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 $a>0$, the minimum value of the function $f(x) = e^{x-a} - \\ln(x+a) - 1$ $(x>0)$ is 0. Determine the range of values for the real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\{\\frac{1}{2}\\}"
} | {
"index": 14900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a\\in (\\frac{\\pi }{2},\\pi )$ and $\\sin \\alpha =\\frac{1}{3}$,\n\n(1) Find the value of $\\sin 2\\alpha$;\n\n(2) If $\\sin (\\alpha +\\beta )=-\\frac{3}{5}$, $\\beta \\in (0,\\frac{\\pi }{2})$, find the value of $\\sin \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2}+4}{15}"
} | {
"index": 14901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a\\in\\{0,1,2\\}$ and $b\\in\\{-1,1,3,5\\}$, find the probability that the function $f(x)=ax^{2}-2bx$ is increasing on the interval $(1,+\\infty)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 14902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ are five different integers satisfying the condition $a_1 + a_2 + a_3 + a_4 + a_5 = 9$, if $b$ is an integer root of the equation $(x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) = 2009$, then the value of $b$ is."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ are distinct positive integers whose sum equals 159, find the maximum value of the smallest number $a_1$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 14904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a_1, a_2, b_1, b_2, b_3$ are real numbers, and $-1, a_1, a_2, -4$ form an arithmetic sequence, $-4, b_1, b_2, b_3, -1$ form a geometric sequence, calculate the value of $\\left(\\frac{a_2 - a_1}{b_2}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a_{1}, a_{2}, \\cdots, a_{10}$ are ten different positive integers satisfying the equation $\\left|a_{i+1}-a_{i}\\right|=2 \\text { or } 3$, where $i=1,2, \\cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximu... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 14906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $a_{1}, a_{2}, b_{1}, b_{2}, \\cdots, b_{k}$ are vectors in the plane that are pairwise non-parallel, with $\\left|a_{1}-a_{2}\\right|=1$, and $\\left|a_{i}-b_{j}\\right| \\in\\{1,2,3\\} (i=1,2; j=1,2, \\cdots, k)$, determine the maximum value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $ab= \\frac{1}{4}$, $a$, $b \\in (0,1)$, find the minimum value of $\\frac{1}{1-a}+ \\frac{2}{1-b}$."
}
] | math | {
"style": "rule",
"ground_truth": "4+ \\frac{4 \\sqrt{2}}{3}"
} | {
"index": 14908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $abc$ represents a three-digit number, if it satisfies $a \\lt b$ and $b \\gt c$, then we call this three-digit number a \"convex number\". The number of three-digit \"convex\" numbers without repeated digits is ______."
}
] | math | {
"style": "rule",
"ground_truth": "204"
} | {
"index": 14909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less tha... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 14910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $b$ is a multiple of $570$, find the greatest common divisor of $4b^3 + 2b^2 + 5b + 171$ and $b$."
