data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given $f\\left( \\alpha \\right)=\\frac{\\cos \\left( \\frac{\\pi }{2}+\\alpha \\right)\\cdot \\cos \\left( 2\\pi -\\alpha \\right)\\cdot \\sin \\left( -\\alpha +\\frac{3}{2}\\pi \\right)}{\\sin \\left( -\\pi -\\alpha \\right)\\sin \\left( \\frac{3}{2}\\pi +\\alpha \\right)}$.\... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{6}}{5}"
} | {
"index": 12800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f\\left(\\alpha \\right)=\\frac{\\mathrm{sin}\\left(\\pi -\\alpha \\right)\\mathrm{cos}\\left(2\\pi -\\alpha \\right)\\mathrm{cos}\\left(-\\alpha +\\frac{3\\pi }{2}\\right)}{\\mathrm{cos}\\left(\\frac{\\pi }{2}-\\alpha \\right)\\mathrm{sin}\\left(-\\pi -\\alpha \\right)}... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}}{5}"
} | {
"index": 12801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f\\left(x\\right)=(e^{x-a}-1)\\ln \\left(x+2a-1\\right)$, if $f\\left(x\\right)\\geqslant 0$ always holds for $x\\in \\left(1-2a,+\\infty \\right)$, then the real number $a=$____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f\\left(x\\right)=\\left(1+2x\\right)^{n}$, where the sum of the binomial coefficients in the expansion is $64$, and ${\\left(1+2x\\right)^n}={a_0}+{a_1}x+{a_2}{x^2}+…+{a_n}{x^n}$. \n$(1)$ Find the value of $a_{2}$; \n$(2)$ Find the term with the largest binomial coeff... | math | {
"style": "rule",
"ground_truth": "2916"
} | {
"index": 12803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $f\\left(x\\right)=e^{x}-ax+\\frac{1}{2}{x}^{2}$, where $a \\gt -1$.<br/>$(Ⅰ)$ When $a=0$, find the equation of the tangent line to the curve $y=f\\left(x\\right)$ at the point $\\left(0,f\\left(0\\right)\\right)$;<br/>$(Ⅱ)$ When $a=1$, find the extreme values of the func... | math | {
"style": "rule",
"ground_truth": "1 + \\frac{1}{e}"
} | {
"index": 12804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $l_{1}$: $ρ \\sin (θ- \\frac{π}{3})= \\sqrt {3}$, $l_{2}$: $ \\begin{cases} x=-t \\\\ y= \\sqrt {3}t \\end{cases}(t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, points $A$, $B$, and $C$ are on the ellipse... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"index": 12805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m \\gt 0$, $n \\gt 0$, and $m+2n=1$, find the minimum value of $\\frac{(m+1)(n+1)}{mn}$."
}
] | math | {
"style": "rule",
"ground_truth": "8+4\\sqrt{3}"
} | {
"index": 12806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m \\gt 1$, $n \\gt 0$, and $m^{2}-3m+n=0$, find the minimum value of $\\frac{4}{m-1}+\\frac{m}{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 12807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m$ and $n∈\\{lg2+lg5,lo{g}_{4}3,{(\\frac{1}{3})}^{-\\frac{3}{5}},tan1\\}$, the probability that the function $f\\left(x\\right)=x^{2}+2mx+n^{2}$ has two distinct zeros is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 12808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m$ points on a plane, where no three points are collinear, and their convex hull is an $n$-gon. Connecting the points appropriately can form a mesh region composed of triangles. Let $f(m, n)$ represent the number of non-overlapping triangles in this region. Find $f(2016,... | math | {
"style": "rule",
"ground_truth": "4000"
} | {
"index": 12809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m+n=2$ and $mn=-2$. Find the value of:\n\n1. $2^{m}\\cdot 2^{n}-(2^{m})^{n}$\n2. $(m-4)(n-4)$\n3. $(m-n)^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m, n \\in \\mathbb{R}$, if the line $(m+1)x + (n+1)y - 2 = 0$ is tangent to the circle $x^2 + y^2 = 1$, find the maximum value of $m - n$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 12811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m= \\int_{0}^{\\frac{\\pi}{2}} 9\\cos xdx$, then the constant term in the expansion of $\\left( \\frac{1}{\\sqrt{x}}-x\\right)^{m}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-84"
} | {
