data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given point $M(\\sqrt{6}, \\sqrt{2})$ on the ellipse $G$: $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{6}}{3}$.\n\n1. Find the equation of ellipse $G$.\n2. If the line $l$ with a slope of $1$ intersects ellipse $G$ at poin... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 14200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $O$ inside $\\triangle ABC$, and $\\overrightarrow{OA}+\\overrightarrow{OC}+2 \\overrightarrow{OB}=0$, calculate the ratio of the area of $\\triangle AOC$ to the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 14201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $P$ is a moving point on the ellipse $\\frac{x^2}{8} + \\frac{y^2}{4} = 1$ in the first quadrant, through point $P$, two tangents $PA$ and $PB$ are drawn to the circle $x^2 + y^2 = 4$, with the points of tangency being $A$ and $B$ respectively. The line $AB$ interse... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $P(-2,0)$ and the parabola $C$: $y^{2}=4x$, the line passing through $P$ intersects $C$ at points $A$ and $B$, where $|PA|= \\frac {1}{2}|AB|$. Determine the distance from point $A$ to the focus of parabola $C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 14203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $P(0,-2)$, and an ellipse $E$: $\\dfrac{x^{2}}{a^{2}} + \\dfrac{y^{2}}{b^{2}} = 1$ $(a > b > 0)$ with eccentricity $\\dfrac{\\sqrt{2}}{2}$, let $F$ be the right focus of ellipse $E$, and the slope of line $PF$ is $2$, with $O$ being the origin. \n$(1)$ Find the equ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"index": 14204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $P(1, 2, 3)$, the symmetric point of $P$ about the $y$-axis is $P_1$, and the symmetric point of $P$ about the coordinate plane $xOz$ is $P_2$. Find the distance $|P_1P_2|$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{14}"
} | {
"index": 14205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $P(2,-1)$,\n(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.\n(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the ma... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 14206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.\n\n$(1)$ Find the equation of the trajectory of point $M$;\n... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{5}"
} | {
"index": 14207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point A (-2, 1) and circle C: $(x-2)^2+(y-2)^2=1$, a ray of light is emitted from point A to the x-axis and then reflects in the direction of the tangent to the circle. The distance traveled by the ray of light from point A to the tangent point is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"index": 14208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point O in the plane of △ABC, such that $|$$\\overrightarrow {OA}$$|=|$$\\overrightarrow {OB}$$|=|$$\\overrightarrow {OC}$$|=1, and 3$$\\overrightarrow {OA}$$+4$$\\overrightarrow {OB}$$+5$$\\overrightarrow {OC}$$= $$\\overrightarrow {0}$$, find the value of $$\\overrighta... | math | {
"style": "rule",
"ground_truth": "\\frac {4}{5}"
} | {
"index": 14209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point O is the circumcenter of triangle ABC, and |BA|=2, |BC|=6, calculate the value of the dot product of the vectors BO and AC."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point O is the circumcenter of ∆ABC, and |→BA|=2, |→BC|=6, calculate →BO⋅→AC."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point P(-2,0) and line l: (1+3λ)x + (1+2λ)y = 2+5λ (λ ∈ ℝ), find the maximum value of the distance from point P to line l."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 14212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point P(-2,0) and the parabola C: y^2=4x, let A and B be the intersection points of the line passing through P and the parabola. If |PA|= 1/2|AB|, find the distance from point A to the focus of parabola C."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 14213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point P(2, 1) is on the parabola $C_1: x^2 = 2py$ ($p > 0$), and the line $l$ passes through point Q(0, 2) and intersects the parabola $C_1$ at points A and B.\n(1) Find the equation of the parabola $C_1$ and the equation for the trajectory $C_2$ of the midpoint M of chor... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point P(a, -1) (a∈R), draw the tangent line to the parabola C: $y=x^2$ at point P, and let the tangent points be A($x_1$, $y_1$) and B($x_2$, $y_2$) (where $x_1<x_2$).\n\n(Ⅰ) Find the values of $x_1$ and $x_2$ (expressed in terms of a);\n\n(Ⅱ) If a circle E with center at... | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"index": 14215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point \\( A(2,0) \\), point \\( B \\) lies on the curve \\( y = \\sqrt{1 - x^2} \\), and the triangle \\( \\triangle ABC \\) is an isosceles right triangle with \\( A \\) as the right angle vertex. Determine the maximum value of \\( |OC| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} + 1"
} | {
"index": 14216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point \\( A(4,0) \\) and \\( B(2,2) \\), while \\( M \\) is a moving point on the ellipse \\(\\frac{x^{2}}{25} + \\frac{y^{2}}{9} = 1\\), the maximum value of \\( |MA| + |MB| \\) is ______."
