data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 14500,
"solution": "The smallest distance is 3, achieved by $A = (0,0)$, $B = (3,0)$, $C = (0,4)$.\nTo check this, it suffices to check that $AB$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triangle inequality implie... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$, $B$, and $C$ are the three interior angles of $\\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\\cos C=-\\frac{1}{4}$. \n$(1)$ If $\\sin A=2\\sin B$, find $b$ and $c$; \n$(2)$ If $\\cos (A-\\frac{π}{4})=\\frac{4}{5}$, find $c$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{30}}{2}"
} | {
"index": 14501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$, $B$, and $C$ are the three interior angles of $\\triangle ABC$, let vectors $\\overrightarrow{m}=(\\cos B,\\sin C)$ and $\\overrightarrow{n}=(\\cos C,-\\sin B)$, and $\\overrightarrow{m}\\cdot \\overrightarrow{n}=\\frac{1}{2}$.\n\n$(1)$ Find the magnitude of the... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}+4"
} | {
"index": 14502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\\frac{\\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region encl... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"index": 14503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\\frac{{y^2}}{4}=1$, and they satisfy $\\overrightarrow{PA}+\\overrightarrow{PB}=2\\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respect... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 14504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A, B, C, D, E, F$ are the vertices of a regular hexagon with a side length of 2, and a parabola passes through the points $A, B, C, D$, find the distance from the focus of the parabola to its directrix."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 14505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $ABCD$ is a rectangle with $AD = 10$ and the shaded area is $100, calculate the shortest distance between the semicircles."
}
] | math | {
"style": "rule",
"ground_truth": "2.5 \\pi"
} | {
"index": 14506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $AC$ and $CE$ are two diagonals of a regular hexagon $ABCDEF$, and points $M$ and $N$ divide $AC$ and $CE$ internally such that $\\frac{AM}{AC}=\\frac{CN}{CE}=r$. If points $B$, $M$, and $N$ are collinear, find the value of $r$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"index": 14507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 14508,
"solution": "Though the problem may appear to be quite daunting, it is actually not that difficult. $\\frac {k(k-1)}2$ always evaluates to an integer (triangular number), and the cosine of $n\\pi$ where $n \\in \\mathbb{Z}$ is 1 if $n$ is even and -1 if $n$ is odd. $\\frac {k(k-1)}2$ will be even ... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $BDEF$ is a square and $AB = BC = 1$, find the number of square units in the area of the regular octagon.\n\n[asy]\nreal x = sqrt(2);\npair A,B,C,D,E,F,G,H;\nF=(0,0); E=(2,0); D=(2+x,x); C=(2+x,2+x);\nB=(2,2+2x); A=(0,2+2x); H=(-x,2+x); G=(-x,x);\ndraw(A--B--C--D--E-... | math | {
"style": "rule",
"ground_truth": "4+4\\sqrt{2}"
} | {
"index": 14509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $C$ is an interior angle of $\\triangle ABC$, and the vectors $\\overrightarrow{m}=(2\\cos C-1,-2)$, $\\overrightarrow{n}=(\\cos C,\\cos C+1)$. If $\\overrightarrow{m}\\perp \\overrightarrow{n}$, calculate the value of $\\angle C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\pi}{3}"
} | {
"index": 14510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 14511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $D$ is a point on the side $AB$ of $\\triangle ABC$, and $\\overrightarrow{CD} = \\frac{1}{3}\\overrightarrow{AC} + \\lambda \\cdot \\overrightarrow{BC}$, determine the value of the real number $\\lambda$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"index": 14512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $D$ is the midpoint of side $AB$ of $\\triangle ABC$ with an area of $1$, $E$ is any point on side $AC$, and $DE$ is connected. Point $F$ is on segment $DE$ and $BF$ is connected. Let $\\frac{DF}{DE} = \\lambda_{1}$ and $\\frac{AE}{AC} = \\lambda_{2}$, with $\\lambda... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"index": 14513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $E$ and $F$ are a chord of the circle $C: x^{2}+y^{2}-2x-4y+1=0$, and $CE\\bot CF$, $P$ is the midpoint of $EF$. When the chord $EF$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $l: x-y-3=0$ such that $∠APB≥\\frac{π}{2}$ always holds. Deter... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 14514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \\frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\\triangle DEF$ to the area of quadrilateral $ABEF$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "1: 5"
