data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the circle $x^2+y^2=18$, and $PQ \\perp x$-axis at point $Q$, if the moving point $M$ satisfies $\\overrightarrow{OM}=\\frac{1}{3}\\overrightarrow{OP}+\\frac{2}{3}\\overrightarrow{OQ}$.\n\n(Ⅰ) Find the equation of the trajectory $C$ of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"index": 15500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the curve $y= \\frac {3-e^{x}}{e^{x}+1}$, find the minimum value of the slant angle $\\alpha$ of the tangent line at point $P$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 15501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $x^{2}=2y$, and the focus is $F$. If the fixed point is $M(1,2)$, calculate the minimum value of $(|PM|+|PF|)$ when point $P$ moves on the parabola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 15502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y=\\frac{1}{4}x^2$, determine the minimum value of the sum of the distance from point $P$ to the line $x+2y+4=0$ and the $x$-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{5}}{5}-1"
} | {
"index": 15503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{2}"
} | {
"index": 15504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $(0,2)$ and the distance from $P$ to the directrix of the parabola."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {17}}{2}"
} | {
"index": 15505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $D(2, \\frac{3}{2} \\sqrt{3})$ and the distance from point $P$ to the $y$-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 15506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}-1"
} | {
"index": 15507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is an intersection point of the ellipse $\\frac{x^{2}}{a_{1}^{2}} + \\frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\\frac{x^{2}}{a_{2}^{2}} - \\frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{3}"
} | {
"index": 15508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \\geq 2$, find the minimum value of $x+ \\sqrt {3}y$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3} - 2"
} | {
"index": 15509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is inside isosceles triangle $ABC$, with $AB = BC$ and $\\angle BPC = 108^{\\circ}$. Let $D$ be the midpoint of side $AC$, and let $BD$ intersect $PC$ at point $E$. If $P$ is the incenter of $\\triangle ABE$, find $\\angle PAC$."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 15510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is on curve $C_1: y^2 = 8x$ and point $Q$ is on curve $C_2: (x-2)^2 + y^2 = 1$. If $O$ is the coordinate origin, find the maximum value of $\\frac{|OP|}{|PQ|}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{7}}{7}"
} | {
"index": 15511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is on the line $y=2x+1$, and point $Q$ is on the curve $y=x+\\ln x$, determine the minimum distance between points $P$ and $Q$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 15512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is the circumcenter of $\\triangle ABC$, and $\\overrightarrow{PA} + \\overrightarrow{PB} + \\lambda \\overrightarrow{PC} = 0$, $\\angle C = 120^{\\circ}$, determine the value of the real number $\\lambda$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\\in R$, $m^{2}+n^{2}\\neq 0$), and point $Q$ is a moving point on the circle $C$: $\\left(x+1\\right)^{2}+y^{2}=1$, calculate the maximum value of $|PQ|$."
... | math | {
"style": "rule",
"ground_truth": "6 + 2\\sqrt{2}"
} | {
"index": 15514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ lies on the ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, and $F\\_1$, $F\\_2$ are the foci of the ellipse with $\\angle F\\_1 P F\\_2 = 60^{\\circ}$, find the area of $\\triangle F\\_1 P F\\_2$."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3}"
} | {
"index": 15515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ lies on the line $3x+4y+8=0$, and $PA$ and $PB$ are the two tangents drawn from $P$ to the circle $x^{2}+y^{2}-2x-2y+1=0$. Let $A$ and $B$ be the points of tangency, and $C$ be the center of the circle. Find the minimum possible area of the quadrilateral $P... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 15516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{2} + 1"
} | {
"index": 15517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ moves on the ellipse $\\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\\sqrt{5}=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"index": 15518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P(-15a, 8a)$ is on the terminal side of angle $\\alpha$, where $a \\in \\mathbb{R}$ and $a \\neq 0$, find the values of the six trigonometric functions of $\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{8}"
} | {
"index": 15519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P(-4,3)$ lies on the terminal side of angle $\\alpha$, find the value of $$\\frac{3\\sin^{2}\\frac{\\alpha}{2}+2\\sin\\frac{\\alpha}{2}\\cos\\frac{\\alpha}{2}+\\cos^{2}\\frac{\\alpha}{2}-2}{\\sin(\\frac{\\pi}{2}+\\alpha)\\tan(-3\\pi+\\alpha)+\\cos(6\\pi-\\alph... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 15520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P(x,y)$ is a moving point on the circle $x^{2}+y^{2}=2y$,\n(1) Find the range of $z=2x+y$;\n(2) If $x+y+a\\geqslant 0$ always holds, find the range of real numbers $a$;\n(3) Find the maximum and minimum values of $x^{2}+y^{2}-16x+4y$."