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 14911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 14912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $cos({\\frac{π}{4}-α})=\\frac{3}{5}$, $sin({\\frac{{5π}}{4}+β})=-\\frac{{12}}{{13}}$, $α∈({\\frac{π}{4},\\frac{{3π}}{4}})$, $β∈({0,\\frac{π}{4}})$, calculate the value of $\\sin \\left(\\alpha +\\beta \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 14913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $cos\\left( \\frac{\\pi}{6}-\\alpha\\right) = \\frac{\\sqrt{3}}{2}$, find the value of $cos\\left( \\frac{5\\pi}{6}+\\alpha\\right) - sin^2\\left(-\\alpha+\\frac{7\\pi}{6}\\right)$.\nIf $cos\\alpha = \\frac{2}{3}$ and $\\alpha$ is an angle in the fourth quadrant, fin... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 14914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f'(x_0)=2$, find the value of $\\lim_{k\\rightarrow 0} \\frac{f(x_0-k)-f(x_0)}{2k}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(\\alpha) = \\cos\\alpha \\sqrt{\\frac{\\cot\\alpha - \\cos\\alpha}{\\cot\\alpha + \\cos\\alpha}} + \\sin\\alpha \\sqrt{\\frac{\\tan\\alpha - \\sin\\alpha}{\\tan\\alpha + \\sin\\alpha}}$, and $\\alpha$ is an angle in the second quadrant. \n(1) Simplify $f(\\alpha)... | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{12}"
} | {
"index": 14916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(\\alpha) = \\left(\\sqrt{\\frac{1 - \\sin{\\alpha}}{1 + \\sin{\\alpha}}} + \\sqrt{\\frac{1 + \\sin{\\alpha}}{1 - \\sin{\\alpha}}}\\right)\\cos^3{\\alpha} + 2\\sin{\\left(\\frac{\\pi}{2} + \\alpha\\right)}\\cos{\\left(\\frac{3\\pi}{2} + \\alpha\\right)}$ (where $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(\\alpha)= \\frac{\\sin(\\pi - \\alpha)\\cos(-\\alpha)\\cos(-\\alpha + \\frac{3\\pi}{2})}{\\cos(\\frac{\\pi}{2} - \\alpha)\\sin(-\\pi - \\alpha)}$.\n(1) Find the value of $f(-\\frac{41\\pi}{6})$;\n(2) If $\\alpha$ is an angle in the third quadrant and $\\cos(\\alph... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"index": 14918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(n) = \\left\\{\\begin{matrix}\\log_{4}{n}, &\\text{if }\\log_{4}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise,}\\end{matrix}\\right.$, evaluate the sum $\\sum_{n = 1}^{1023}{f(n)}$."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 14919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x) = \\frac{1}{3^x + \\sqrt{3}}$, find the value of $f(-12) + f(-11) + f(-10) + ... + f(0) + ... + f(11) + f(12) + f(13)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13\\sqrt{3}}{3}"
} | {
"index": 14920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x) = x^k$ where $k < 0$, what is the range of $f(x)$ on the interval $[1, \\infty)$?"
}
] | math | {
"style": "rule",
"ground_truth": "(0,1]"
} | {
"index": 14921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x) = x^{-1} + \\frac{x^{-1}}{1+x^{-1}}$, what is $f(f(-2))$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac83"
} | {
"index": 14922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x) = x^{2}-2x+5$ and $g(x) =x+3$, what is the value of $f(g(5)) -g(f(5))$?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 14923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ and $g(x)$ are functions defined on $\\mathbb{R}$ with $g(x) \\neq 0$, $f'(x)g(x) > f(x)g'(x)$. Let $a$ and $b$ be the points obtained from two consecutive throws of the same die. If $f(x) - a^x g(x) = 0$ and $\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} \\geq \\... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{36}"
} | {
"index": 14924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ and $g(x)$ are functions defined on $\\mathbb{R}$, and $g(x) \\neq 0$, $f''(x)g(x) < f(x)g''(x)$, $f(x)=a^{x}g(x)$, $\\frac{f(1)}{g(1)}+ \\frac{f(-1)}{g(-1)}= \\frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\\left\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ and $g(x)$ are functions defined on $\\mathbb{R}$, with $g(x) \\neq 0$, $f(x)g'(x) > f'(x)g(x)$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \\neq 1$), $\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}$. For the finite sequence $\\frac{f(n)}{g(n)} = (n = 1,... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ has a derivative and satisfies $\\lim_{\\Delta x \\to 0} \\, \\frac{f(1)-f(1-2\\Delta x)}{2\\Delta x}=-1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is a function defined on $[1,+\\infty)$, and\n$$\nf(x)=\\begin{cases}\n1-|2x-3|, & 1\\leqslant x < 2, \\\\\n\\frac{1}{2}f\\left(\\frac{1}{2}x\\right), & x\\geqslant 2,\n\\end{cases}\n$$\nthen the number of zeros of the function $y=2xf(x)-3$ in the interval $(1... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 14928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is a function defined on $\\mathbb{R}$ with a period of $2$, in the interval $[1,3]$, $f(x)= \\begin{cases}x+ \\frac {a}{x}, & 1\\leqslant x < 2 \\\\ bx-3, & 2\\leqslant x\\leqslant 3\\end{cases}$, and $f( \\frac {7}{2})=f(- \\frac {7}{2})$, find the value of ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 14929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is a function defined on $\\mathbb{R}$, and for any $x \\in \\mathbb{R}$, it holds that $f(x+2) = f(2-x) + 4f(2)$, if the graph of the function $y=f(x+1)$ is symmetric about the point $(-1,0)$ and $f(1)=3$, then find $f(2015)$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is a periodic function defined on $\\mathbb{R}$ with a period of $2$, when $x \\in (-1, 1]$, $f(x)=\\begin{cases} -4x^{2}+ \\frac{9}{8},-1 < x\\leqslant 0, \\\\ \\log _{2}x,0 < x\\leqslant 1, \\end{cases}$, find the value of $f(f( \\frac{7}{2}))=\\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an even function defined on $\\mathbb{R}$ and satisfies $f(x+2)=- \\frac{1}{f(x)}$. When $1 \\leq x \\leq 2$, $f(x)=x-2$. Find $f(6.5)$."
