data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"content": "In an isosceles right triangle \\( \\triangle ABC \\), \\( \\angle A = 90^\\circ \\), \\( AB = 1 \\). \\( D \\) is the midpoint of \\( BC \\), \\( E \\) and \\( F \\) are two other points on \\( BC \\). \\( M \\) is the other intersection point of the circumcircles of \\( \\triangle ADE \\) and \\... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 29620,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Groups of 4 random numbe... | math | {
"ground_truth": "0.75",
"style": "rule"
} | {
"index": 29623,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "If $cos2α=-\\frac{{\\sqrt{10}}}{{10}}$, $sin({α-β})=\\frac{{\\sqrt{5}}}{5}$, and $α∈({\\frac{π}{4},\\frac{π}{2}})$, $β∈({-π,-\\frac{π}{2}})$, then $\\alpha +\\beta =$____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": 29626,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the following four conclusions: \n\\\\((1)\\\\) The center of symmetry of the function \\\\(f(x)= \\dfrac {x-1}{2x+1}\\\\) is \\\\((- \\dfrac {1}{2},- \\dfrac {1}{2})\\\\); \n\\\\((2)\\\\) If the equation in \\\\(x\\\\), \\\\(x- \\dfrac {1}{x}+k=0\\\\), has no real roots for \\\\(x \\in (0... | math | {
"ground_truth": "(3)(4)",
"style": "rule"
} | {
"index": 29628,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function \\( f(x) = \\sin^4 x \\),\n1. Let \\( g(x) = f(x) + f\\left(\\frac{\\pi}{2} - x\\right) \\). Find the maximum and minimum values of \\( g(x) \\) in the interval \\(\\left[\\frac{\\pi}{6}, \\frac{3\\pi}{8}\\right]\\).\n2. Find the value of \\(\\sum_{k=1}^{89} f\\left(\\frac{k\\pi... | math | {
"ground_truth": "\\frac{133}{4}",
"style": "rule"
} | {
"index": 29629,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $XYZ$, which is equilateral with a side length $s$, lines $\\overline{LM}$, $\\overline{NO}$, and $\\overline{PQ}$ are parallel to $\\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$.\nPlease reason step by step, ... | math | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": 29631,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given vectors $\\overrightarrow{a}=(\\sin \\theta+\\cos \\theta,1)$ and $\\overrightarrow{b}=(5,1)$, which are orthogonal, and $\\theta \\in (0,\\pi)$, find the value of $\\tan \\theta$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": 29634,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Simplify: \\\\( \\dfrac {\\sin 7 ^{\\circ} + \\cos 15 ^{\\circ} \\sin 8 ^{\\circ} }{\\cos 7 ^{\\circ} - \\sin 15 ^{\\circ} \\sin 8 ^{\\circ} }= \\) \\_\\_\\_\\_\\_\\_ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2- \\sqrt {3}",
"style": "rule"
} | {
"index": 29635,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The graph of the function $y=\\sin 2x-\\sqrt{3}\\cos 2x$ can be obtained by shifting the graph of the function $y=\\sin 2x+\\sqrt{3}\\cos 2x$ to the right by $\\frac{\\pi}{3}$ units.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": 29636,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In an acute-angled triangle, the sides $a$ and $b$ are the roots of the equation $x^{2}-2 \\sqrt {3}x+2=0$. The angles $A$ and $B$ satisfy the equation $2\\sin (A+B)- \\sqrt {3}=0$. Find the value of the side $c$ and the area of $\\triangle ABC$.\nPlease reason step by step, and put your final ans... | math | {
"ground_truth": "\\dfrac { \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": 29639,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Calculate:<br/>$(1)64.83-5\\frac{18}{19}+35.17-44\\frac{1}{19}$;<br/>$(2)(+2.5)+(-3\\frac{1}{3})-(-1)$;<br/>$(3)\\frac{(0.125+\\frac{3}{5})×\\frac{33}{87}}{12.1×\\frac{1}{11}$;<br/>$(4)41\\frac{1}{3}×\\frac{3}{4}+52\\frac{1}{2}÷1\\frac{1}{4}+63\\frac{3}{5}×\\frac{5}{6}$;<br/>$(5)3\\frac{2}{3}×2\\f... | math | {
"ground_truth": "4044\\frac{2022}{2023}",
"style": "rule"
} | {
"index": 29640,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": 29641,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the hyperbola $C$: $\\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ and the line $l$: $x+y=1$ intersect at two distinct points $A$ and $B$.\n(I) Find the range of the eccentricity $e$ of the hyperbola $C$.\n(II) Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\\overrightarrow{PA} ... | math | {
"ground_truth": "\\frac{17}{13}",
"style": "rule"
} | {
