\\right]\\), find the maximum value of \\(\\sin(\\alpha - \\beta) + 2 \\sin(\\alpha + \\beta)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 13056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\frac{\\sin (\\beta+\\gamma) \\sin (\\gamma+\\alpha)}{\\cos \\alpha \\cos \\gamma}=\\frac{4}{9}\\), find the value of \\(\\frac{\\sin (\\beta+\\gamma) \\sin (\\gamma+\\alpha)}{\\cos (\\alpha+\\beta+\\gamma) \\cos \\gamma}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 13057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\frac{x+ \\sqrt{2}i}{i}=y+i\\), where \\(x\\), \\(y\\in\\mathbb{R}\\), and \\(i\\) is the imaginary unit, find \\(|x-yi|\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\lg 2 = 0.30103\\), calculate the number of digits in \\( M = 1 + 10^4 + \\frac{10^4 (10^4 - 1)}{1 \\cdot 2} + \\frac{10^4 (10^4-1)(10^4-2)}{1 \\cdot 2 \\cdot 3} + \\cdots + \\frac{10^4 (10^4 - 1)}{1 \\cdot 2} + 10^4 + 1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3011"
} | {
"index": 13059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\omega = -\\frac{1}{2} + \\frac{1}{2}i\\sqrt{3}\\), representing a cube root of unity, specifically \\(\\omega = e^{2\\pi i / 3}\\). Let \\(T\\) denote all points in the complex plane of the form \\(a + b\\omega + c\\omega^2\\), where \\(0 \\leq a \\leq 2\\), \\(0 \\l... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 13060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\sin \\alpha + \\sin (\\alpha + \\beta) + \\cos (\\alpha + \\beta) = \\sqrt{3}\\), where \\(\\beta \\in \\left[\\frac{\\pi}{4}, \\pi\\right]\\), find the value of \\(\\beta\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 13061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\sin x + \\sin y = 0.6\\) and \\(\\cos x + \\cos y = 0.8\\), find \\(\\cos x \\cdot \\cos y\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{100}"
} | {
"index": 13062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\sin^{2}(18^{\\circ}) + \\cos^{2}(63^{\\circ}) + \\sqrt{2} \\sin(18^{\\circ}) \\cdot \\cos(63^{\\circ}) =\\)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j)\\), find the value of the expression."
}
] | math | {
"style": "rule",
"ground_truth": "1010000"
} | {
"index": 13064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\tan \\theta = \\frac{5}{12}\\), where \\(180^{\\circ} \\leq \\theta \\leq 270^{\\circ}\\). If \\(A = \\cos \\theta + \\sin \\theta\\), find the value of \\(A\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}{13}"
} | {
"index": 13065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\theta = \\arctan \\frac{5}{12}\\), find the principal value of the argument of the complex number \\(z = \\frac{\\cos 2\\theta + i \\sin 2\\theta}{239 + i}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 13066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\theta_{1}, \\theta_{2}, \\theta_{3}, \\theta_{4} \\in \\mathbf{R}^{+}\\) and \\(\\theta_{1}+\\theta_{2}+\\theta_{3}+\\theta_{4}=\\pi\\), find the minimum value of \\(\\left(2 \\sin ^{2} \\theta_{1}+\\frac{1}{\\sin ^{2} \\theta_{1}}\\right)\\left(2 \\sin ^{2} \\theta_... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 13067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(\\triangle DEF\\), where \\(DE=28\\), \\(EF=30\\), and \\(FD=16\\), calculate the area of \\(\\triangle DEF\\)."