}
] | math | {
"style": "rule",
"ground_truth": "-0.5"
} | {
"index": 14932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an even function defined on $\\mathbb{R}$, and $g(x)$ is an odd function defined on $\\mathbb{R}$ that passes through the point $(-1, 3)$ and $g(x) = f(x-1)$, find the value of $f(2007) + f(2008)$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$ and satisfies $f(x+2)=- \\frac{1}{f(x)}$, and $f(x)=x$ when $1 \\leq x \\leq 2$, find $f(- \\frac{11}{2})$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{3}{2}"
} | {
"index": 14934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$ with a minimal positive period of $3$, and $f(x)=\\log_2(1-x)$ when $x \\in \\left(-\\frac{3}{2}, 0\\right)$, find the value of $f(2014) + f(2016)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$ with a minimum positive period of 3, and for $x \\in \\left(-\\frac{3}{2}, 0\\right)$, $f(x)=\\log_{2}(-3x+1)$. Find $f(2011)$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, $f(x) = \\begin{cases} \\log_{2}(x+1) & \\text{for } x \\geq 0 \\\\ g(x) & \\text{for } x < 0 \\\\ \\end{cases}$. Find the value of $g\\left(f(-7)\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, and for $x \\geqslant 0$, $f(x) = \\begin{cases} \\log_{\\frac{1}{2}}(x+1), & 0 \\leqslant x < 1 \\\\ 1-|x-3|, & x \\geqslant 1 \\end{cases}$, determine the sum of all zeros of the function $y = f(x) + \\frac{1}{2}$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"index": 14938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x > 0$, $f(x) = 2 + f\\left(\\frac{1}{2}\\right)\\log_{2}x$, evaluate $f(-2)$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, when $x > 0$, $f(x)=2^{x}+ \\ln \\frac{x}{4}$. Let $a_{n}=f(n-5)$, then the sum of the first $8$ terms of the sequence $\\{a_{n}\\}$ is $\\_\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 14940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$ and satisfies $f(x+4)=f(x)$, when $x \\in (0,2)$, $f(x)=2x^2$. Evaluate $f(2015)$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$, when $x\\geqslant 0$, $f(x)= \\begin{cases} \\log _{\\frac {1}{2}}(x+1),0\\leqslant x < 1 \\\\ 1-|x-3|,x\\geqslant 1\\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\\frac {1}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}-1"
} | {
"index": 14942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x)= \\frac {4}{4^{x}+2}$, $S\\_n$ is the sum of the first $n$ terms of the sequence $\\{a\\_n\\}$, and $\\{a\\_n\\}$ satisfies $a\\_1=0$, and when $n \\geqslant 2$, $a\\_n=f( \\frac {1}{n})+f( \\frac {2}{n})+f( \\frac {3}{n})+…+f( \\frac {n-1}{n})$, find the maxim... | math | {
"style": "rule",
"ground_truth": "\\frac {2}{7}"
} | {
"index": 14943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "-206"
} | {
"index": 14944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $f: x \\rightarrow \\sqrt{x}$ is a function from set $A$ to set $B$.\n1. If $A=[0,9]$, then the range of the function $f(x)$ is ________.\n2. If $B={1,2}$, then $A \\cap B =$ ________."