}
] | math | {
"style": "rule",
"ground_truth": "10 + 2\\sqrt{10}"
} | {
"index": 14217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\\cos \\alpha,3\\sin \\alpha)$ respectively, and $\\alpha\\in\\left( \\frac {\\pi}{2}, \\frac {3\\pi}{4}\\right)$. If $\\overrightarrow{AC} \\perp \\overrightarrow{BC}$, find the value of $\\frac {2\\sin ^{2}\\alp... | math | {
"style": "rule",
"ground_truth": "- \\frac {7 \\sqrt {23}}{48}"
} | {
"index": 14218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A$, $B$, and $C$ are on the curve $y=\\sqrt{x}$ $(x \\geqslant 0)$, with x-coordinates $1$, $m$, and $4$ $(1 < m < 4)$, find the value of $m$ that maximizes the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 14219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A$, $B$, and $C$ are on the curve $y=\\sqrt{x}$ $(x \\geqslant 0)$, with x-coordinates $1$, $m$, and $4$ $(1 < m < 4)$, when the area of $\\triangle ABC$ is maximized, $m$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{9}{4}$"
} | {
"index": 14220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-1, 2, 0)$, $B(5, 2, -1)$, $C(2, -1, 4)$, and $D(-2, 2, -1)$ in space, find:\n\na) the distance from vertex $D$ of tetrahedron $ABCD$ to the intersection point of the medians of the base $ABC$;\n\nb) the equation of the plane $ABC$;\n\nc) the height of the tetra... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\\overrightarrow{AB}$ in the direction of $\\overrightarrow{CD}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{5}"
} | {
"index": 14222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-2, -3)$ and $B(5, 3)$ on the $xy$-plane, find the point $C(x, n)$ such that $AC + CB$ is minimized, where $x = 2$. Find the value of $n$.\nA) $\\frac{6}{7}$\nB) $\\frac{12}{7}$\nC) $6.5$\nD) $\\frac{25}{6}$\nE) $\\frac{13}{2}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2}"
} | {
"index": 14223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum are... | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{2}"
} | {
"index": 14224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-2,-2)$, $B(-2,6)$, $C(4,-2)$, and point $P$ moving on the circle $x^{2}+y^{2}=4$, find the maximum value of $|PA|^{2}+|PB|^{2}+|PC|^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 14225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-2,0)$ and $B(0,2)$, let point $C$ be a moving point on the circle $x^{2}-2x+y^{2}=0$. Determine the minimum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "3-\\sqrt{2}"
} | {
"index": 14226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\\frac{3}{4}$.\n$(1)$ Find the equation of the locus $C$ for point $P$.\n$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-2,0)$ and $P(1, \\frac{3}{2})$ on the ellipse $M: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a>b>0)$, and two lines with slopes $k$ and $-k (k>0)$ passing through point $P$ intersect ellipse $M$ at points $B$ and $C$.\n(I) Find the equation of ellipse $M$ an... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(-3, -4)$ and $B(6, 3)$ in the xy-plane; point $C(1, m)$ is taken so that $AC + CB$ is a minimum. Find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{9}"
} | {
"index": 14229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(2, 0)$, $B(0, 2)$, $C(\\cos\\alpha, \\sin\\alpha)$ and $0 < \\alpha < \\pi$:\n1. If $|\\vec{OA} + \\vec{OC}| = \\sqrt{7}$, find the angle between $\\vec{OB}$ and $\\vec{OC}$.\n2. If $\\vec{AC} \\perp \\vec{BC}$, find the value of $\\cos\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{7}}{4}"
} | {
"index": 14230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(2,-1,1)$, $B(1,-2,1)$, $C(0,0,-1)$, the distance from $A$ to $BC$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{3}"
} | {
"index": 14231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(2,-5,1)$, $B(2,-2,4)$, $C(1,-4,1)$, the angle between vectors $\\overrightarrow{AB}$ and $\\overrightarrow{AC}$ is equal to ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(\\cos\\alpha, \\sin\\alpha)$ and $B(\\cos\\beta, \\sin\\beta)$, where $\\alpha, \\beta$ are acute angles, and that $|AB| = \\frac{\\sqrt{10}}{5}$:\n(1) Find the value of $\\cos(\\alpha - \\beta)$;\n(2) If $\\tan \\frac{\\alpha}{2} = \\frac{1}{2}$, find the value... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{25}"