"index": 29642,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2$, $\\overrightarrow{b}=(4\\cos \\alpha,-4\\sin \\alpha)$, and $\\overrightarrow{a}\\perp (\\overrightarrow{a}- \\overrightarrow{b})$, let the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$... | math | {
"ground_truth": "\\dfrac {\\pi}{3}",
"style": "rule"
} | {
"index": 29643,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that $\\{a_n\\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.\n\n(Ⅰ) Find the value of $q$; \n(Ⅱ) Let the sum of the first $n$ terms of the sequence $\\{a_n\\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form a... | math | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": 29644,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $ABC$, $AB = 8$, $BC = 8$, and $CA = 6$.\n\nPoint $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than it is to either vertex $A$ or $B$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "use... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": 29646,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the ellipse $\\Gamma$: $\\dfrac {x^{2}}{a^{2}}+y^{2}=1(a > 1)$, its left focus is $F_{1}$, the right vertex is $A_{1}$, and the top vertex is $B_{1}$. The circle $P$ that passes through points $F_{1}$, $A_{1}$, and $B_{1}$ has its center coordinates at $\\left( \\dfrac { \\sqrt {3}- \\sqrt {... | math | {
"ground_truth": "\\dfrac { \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": 29647,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f(x) = \\cos x \\cdot \\sin \\left( x + \\frac{\\pi}{3} \\right) - \\sqrt{3}\\cos^2 x + \\frac{\\sqrt{3}}{4}, x \\in \\mathbb{R}$,\n\n$(1)$ Find the smallest positive period of $f(x)$;\n\n$(2)$ Find the maximum and minimum values of $f(x)$ on the closed interval $\\left[ -\\fra... | math | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": 29648,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given a function $f(x)$ whose domain is $D$, if for any $x_1, x_2 \\in D$, it holds that $f(x_1) \\leq f(x_2)$ whenever $x_1 < x_2$, then the function $f(x)$ is called a non-decreasing function on $D$. Suppose $f(x)$ is a non-decreasing function on the interval $[0,1]$ and satisfies the following ... | math | {
"ground_truth": "\\frac{11}{4}",
"style": "rule"
} | {
"index": 29650,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the differences between the scores of 14 students in a group and the class average score of 85 are 2, 3, -3, -5, 12, 12, 8, 2, -1, 4, -10, -2, 5, 5, find the average score of this group.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "87.29",
"style": "rule"
} | {
"index": 29653,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $\\sigma (n)$ denote the sum and $\\tau (n)$ denote the amount of natural divisors of number $n$ (including $1$ and $n$ ). Find the greatest real number $a$ such that for all $n>1$ the following inequality is true: $$ \\frac{\\sigma (n)}{\\tau (n)}\\geq a\\sqrt{n} $$\nPlease rea... | math | {
"ground_truth": "\\frac{3 \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": 29654,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Suppose that \\( a^3 \\) varies inversely with \\( b^2 \\). If \\( a = 5 \\) when \\( b = 2 \\), find the value of \\( a \\) when \\( b = 8 \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": 29655,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\\frac{AP}{AB} =\\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ intersects the circl... | math | {
"ground_truth": "205/1681",
"style": "rule"
} | {
"index": 29658,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given two-dimensional vectors $\\vec{a}$, $\\vec{b}$, with $|\\vec{a}|=1$, $|\\vec{b}|=2$, and $\\vec{a} \\cdot \\vec{b}=1$. If $\\vec{e}$ is a two-dimensional unit vector, find the maximum value of $|\\vec{a} \\cdot \\vec{e}| + |\\vec{b} \\cdot \\vec{e}|$.\nPlease reason step by step, and put you... | math | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": 29660,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $XYZ$, $\\angle Y = 90^\\circ$, $YZ = 4$, and $XY = 5$. What is $\\tan X$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": 29661,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $z_1$, $z_2$, $z_3$, $\\dots$, $z_{8}$ be the 8 zeros of the polynomial $z^{8} - 16^8$. For each $j$, let $w_j$ be either $z_j$, $-z_j$, or $iz_j$. Find the maximum possible value of the real part of\n\\[\\sum_{j = 1}^{8} w_j.\\]\nPlease reason step by step, and put your final answer within \\... | math | {