} | {
"index": 14515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $E, U, L, S, R, T$ represent the numbers $1, 2, 3, 4, 5, 6$ (with each letter representing a different number), and satisfying the following conditions:\n(1) $E + U + L = 6$\n(2) $S + R + U + T = 18$\n(3) $U \\times T = 15$\n(4) $S \\times L = 8$\n\nWhat is the six-d... | math | {
"style": "rule",
"ground_truth": "132465"
} | {
"index": 14516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $E, U, L, S, R,$ and $T$ represent the digits $1, 2, 3, 4, 5, 6$ (each letter represents a unique digit), and the following conditions are satisfied:\n1. $E + U + L = 6$\n2. $S + R + U + T = 18$\n3. $U \\times T = 15$\n4. $S \\times L = 8$\n\nDetermine the six-digit ... | math | {
"style": "rule",
"ground_truth": "132465"
} | {
"index": 14517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the focus of the parabola $4y^{2}=x$, and points $A$ and $B$ are on the parabola and located on both sides of the $x$-axis. If $\\overrightarrow{OA} \\cdot \\overrightarrow{OB} = 15$ (where $O$ is the origin), determine the minimum value of the sum of the area... | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt{65}}{2}"
} | {
"index": 14518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2ρx (ρ > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\\frac{{{x}^{2}}}{{{a}^{2}}}-\\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \\perp x$-axis, find the eccentricit... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 14519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the focus of the parabola $x^{2}=8y$, $P$ is a moving point on the parabola, and the coordinates of $A$ are $(0,-2)$, find the minimum value of $\\frac{|PF|}{|PA|}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 14520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the focus of the parabola $y^{2}=4x$, and a perpendicular line to the directrix is drawn from a point $M$ on the parabola, with the foot of the perpendicular being $N$. If $|MF|= \\frac{4}{3}$, then $\\angle NMF=$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 14521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the left focus of the ellipse $C:\\frac{{x}^{2}}{3}+\\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "5 - 2\\sqrt{3}"
} | {
"index": 14522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the ellipse $C:\\frac{x^2}{4}+\\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\\sqrt{2})$, find the maximum value of $|PA|+|PF|$."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{3}"
} | {
"index": 14523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \\frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \\sqrt {6})$. When the perimeter of $\\triangle APF$ is minimized, the area of this triangle is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "12 \\sqrt {6}"
} | {
"index": 14524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \\sqrt {6})$ is a point on the $y$-axis. The minimum area of $\\triangle APF$ is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "6+9 \\sqrt {6}"
} | {
"index": 14525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\\Delta APF$ is minimized, the area of this triangle is \\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{36}{7}"
} | {
"index": 14526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C: x^{2}- \\frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \\sqrt {6})$, when the perimeter of $\\triangle APF$ is minimized, the ordinate of point $P$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {6}"
} | {
"index": 14527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C:x^{2}-\\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6\\sqrt{6})$, the minimum perimeter of $\\triangle APF$ is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, and the line $l$ passing through the origin intersects the hyperbola at points $M$ and $N$, with $\\overrightarrow{MF} \\cdot \\overrightarrow{NF} = 0$. If the a... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F(1,0)$ is the focus of the ellipse $\\frac{x^2}{9} + \\frac{y^2}{m} = 1$, $P$ is a moving point on the ellipse, and $A(1,1)$, find the minimum value of $|PA| + |PF|$."