} | {
"index": 14233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A(\\sin\\theta, 1)$, $B(\\cos\\theta, 0)$, $C(-\\sin\\theta, 2)$, and $\\overset{→}{AB}=\\overset{→}{BP}$.\n(I) Consider the function $f\\left(\\theta\\right)=\\overset{→}{BP}\\cdot\\overset{→}{CA}$, $\\theta\\in\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$, d... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{74}}{5}"
} | {
"index": 14234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC$, $AB>r$, and the length of minor arc $BC$ is $r$, calculate the ratio of the length of $AB$ to the length of $BC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}\\csc(\\frac{1}{4})"
} | {
"index": 14235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A=(4,10)$ and $B=(10,8)$ lie on circle $\\omega$ in the plane, and the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis, find the area of $\\omega$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100\\pi}{9}"
} | {
"index": 14236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $A\\left(\\begin{matrix} \\cos \\alpha , & \\sin \\alpha \\end{matrix}\\right)$ and $B\\left(\\begin{matrix} \\cos \\beta , & \\sin \\beta \\end{matrix}\\right)$, where $\\alpha$ and $\\beta$ are acute angles, and the distance between $A$ and $B$ is $\\frac{\\sqrt{... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{25}"
} | {
"index": 14237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $F_{1}$, $F_{2}$ are the foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and point $M(x_{0},y_{0})$ ($x_{0} > 0$, $y_{0} > 0$) lies on the asymptote of the hyperbola such that $MF_{1} \\perp MF_{2}$. If there is a parabola with f... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{5}"
} | {
"index": 14238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\\overset{→}{MN} \\cdot \\overset{→}{MP} = 6|\\overset{→}{PN}|$.\n\n(I) Find the trajectory equation of point $P$;\n\n(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$... | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{3}"
} | {
"index": 14239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"index": 14240,
"solution": "1. **Identify the Goal:**\n We need to find the value of $m$ such that the sum of the distances $PR + RQ$ is minimized, where $R(1,m)$.\n\n2. **Use the Triangle Inequality:**\n The Triangle Inequality states that for any triangle, the sum of the lengths of any two sides must be gr... |
DeepScaleR | [
{
"role": "user",
"content": "Given points $P(-2,-3)$ and $Q(5, 3)$ in the $xy$-plane; point $R(2,m)$ is taken so that $PR+RQ$ is minimized. Determine the value of $m$.\nA) $\\frac{3}{5}$\nB) $\\frac{2}{5}$\nC) $\\frac{3}{7}$\nD) $\\frac{1}{5}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"index": 14241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $P(-2,7)$ and $Q(4,y)$ in a coordinate plane, for what value of $y$ is the slope of the line through $P$ and $Q$ equal to $\\frac{-3}{2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $P(\\cos \\alpha, \\sin \\alpha)$, $Q(\\cos \\beta, \\sin \\beta)$, and $R(\\cos \\alpha, -\\sin \\alpha)$ in a two-dimensional space, where $O$ is the origin, if the cosine distance between $P$ and $Q$ is $\\frac{1}{3}$, and $\\tan \\alpha \\cdot \\tan \\beta = \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $P(\\sqrt{3}, 1)$, $Q(\\cos x, \\sin x)$, and $O$ as the origin of coordinates, the function $f(x) = \\overrightarrow{OP} \\cdot \\overrightarrow{QP}$\n\n(Ⅰ) Find the smallest positive period of the function $f(x)$;\n\n(Ⅱ) If $A$ is an internal angle of $\\triangle... | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{3}"
} | {
"index": 14244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $S$, $A$, $B$, $C$ on the surface of a sphere $O$, $SA \\perp$ plane $ABC$, $AB \\perp BC$, $SA = AB = 1$, $BC = \\sqrt{2}$, calculate the surface area of sphere $O$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 14245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points $a$ and $b$ in the plane, let $a \\oplus b$ be the unique point $c$ such that $a b c$ is an equilateral triangle with $a, b, c$ in the clockwise orientation. Solve $(x \\oplus(0,0)) \\oplus(1,1)=(1,-1)$ for $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{1-\\sqrt{3}}{2}, \\frac{3-\\sqrt{3}}{2}\\right)"
