data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given $a, b \\in \\mathbb{R}$ and $a^{2}+2b^{2}=6$, find the minimum value of $a+ \\sqrt{2}b$."
}
] | math | {
"style": "rule",
"ground_truth": "-2\\sqrt{3}"
} | {
"index": 12700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a, b \\in \\mathbb{R}$, $m = ab + 1$, $n = a + b$.\n- $(1)$ If $a > 1$, $b < 1$, compare the sizes of $m$ and $n$.\n- $(2)$ If $a > 1$, $b > 1$, and $m - n = 49$, find the minimum value of $a + b$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a, b, c > 0$ and $(a+b)bc = 5$, find the minimum value of $2a+b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 12702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a, b, c, d \\in \\mathbf{N}$ such that $342(abcd + ab + ad + cd + 1) = 379(bcd + b + d)$, determine the value of $M$ where $M = a \\cdot 10^{3} + b \\cdot 10^{2} + c \\cdot 10 + d$."
}
] | math | {
"style": "rule",
"ground_truth": "1949"
} | {
"index": 12703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a-b=4$ and $b+c=2$, determine the value of $a^2+b^2+c^2-ab+bc+ca$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 12704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a= \\int_{ 0 }^{ \\pi }(\\sin x-1+2\\cos ^{2} \\frac {x}{2})dx$, find the constant term in the expansion of $(a \\sqrt {x}- \\frac {1}{ \\sqrt {x}})^{6}\\cdot(x^{2}+2)$."
}
] | math | {
"style": "rule",
"ground_truth": "-332"
} | {
"index": 12705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a=(2,4,x)$ and $b=(2,y,2)$, if $|a|=6$ and $a \\perp b$, then the value of $x+y$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a=1$, $b=2$, $C=\\frac{2π}{3}$ in triangle $\\triangle ABC$, calculate the value of $c$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{9}"
} | {
"index": 12707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a=\\frac{\\sqrt{5}+\\sqrt{3}}{\\sqrt{5}-\\sqrt{3}}$ and $b=\\frac{\\sqrt{5}-\\sqrt{3}}{\\sqrt{5}+\\sqrt{3}$, find $\\frac{b}{a}+\\frac{a}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 12708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\in R$, $b \\gt 0$, $a+b=2$, then the minimum value of $\\frac{1}{2|a|}+\\frac{|a|}{b}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 12709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\in R$, $b\\in R$, if the set $A=\\{a, \\frac{b}{a}, 1\\}$, $B=\\{a^{2}, a+b, 0\\}$, $A\\subseteq B$, and $B\\subseteq A$, then the value of $a^{2023}+b^{2023}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\in \\mathbb{R}$, $b\\in \\mathbb{R}$, if the set $\\{a, \\frac{b}{a}, 1\\} = \\{a^{2}, a-b, 0\\}$, calculate the value of $a^{2019}+b^{2019}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\in \\mathbb{R}$, $b\\in \\mathbb{R}$, if the set $\\{a,\\frac{b}{a},1\\}=\\{a^{2},a+b,0\\}$, determine the value of $a^{2023}+b^{2024}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\in \\mathbb{R}$, $i$ is the imaginary unit. If $z=1+ai$ and $z\\cdot \\overline{z}=4$, solve for the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"index": 12713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\in\\{1,3,5\\}$ and $b\\in\\{2,4,8\\}$, find the probability that the function $y=\\log_{\\frac{b}{a}}{\\frac{1}{x}}$ is an increasing function."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a\\ln a=be^{b}$, where $b > 0$, find the maximum value of $\\frac{b}{{{a^2}}}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2e}"
} | {
"index": 12715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a^2 = 16$, $|b| = 3$, $ab < 0$, find the value of $(a - b)^2 + ab^2$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a^{x}=2$ and $a^{y}=3$, find the values of $a^{x+y}$ and $a^{2x-3y}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{27}"
} | {
"index": 12717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a_1 + a_2 = 1$, $a_2 + a_3 = 2$, $a_3 + a_4 = 3$, ..., $a_{99} + a_{100} = 99$, $a_{100} + a_1 = 100$, find the value of $a_1 + a_2 + a_3 + \\ldots + a_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "2525"
} | {
"index": 12718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a_{1}+a_{2}=1$, $a_{2}+a_{3}=2$, $a_{3}+a_{4}=-3$, $a_{4}+a_{5}=-4$, $a_{5}+a_{6}=5$, $a_{6}+a_{7}=6$, $a_{7}+a_{8}=-7$, $a_{8}+a_{9}=-8$, $\\ldots $, $a_{99}+a_{100}=-99$, $a_{100}+a_{1}=-100$, calculate the value of $a_{1}+a_{2}+a_{3}+\\ldots +a_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "-50"
} | {
"index": 12719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $ab+bc+cd+da = 30$ and $b+d = 5$, find $a+c$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 12720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $b-a=-6$ and $ab=7$, find the value of $a^2b-ab^2$."