}
] | math | {
"style": "rule",
"ground_truth": "6 - \\sqrt{5}"
} | {
"index": 14530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F(c,0)$ is the right focus of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > b > 0)$, a line perpendicular to one asymptote of the hyperbola passes through $F$ and intersects the two asymptotes at points $A$ and $B$. $O$ is the coordinate origin, and th... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 14531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the foci of a hyperbola, a line passing through $F\\_2$ perpendicular to the real axis intersects the hyperbola at points $A$ and $B$. If $BF\\_1$ intersects the $y$-axis at point $C$, and $AC$ is perpendicular to $BF\\_1$, determine the eccen... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $(E)$: $\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\\_1NF\\_2$ is $4$, let line $(l)$ pass... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, and point $A(1, \\frac{3}{2})$ on ellipse $C$ has a sum of distances to these two points equal to $4$.\n(I) Find the equation of the ellipse $C$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 14534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, $D$ and $E$ are the upper and right vertices of the ellipse $C$, and $S_{\\triangle DEF_2} = \\frac{\\sqrt{3}}{2}$, eccentricity $e = \\frac{1}{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the ellipse such that $PF\\_2$ is perpendicular to the $x$-axis. If $|F\\_1F\\_2| = 2|PF\\_2|$, calculate the eccentricity of t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 14536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, perpendicular lines to the $x$-axis are drawn through points $F\\_1$ and $F\\_2$ intersecting the ellipse at four points to form a square, de... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 14537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $(x^{2}- \\dfrac {y^{2}}{24}=1)$, and $P$ is a point on the hyperbola, find the area of $\\triangle F\\_1PF\\_2$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 14538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $C: \\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\\_1|=3|PF\\_2|$ and $\\overrightarrow{PF\\_1} \\cdot \\overrighta... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $E$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$, with $MF\\_1$ perpendicular to the $x$-axis and $\\sin \\angle MF\\_2F\\_1 = \\frac{1}{3}$. Find the eccentricity of $E$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, a line parallel to one of the hyperbola's asymptotes passes through $F\\_2$ and intersects the hyperbola at point $P$. If $|PF\\_1| = 3|PF\\_2|$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\\triangle PF\\_1F\\_2$, satisfying $S\\_{\\triangle MPF\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and there exists a point $P$ on the hyperbola such that $\\angle F\\_1PF\\_2 = 60^{\\circ}$ and $|OP| = 3b$ (where $O$ is the origin), find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{42}}{6}"
} | {
"index": 14543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the two foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the hyperbola such that $\\overrightarrow{PF\\_1} \\cdot \\overrightarrow{PF\\_2} = 0$ and $|\\overrightarrow{PF\\_1}| \\cdot |\\overrig... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 14544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\\angle F\\_1PF\\_2=60^{\\circ}$, determine the area of $\\triangle F\\_1PF\\_2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1$ is the left focus of the hyperbola $C$: $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\\overrightarrow{QP}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F\\_1(-4,0)$ and $F\\_2(4,0)$ are the two foci of the ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, and $P$ is a point on the ellipse such that the area of $\\triangle PF\\_1F\\_2$ is $3\\sqrt{3}$, find the value of $\\cos\\angle{F\\_1PF\\_2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the common foci of the ellipse $C_1: \\frac{x^2}{4} + y^2 = 1$ and the hyperbola $C_2$, and $A, B$ are the common points of $C_1$ and $C_2$ in the second and fourth quadrants, respectively. If the quadrilateral $AF_1BF_2$ is a rectangle, determine... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 14548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of a hyperbola $E$, and point $P$ is on $E$, with $\\angle F_1 P F_2 = \\frac{\\pi}{6}$ and $(\\overrightarrow{F_2 F_1} + \\overrightarrow{F_2 P}) \\cdot \\overrightarrow{F_1 P} = 0$, determine the eccentricity $e$ of hyper... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}"
} | {