}
] | math | {
"style": "rule",
"ground_truth": "6-2\\sqrt{73}"
} | {
"index": 15521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point M $(3n-2, 2n+7)$ is on the angle bisector of the second and fourth quadrants, then $n=$ ."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point O is the center of the regular octagon ABCDEFGH, and Y is the midpoint of the side CD, determine the fraction of the area of the octagon that is shaded if the shaded region includes triangles DEO, EFO, and half of triangle CEO."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 15523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point P is a point on the graph of the function f(x)=e^(2x), find the minimum distance from point P to the line l:y=2x."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 15524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point P is any point on the graph of the function $f(x) = 2\\sqrt{2x}$, and a tangent line is drawn from point P to circle D: $x^2 + y^2 - 4x + 3 = 0$, with the points of tangency being A and B, find the minimum value of the area of quadrilateral PADB."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 15525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point P is on the curve $y=\\frac{1}{2}{{e}^{x}}$ and point Q is on the curve $y=x$, determine the minimum value of the distance $|PQ|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}(1 - \\ln 2)"
} | {
"index": 15526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point P is on the curve \\( y = -x^{2} - 1 \\) and point Q is on the curve \\( x = 1 + y^{2} \\), find the minimum value of \\( |PQ| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"index": 15527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point P is on the left branch of the hyperbola $x^2-y^2=4$, and $F_1$, $F_2$ are the left and right foci of the hyperbola, respectively, then $|PF_1|-|PF_2|$ equals to __."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 15528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point P(x, y) satisfies the equation (x-4 cos θ)^{2} + (y-4 sin θ)^{2} = 4, where θ ∈ R, find the area of the region that point P occupies."
}
] | math | {
"style": "rule",
"ground_truth": "32 \\pi"
} | {
"index": 15529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( F \\) is the right focus of the ellipse \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) (\\(a > b > 0\\)), and the eccentricity of the ellipse is \\(\\frac{\\sqrt{3}}{2}\\), a line \\( l \\) passing through point \\( F \\) intersects the ellipse at points... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"index": 15530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P \\) lies in the plane of triangle \\( \\triangle ABC \\) and satisfies the condition \\( PA - PB - PC = \\overrightarrow{BC} \\), determine the ratio of the area of \\( \\triangle ABP \\) to the area of \\( \\triangle ABC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"index": 15531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\( \\Gamma: \\frac{x^{2}}{463^{2}} - \\frac{y^{2}}{389^{2}} = 1 \\). A line \\( l \\) passes through point \\( P \\) and intersects the asymptotes of hyperbola \\( \\Gamma \\) at points \\( A \\) and \\( B \\), respectively. If ... | math | {
"style": "rule",
"ground_truth": "180107"
} | {
"index": 15532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\( \\frac{x^2}{16} - \\frac{y^2}{9} = 1 \\), and the distance from \\( P \\) to the right directrix of this hyperbola is the arithmetic mean of the distances from \\( P \\) to the two foci of the hyperbola, find the x-coordinate... | math | {
"style": "rule",
"ground_truth": "-\\frac{64}{5}"
} | {
"index": 15533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\(\\frac{x^{2}}{16} - \\frac{y^{2}}{9} = 1\\), and the distance from \\( P \\) to the right directrix of the hyperbola is the arithmetic mean of the distances from \\( P \\) to the two foci of the hyperbola, find the x-coordinat... | math | {
"style": "rule",
"ground_truth": "-\\frac{64}{5}"
} | {