} | {
"index": 14246,
"solution": "It is clear from the definition of $\\oplus$ that $b \\oplus(a \\oplus b)=a$ and if $a \\oplus b=c$ then $b \\oplus c=a$ and $c \\oplus a=b$. Therefore $x \\oplus(0,0)=(1,1) \\oplus(1,-1)=(1-\\sqrt{3}, 0)$. Now this means $x=(0,0) \\oplus(1-\\sqrt{3}, 0)=\\left(\\frac{1-\\sqrt{3}}{2},... |
DeepScaleR | [
{
"role": "user",
"content": "Given points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{13}"
} | {
"index": 14247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points A(1, 2, 1), B(-2, $$\\frac{7}{2}$$, 4), and D(1, 1, 1), if $$\\vec{AP} = 2\\vec{PB}$$, then the value of |$$\\vec{PD}$$| is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 14248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points A, B, and C on the surface of a sphere O, the height of the tetrahedron O-ABC is 2√2 and ∠ABC=60°, with AB=2 and BC=4. Find the surface area of the sphere O."
}
] | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"index": 14249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points E and D are on sides AB and BC of triangle ABC, where AE:EB=1:3 and CD:DB=1:2, find the value of EF/FC + AF/FD."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points F₁(-1, 0), F₂(1, 0), line l: y = x + 2. If the ellipse C, with foci at F₁ and F₂, intersects with line l, calculate the maximum eccentricity of ellipse C."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {10}}{5}"
} | {
"index": 14251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points O(0,0) and A(1,1), the slope angle of line OA is $\\boxed{\\text{answer}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 14252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points P and Q are on a circle of radius 7 and PQ = 8, find the length of the line segment PR, where R is the midpoint of the minor arc PQ."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{98 - 14\\sqrt{33}}"
} | {
"index": 14253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points P and Q are on a circle of radius 7 and the length of chord PQ is 8. Point R is the midpoint of the minor arc PQ. Find the length of the line segment PR."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{98 - 14\\sqrt{33}}"
} | {
"index": 14254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points P(-2, -2), Q(0, -1), and a point R(2, m) is chosen such that PR + PQ is minimized. What is the value of the real number $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"index": 14256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points \\( A(3,1) \\) and \\( B\\left(\\frac{5}{3}, 2\\right) \\), and the four vertices of quadrilateral \\( \\square ABCD \\) are on the graph of the function \\( f(x)=\\log _{2} \\frac{a x+b}{x-1} \\), find the area of \\( \\square ABCD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{26}{3}"
} | {
"index": 14258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points \\( A(4,0) \\) and \\( B(2,2) \\) are inside the ellipse \\( \\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1 \\), and \\( M \\) is a point on the ellipse, find the maximum value of \\( |MA| + |MB| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10 + 2\\sqrt{10}"
} | {
"index": 14259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points \\(A(4,5)\\), \\(B(4,0)\\) and \\(C(0,5)\\), compute the line integral of the second kind \\(\\int_{L}(4 x+8 y+5) d x+(9 x+8) d y\\) where \\(L:\\)\na) the line segment \\(OA\\);\nb) the broken line \\(OCA\\);\nc) the parabola \\(y=k x^{2}\\) passing through the po... | math | {
"style": "rule",
"ground_truth": "\\frac{796}{3}"
} | {
"index": 14260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given points \\(A=(8,15)\\) and \\(B=(16,9)\\) are on a circle \\(\\omega\\), and the tangent lines to \\(\\omega\\) at \\(A\\) and \\(B\\) meet at a point \\(P\\) on the x-axis, calculate the area of the circle \\(\\omega\\)."