}
] | math | {
"style": "rule",
"ground_truth": "-42"
} | {
"index": 12721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $cos(\\frac{π}{4}-α)=\\frac{3}{5}$ and $sin(\\frac{5π}{4}+β)=-\\frac{12}{13}$, where $α∈(\\frac{π}{4},\\frac{3π}{4})$ and $β∈(0,\\frac{π}{4})$, find $\\frac{tanα}{tanβ}$."
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"index": 12722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $cos(x+\\frac{π}{6})+sin(\\frac{2π}{3}+x)=\\frac{1}{2}$, calculate $sin(2x-\\frac{π}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"index": 12723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $cos({\\frac{π}{4}+α})=\\frac{{\\sqrt{2}}}{3}$, then $\\frac{{sin2α}}{{1-sinα+cosα}}=$____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $cos({α-\\frac{π}{6}})=\\frac{1}{3}$, find $sin({2α+\\frac{π}{6}})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"index": 12725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $cosθ+cos(θ+\\frac{π}{3})=\\frac{\\sqrt{3}}{3},θ∈(0,\\frac{π}{2})$, find $\\sin \\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + 2\\sqrt{6}}{6}"
} | {
"index": 12726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $e^{i \\theta} = \\frac{3 + i \\sqrt{8}}{5}$, find $\\cos 4 \\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{287}{625}"
} | {
"index": 12727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha) = \\frac {\\sin(\\pi-\\alpha)\\cos(2\\pi-\\alpha)\\tan(-\\alpha+\\pi)}{-\\tan(-\\alpha -\\pi )\\cos( \\frac {\\pi}{2}-\\alpha )}$:\n1. Simplify $f(\\alpha)$.\n2. If $\\alpha$ is an angle in the third quadrant and $\\cos(\\alpha- \\frac {3\\pi}{2}) = \\frac {1}... | math | {
"style": "rule",
"ground_truth": "\\frac {2\\sqrt{6}}{5}"
} | {
"index": 12728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha) = \\frac{\\sin(\\frac{\\pi}{2} + \\alpha) + 3\\sin(-\\pi - \\alpha)}{2\\cos(\\frac{11\\pi}{2} - \\alpha) - \\cos(5\\pi - \\alpha)}$.\n\n(I) Simplify $f(\\alpha)$;\n(II) If $\\tan \\alpha = 3$, find the value of $f(\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha) = \\frac{\\sin(\\pi - \\alpha)\\cos(\\pi + \\alpha)\\sin(-\\alpha + \\frac{3\\pi}{2})}{\\cos(-\\alpha)\\cos(\\alpha + \\frac{\\pi}{2})}$.\n\n$(1)$ Simplify $f(\\alpha)$; \n\n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos(\\alpha - \\frac{3\\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}}{5}"
} | {
"index": 12730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha)= \\dfrac {\\sin (\\alpha- \\dfrac {5\\pi}{2})\\cos ( \\dfrac {3\\pi}{2}+\\alpha)\\tan (\\pi-\\alpha)}{\\tan (-\\alpha-\\pi)\\sin (\\pi-\\alpha)}$.\n(1) Simplify $f(\\alpha)$\n(2) If $\\cos (\\alpha+ \\dfrac {3\\pi}{2})= \\dfrac {1}{5}$ and $\\alpha$ is an angl... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\sqrt{6}}{5}"
} | {
"index": 12731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha)= \\dfrac {\\sin (\\pi-\\alpha)\\cos (2\\pi-\\alpha)\\cos (-\\alpha+ \\dfrac {3}{2}\\pi)}{\\cos ( \\dfrac {\\pi}{2}-\\alpha)\\sin (-\\pi-\\alpha)}$ \n$(1)$ Simplify $f(\\alpha)$; \n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos (\\alpha- \\d... | math | {
"style": "rule",
"ground_truth": "-\\dfrac {1}{2}"
} | {
"index": 12732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha)= \\frac {\\sin (\\pi-\\alpha)\\cos (2\\pi-\\alpha)\\tan (-\\alpha+ \\frac {3\\pi}{2})}{\\cot (-\\alpha-\\pi)\\sin (-\\pi-\\alpha)}$. \n$(1)$ Simplify $f(\\alpha)$; \n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos (\\alpha- \\frac {3\\pi}{2}... | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {6}}{5}"
} | {