}
] | math | {
"style": "rule",
"ground_truth": "221.25"
} | {
"index": 13068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(a > 0\\), \\(b > 0\\), \\(c > 1\\), and \\(a + b = 1\\). Find the minimum value of \\(\\left(\\frac{2a + b}{ab} - 3\\right)c + \\frac{\\sqrt{2}}{c - 1}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"index": 13069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(a, b, c \\in (0, 1]\\) and \\(\\lambda \\) is a real number such that\n\\[ \\frac{\\sqrt{3}}{\\sqrt{a+b+c}} \\geqslant 1+\\lambda(1-a)(1-b)(1-c), \\]\nfind the maximum value of \\(\\lambda\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{27}"
} | {
"index": 13070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(a, b, c, x, y, z \\in \\mathbf{R}_{+}\\) satisfying the equations:\n\\[\ncy + bz = a, \\quad az + cx = b, \\quad bx + ay = c,\n\\]\nfind the minimum value of the function\n\\[\nf(x, y, z) = \\frac{x^2}{1 + x} + \\frac{y^2}{1 + y} + \\frac{z^2}{1 + z}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(a\\) and \\(b\\) are real numbers, satisfying:\n\\[\n\\sqrt[3]{a} - \\sqrt[3]{b} = 12, \\quad ab = \\left( \\frac{a + b + 8}{6} \\right)^3.\n\\]\n\nFind \\(a - b\\)."
}
] | math | {
"style": "rule",
"ground_truth": "468"
} | {
"index": 13072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(w\\) and \\(z\\) are complex numbers such that \\(|w+z|=2\\) and \\(|w^2+z^2|=8,\\) find the smallest possible value of \\(|w^3+z^3|.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(x \\geqslant 1\\), the minimum value of the function \\(y=f(x)= \\frac {4x^{2}-2x+16}{2x-1}\\) is \\_\\_\\_\\_\\_\\_, and the corresponding value of \\(x\\) is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {5}{2}"
} | {
"index": 13074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\(x\\) and \\(y\\) are real numbers, and:\n\\[\nz_1 = x + (y + 2)i,\n\\]\n\\[\nz_2 = (x - 2) + yi,\n\\]\nwith the condition \\(\\left|z_1\\right| + \\left|z_2\\right| = 4\\),\n\nFind the maximum value of \\(|x + y|\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 13075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\((a+b-c)(a+b+c)=3ab\\\\) and \\\\(c=4\\\\), the maximum area of \\\\(\\Delta ABC\\\\) is \\_\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 13076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(-\\pi < 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 + ... | math | {
"style": "rule",
"ground_truth": "-\\frac{108}{125}"
} | {
"index": 13077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(\\alpha \\in (0, \\frac{\\pi}{2})\\\\) and \\\\(\\beta \\in (\\frac{\\pi}{2}, \\pi)\\\\) with \\\\(\\sin(\\alpha + \\beta) = \\frac{3}{5}\\\\) and \\\\(\\cos \\beta = -\\frac{5}{13}\\\\), find the value of \\\\(\\sin \\alpha\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"index": 13078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(\\alpha \\in (0^{\\circ}, 90^{\\circ})\\\\) and \\\\(\\sin (75^{\\circ} + 2\\alpha) = -\\frac{3}{5}\\\\), calculate \\\\(\\sin (15^{\\circ} + \\alpha) \\cdot \\sin (75^{\\circ} - \\alpha)\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{20}"
} | {
"index": 13079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(\\cos \\left(\\alpha+ \\frac{\\pi}{6}\\right)-\\sin \\alpha= \\frac{2 \\sqrt{3}}{3} \\\\), determine the value of \\\\(\\sin \\left(\\alpha- \\frac{7\\pi}{6}\\right) \\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 13080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(\\sin \\theta + \\cos \\theta = \\frac{3}{4}\\\\), where \\\\(\\theta\\\\) is an angle of a triangle, the value of \\\\(\\sin \\theta - \\cos \\theta\\\\) is \\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{23}}{4}"
} | {
"index": 13081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(\\{x_{1},x_{2},x_{3},x_{4}\\} \\subseteq \\{x | (x-3) \\cdot \\sin \\pi x = 1, x > 0\\}\\\\), find the minimum value of \\\\(x_{1}+x_{2}+x_{3}+x_{4}\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(a < 0\\\\), \\\\((3x^{2}+a)(2x+b) \\geqslant 0\\\\) holds true over the interval \\\\((a,b)\\\\), then the maximum value of \\\\(b-a\\\\) is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"index": 13083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(a > 0\\\\), \\\\(b > 0\\\\), and \\\\(a+4b={{(ab)}^{\\\\frac{3}{2}}}\\\\).\n\n\\\\((\\\\)I\\\\()\\\\) Find the minimum value of \\\\(a^{2}+16b^{2}\\\\);\n\n\\\\((\\\\)II\\\\()\\\\) Determine whether there exist \\\\(a\\\\) and \\\\(b\\\\) such that \\\\(a+3b=6\\\\), ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 13084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(a > 0\\\\), the function \\\\(f(x)= \\frac {1}{3}x^{3}+ \\frac {1-a}{2}x^{2}-ax-a\\\\).\n\\\\((1)\\\\) Discuss the monotonicity of \\\\(f(x)\\\\);\n\\\\((2)\\\\) When \\\\(a=1\\\\), let the function \\\\(g(t)\\\\) represent the difference between the maximum and mini... | math | {