"index": 12812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m=(\\sqrt{3}\\sin \\omega x,\\cos \\omega x)$, $n=(\\cos \\omega x,-\\cos \\omega x)$ ($\\omega > 0$, $x\\in\\mathbb{R}$), $f(x)=m\\cdot n-\\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\\frac{\\pi}{2}$.\n\n$(1)$ Find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"index": 12813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m>0$ and $|x+1|+|2x-1|\\geq m$ always holds, $a, b, c \\in \\mathbb{R}$ satisfy $a^2+2b^2+3c^2=m$. Find the minimum value of $a+2b+3c$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $m\\in R$, determine the condition for the lines $l_{1}$: $mx+2y-1=0$ and $l_{2}$: $3x+\\left(m+1\\right)y+1=0$ to be perpendicular."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"index": 12815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $n \\in \\mathbb{N}^*$, in the expansion of $(x+2)^n$, the coefficient of the second term 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": 12816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sin( \\frac {\\pi}{6}-\\alpha)-cos\\alpha= \\frac {1}{3}$, find $cos(2\\alpha+ \\frac {\\pi}{3})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {7}{9}"
} | {
"index": 12817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sin(\\alpha-\\beta)=\\frac{1}{3}$ and $cos\\alpha sin\\beta=\\frac{1}{6}$, find $\\cos \\left(2\\alpha +2\\beta \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 12818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sin(\\alpha-\\frac{\\pi}{6})=\\frac{3}{5}$, calculate $cos(\\frac{2\\pi}{3}-\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 12819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sin(x+\\frac{π}{12})=-\\frac{1}{4}$, find the value of $cos(\\frac{5π}{6}-2x)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{8}"
} | {
"index": 12820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sin({α+\\frac{π}{4}})=\\frac{{12}}{{13}}$, and $\\frac{π}{4}＜α＜\\frac{{3π}}{4}$, find the value of $\\cos \\alpha$____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{2}}{26}"
} | {
"index": 12821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sin2θ=-\\frac{1}{3}$, if $\\frac{π}{4}＜θ＜\\frac{3π}{4}$, then $\\tan \\theta =$____."
}
] | math | {
"style": "rule",
"ground_truth": "-3-2\\sqrt{2}"
} | {
"index": 12822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sinα+cosα=-\\frac{{\\sqrt{10}}}{5}, α∈(-\\frac{π}{2},\\frac{π}{2})$. \n$(1)$ Find the value of $\\tan \\alpha$; \n$(2)$ Find the value of $2\\sin ^{2}\\alpha +\\sin \\alpha \\cos \\alpha -1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sinα-\\sqrt{3}cosα=1$, then the value of $sin({\\frac{{7π}}{6}-2α})$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sinα-cosα=\\frac{1}{5},0≤α≤π$, calculate $sin(2α-\\frac{π}{4})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31\\sqrt{2}}{50}"
} | {
"index": 12825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $sn(α+ \\frac {π}{6})= \\frac {1}{3}$, and $\\frac {π}{3} < α < \\pi$, find $\\sin ( \\frac {π}{12}-α)$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {4+ \\sqrt {2}}{6}"
} | {
"index": 12826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $tan({θ+\\frac{π}{4}})=2tanθ-7$, determine the value of $\\sin 2\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 12827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $tan({θ+\\frac{π}{{12}}})=2$, find $sin({\\frac{π}{3}-2θ})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"index": 12828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $tanA=\\frac{2}{3}$, find the value of $\\cos A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{13}}{13}"
} | {
"index": 12829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $u$ and $v$ are complex numbers such that $|u+v|=2$ and $|u^2+v^2|=8,$ find the smallest possible value of $|u^3+v^3|$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14,$ find the smallest possible value of $|w^3+z^3|.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{41}2"
} | {
"index": 12831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^3+z^3|$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 12832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^4+z^4|$."