}
] | math | {
"style": "rule",
"ground_truth": "250\\pi"
} | {
"index": 14261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given polynomials $A=ax^{2}+3xy+2|a|x$ and $B=2x^{2}+6xy+4x+y+1$. \n$(1)$ If $2A-B$ is a quadratic trinomial in terms of $x$ and $y$, find the value of $a$. \n$(2)$ Under the condition in $(1)$, simplify and evaluate the expression $3(-3a^{2}-2a)-[a^{2}-2(5a-4a^{2}+1)-2a]$."
... | math | {
"style": "rule",
"ground_truth": "-22"
} | {
"index": 14262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\\cdots p_r^{i_r} (0 \\le i_1,i_2,\\cdots,i_r \\le n).$\n\nAlice and Bob play a game by making a move by turns... | math | {
"style": "rule",
"ground_truth": "M^{\\lfloor \\frac{n}{2} \\rfloor}"
} | {
"index": 14263,
"solution": "\nGiven positive integer \\( n \\) and \\( r \\) pairwise distinct primes \\( p_1, p_2, \\cdots, p_r \\). Initially, there are \\( (n+1)^r \\) numbers written on the blackboard: \\( p_1^{i_1} p_2^{i_2} \\cdots p_r^{i_r} \\) where \\( 0 \\le i_1, i_2, \\cdots, i_r \\le n \\).\n\nAlice ... |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $a$ and $b$ are members of a set where $a \\in \\{2, 3, 5, 7\\}$ and $b \\in \\{2, 4, 6, 8\\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \\cdot a - a \\cdot b$."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 14264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $k, m, n$ such that $1 \\leq k \\leq m \\leq n$. Evaluate\n\n\\[\\sum^{n}_{i=0} \\frac{(-1)^i}{n+k+i} \\cdot \\frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 14265,
"solution": "\nGiven positive integers \\( k, m, n \\) such that \\( 1 \\leq k \\leq m \\leq n \\), we aim to evaluate the sum\n\\[\n\\sum_{i=0}^n \\frac{(-1)^i}{n+k+i} \\cdot \\frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\n\\]\n\nTo solve this, we employ a calculus-based approach. We start by expressing the s... |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \\times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns.\nDetermine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, ... | math | {
"style": "rule",
"ground_truth": "n(k-1)^2"
} | {
"index": 14266,
"solution": "\nGiven positive integers \\( n \\) and \\( k \\) such that \\( n > k^2 > 4 \\), we aim to determine the maximal possible \\( N \\) such that one can choose \\( N \\) unit squares in an \\( n \\times n \\) grid and color them, with the condition that in any \\( k \\)-group from the co... |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $n, k$ such that $n\\ge 4k$, find the minimal value $\\lambda=\\lambda(n,k)$ such that for any positive reals $a_1,a_2,\\ldots,a_n$, we have\n\\[ \\sum\\limits_{i=1}^{n} {\\frac{{a}_{i}}{\\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\\cdots}{{+}}{a}_{{i}{+}{k}}^... | math | {
"style": "rule",
"ground_truth": "n - k"
} | {
"index": 14267,
"solution": "\nGiven positive integers \\( n \\) and \\( k \\) such that \\( n \\geq 4k \\), we aim to find the minimal value \\( \\lambda = \\lambda(n, k) \\) such that for any positive reals \\( a_1, a_2, \\ldots, a_n \\), the following inequality holds:\n\\[\n\\sum_{i=1}^{n} \\frac{a_i}{\\sqrt{... |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $\\frac{1}{x} + \\frac{1}{2y} = \\frac{1}{7}$, what is the least possible value of $xy$?"