"style": "rule",
"ground_truth": "\\frac {4}{3}"
} | {
"index": 13085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(a > b\\\\), the quadratic trinomial \\\\(a{x}^{2}+2x+b \\geqslant 0 \\\\) holds for all real numbers, and there exists \\\\(x_{0} \\in \\mathbb{R}\\\\), such that \\\\(ax_{0}^{2}+2{x_{0}}+b=0\\\\), then the minimum value of \\\\(\\dfrac{a^{2}+b^{2}}{a-b}\\\\) is \\_\... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}"
} | {
"index": 13086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(a, b, c > 0\\\\), the minimum value of \\\\(\\frac{a^{2} + b^{2} + c^{2}}{ab + 2bc}\\\\) is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{5}}{5}"
} | {
"index": 13087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(f(x)=2x^{5}+3x^{3}-2x^{2}+x-1\\\\), when calculating the value of the function at \\\\(x=2\\\\) using the Horner's method, find \\\\(v_{3}=\\\\) \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(x \\geqslant 0\\\\), \\\\(y \\geqslant 0\\\\), \\\\(x\\\\), \\\\(y \\in \\mathbb{R}\\\\), and \\\\(x+y=2\\\\), find the minimum value of \\\\( \\dfrac {(x+1)^{2}+3}{x+2}+ \\dfrac {y^{2}}{y+1}\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {14}{5}"
} | {
"index": 13089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\\\(|a|=1\\\\), \\\\(|b|= \\sqrt{2}\\\\), and \\\\(a \\perp (a-b)\\\\), the angle between vector \\\\(a\\\\) and vector \\\\(b\\\\) is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 13090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a \"double-single\" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 13091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 13092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $3 \\times 3$ grid (like a Tic-Tac-Toe board), four randomly selected cells have been randomly placed with four tokens.\n\nFind the probability that among these four tokens, three are aligned in a row either vertically, horizontally, or diagonally."
}
] | math | {
"style": "rule",
"ground_truth": "8/21"
} | {
"index": 13093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $4 \\times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a blac... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{65536}"
} | {
"index": 13094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $4 \\times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\\circ}$ clockwise rotation and any white square landing in... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{65536}"
} | {
"index": 13095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $4 \\times 4$ square grid, where each unit square is painted white or black with equal probability and then rotated $180\\,^{\\circ}$ clockwise, calculate the probability that the grid becomes entirely black after this operation."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{65536}"
} | {
"index": 13096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $4\\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15... | math | {
"style": "rule",
"ground_truth": "\\frac{43}{3}"
} | {
"index": 13097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $5 \\times 5$ grid where the number in the $i$-th row and $j$-th column is denoted by \\( a_{ij} \\) (where \\( a_{ij} \\in \\{0, 1\\} \\)), with the condition that \\( a_{ij} = a_{ji} \\) for \\( 1 \\leq i, j \\leq 5 \\). Calculate the total number of ways to fill the ... | math | {
"style": "rule",
"ground_truth": "326"
} | {
"index": 13098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $5\\times 5$ chess board, how many ways can you place five distinct pawns on the board such that each column and each row contains exactly one pawn and no two pawns are positioned as if they were \"attacking\" each other in the manner of queens in chess?"
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"index": 13099,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.