"index": 14549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$, with $A$ being the left vertex, and $P$ is a point on the ellipse $E$ such that the circle with diameter $PF_1$ passes through $F_2$ and $|PF_{2}|= \\frac {1}{4}|AF_{2}|$, determine the eccentricity of t... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $x^2 + \\frac{y^2}{b^2} = 1 (0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the sequence $|AF_2|, |AB|, |BF_2|$ forms an arithmetic progression, then:\n\n(1)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 14551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\\overrightarrow{PF_{1}} \\cdot (\\overrightarrow{OF_{1}} + \\overrightarrow{OP}) = 0$, if $|\\overrightarrow{... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6} - \\sqrt{3}"
} | {
"index": 14552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\\frac{x^{2}}{16}{+}\\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$ with $MF_1$ perpendicular to the x-axis and $\\sin \\angle MF_2F_1 = \\frac{1}{3}$, find the eccentricity of $E$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and line $l$ passes through points $(a, 0)$ and $(0, b)$. The sum of the distances from $F_1$ and $F_2$ to line $l$ is $\\frac{4c}{5}$. Determine th... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{21}}{21}"
} | {
"index": 14555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$. If the point $F_2$ is symmetric with respect to the asymptote line and lies on the hyperbola, calculate the eccentricity of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 14556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 14557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the two foci of the hyperbola $x^2 - \\frac{y^2}{24} = 1$, and $P$ is a common point of the hyperbola and the ellipse $\\frac{x^2}{49} + \\frac{y^2}{24} = 1$, find the area of the triangle $PF_1F_2$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 14558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\ (a > b > 0)$, a circle with the origin as its center and the semi-focal distance as its radius intersects the right branch of the hyperbola at po... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 1"
} | {
"index": 14559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\\sin \\angle PF_{1}F_{2} = \\frac {1}{3}$, dete... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\\frac {x^{2}}{16}+ \\frac {y^{2}}{9}=1$, and a line passing through point $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|AB|=5$, calculate $|AF_{1}|+|BF_{1}|$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 14561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\\frac{x^{2}}{20} + \\frac{y^{2}}{4} = 1$, a line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|F_{1}A| + |F_{1}B| = 5\\sqrt{5}$, then $|AB| = $ ______."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5}"
} | {
"index": 14562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are two foci of ellipse $C$, $P$ is a point on $C$, and $\\angle F_{1}PF_{2}=60^{\\circ}$, $|PF_{1}|=3|PF_{2}|$, calculate the eccentricity of $C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{4}"
} | {
"index": 14563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are two foci of the ellipse $\\frac{x^2}{9}+\\frac{y^2}{7}=1$, $A$ is a point on the ellipse, and $\\angle AF_{1}F_{2}=45^{\\circ}$, calculate the area of triangle $AF_{1}F_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 14564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $C: x^{2}-\\frac{{y}^{2}}{3}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $\\angle PF_{2}Q=120^{\\circ}$, find the area of quadrilateral $PF_{1}QF_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"index": 14565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\\left(0 \\lt m \\lt 1\\right)$, point $A\\left(0,2\\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\\frac{4}{3}$, then $m=$____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 14566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $M$ and $N$ are the common points of circle $A$: $x^{2}+y^{2}-2x=0$ and circle $B$: $x^{2}+y^{2}+2x-4y=0$, find the area of $\\triangle BMN$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $M(2,5)$ is the midpoint of $\\overline{AB}$ and $A(3,1)$ is one endpoint, what is the product of the coordinates of point $B$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 14568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $M(m, n)$ is any point on the circle $C: x^2+y^2-4x-14y+45=0$, find the maximum and minimum values of $\\frac{n-3}{m+2}$."
}
] | math | {
"style": "rule",
"ground_truth": "2- \\sqrt{3}"
} | {
"index": 14569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $M(m,n)$ is any point on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.\n\n(I) Find the maximum and minimum values of $|MQ|$;\n\n(II) Find the maximum and minimum values of $\\frac{n-3}{m+2}$."