"index": 12733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha)= \\frac{\\sin (\\alpha-3\\pi)\\cos (2\\pi-\\alpha)\\cdot\\sin (-\\alpha+ \\frac{3}{2}\\pi)}{\\cos (-\\pi-\\alpha)\\sin (-\\pi-\\alpha)}$. \n$(1)$ Simplify $f(\\alpha)$. \n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos \\left(\\alpha- \\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}}{5}"
} | {
"index": 12734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha)=\\dfrac{\\sin(\\alpha-3\\pi)\\cdot \\cos(2\\pi-\\alpha)\\cdot \\sin(-\\alpha+\\frac{3}{2}\\pi)}{\\cos(-\\pi-\\alpha)\\cdot \\sin(-\\pi-\\alpha)}$,\n(1) Simplify $f(\\alpha)$;\n(2) If $\\sin(\\alpha-\\frac{3}{2}\\pi)=\\frac{1}{5}$, find the value of $f(\\alpha)... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5}"
} | {
"index": 12735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\alpha)=\\frac{2\\sin(2\\pi-\\alpha)\\cos(2\\pi+\\alpha)-\\cos(-\\alpha)}{1+\\sin^{2}\\alpha+\\sin(2\\pi+\\alpha)-\\cos^{2}(4\\pi-\\alpha)}$, find the value of $f\\left(-\\frac{23}{6}\\pi \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"index": 12736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(\\sin \\alpha + \\cos \\alpha) = \\sin \\alpha \\cdot \\cos \\alpha$, determine the domain of $f(x)$ and the value of $f\\left(\\sin \\frac{\\pi}{6}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"index": 12737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(n) = n^2 \\cos(n\\pi)$ and $a_n = f(n) + f(n+1)$, find the sum of $a_1 + a_2 + a_3 + \\cdots + a_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"index": 12738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = (m^2+2m)x^{m^2+m-1}$, for what value(s) of $m$ does: \n(Ⅰ) $f(x)$ become a power function; \n(Ⅱ) $f(x)$ become a direct proportionality function; \n(Ⅲ) $f(x)$ become an inverse proportionality function."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = -4x^2 + 4ax - 4a - a^2$ has a maximum value of $-5$ in the interval $[0, 1]$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 12740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = 2\\cos^{2}x + \\sqrt{3}\\sin2x + a$, where $a$ is a real constant, find the value of $a$, given that the function has a minimum value of $-4$ on the interval $\\left[0, \\frac{\\pi}{2}\\right]$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 12741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = 2\\sin x (\\sin x + \\cos x)$, where $x \\in \\mathbb{R}$.\n\n(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.\n\n(Ⅱ) If $f\\left( \\frac{a}{2} \\right) = 1 + \\frac{3\\sqrt{2}}{5}$, and $\\frac{3\\pi}{4} < a < \\frac{5\\pi}{4}$, find the valu... | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\sqrt{2}}{10}"
} | {
"index": 12742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = 2\\sqrt{3}\\sin x \\cos x + 2\\cos^2x - 1$,\n\n(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;\n\n(2) In $\\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 12743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = 3x^2 + 2x + 1$, if $\\int_{-1}^{1} f(x)\\,dx = 2f(a)$, then $a = \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = 4\\cos x\\sin \\left(x+ \\frac{\\pi}{6}\\right)-1$.\n(Ⅰ) Determine the smallest positive period of $f(x)$;\n(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\\left[- \\frac{\\pi}{6}, \\frac{\\pi}{4}\\right]$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\begin{cases} x+2 & (x\\leq-1) \\\\ x^{2} & (-1<x<2) \\\\ 2x & (x\\geq2) \\end{cases}$, determine the value of $x$ if $f(x)=3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 12746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\begin{cases} x^{2}+1 & (x>0) \\\\ 2f(x+1) & (x\\leq 0) \\end{cases}$, find $f(2)$ and $f(-2)$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\frac {\\log_{2}x-1}{2\\log_{2}x+1}$ (where $x > 2$), and $f(x_1) + f(2x_2) = \\frac {1}{2}$, find the minimum value of $f(x_1x_2)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{3}"