"ground_truth": "32 + 32 \\sqrt{2}",
"style": "rule"
} | {
"index": 29664,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\\leq4$, calculate the value of $2x+y+z$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 29665,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "$2:5$",
"style": "rule"
} | {
"index": 29667,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "If the function \\( y = \\sin(w x) \\) with \\( w > 0 \\) attains its maximum value at least 50 times in the interval \\([0,1]\\), what is the minimum value of \\( w \\)?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "100 \\pi",
"style": "rule"
} | {
"index": 29669,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. What is the ratio of the perimeter of the rectangle to the perimeter of the rhombus?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role":... | math | {
"ground_truth": "1: 1",
"style": "rule"
} | {
"index": 29670,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Suppose $\\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\\circ$ clockwise. Let $P$ be a point wi... | math | {
"ground_truth": "\\frac{1 + \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": 29672,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In rectangle $ABCD$, $AB = 4$ cm, $BC = 10$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fourth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.\nPlease reason step by step, and put your final answer within \\box... | math | {
"ground_truth": "2\\sqrt{10}",
"style": "rule"
} | {
"index": 29674,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f(x) = 2\\sin x\\cos x - 2\\sin^2 x$ \n(1) Find the smallest positive period of the function $f(x)$. \n(2) Let $\\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$, respectively, and satisfy $f(A) = 0$, $c = 1$, $b = \\sqrt{2}$. Find the area of $\... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 29675,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": 29676,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $c$ and $d$ be real numbers such that\n\\[\\frac{c}{d} + \\frac{c}{d^3} + \\frac{c}{d^6} + \\dots = 9.\\]\nFind\n\\[\\frac{c}{c + 2d} + \\frac{c}{(c + 2d)^2} + \\frac{c}{(c + 2d)^3} + \\dotsb.\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
... | math | {
"ground_truth": "\\frac{9}{11}",
"style": "rule"
} | {
"index": 29677,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors and satisfy $\\overrightarrow{a} \\cdot \\overrightarrow{b} = 0$, find the maximum value of $(\\overrightarrow{a} + \\overrightarrow{b} + \\overrightarrow{c}) \\cdot (\\overrightarrow{a} + \\overrightarrow{c})$.\nPlease re... | math | {
"ground_truth": "2 + \\sqrt{5}",
"style": "rule"
} | {
"index": 29678,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Over the summer, a one-room apartment increased in price by 21%, a two-room apartment by 11%, and the total cost of both apartments by 15%. How many times cheaper is the one-room apartment compared to the two-room apartment?\nPlease reason step by step, and put your final answer within \\boxed{}."... | math | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": 29679,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "\\( z_{1}, z_{2}, z_{3} \\) are the three roots of the polynomial\n\\[ P(z) = z^{3} + a z + b \\]\nand satisfy the condition\n\\[ \\left|z_{1}\\right|^{2} + \\left|z_{2}\\right|^{2} + \\left|z_{3}\\right|^{2} = 250 \\]\nMoreover, the three points \\( z_{1}, z_{2}, z_{3} \\) in the complex plane fo... | math | {
"ground_truth": "5\\sqrt{15}",
"style": "rule"
} | {
"index": 29680,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Three of the four vertices of a rectangle are \\((1, 7)\\), \\((14, 7)\\), and \\((14, -4)\\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \\((x - 1)^2 + (y + 4)^2 = 16\\)?\nPlease reason step by step, and put your final answer wi... | math | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": 29681,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who also did not attemp... | math | {
"ground_truth": "\\frac{359}{600}",
"style": "rule"
} | {