}
] | math | {
"style": "rule",
"ground_truth": "2-\\sqrt{3}"
} | {
"index": 14570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 14571,
"solution": "Let the octahedron have vertices $(\\pm 3, 0, 0), (0, \\pm 3, 0), (0, 0, \\pm 3)$. Then the vertices of the cube lie at the centroids of the faces, which have coordinates $(\\pm 1, \\pm 1, \\pm 1)$. The cube has volume 8. The region of the octahedron lying in each octant is a tetrahed... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is any point in space, and $A$, $B$, $C$, $D$ are four points such that no three of them are collinear, but they are coplanar, and $\\overrightarrow{OA}=2x\\cdot \\overrightarrow{BO}+3y\\cdot \\overrightarrow{CO}+4z\\cdot \\overrightarrow{DO}$, find the value of ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is the center of the circumcircle of $\\triangle ABC$, $D$ is the midpoint of side $BC$, and $BC=4$, and $\\overrightarrow{AO} \\cdot \\overrightarrow{AD} = 6$, find the maximum value of the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 14573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is the circumcenter of $\\triangle ABC$, $AC \\perp BC$, $AC = 3$, and $\\angle ABC = \\frac{\\pi}{6}$, find the dot product of $\\overrightarrow{OC}$ and $\\overrightarrow{AB}$."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 14574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is the circumcenter of $\\triangle ABC$, and $D$ is the midpoint of $BC$, if $\\overrightarrow{AO} \\cdot \\overrightarrow{AD} = 4$ and $BC = 2\\sqrt{6}$, find the length of $AD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is the coordinate origin, and vectors $\\overrightarrow{OA}=(\\sin α,1)$, $\\overrightarrow{OB}=(\\cos α,0)$, $\\overrightarrow{OC}=(-\\sin α,2)$, and point $P$ satisfies $\\overrightarrow{AB}=\\overrightarrow{BP}$.\n(I) Denote function $f(α)=\\overrightarrow{PB}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{74}}{5}"
} | {
"index": 14576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is the coordinate origin, the complex numbers $z_1$ and $z_2$ correspond to the vectors $\\overrightarrow{OZ_1}$ and $\\overrightarrow{OZ_2}$, respectively. $\\bar{z_1}$ is the complex conjugate of $z_1$. The vectors are represented as $\\overrightarrow{OZ_1} = (... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{130}}{8}"
} | {
"index": 14577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $O$ is the origin of coordinates, and $M$ is a point on the ellipse $\\frac{x^2}{2} + y^2 = 1$. Let the moving point $P$ satisfy $\\overrightarrow{OP} = 2\\overrightarrow{OM}$.\n- (I) Find the equation of the trajectory $C$ of the moving point $P$;\n- (II) If the lin... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is a moving point on the curve $y= \\frac {1}{4}x^{2}- \\frac {1}{2}\\ln x$, and $Q$ is a moving point on the line $y= \\frac {3}{4}x-1$, then the minimum value of $PQ$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2-2\\ln 2}{5}"
} | {
"index": 14579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is a moving point on the line $l: x-2y+4=0$, two tangents are drawn from point $P$ to the circle $C: x^{2}+y^{2}-2x=0$, with tangents intersecting at points $A$ and $B$. Find the minimum area of the circumcircle of quadrilateral $PACB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{4}"
} | {
"index": 14580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, find the minimum value of the sum of the distance between points $P$ and $Q$ and the distance between point $P$ and the axis of symmetry of the parabola."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{17}-1"
} | {
"index": 14581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{17}-1"
} | {
"index": 14582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"index": 14583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is a point on the ellipse $\\frac{x^2}{25}+\\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\\frac{{\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}}}{{|\\overrightarrow{PF_1}| \\cdot |\\overrightarrow{PF_2}... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 14584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is any point on the circle $C$: $(x-2)^{2}+(y-2)^{2}=1$, and $Q$ is any point on the line $l$: $x+y=1$, find the minimum value of $| \\overrightarrow{OP}+ \\overrightarrow{OQ}|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{2}-2}{2}"
} | {
"index": 14585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $P$ is any point on the hyperbola $\\frac{x^2}{3} - y^2 = 1$, a line perpendicular to each asymptote of the hyperbola is drawn through point $P$, with the feet of these perpendiculars being $A$ and $B$. Determine the value of $\\overrightarrow{PA} \\cdot \\overrighta... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"index": 14586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S = 6 \\times 10,000 + 5 \\times 1000 + 4 \\times 10 + 3 \\times 1$, calculate the value of $S$."