} | {
"index": 12748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\frac{x^2 + 33}{x}, (x \\in \\mathbb{N}^*)$, find the minimum value of $f(x)$ in its domain."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{2}"
} | {
"index": 12749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\log_a x$, its inverse function is $g(x)$. \n(1) Solve the equation for $x$: $f(x-1) = f(a-x) - f(5-x)$; \n(2) Let $F(x) = (2m-1)g(x) + \\left( \\frac{1}{m} - \\frac{1}{2} \\right)g(-x)$. If $F(x)$ has a minimum value, find the expression for $h(m)$; \n(3) Fin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 12750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\sin \\left( \\frac{\\pi}{3}x \\right)$, and the set $A = \\{1, 2, 3, 4, 5, 6, 7, 8\\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\\cdot f(t) = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = \\sin x + a\\cos x$, \n(1) If $a= \\sqrt{3}$, find the maximum value of $f(x)$ and the corresponding value of $x$. \n(2) If $f\\left(\\frac{\\pi}{4}\\right) = 0$ and $f(x) = \\frac{1}{5}$ $(0 < x < \\pi)$, find the value of $\\tan x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 12752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = ax^3 + bx^9 + 2$ has a maximum value of 5 on the interval $(0, +\\infty)$, find the minimum value of $f(x)$ on the interval $(-\\infty, 0)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = e^{-x}$, calculate the limit $$\\lim_{\\Delta x \\to 0} \\frac{f(1 + \\Delta x) - f(1 - 2\\Delta x)}{\\Delta x}$$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{e}"
} | {
"index": 12754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = kx + \\frac {2}{x^{3}} - 3$ $(k \\in \\mathbb{R})$, and it is known that $f(\\ln 6) = 1$. Find $f\\left(\\ln \\frac {1}{6}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 12755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = x^2$ and $g(x) = |x - 1|$, let $f_1(x) = g(f(x))$, $f_{n+1}(x) = g(f_n(x))$, calculate the number of solutions to the equation $f_{2015}(x) = 1$."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 12756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = x^5 + 2x^3 + 3x^2 + x + 1$, calculate the value of $v_3$ when evaluating the value of $f(3)$ using Horner's method."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x) = x^{3} + 3xf''(2)$, then $f(2) = \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-28"
} | {
"index": 12758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)$ be a differentiable function, and $\\lim_{\\Delta x \\to 0} \\frac{{f(1)-f(1-2\\Delta x)}}{{\\Delta x}}=-1$, determine the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 12759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)$ be a differentiable function, satisfying $\\lim_{x \\to 0} \\frac{f(1)-f(1-x)}{2x} = -1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)$ is an even function defined on $\\mathbb{R}$ and satisfies $f(x+2)=-\\frac{1}{f(x)}$. If $f(x)=x$ for $2\\leq x \\leq 3$, find the value of $f\\left(-\\frac{11}{2}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 12761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\begin{cases} 2a-(x+ \\frac {4}{x}),x < a\\\\x- \\frac {4}{x},x\\geqslant a\\\\end{cases}$.\n(1) When $a=1$, if $f(x)=3$, then $x=$ \\_\\_\\_\\_\\_\\_;\n(2) When $a\\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=... | math | {