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"index": 12833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 12834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=28,$ find the smallest possible value of $|w^3+z^3|.$"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 12835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x < 0$, the maximum value of $3x +\\dfrac{4}{x}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-4\\sqrt{3}"
} | {
"index": 12836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x = \\frac{2}{3}$ and $y = \\frac{5}{2}$, find the value of $\\frac{1}{3}x^8y^9$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5^9}{2 \\cdot 3^9}"
} | {
"index": 12837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x > 0$, $y > 0$, and $2x+8y-xy=0$, find the minimum value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 12838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x > 0$, $y > 0$, and the inequality $2\\log_{\\frac{1}{2}}[(a-1)x+ay] \\leq 1 + \\log_{\\frac{1}{2}}(xy)$ always holds, find the minimum value of $4a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}+\\sqrt{2}"
} | {
"index": 12839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x > 0, y > 0$, and $2x + 8y - xy = 0$, find the minimum value of $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 12840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x \\gt -1$, $y \\gt 0$, and $x+2y=1$, find the minimum value of $\\frac{1}{x+1}+\\frac{1}{y}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+2\\sqrt{2}}{2}"
} | {
"index": 12841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x \\gt 0$, $y \\gt 0$, $x+2y=1$, calculate the minimum value of $\\frac{{(x+1)(y+1)}}{{xy}}$."
}
] | math | {
"style": "rule",
"ground_truth": "8+4\\sqrt{3}"
} | {
"index": 12842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x \\gt 0$, $y \\gt 0$, and $x+y=1$, find the minimum value of $\\frac{2{x}^{2}-x+1}{xy}$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}+1"
} | {
"index": 12843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x \\gt 0$, $y \\gt 0$, when $x=$______, the maximum value of $\\sqrt{xy}(1-x-2y)$ is _______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{16}"
} | {
"index": 12844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x$ and $y \\in \\mathbb{R}$, if $x^2 + y^2 + xy = 315$, find the minimum value of $x^2 + y^2 - xy$."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 12845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x$, $y \\in \\mathbb{R}^{+}$ and $2x+3y=1$, find the minimum value of $\\frac{1}{x}+ \\frac{1}{y}$."
}
] | math | {
"style": "rule",
"ground_truth": "5+2 \\sqrt{6}"
} | {
"index": 12846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x$, $y$, and $a \\in R^*$, and when $x + 2y = 1$, the minimum value of $\\frac{3}{x} + \\frac{a}{y}$ is $6\\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\\frac{1}{x} + \\frac{2}{y} = 1$."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"index": 12847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x+m≤{e}^{\\frac{2x}{m}+n}$ holds for any $x\\in \\left(-m,+\\infty \\right)$, then the minimum value of $m\\cdot n$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{e^2}"
} | {
"index": 12848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x+x^{-1}=3$, calculate the value of $x^{ \\frac {3}{2}}+x^{- \\frac {3}{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 12849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x, y \\in (-1, 1)$, find the minimum value of the expression $$\\sqrt {(x+1)^{2}+(y-1)^{2}}+\\sqrt {(x+1)^{2}+(y+1)^{2}}+\\sqrt {(x-1)^{2}+(y+1)^{2}}+\\sqrt {(x-1)^{2}+(y-1)^{2}}.$$"
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 12850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x, y \\in \\mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:\n$$\n\\frac{9}{17}<\\frac{x}{x+y}<\\frac{8}{15}.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 12851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x, y, z \\in \\mathbb{R}$ and $x^{2}+y^{2}+z^{2}=25$, find the maximum and minimum values of $x-2y+2z$."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"index": 12852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x,y \\in (0, +\\infty)$, and satisfying $\\frac{1}{x} + \\frac{1}{2y} = 1$, determine the minimum value of $x+4y$."
}
] | math | {
"style": "rule",
"ground_truth": "3+2\\sqrt{2}"
} | {
"index": 12853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x-y=1$ and $x^3-y^3=2$, find the values of $x^4+y^4$ and $x^5-y^5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{29}{9}"
} | {
"index": 12854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x= \\frac {\\pi}{12}$ is a symmetry axis of the function $f(x)= \\sqrt {3}\\sin(2x+\\varphi)+\\cos(2x+\\varphi)$ $(0<\\varphi<\\pi)$, after shifting the graph of function $f(x)$ to the right by $\\frac {3\\pi}{4}$ units, find the minimum value of the resulting function $... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x=1-2a$, $y=3a-4$. \n$(1)$ Given that the arithmetic square root of $x$ is $3$, find the value of $a$; \n$(2)$ If $x$ and $y$ are both square roots of the same number, find this number."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 12856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\\%$."