}
] | math | {
"style": "rule",
"ground_truth": "65043"
} | {
"index": 14587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_n$ and $T_n$ represent the sum of the first n terms of the arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$ respectively, and $\\frac{S_n}{T_n} = \\frac{2n+1}{n+3}$, determine the value of $\\frac{a_7}{b_7}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{16}"
} | {
"index": 14588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of a positive geometric sequence $\\{a_{n}\\}$, and $S_{3}=10$, find the minimum value of $2S_{9}-3S_{6}+S_{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{4}"
} | {
"index": 14589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$, $S_{1} < 0$, $2S_{21}+S_{25}=0$, find the value of $n$ when $S_{n}$ is minimized."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 14590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, and ${S}_{n}=\\frac{{n}^{2}+3n}{2}$. \n$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$; \n$(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\\{\\frac{1}{{a}_{n... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, if ${a_1}=\\frac{5}{2}$, and ${a_{n+1}}({2-{a_n}})=2$ for $n\\in\\mathbb{N}^*$, then $S_{22}=$____."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"index": 14592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, with $a_{1}=1$, $a_{2}=2$, $a_{3}=3$, and the sequence $\\{a_{n}+a_{n+1}+a_{n+2}\\}$ is an arithmetic sequence with a common difference of $2$, calculate the value of $S_{25}$."
}
] | math | {
"style": "rule",
"ground_truth": "269"
} | {
"index": 14593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{n}=n^{2}-4n+4$.\n(1) Find the general term formula of the sequence ${a_{n}}$;\n(2) Let ${c_{n}}$ be a sequence where all $c_{n}$ are non-zero, and the number of positive integers $k$ satisfy... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 14594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $S_{n}=\\sin \\frac{\\pi}{7}+\\sin \\frac{2\\pi}{7}+\\cdots+\\sin \\frac{n\\pi}{7}$, determine the number of positive values in the sequence $S_{1}$, $S_{2}$, …, $S_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 14595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $W,X,Y$ and $Z$ are four different digits selected from the set ${1,2,3,4,5,6,7,8,9}$, find the value of $\\frac{W}{X} + \\frac{Y}{Z}$ when it is as small as possible."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{72}"
} | {
"index": 14596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $[x]$ is the greatest integer less than or equal to $x$, calculate $\\sum_{N=1}^{1024}\\left[\\log _{2} N\\right]$."
}
] | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"index": 14597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $[x]$ represents the greatest integer less than or equal to $x$, if\n$$\n[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104\n$$\n\nthen the smallest value of $x$ is ( )."
}
] | math | {
"style": "rule",
"ground_truth": "11.5"
} | {
"index": 14598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\alpha \\in (-\\frac{\\pi }{2},\\frac{\\pi }{2})$, $\\beta \\in (-\\frac{\\pi }{2},\\frac{\\pi }{2})$, and $\\tan \\alpha$ and $\\tan \\beta$ are the two real roots of the equation $x^{2}+3\\sqrt{3}x+4=0$, find the value of $\\alpha + \\beta$ = \\_\\_\\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "- \\frac{2\\pi}{3}"
} | {
"index": 14599,
"solution": ""
} |
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