"style": "rule",
"ground_truth": "- \\frac {11}{6}"
} | {
"index": 12762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\begin{cases} \\sin \\frac{\\pi}{3}x, & x\\leqslant 2011, \\\\ f(x-4), & x > 2011, \\end{cases}$ find $f(2012)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"index": 12763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\begin{cases} x+3, x > 10 \\\\ f(f(x+5)), x\\leqslant 10 \\end{cases}$, evaluate $f(5)$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\frac{1}{2^{x}+ \\sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {2}"
} | {
"index": 12765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\frac{1}{4^{x}+2}$, use the method of deriving the sum formula for an arithmetic sequence to find the value of $f( \\frac{1}{10})+f( \\frac{2}{10})+…+f( \\frac{9}{10})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 12766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\frac{2x}{x+1}$, calculate the value of the expression $f\\left( \\frac{1}{2016}\\right)+f\\left( \\frac{1}{2015}\\right)+f\\left( \\frac{1}{2014}\\right)+\\ldots+f\\left( \\frac{1}{2}\\right)+f(1)+f(2)+\\ldots+f(2014)+f(2015)+f(2016)$."
}
] | math | {
"style": "rule",
"ground_truth": "4031"
} | {
"index": 12767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\frac{\\ln x+2^{x}}{x^{2}}$, find $f'(1)=$ ___."
}
] | math | {
"style": "rule",
"ground_truth": "2\\ln 2 - 3"
} | {
"index": 12768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\frac{a\\cdot 2^{x}+a-2}{2^{x}+1}$ ($x\\in\\mathbb{R}$), if $f(x)$ satisfies $f(-x)+f(x)=0$,\n(1) find the value of the real number $a$ and $f(3)$;\n(2) determine the monotonicity of the function and provide a proof."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 12769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\sqrt {3}\\sin \\dfrac {x}{4}\\cos \\dfrac {x}{4}+ \\cos ^{2} \\dfrac {x}{4}+ \\dfrac {1}{2}$.\n(1) Find the period of $f(x)$;\n(2) In $\\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\\cos B=b\\cos ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{2} + 1"
} | {
"index": 12770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\sqrt {3}\\sin x\\cos (x+ \\dfrac {π}{6})+\\cos x\\sin (x+ \\dfrac {π}{3})+ \\sqrt {3}\\cos ^{2}x- \\dfrac { \\sqrt {3}}{2}$.\n(I) Find the range of $f(x)$ when $x\\in(0, \\dfrac {π}{2})$;\n(II) Given $\\dfrac {π}{12} < α < \\dfrac {π}{3}$, $f(α)= \\dfrac {6}{5}$,... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{33}{65}"
} | {
"index": 12771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)= \\sqrt{2}\\sin \\left( 2x+ \\frac{π}{4} \\right)$.\n\n(1) Find the equation of the axis of symmetry of the graph of the function $f(x)$;\n\n(2) Find the interval(s) where $f(x)$ is monotonically increasing;\n\n(3) Find the maximum and minimum values of the function... | math | {
"style": "rule",
"ground_truth": "- \\sqrt{2}"
} | {
"index": 12772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=1-2x^{2}$ and $g(x)=x^{2}-2x$, let $F(x) = \\begin{cases} f(x), & \\text{if } f(x) \\geq g(x) \\\\ g(x), & \\text{if } f(x) < g(x) \\end{cases}$. Determine the maximum value of $F(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 12773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=2x^3-6x^2+a$ (where $a$ is a constant), the function has a maximum value of 3 on the interval $[-2, 2]$. Find the minimum value of $f(x)$ on the interval $[-2, 2]$."