}
] | math | {
"style": "rule",
"ground_truth": "{1}"
} | {
"index": 14946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $i$ is the imaginary unit, $a\\in\\mathbb{R}$, if $\\frac{1-i}{a+i}$ is a pure imaginary number, calculate the modulus of the complex number $z=(2a+1)+ \\sqrt{2}i$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{11}"
} | {
"index": 14947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 14948,
"solution": "1. **Expand the expression**: We start by expanding $(n+i)^4$ using the binomial theorem:\n \\[\n (n+i)^4 = \\binom{4}{0}n^4i^0 + \\binom{4}{1}n^3i^1 + \\binom{4}{2}n^2i^2 + \\binom{4}{3}n^1i^3 + \\binom{4}{4}n^0i^4.\n \\]\n Simplifying each term, we get:\n \\[\n (n+i)^4 =... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $k$ is a composite number, and $1 < k < 100$, when the sum of the digits of $k$ is a prime number, this prime number is called the \"derived prime\" of $k$. \n(1) If the \"derived prime\" of $k$ is 2, then $k=$ \n(2) Let set $A=\\{P(k) | P(k)$ is the \"derived prim... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 14949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $l$ is the incenter of $\\triangle ABC$, with $AC=2$, $BC=3$, and $AB=4$. If $\\overrightarrow{AI}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, then $x+y=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2}{3}"
} | {
"index": 14950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $m \\angle A= 60^\\circ$, $BC=12$ units, $\\overline{BD} \\perp \\overline{AC}$, $\\overline{CE} \\perp \\overline{AB}$ and $m \\angle DBC = 3m \\angle ECB$, the length of segment $EC$ can be expressed in the form $a(\\sqrt{b}+\\sqrt{c})$ units where $b$ and $c$ have... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 14951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $m$ and $n$ are two non-coincident lines, and $\\alpha$, $\\beta$, $\\gamma$ are three pairwise non-coincident planes, consider the following four propositions: \n$(1)$ If $m \\perp \\alpha$ and $m \\perp \\beta$, then $\\alpha \\parallel \\beta$; \n$(2)$ If $\\alp... | math | {
"style": "rule",
"ground_truth": "(2)(3)(4)"
} | {
"index": 14952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40, m] = 120$ and $\\mathop{\\text{lcm}}[m, 45] = 180$, what is $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 14953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40,m]=120$ and $\\mathop{\\text{lcm}}[m,45]=180$, what is $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 14954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $m$ is an integer and $0 < 3m < 27$, what is the sum of all possible integer values of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $m$ is an integer and $0<3m<27$, what is the sum of all possible integer values of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $min\\{ a,b\\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ satisfy $(\\vert\\overrightarrow{a}\\vert=1,\\vert\\overrightarrow{b}\\vert=2,\\overrightarrow{a}\\cdot\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 14957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n \\in \\mathbb{N}^*$, the coefficient of the second term in the expansion of $(x+2)^n$ is $\\frac{1}{5}$ of the coefficient of the third term.\n(1) Find the value of $n$;\n(2) Find the term with the maximum binomial coefficient in the expansion;\n(3) If $(x+2)^n = ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 14958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n$ is a positive integer, and $4^7 + 4^n + 4^{1998}$ is a perfect square, then one value of $n$ is."
}
] | math | {
"style": "rule",
"ground_truth": "3988"
} | {
"index": 14959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n$ is a positive integer, and given that $\\mathop{\\text{lcm}}[24,n]=72$ and $\\mathop{\\text{lcm}}[n,27]=108$, what is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n$ is a positive integer, find the minimum value of $|n-1| + |n-2| + \\cdots + |n-100|$."