}
] | math | {
"style": "rule",
"ground_truth": "42.13\\%"
} | {
"index": 12857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x^{2}+y^{2}=4$, find the minimum value of $\\sqrt{2-y}+\\sqrt{5-2x}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 12858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x^{2}-5x-2006=0$, evaluate the algebraic expression $\\dfrac {(x-2)^{3}-(x-1)^{2}+1}{x-2}$."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 12859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x^{3}=4$, solve for $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{4}"
} | {
"index": 12860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x_1$ and $x_2$ are the two real roots of the quadratic equation in $x$: $x^2 - 2(m+2)x + m^2 = 0$.\n\n(1) When $m=0$, find the roots of the equation; \n(2) If $(x_1 - 2)(x_2 - 2) = 41$, find the value of $m$; \n(3) Given an isosceles triangle $ABC$ with one side length... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 12861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $x_{1}=4$, $x_{2}=5$, $x_{3}=6$, calculate the standard deviation of this sample."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 12862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, find the value of $g(-1)$, where $g(x) = f(x) + 2$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $y=f(x)$ is a quadratic function, and $f(0)=-5$, $f(-1)=-4$, $f(2)=-5$,\n\n(1) Find the analytical expression of this quadratic function. \n(2) Find the maximum and minimum values of the function $f(x)$ when $x \\in [0,5]$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {16}{3}"
} | {
"index": 12864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, if $g(x)=f(x)+2$, then $g(-1)=$ ."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, then $f(-1)=$?"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $y=y_{1}+y_{2}$, where $y_{1}$ is directly proportional to $(x+1)$, and $y_{2}$ is inversely proportional to $(x+1)$. When $x=0$, $y=-5$; when $x=2$, $y=-7$. Find the value of $x$ when $y=5$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{5}{2}"
} | {
"index": 12867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $y_1 = x^2 - 7x + 6$, $y_2 = 7x - 3$, and $y = y_1 + xy_2$, find the value of $y$ when $x = 2$."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 12868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $| \\mathbf{e} |=1$, and it satisfies $|\\mathbf{a} + \\mathbf{e}|=|\\mathbf{a} - 2\\mathbf{e}|$, then the projection of vector $\\mathbf{a}$ in the direction of $\\mathbf{e}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\overrightarrow {a}|=4$, $|\\overrightarrow {b}|=2$, and the angle between $\\overrightarrow {a}$ and $\\overrightarrow {b}$ is $120^{\\circ}$, find:\n\n1. $\\left(\\overrightarrow {a}-2\\overrightarrow {b}\\right)\\cdot \\left(\\overrightarrow {a}+\\overrightarrow {b}... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{6}"
} | {
"index": 12870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\overrightarrow {a}|=\\sqrt {2}$, $|\\overrightarrow {b}|=2$, and $(\\overrightarrow {a}-\\overrightarrow {b})\\bot \\overrightarrow {a}$, determine the angle between $\\overrightarrow {a}$ and $\\overrightarrow {b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 12871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, $\\overrightarrow{a}\\cdot (\\overrightarrow{b}-\\overrightarrow{a})=0$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 12872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}⊥(\\overrightarrow{a}+\\overrightarrow{b})$, determine the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 12873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}⊥(\\overrightarrow{a}+\\overrightarrow{b})$, find the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 12874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\vec{a}| = |\\vec{b}| = 2$, and $(\\vec{a} + 2\\vec{b}) \\cdot (\\vec{a} - \\vec{b}) = -2$, find the angle between $\\vec{a}$ and $\\vec{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 12875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|= \\sqrt{2}$, and $\\vec{a} \\perp (\\vec{a} - \\vec{b})$, find the angle between the vectors $\\vec{a}$ and $\\vec{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 12876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|=2$, and the angle between $\\vec{a}$ and $\\vec{b}$ is $60^{\\circ}$.\n$(1)$ Find $\\vec{a}\\cdot \\vec{b}$, $(\\vec{a}- \\vec{b})\\cdot(\\vec{a}+ \\vec{b})$;\n$(2)$ Find $|\\vec{a}- \\vec{b}|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 12877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|=6$, and $\\vec{a}\\cdot(\\vec{b}-\\vec{a})=2$, calculate the angle between $\\vec{a}$ and $\\vec{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{3}"