"index": 29682,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "What is the probability that $abc + ab + a$ is divisible by $4$, where positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3, \\dots, 1024\\}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "us... | math | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": 29684,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a,b>0)\\) with left and right foci as \\(F_{1}\\) and \\(F_{2}\\), a line passing through \\(F_{2}\\) with an inclination angle of \\(\\frac{\\pi}{4}\\) intersects the hyperbola at a point \\(A\\). If the triangle \\(\\triangle F_{... | math | {
"ground_truth": "\\sqrt{2}+1",
"style": "rule"
} | {
"index": 29685,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Arrange numbers $ 1,\\ 2,\\ 3,\\ 4,\\ 5$ in a line. Any arrangements are equiprobable. Find the probability such that the sum of the numbers for the first, second and third equal to the sum of that of the third, fourth and fifth. Note that in each arrangement each number are used one time witho... | math | {
"ground_truth": "1/15",
"style": "rule"
} | {
"index": 29686,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"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 (\\overrightarrow{c} ... | math | {
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}",
"style": "rule"
} | {
"index": 29688,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In rectangle $JKLM$, $P$ is a point on $LM$ so that $\\angle JPL=90^{\\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$. [asy]\nsize(7cm);d... | math | {
"ground_truth": "\\dfrac{28}{3}",
"style": "rule"
} | {
"index": 29691,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{6}{35}",
"style": "rule"
} | {
"index": 29692,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given a triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $b^2 + c^2 - a^2 = \\sqrt{3}bc$.\n\n(1) If $\\tan B = \\frac{\\sqrt{6}}{12}$, find $\\frac{b}{a}$;\n\n(2) If $B = \\frac{2\\pi}{3}$ and $b = 2\\sqrt{3}$, find the length of the ... | math | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": 29693,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $A$ be a point on the parabola $y = x^2 - 4x + 4,$ and let $B$ be a point on the line $y = 2x - 3.$ Find shortest possible distance $AB.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 29695,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $\\triangle ABC$, $2b\\cos A+a=2c$, $c=8$, $\\sin A=\\frac{{3\\sqrt{3}}}{{14}}$. Find:\n\n$(Ⅰ)$ $\\angle B$;\n\n$(Ⅱ)$ the area of $\\triangle ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6\\sqrt{3}",
"style": "rule"
} | {
"index": 29699,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "You, your friend, and two strangers are sitting at a table. A standard $52$ -card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand cont... | math | {
"ground_truth": "22/703",
"style": "rule"
} | {
"index": 29701,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Calculate the radius of the circle where all the complex roots of the equation $(z - 1)^6 = 64z^6$ lie when plotted in the complex plane.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 29708,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{7}{64}",
"style": "rule"
} | {
"index": 29709,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Calculate the distance between the points $(5, -3)$ and $(-7, 4)$. Additionally, determine if the points $(5, -3)$, $(-7, 4)$, and $(5, 4)$ form a right triangle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{193}",
"style": "rule"
} | {
"index": 29712,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A right triangle has one angle measuring $30^\\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\\circ$, find the area of the quadrilateral given that... | math | {
"ground_truth": "\\frac{25\\sqrt{3} + 50}{2}",
"style": "rule"
} | {
"index": 29713,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "(1) Solve the inequality $|x+1| + 2|x-1| < 3x + 5$.\n(2) Given $a, b \\in [0, 1]$, find the maximum value of $ab + (1-a-b)(a+b)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": 29715,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Find the maximum real number \\( k \\) such that for any simple graph \\( G \\) with \\( n \\) vertices (\\( n \\geq 3 \\)), the inequality \\( x^3 \\geq k y^2 \\) holds, where \\( x \\) is the number of edges of \\( G \\) and \\( y \\) is the number of triangles in \\( G \\).\nPlease reason step ... | math | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": 29716,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Regular hexagon $GHJKLMN$ is the base of a right pyramid $QGHJKLMN$. If $QGM$ is an equilateral triangle with side length 10, then what is the volume of the pyramid?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "187.5",