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 14961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 14962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n$ is an integer between $1$ and $60$, inclusive, determine for how many values of $n$ the expression $\\frac{((n+1)^2 - 1)!}{(n!)^{n+1}}$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 14963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n$ represents a positive integer less than $100$, determine the number of integers with an odd number of positive divisors and the number of integers with an even number of divisors."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 14964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $n\\in N^{*}$, select $k(k\\in N, k\\geqslant 2)$ numbers $j\\_1$, $j\\_2$, $...$, $j\\_k$ from the set ${1,2,3,...,n}$ such that they simultaneously satisfy the following two conditions: $①1\\leqslant j\\_1 < j\\_2 < ...j\\_k\\leqslant n$; $②j_{i+1}-j_{i}\\geqslant ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $r$ and $s$ are relatively prime positive integers such that $\\frac{r}{s} = \\frac{2(\\sqrt{2} + \\sqrt{10})}{5(\\sqrt{3 + \\sqrt{5}})}$, find $r$ and $s$."
}
] | math | {
"style": "rule",
"ground_truth": "r = 4, s = 5"
} | {
"index": 14966,
"solution": "Squaring both sides of the given equation yields $\\frac{r^{2}}{s^{2}} = \\frac{4(12 + 4 \\sqrt{5})}{25(3 + \\sqrt{5})} = \\frac{16(3 + \\sqrt{5})}{25(3 + \\sqrt{5})} = \\frac{16}{25}$. Because $r$ and $s$ are positive and relatively prime, then by inspection, $r = 4$ and $s = 5$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $sin(x- \\frac {π}{4})= \\frac {2}{3}$, find the value of $sin2x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 14967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $sinα + cosα = \\frac{\\sqrt{2}}{3}, α ∈ (0, π)$, find the value of $sin(α + \\frac{π}{12})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + \\sqrt{3}}{6}"
} | {
"index": 14968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $w$ and $z$ are complex numbers such that $|w+z|=1$ and $\\left|w^{2}+z^{2}\\right|=14$, find the smallest possible value of $\\left|w^{3}+z^{3}\\right|$. Here, $|\\cdot|$ denotes the absolute value of a complex number, given by $|a+b i|=\\sqrt{a^{2}+b^{2}}$ whenever... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{2}"
} | {
"index": 14969,
"solution": "We can rewrite $\\left|w^{3}+z^{3}\\right|=|w+z|\\left|w^{2}-w z+z^{2}\\right|=\\left|w^{2}-w z+z^{2}\\right|=\\left|\\frac{3}{2}\\left(w^{2}+z^{2}\\right)-\\frac{1}{2}(w+z)^{2}\\right|$$ By the triangle inequality, $\\left|\\frac{3}{2}\\left(w^{2}+z^{2}\\right)-\\frac{1}{2}(w+z)^{2}+... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x - \\frac{1}{x} = 4$, what is $x^3 - \\frac{1}{x^3}$?"
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 14970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x - \\frac{1}{x} = i \\sqrt{2},$ find $x^{2187} - \\frac{1}{x^{2187}}.$"
}
] | math | {
"style": "rule",
"ground_truth": "i \\sqrt{2}"
} | {
"index": 14971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x = -1$ is a solution to the equation $7x^3 - 3x^2 + kx + 5 = 0$, find the value of $k^3 + 2k^2 - 11k - 85$."