} | {
"index": 12878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|a|=1$, $|b|=2$, and $a+b=(1, \\sqrt{2})$, the angle between vectors $a$ and $b$ is _______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 12879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|a|=3$, $|b-2|=9$, and $a+b > 0$, find the value of $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "-33"
} | {
"index": 12880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 12881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|m|=4$, $|n|=3$.\n\n(1) When $m$ and $n$ have the same sign, find the value of $m-n$.\n\n(2) When $m$ and $n$ have opposite signs, find the value of $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $|x|=4$, $|y|=2$, and $x<y$, then the value of $x\\div y$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $α \\in \\left( \\frac{π}{2}, π \\right)$, and $\\sin α = \\frac{1}{3}$.\n\n$(1)$ Find the value of $\\sin 2α$; \n$(2)$ If $\\sin (α+β) = -\\frac{3}{5}$, and $β \\in (0, \\frac{π}{2})$, find the value of $\\sin β$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2}+4}{15}"
} | {
"index": 12885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $α \\in \\left(0, \\frac{\\pi}{2}\\right)$, $\\cos \\left(α+ \\frac{\\pi}{3}\\right) = -\\frac{2}{3}$, then $\\cos α =$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}-2}{6}"
} | {
"index": 12886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $α-β=\\frac{π}{3}$ and $tanα-tanβ=3\\sqrt{3}$, calculate the value of $\\cos \\left(\\alpha +\\beta \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{6}"
} | {
"index": 12887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $α∈(\\frac{\\pi}{2},π)$, $\\sin α =\\frac{\\sqrt{5}}{5}$.\n\n(Ⅰ) Find the value of $\\tan\\left( \\frac{\\pi}{4}+2α \\right)$;\n\n(Ⅱ) Find the value of $\\cos\\left( \\frac{5\\pi}{6}-2α \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\sqrt{3}+4}{10}"
} | {
"index": 12888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $α∈(\\frac{π}{2},π)$, and $sin(α+\\frac{π}{3})=\\frac{12}{13}$, determine the value of $sin(\\frac{π}{6}-α)+sin(\\frac{2π}{3}-α)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{13}"
} | {
"index": 12889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $α∈({\\frac{π}{2},π})$, $sinα=\\frac{3}{5}$, find $\\tan 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{7}"
} | {
"index": 12890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $π < α < 2π$, $\\cos (α-9π)=- \\dfrac {3}{5}$, find the value of $\\cos (α- \\dfrac {11π}{2})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{5}"
} | {
"index": 12891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given (1+i)x=1+yi, find |x+yi|."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 12892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 12893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 100 real numbers, with their sum equal to zero. What is the minimum number of pairs that can be selected from them such that the sum of the numbers in each pair is non-negative?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 12894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 12 points in a diagram, calculate the number of groups of 3 points that can be formed to create a triangle."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 12895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 15 points in space, 5 of which are collinear, what is the maximum possible number of unique planes that can be determined by these points?"
}
] | math | {
"style": "rule",
"ground_truth": "445"
} | {
"index": 12896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 18 parking spaces in a row, 14 cars arrive and occupy spaces at random, followed by Auntie Em, who requires 2 adjacent spaces, determine the probability that the remaining spaces are sufficient for her to park."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{113}{204}"
} | {
"index": 12897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets?"
}
] | math | {
"style": "rule",
"ground_truth": "87341"
} | {
"index": 12898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed."
}
] | math | {
"style": "rule",
"ground_truth": "4019"
} | {
"index": 12899,
"solution": ""
} |
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