"style": "rule"
} | {
"index": 29718,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A modified deck of cards contains 60 cards consisting of 15 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Prince, and Princess) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, $\\clubsuit$). Hence, each suit now has 15 cards. Four suits are still divided into two colors wi... | math | {
"ground_truth": "\\frac{7}{20}",
"style": "rule"
} | {
"index": 29723,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given points O(0,0) and A(1,1), the slope angle of line OA is $\\boxed{\\text{answer}}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": 29725,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A valuable right-angled triangular metal plate $A B O$ is placed in a plane rectangular coordinate system (as shown in the diagram), with $A B = B O = 1$ (meter) and $A B \\perp O B$. Due to damage in the shaded part of the triangular plate, a line $M N$ passing through the point $P\\left(\\frac{1... | math | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": 29726,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $P$ and $Q$ be points on the circle $(x-0)^2+(y-6)^2=2$ and the ellipse $\\frac{x^2}{10}+y^2=1$, respectively. What is the maximum distance between $P$ and $Q$?\nA) $5\\sqrt{2}$\nB) $\\sqrt{46}+\\sqrt{2}$\nC) $7+\\sqrt{2}$\nD) $6\\sqrt{2}$\nPlease reason step by step, and put your final answer... | math | {
"ground_truth": "6\\sqrt{2}",
"style": "rule"
} | {
"index": 29728,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f(x)=\\sin (2x+ \\frac {π}{3})- \\sqrt {3}\\sin (2x- \\frac {π}{6})$\n(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;\n(2) When $x\\in\\[- \\frac {π}{6}, \\frac {π}{3}\\]$, find the maximum and minimum values of $f(x)$, an... | math | {
"ground_truth": "-\\sqrt {3}",
"style": "rule"
} | {
"index": 29729,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In a bag, there are three balls of different colors: red, yellow, and blue, each color having one ball. Each time a ball is drawn from the bag, its color is recorded and then the ball is put back. The drawing stops when all three colors of balls have been drawn, what is the probability of stopping... | math | {
"ground_truth": "\\frac{14}{81}",
"style": "rule"
} | {
"index": 29730,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Express $0.6\\overline{03}$ as a common fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{104}{165}",
"style": "rule"
} | {
"index": 29732,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Arrange the numbers 3, 4, 5, 6 into two natural numbers A and B, so that the product A×B is maximized. Find the value of A×B.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\left( 3402 \\right)",
"style": "rule"
} | {
"index": 29733,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function \\( f(x) = ax^2 + 8x + 3 \\) where \\( a < 0 \\). For a given negative \\( a \\), there exists a largest positive real number \\( L(a) \\) such that the inequality \\( |f(x)| \\leq 5 \\) holds for the entire interval \\([0, L(a)]\\). Find the value of \\( a \\) that maximizes \\... | math | {
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}",
"style": "rule"
} | {
"index": 29736,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "If $x$ is a real number and $k$ is a nonnegative integer, compute the value of\n\\[\n \\frac{\\binom{1/2}{2015} \\cdot 4^{2015}}{\\binom{4030}{2015}} \\, .\n\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{1}{4030 \\cdot 4029 \\cdot 4028}",
"style": "rule"
} | {
"index": 29742,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"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 $m$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{8}{9}",
"style": "rule"
} | {
"index": 29743,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Compute $\\arccos (\\cos 3).$ All functions are in radians.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3 - 2\\pi",
"style": "rule"
} | {