}
] | math | {
"style": "rule",
"ground_truth": "-29"
} | {
"index": 12774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=3x^{2}+2x+1$, if $\\int_{-1}^{1}f(x)\\,dx=2f(a)$, then $a=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=6-12x+x\\,^{3},x\\in\\left[-\\frac{1}{3},1\\right]$, find the maximum and minimum values of the function."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 12776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=9^{x}-2×3^{x}+4$, where $x\\in\\[-1,2\\]$:\n1. Let $t=3^{x}$, with $x\\in\\[-1,2\\}$, find the maximum and minimum values of $t$.\n2. Find the maximum and minimum values of $f(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 12777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\cos x(\\sqrt{3}\\sin x-\\cos x)+\\frac{3}{2}$. \n$(1)$ Find the interval on which $f(x)$ is monotonically decreasing on $[0,\\pi]$. \n$(2)$ If $f(\\alpha)=\\frac{2}{5}$ and $\\alpha\\in(\\frac{\\pi}{3},\\frac{5\\pi}{6})$, find the value of $\\sin 2\\alpha$."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{-3\\sqrt{3} - 4}{10}"
} | {
"index": 12778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\cos x+\\cos (x+ \\frac {π}{2}).$\n(1) Find $f( \\frac {π}{12})$;\n(2) Suppose $α$ and $β∈(- \\frac {π}{2},0)$, $f(α+ \\frac {3π}{4})=- \\frac {3 \\sqrt {2}}{5}$, $f( \\frac {π}{4}-β)=- \\frac {5 \\sqrt {2}}{13}$, find $\\cos (α+β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {16}{65}"
} | {
"index": 12779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\cos x\\cdot\\ln x$, $f(x_{0})=f(x_{1})=0(x_{0}\\neq x_{1})$, find the minimum value of $|x_{0}-x_{1}|$ ___."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{2}-1"
} | {
"index": 12780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\frac{1}{2}\\cos^{2}x-\\frac{1}{2}\\sin^{2}x+1-\\sqrt{3}\\sin x \\cos x$. \n$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing. \n$(2)$ Find the minimum value of $f(x)$ on $[0,\\frac{\\pi}{2}]$ and the corresponding set of independent... | math | {
"style": "rule",
"ground_truth": "\\left\\{\\frac{\\pi}{3}\\right\\}"
} | {
"index": 12781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\frac{1}{x}$, calculate the limit of $\\frac{f(2+3\\Delta x)-f(2)}{\\Delta x}$ as $\\Delta x$ approaches infinity."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 12782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\ln x$, $g(x)= \\frac {1}{3}x^{3}+ \\frac {1}{2}x^{2}+mx+n$, and line $l$ is tangent to both the graphs of $f(x)$ and $g(x)$ at point $(1,0)$.\n1. Find the equation of line $l$ and the expression for $g(x)$.\n2. If $h(x)=f(x)-g′(x)$ (where $g′(x)$ is the derivative... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}- \\ln 2"
} | {
"index": 12783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\sin (2017x+\\frac{\\pi }{6})+\\cos (2017x-\\frac{\\pi }{3})$, find the minimum value of $A|x_{1}-x_{2}|$ where $x_{1}$ and $x_{2}$ are real numbers such that $f(x_{1})\\leq f(x)\\leq f(x_{2})$ for all real $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{2017}"
} | {
"index": 12784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\sin (\\omega x+\\dfrac{\\pi }{3})$ ($\\omega > 0$), $f(\\dfrac{\\pi }{6})=f(\\dfrac{\\pi }{3})$, and $f(x)$ has a minimum value but no maximum value in the interval $(\\dfrac{\\pi }{6},\\dfrac{\\pi }{3})$, find the value of $\\omega$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{14}{3}"
} | {