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 14268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $\\frac{1}{x} + \\frac{1}{3y} = \\frac{1}{8}$, find the least possible value of $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 14269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{12}$, what is the smallest possible value for $x + y$?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 14270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{15}$, what is the smallest possible value for $x + y$?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 14271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{18}$, what is the smallest possible value for $x + y$?"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 14272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\( N \\) and \\( k \\), we counted how many different ways the number \\( N \\) can be written in the form \\( a + b + c \\), where \\( 1 \\leq a, b, c \\leq k \\), and the order of the summands matters. Could the result be 2007?"
}
] | math | {
"style": "rule",
"ground_truth": "2007"
} | {
"index": 14273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\( a, b, \\) and \\( c \\) such that \\( a < b < c \\). If the product of any two numbers minus 1 is divisible by the third number, what is \\( a^{2} + b^{2} + c^{2} \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 14274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\( a, b, c, \\) and \\( d \\) such that \\( a > b > c > d \\) and \\( a + b + c + d = 2004 \\), as well as \\( a^2 - b^2 + c^2 - d^2 = 2004 \\), what is the minimum value of \\( a \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "503"
} | {
"index": 14275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\( n \\) and \\( m \\), let \\( A = \\{1, 2, \\cdots, n\\} \\) and define \\( B_{n}^{m} = \\left\\{\\left(a_{1}, a_{2}, \\cdots, a_{m}\\right) \\mid a_{i} \\in A, i=1,2, \\cdots, m\\} \\right. \\) satisfying:\n1. \\( \\left|a_{i} - a_{i+1}\\right| \\neq... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 14276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\( x, y, z \\) that satisfy the condition \\( x y z = (14 - x)(14 - y)(14 - z) \\), and \\( x + y + z < 28 \\), what is the maximum value of \\( x^2 + y^2 + z^2 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "219"
} | {
"index": 14277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\(a, b,\\) and \\(c\\) that satisfy \\(2017 \\geq 10a \\geq 100b \\geq 1000c\\), determine how many such ordered triples \\((a, b, c)\\) exist."
}
] | math | {
"style": "rule",
"ground_truth": "574"
} | {
"index": 14278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\(a\\) and \\(b\\) are each less than 10, find the smallest possible value for \\(2 \\cdot a - a \\cdot b\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-63"
} | {
"index": 14279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\(a\\) and \\(b\\) such that \\(15a + 16b\\) and \\(16a - 15b\\) are both perfect squares, find the smallest possible value of these two perfect squares."