"index": 15534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\(\\frac{x^{2}}{16} - \\frac{y^{2}}{9} = 1\\), and the distance from \\( P \\) to the right directrix of this hyperbola is the arithmetic mean of the distances from \\( P \\) to the two foci of this hyperbola, find the x-coordin... | math | {
"style": "rule",
"ground_truth": "-\\frac{64}{5}"
} | {
"index": 15535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P \\) moves on the circle \\( C: x^{2}+(y+2)^{2}=\\frac{1}{4} \\), and point \\( Q \\) moves on the curve \\( y=a x^{2} \\) (where \\( a > 0 \\) and \\( -1 \\leq x \\leq 2 \\)), if the maximum value of \\(|PQ|\\) is \\(\\frac{9}{2}\\), then find \\( a \\)."... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} - 1}{2}"
} | {
"index": 15536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\( P(x, y) \\) satisfies \\( |x| + |y| \\leq 2 \\), find the probability for point \\( P \\) to have a distance \\( d \\leq 1 \\) from the \\( x \\)-axis."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"index": 15537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\(Z\\) moves on \\(|z| = 3\\) in the complex plane, and \\(w = \\frac{1}{2}\\left(z + \\frac{1}{z}\\right)\\), where the trajectory of \\(w\\) is the curve \\(\\Gamma\\). A line \\(l\\) passes through point \\(P(1,0)\\) and intersects the curve \\(\\Gamma\\) a... | math | {
"style": "rule",
"ground_truth": "-\\frac{25}{8}"
} | {
"index": 15538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point \\\\(A\\\\) on the terminal side of angle \\\\(\\alpha\\\\) has coordinates \\\\(\\left( \\sqrt{3}, -1\\right)\\\\), \n\\\\((1)\\\\) Find the set of angle \\\\(\\alpha\\\\) \n\\\\((2)\\\\) Simplify the following expression and find its value: \\\\( \\dfrac{\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"index": 15539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points $ D$ and $ E$ lie on $ \\overline{BC}$ and $ \\overline{AC}$ respectively, if $ \\overline{AD}$ and $ \\overline{BE}$ intersect at $ T$ so that $ AT/DT \\equal{} 3$ and $ BT/ET \\equal{} 4$, calculate the value of $ CD/BD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"index": 15540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points $A$ and $B$ lie on the curves $C_{1}: x^{2}-y+1=0$ and $C_{2}: y^{2}-x+1=0$ respectively, what is the minimum value of the distance $|AB|$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"index": 15541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points $E$ and $F$ are on the same side of diameter $\\overline{GH}$ in circle $P$, $\\angle GPE = 60^\\circ$, and $\\angle FPH = 90^\\circ$, find the ratio of the area of the smaller sector $PEF$ to the area of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 15542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points $P$ and $Q$ are moving points on the curve $y=xe^{-2x}$ and the line $y=x+2$ respectively, find the minimum distance between points $P$ and $Q$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 15543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points $P$ and $Q$ are on the circle $x^2 + (y-6)^2 = 2$ and the ellipse $\\frac{x^2}{10} + y^2 = 1$, respectively, what is the maximum distance between $P$ and $Q$?\n\nA) $5\\sqrt{2}$\nB) $\\sqrt{46} + \\sqrt{2}$\nC) $7 + \\sqrt{2}$\nD) $6\\sqrt{2}$"
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}"
} | {
"index": 15544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points $\\mathbf{A}$ and $\\mathbf{B}$ lie on the curves $C_{1}: x^{2} - y + 1 = 0$ and $C_{2}: y^{2} - x + 1 = 0$ respectively, determine the minimum value of $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"index": 15545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points A and B are on the x-axis, and the two circles with centers at A and B intersect at points M $(3a-b, 5)$ and N $(9, 2a+3b)$, find the value of $a^{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 15546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points A and B lie on the graph of y = \\frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 15547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points P and Q are on the curve $f(x) = x^2 - \\ln x$ and the line $x-y-2=0$ respectively, find the minimum distance between points P and Q."