}
] | math | {
"style": "rule",
"ground_truth": "-105"
} | {
"index": 14972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x = \\frac{3}{4}$ is a solution to the equation $108x^2 + 61 = 145x - 7,$ what is the other value of $x$ that solves the equation? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{68}{81}"
} | {
"index": 14973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x = \\frac{3}{4}$ is a solution to the equation $108x^2 - 35x - 77 = 0$, what is the other value of $x$ that will solve the equation? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{23}{54}"
} | {
"index": 14974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x = \\frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 14975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x = \\frac{5}{7}$ is a solution to the equation $56 x^2 + 27 = 89x - 8,$ what is the other value of $x$ that will solve the equation? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"index": 14976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x > 0$ and $y > 0$, and $\\frac{4}{x} + \\frac{3}{y} = 1$.\n(I) Find the minimum value of $xy$ and the values of $x$ and $y$ when the minimum value is obtained.\n(II) Find the minimum value of $x + y$ and the values of $x$ and $y$ when the minimum value is obtained.... | math | {
"style": "rule",
"ground_truth": "7 + 4\\sqrt{3}"
} | {
"index": 14977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x > 0$, $y > 0$ and $x + y = 4$, find the minimum value of $$\\frac{x^2}{x + 1} + \\frac{y^2}{y + 2}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{7}"
} | {
"index": 14978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x > 0$, $y > 0$, and $x + 2y = 2$, find the minimum value of $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x > 0$, $y > 0$, and $x+y=1$, find the minimum value of $\\frac{x^{2}}{x+2}+\\frac{y^{2}}{y+1}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 14980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x > 0$, $y > 0$, and ${\\!\\!}^{2x+2y}=2$, find the minimum value of $\\frac{1}{x}+\\frac{1}{y}$."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"index": 14981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x \\in (0, \\frac{1}{2})$, find the minimum value of $\\frac{2}{x} + \\frac{9}{1-2x}$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 14982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x \\in (1,5)$, find the minimum value of the function $y= \\frac{2}{x-1}+ \\frac{1}{5-x}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+2 \\sqrt{2}}{4}"
} | {
"index": 14983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x \\sim N(-1,36)$ and $P(-3 \\leqslant \\xi \\leqslant -1) = 0.4$, calculate $P(\\xi \\geqslant 1)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"index": 14984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are distinct nonzero real numbers such that $x - \\tfrac{2}{x} = y - \\tfrac{2}{y}$, determine the product $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are distinct nonzero real numbers such that $x+\\frac{2}{x} = y + \\frac{2}{y}$, what is $xy$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 14986,
"solution": "1. **Identify the Equation**: Given that $x$ and $y$ are distinct nonzero real numbers such that \n \\[\n x + \\frac{2}{x} = y + \\frac{2}{y},\n \\]\n we denote this common value by $A$. Thus, we have\n \\[\n A = x + \\frac{2}{x} = y + \\frac{2}{y}.\n \\]\n\n2. **Formula... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are nonzero real numbers such that $x+\\frac{1}{y}=10$ and $y+\\frac{1}{x}=\\frac{5}{12},$ find all possible values for $x.$\n\n(Enter your answer as a comma-separated list.)"
}
] | math | {
"style": "rule",
"ground_truth": "4, 6"
} | {
"index": 14987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are positive integers, and $x^2 - y^2 = 53$, find the value of $x^3 - y^3 - 2(x + y) + 10$."
}
] | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 14988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are positive numbers satisfying the equation $xy = \\frac{x-y}{x+3y}$, find the maximum value of $y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are positive numbers, $\\theta \\in \\left( \\frac{\\pi}{4}, \\frac{\\pi}{2} \\right)$, and it satisfies $\\frac{\\sin\\theta}{x} = \\frac{\\cos\\theta}{y}$ and $\\frac{\\cos^2\\theta}{x^2} + \\frac{\\sin^2\\theta}{y^2} = \\frac{10}{3(x^2+y^2)}$, determin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ are real numbers, and they satisfy $xy + x + y = 17$, $x^2y + xy^2 = 66$, find the value of $x^4 + x^3y + x^2y^2 + xy^3 + y^4$."
}
] | math | {
"style": "rule",
"ground_truth": "12499"
} | {
"index": 14991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ satisfy the equation $(x-1)^{2}+(y+2)^{2}=4$, find the maximum and minimum values of $S=3x-y$."
}
] | math | {
"style": "rule",
"ground_truth": "5 - 2\\sqrt{10}"
} | {
"index": 14992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ and $y$ satisfy the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, find the minimum value of $x^2 + y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "14 - 2\\sqrt{13}"
} | {
"index": 14993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 14994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $15336$, what is the greatest common divisor of $f(x)=(3x+4)(7x+1)(13x+6)(2x+9)$ and $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 14995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $18720$, what is the greatest common divisor of $f(x)=(5x+3)(8x+2)(12x+7)(3x+11)$ and $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "462"
} | {
"index": 14996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $23478$, what is the greatest common divisor of $f(x)=(2x+3)(7x+2)(13x+7)(x+13)$ and $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "546"
} | {
"index": 14997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"index": 14998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 14999,
"solution": ""
} |
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