}
] | math | {
"style": "rule",
"ground_truth": "231361"
} | {
"index": 14280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\(a\\), \\(b\\) \\((a \\leq b)\\), a sequence \\(\\{ f_{n} \\}\\) satisfies:\n\\[ f_{1} = a, \\, f_{2} = b, \\, f_{n+2} = f_{n+1} + f_{n} \\text{ for } n = 1, 2, \\ldots \\]\n\nIf for any positive integer \\(n\\), it holds that \n\\[ \\left( \\sum_{k=1}... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{5}"
} | {
"index": 14281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive integers \\(a_{1}, a_{2}, \\ldots, a_{2023}\\) such that \\(a_{k}=\\sum_{i=1}^{2023}\\left|a_{k}-a_{i}\\right|\\) for all \\(1 \\leq k \\leq 2023\\), find the minimum possible value of \\(a_{1}+a_{2}+\\cdots+a_{2023}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2046264"
} | {
"index": 14282,
"solution": "Without loss of generality, let \\(a_{1} \\leq a_{2} \\leq \\cdots \\leq a_{2023}\\). Then, note that \\(a_{k+1}-a_{k}=\\sum_{i=1}^{2023}\\left|a_{k+1}-a_{i}\\right|-\\left|a_{k}-a_{i}\\right|=k(a_{k+1}-a_{k})-(2023-k)(a_{k+1}-a_{k})=(2k-2023)(a_{k+1}-a_{k})\\). Thus, \\(a_{k+1}=a_{k}... |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $a$ and $b$ satisfying $\\log _{6}(2a+3b)=\\log _{3}b+\\log _{6}9-1=\\log _{2}a+\\log _{6}9-\\log _{2}3$, find $\\lg \\left(2a+3b\\right)-\\lg \\left(10a\\right)-\\lg \\left(10b\\right)=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $a$ and $b$ satisfying $a+b=1$, $c\\in R$, find the minimum value of $\\frac{3a}{b{c}^{2}+b}+\\frac{1}{ab{c}^{2}+ab}+3c^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} - 3"
} | {
"index": 14284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $a$ and $b$ satisfying $a+b=1$, determine the minimum value of $\\sqrt{ab}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $a$, $b$, $c$ satisfying: $a^2+ab+ac+bc=6+2\\sqrt{5}$, find the minimum value of $3a+b+2c$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}+2\\sqrt{2}"
} | {
"index": 14286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"index": 14287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}-2"
} | {
"index": 14288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $x$ and $y$ satisfying $x+y=2$, calculate the minimum value of $\\frac{4}{x+2}+\\frac{3x-7}{3y+4}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"index": 14289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\\frac {1}{x}+ \\frac {1}{y}$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers $x$, $y$ satisfying $xy= \\frac{x-y}{x+3y}$, find the maximum value of $y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers \\( a \\) and \\( b \\) that satisfy \\( 2 + \\log_{2} a = 3 + \\log_{3} b = \\log_{6}(a+b) \\), find the value of \\( \\frac{1}{a} + \\frac{1}{b} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 14292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers \\(a, b, c, x, y, z\\) satisfying the equations \\(cy + bz = a\\), \\(az + cx = b\\), \\(bx + ay = c\\), find the minimum value of the function\n\\[ f(x, y, z) = \\frac{x^{2}}{1+x} + \\frac{y^{2}}{1+y} + \\frac{z^{2}}{1+z}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers \\(a, b, c, x, y, z\\) that satisfy the equations \\(cy + bz = a\\), \\(az + cx = b\\), and \\(bx + ay = c\\), find the minimum value of the function \n\\[ f(x, y, z) = \\frac{x^2}{1 + x} + \\frac{y^2}{1 + y} + \\frac{z^2}{1 + z}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 14294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive numbers \\(a,b\\) satisfying \\(2ab=\\dfrac{2a-b}{2a+3b},\\) then the maximum value of \\(b\\) is \\_\\_\\_\\_\\_"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"index": 14295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers $a$ and $b$ satisfying $a+b=1$, find the maximum value of $\\dfrac {2a}{a^{2}+b}+ \\dfrac {b}{a+b^{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {2 \\sqrt {3}+3}{3}"
} | {
"index": 14296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers $a$ and $b$ satisfying $a+b=2$, the minimum value of $\\dfrac{1}{a}+\\dfrac{2}{b}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3+2 \\sqrt{2}}{2}"
} | {
"index": 14297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers $a$, $b$, $c$, $d$ satisfying $a^{2}-ab+1=0$ and $c^{2}+d^{2}=1$, find the value of $ab$ when $\\left(a-c\\right)^{2}+\\left(b-d\\right)^{2}$ reaches its minimum."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2} + 1"
} | {
"index": 14298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers $x$, $y$, and $z$, find the minimum value of $\\frac{x^2+2y^2+z^2}{xy+3yz}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 14299,
"solution": ""
} |
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