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 15548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \\sqrt {2}\\sin ωx-\\cos ωx$ $(x\\in\\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 15549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that points \\( B \\) and \\( C \\) are in the fourth and first quadrants respectively, and both lie on the parabola \\( y^2 = 2px \\) where \\( p > 0 \\). Let \\( O \\) be the origin, and \\(\\angle OBC = 30^\\circ\\) and \\(\\angle BOC = 60^\\circ\\). If \\( k \\) is th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 15550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive integers \\( a, b, c \\) (\\( a < b < c \\)) form a geometric sequence, and\n\\[ \\log_{2016} a + \\log_{2016} b + \\log_{2016} c = 3, \\]\nfind the maximum value of \\( a + b + c \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4066273"
} | {
"index": 15551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive numbers $x$ and $y$ satisfy $x + 4y = 2$, find the minimum value of $\\frac{x + 40y + 4}{3xy}$."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 15552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive real numbers $a$ and $b$ satisfy $a \\gt b$ and $ab=\\frac{1}{2}$, find the minimum value of $\\frac{4{a}^{2}+{b}^{2}+3}{2a-b}$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 15553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive real numbers $x$ and $y$ satisfy $4x+3y=4$, find the minimum value of $\\frac{1}{2x+1}+\\frac{1}{3y+2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}+\\frac{\\sqrt{2}}{4}"
} | {
"index": 15554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive real numbers $x$ and $y$ satisfy $e^{x}=y\\ln x+y\\ln y$, then the minimum value of $\\frac{{e}^{x}}{x}-\\ln y$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "e-1"
} | {
"index": 15555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive real numbers \\( a \\) and \\( b \\) satisfy \\( ab(a+b)=4 \\), find the minimum value of \\( 2a + b \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 15556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that positive real numbers a and b satisfy $a^{2}+2ab+4b^{2}=6$, calculate the maximum value of a+2b."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 15557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that quadrilateral \\(ABCD\\) is a right trapezoid with the upper base \\(AD = 8\\) cm, the lower base \\(BC = 10\\) cm, and the right leg \\(CD = 6\\) cm. Point \\(E\\) is the midpoint of \\(AD\\), and point \\(F\\) is on \\(BC\\) such that \\(BF = \\frac{2}{3} BC\\). Po... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 15558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that quadrilateral \\(ABCD\\) is an isosceles trapezoid with \\(AB \\parallel CD\\), \\(AB = 6\\), and \\(CD = 16\\). Triangle \\(ACE\\) is a right-angled triangle with \\(\\angle AEC = 90^\\circ\\), and \\(CE = BC = AD\\). Find the length of \\(AE\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{6}"
} | {
"index": 15559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that real numbers $a$ and $b$ satisfy $\\frac{1}{a} + \\frac{2}{b} = \\sqrt{ab}$, calculate the minimum value of $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 15560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that real numbers \\( a \\) and \\( b \\) are such that the equation \\( a x^{3} - x^{2} + b x - 1 = 0 \\) has three positive real roots, find the minimum value of \\( P = \\frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)} \\) for all such \\( a \\) and \\( b \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12 \\sqrt{3}"
} | {
"index": 15561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that real numbers \\( x \\) and \\( y \\) satisfy the equation \\( 1 + \\cos^{2}(x + y - 1) = \\frac{x^{2} + y^{2} + 2(x + 1)(1 - y)}{x - y + 1} \\), find the minimum value of \\( xy \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 15562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that real numbers x and y satisfy x + y = 5 and x * y = -3, find the value of x + x^4 / y^3 + y^4 / x^3 + y."