"index": 12785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\sin 2x+\\cos 2x$. \n$(1)$ Find the period and the interval of monotonic increase of $f(x)$. \n$(2)$ Find the maximum and minimum values of the function $f(x)$ on $[0,\\frac{π}{2}]$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\sin \\left( 2x+ \\frac{π}{6} \\right)+ \\frac{3}{2}$, $x\\in R$.\n\n(1) Find the minimum positive period of the function $f(x)$;\n\n(2) Find the interval(s) where the function $f(x)$ is monotonically decreasing;\n\n(3) Find the maximum value of the function and th... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 12787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\sin \\omega x \\cos \\omega x - \\sqrt{3} \\cos^2 \\omega x$, where $(\\omega > 0)$, ${x}_{1}$ and ${x}_{2}$ are the two zeros of the function $y=f(x)+\\frac{2+\\sqrt{3}}{2}$, and $|x_{1}-x_{2}|_{\\min }=\\pi$. When $x\\in[0,\\frac{7\\pi}{12}]$, the sum of the min... | math | {
"style": "rule",
"ground_truth": "\\frac{2-3\\sqrt{3}}{2}"
} | {
"index": 12788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=\\sqrt{3}\\cos^2{\\omega}x+\\sin{\\omega}x\\cos{\\omega}x (\\omega>0)$, if there exists a real number $x_{0}$ such that for any real number $x$, $f(x_{0})\\leq f(x)\\leq f(x_{0}+2022\\pi)$ holds, then the minimum value of $\\omega$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4044}"
} | {
"index": 12789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=m\\sin (πx+α)+n\\cos (πx+β)+8$, where $m$, $n$, $α$, $β$ are all real numbers. If $f(2000)=-2000$, find $f(2015)$ = \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 12790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=x^{2005}+ax^{3}- \\frac {b}{x}-8$, and $f(-2)=10$, find $f(2)$."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"index": 12791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=x^{2}-ax$, $g(x)=\\ln x$, $h(x)=f(x)+g(x)$,\n(1) Find the range of values for the real number $a$ such that $f(x) \\geq g(x)$ holds true for any $x$ within their common domain;\n(2) Suppose $h(x)$ has two critical points $x_{1}$, $x_{2}$, with $x_{1} \\in (0, \\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4} - \\ln 2"
} | {
"index": 12792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=x^{3}+3ax^{2}+bx+a^{2}$ has an extremum of $0$ at $x=-1$, find $a-b=\\_\\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 12793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=x^{3}+3x^{2}+6x$, $f(a)=1$, $f(b)=-9$, the value of $a+b$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)=x^{3}+ax^{2}+bx+a^{2}$, the extreme value at $x=1$ is $10$. Find the value of $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 12795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x)={x}^{2023}+a{x}^{3}-\\frac{b}{x}-8$, $f\\left(-2\\right)=10$, find $f\\left(2\\right)=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"index": 12796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(x+1) = x^2 - 1$, \n(1) Find $f(x)$. \n(2) Find the maximum or minimum value of $f(x)$ and the corresponding value of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\\sqrt{n... | math | {
"style": "rule",
"ground_truth": "230"
} | {
"index": 12798,
"solution": "Notice that we must have \\[\\frac{f(f(z))-f(z)}{f(z)-z}=-\\frac{f(f(z))-f(z)}{z-f(z)}\\in i\\mathbb R .\\]However, $f(t)-t=t(t-20)$, so \\begin{align*} \\frac{f(f(z))-f(z)}{f(z)-z}&=\\frac{(z^2-19z)(z^2-19z-20)}{z(z-20)}\\\\ &=\\frac{z(z-19)(z-20)(z+1)}{z(z-20)}\\\\ &=(z-19)(z+1)\\\\... |
DeepScaleR | [
{
"role": "user",
"content": "Given $f(α) = \\frac{\\sin(2π-α)\\cos(π+α)\\cos\\left(\\frac{π}{2}+α\\right)\\cos\\left(\\frac{11π}{2}-α\\right)}{2\\sin(3π + α)\\sin(-π - α)\\sin\\left(\\frac{9π}{2} + α\\right)}$. \n(1) Simplify $f(α)$; \n(2) If $α = -\\frac{25}{4}π$, find the value of $f(α)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{4}"
} | {
"index": 12799,
"solution": ""
} |
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