}
] | math | {
"style": "rule",
"ground_truth": "5 + \\frac{2829}{27}"
} | {
"index": 15563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that rectangle ABCD has dimensions AB = 7 and AD = 8, and right triangle DCE shares the same height as rectangle side DC = 7 and extends horizontally from D towards E, and the area of the right triangle DCE is 28, find the length of DE."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{113}"
} | {
"index": 15564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\\angle CAD = 30^{\\circ}$, find $\\sin(2\\angle BAD)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}."
}
] | math | {
"style": "rule",
"ground_truth": "94"
} | {
"index": 15566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that signals are composed of the digits $0$ and $1$ with equal likelihood of transmission, the probabilities of error in transmission are $0.9$ and $0.1$ for signal $0$ being received as $1$ and $0$ respectively, and $0.95$ and $0.05$ for signal $1$ being received as $1$ ... | math | {
"style": "rule",
"ground_truth": "0.525"
} | {
"index": 15567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that sin(α + $\\frac {π}{4}$) = $\\frac { \\sqrt {3}}{3}$, and α ∈ (0, π), find the value of cosα."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {-2 \\sqrt {3}+ \\sqrt {6}}{6}"
} | {
"index": 15568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that sinα + cosα = $\\frac{7}{5}$, find the value of tanα."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 15569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that six balls are numbered 1, 2, 3, 4, 5, and 6, and the requirement that ball number 1 must be adjacent to ball number 2, and ball number 5 must not be adjacent to ball number 6, calculate the total number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 15570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 15571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that six students are to be seated in three rows of two seats each, with one seat reserved for a student council member who is Abby, calculate the probability that Abby and Bridget are seated next to each other in any row."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 15572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that socks cost $4 per pair and a T-shirt costs $5 more than a pair of socks, and each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games, with a total cost of $2366, determine the number of members in the League... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 15573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that spinner A contains the numbers 4, 5, 6, spinner B contains the numbers 1, 2, 3, 4, 5, and spinner C can take numbers from the set 7, 8, 9 if spinner B lands on an odd number and the set {6, 8} if spinner B lands on an even number, find the probability that the sum of... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 15574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 15575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\\triangle AFD$ to the area of square $ABCE$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 15576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that square PQRS has dimensions 3 × 3, points T and U are located on side QR such that QT = TU = UR = 1, and points V and W are positioned on side RS such that RV = VW = WS = 1, find the ratio of the shaded area to the unshaded area."
}
] | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"index": 15577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that tetrahedron $ABCD$ is inscribed in sphere $O$, and $AD$ is the diameter of sphere $O$. If triangles $\\triangle ABC$ and $\\triangle BCD$ are equilateral triangles with side length 1, what is the volume of tetrahedron $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{\\sqrt{2}}{12}$"
} | {
"index": 15578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that tetrahedron P-ABC is a 'Bie'zhi' with PA⊥ plane ABC, PA=AB=2, and AC=4, and all four vertices of the tetrahedron P-ABC lie on the surface of sphere O, calculate the surface area of the sphere O."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"index": 15579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that tetrahedron PQRS has edge lengths PQ = 3, PR = 4, PS = 5, QR = 5, QS = √34, and RS = √41, calculate the volume of tetrahedron PQRS."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 15580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the 32-digit integer 64312311692944269609355712372657 is the product of 6 consecutive primes, compute the sum of these 6 primes."
}
] | math | {
"style": "rule",
"ground_truth": "1200974"
} | {
"index": 15581,
"solution": "Because the product is approximately $64 \\cdot 10^{30}$, we know the primes are all around 200000. Say they are $200000+x_{i}$ for $i=1, \\ldots, 6$. By expanding $\\prod_{i=1}^{6}\\left(200000+x_{i}\\right)$ as a polynomial in 200000, we see that $$31231 \\cdot 10^{25}=200000^{5}\\l... |
DeepScaleR | [
{
"role": "user",
"content": "Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 15582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the Riemann function defined on the interval $\\left[0,1\\right]$ is: $R\\left(x\\right)=\\left\\{\\begin{array}{l}{\\frac{1}{q}, \\text{when } x=\\frac{p}{q} \\text{(p, q are positive integers, } \\frac{p}{q} \\text{ is a reduced proper fraction)}}\\\\{0, \\text{whe... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 15583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the amount of cultural and tourism vouchers issued is $2.51 million yuan, express this amount in scientific notation."
}
] | math | {
"style": "rule",
"ground_truth": "2.51 \\times 10^{6}"
} | {
"index": 15584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between the generating line and the axis of a cone is $\\frac{\\pi}{3}$, and the length of the generating line is $3$, find the maximum value of the cross-sectional area through the vertex."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 15585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between the unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\\overrightarrow{a}+y\\overrightarrow{b}|=1$ and $xy\\geqslant 0$, the inequality $|x+2y|\\leqslant \\frac{8}{\\sqrt{15}}$ holds. Find t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 15586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between the unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is obtuse, the minimum value of $|\\overrightarrow{b} - t\\overrightarrow{a}| (t \\in \\mathbb{R})$ is $\\frac{\\sqrt{3}}{2}$, and $(\\overrightarrow{c} - \\overrightarrow{a}) \\cdot (... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}"
} | {
"index": 15587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between the unit vectors $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ is $60^{\\circ}$, and $\\overrightarrow{a}=2\\overrightarrow{e_{1}}-\\overrightarrow{e_{2}}$, find the projection of $\\overrightarrow{a}$ in the direction of $\\overrightarrow... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"index": 15588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ is 120°, and the magnitude of $\\overrightarrow {a}$ is 2. If $(\\overrightarrow {a} + \\overrightarrow {b}) \\cdot (\\overrightarrow {a} - 2\\overrightarrow {b}) = 0$, find the projection of... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{33} + 1}{8}"
} | {
"index": 15589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $60^{\\circ}$, $\\overrightarrow{a}=(2,)$, $|\\overrightarrow{b}|=1$, calculate $|\\overrightarrow{a}+2\\overrightarrow{b}|$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 15590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $60^{\\circ}$, $|\\overrightarrow{a}|=1$, and $|\\overrightarrow{b}|=2$,\n(1) Find $(2\\overrightarrow{a}-\\overrightarrow{b})\\cdot\\overrightarrow{a}$;\n(2) Find $|2\\overrightarrow{a}+\\o... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 15591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angles A, B, C of triangle ABC correspond to the sides a, b, c respectively, and vectors $\\overrightarrow {m}$ = (a, $- \\sqrt {3}b$) and $\\overrightarrow {n}$ = (cosA, sinB), and $\\overrightarrow {m}$ is parallel to $\\overrightarrow {n}$.\n(1) Find angle A.\... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{3}}{2}"
} | {
"index": 15592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the angles of a triangle at points \\( A, B \\), and \\( C \\) are such that \\( \\angle ABC = 50^\\circ \\) and \\( \\angle ACB = 30^\\circ \\), calculate the value of \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 15593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 15594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $2 \\sqrt {3}$, $BC=2$, $C=120^{\\circ}$, find the length of side $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {7}"
} | {
"index": 15595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $S$, and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = S$.\n(1) Find the values of $\\sin A$, $\\cos A$, and $\\tan 2A$.\n(2) If $B = \\frac{\\pi}{4}, \\; |\\overrightarrow{CA} - \\overrightarrow{CB}| = 6$, find the area $S$ of $... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 15596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $S$, and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = S$.\n(I) Find the value of $\\tan 2A$;\n(II) If $\\cos C = \\frac{3}{5}$, and $|\\overrightarrow{AC} - \\overrightarrow{AB}| = 2$, find the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 15597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $S$, and $\\overrightarrow{BA} \\cdot \\overrightarrow{CA} = S$.\n\n(1) Find the value of $\\tan A$;\n\n(2) If $B = \\frac{\\pi}{4}, c = 6$, find the area of $\\triangle ABC$, $S$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 15598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $\\frac{1}{2}$, $AB=1$, $BC=\\sqrt{2}$, determine the value of $AC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 15599,
"solution": ""
} |
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