"index": 29745,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\\begin{cases} x=5\\cos \\alpha \\\\ y=\\sin \\alpha \\end{cases}$ where $\\alpha$ is a parameter, and point $P$ has coordinates $(3 \\sqrt {2},0)$.\n(1) Determine the shape of curve $C$;\n(2) Give... | math | {
"ground_truth": "\\frac{7}{13}",
"style": "rule"
} | {
"index": 29746,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?\nPlease reason step by step, and put your final answer... | math | {
"ground_truth": "1/36",
"style": "rule"
} | {
"index": 29750,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "On a busy afternoon, David decides to drink a cup of water every 20 minutes to stay hydrated. Assuming he maintains this pace, how many cups of water does David drink in 3 hours and 45 minutes?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "11.25",
"style": "rule"
} | {
"index": 29751,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $XYZ$, we have $\\angle X = 90^\\circ$, $YZ = 20$, and $\\tan Z = 3 \\sin Y$. Calculate $XY$.\n\n---\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{40\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": 29754,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, find the ratio of the shorter side to the longer side of this frame... | math | {
"ground_truth": "2:3",
"style": "rule"
} | {
"index": 29757,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Convex quadrilateral $ABCD$ has $AB = 10$ and $CD = 15$. Diagonals $AC$ and $BD$ intersect at $E$, where $AC = 18$, and $\\triangle AED$ and $\\triangle BEC$ have equal perimeters. Calculate the length of $AE$.\n**A)** $6$ \n**B)** $7.2$ \n**C)** $7.5$ \n**D)** $9$ \n**E)** $10$\nPlease reason... | math | {
"ground_truth": "7.2",
"style": "rule"
} | {
"index": 29759,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "For each of the possible outcomes of rolling three coins (TTT, THT, TTH, HHT, HTH, HHH), a fair die is rolled for each two heads that appear together. What is the probability that the sum of the die rolls is odd?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role"... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": 29761,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Yan is between his home and the library. To get to the library, he can either walk directly to the library or walk home and then ride his bicycle to the library. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his hom... | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 29762,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the polar coordinate system, the equation of curve $C$ is $\\rho^{2}\\cos 2\\theta=9$. Point $P(2 \\sqrt {3}, \\frac {\\pi}{6})$ is given. Establish a Cartesian coordinate system with the pole $O$ as the origin and the positive half-axis of the $x$-axis as the polar axis. \n$(1)$ Find the para... | math | {
"ground_truth": "\\sqrt {2}",
"style": "rule"
} | {
"index": 29763,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In right triangle $XYZ$ with $\\angle Z = 90^\\circ$, the lengths of sides $XY$ and $XZ$ are 8 and 3 respectively. Find $\\cos Y$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": 29764,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given $f(α) = \\frac{\\sin(2π-α)\\cos(π+α)\\cos\\left(\\frac{π}{2}+α\\right)\\cos\\left(\\frac{11π}{2}-α\\right)}{2\\sin(3π + α)\\sin(-π - α)\\sin\\left(\\frac{9π}{2} + α\\right)}$. \n(1) Simplify $f(α)$; \n(2) If $α = -\\frac{25}{4}π$, find the value of $f(α)$.\nPlease reason step by step, and ... | math | {
"ground_truth": "-\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": 29766,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A gambler starts with $100. During a series of 4 rounds, they bet each time one-third of the amount they have. They win twice and lose twice, but now the wins are twice the staked amount, and losses mean losing the staked amount. Determine the final amount of money the gambler has, assuming the wi... | math | {
"ground_truth": "\\frac{8000}{81}",
"style": "rule"
} | {
"index": 29768,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "16.67\\%",
"style": "rule"
} | {
"index": 29771,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The volume of a regular octagonal prism is 8 cubic meters, and its height is 2.2 meters. Find the lateral surface area of the prism.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "16 \\sqrt{2.2 (\\sqrt{2} - 1)}",
"style": "rule"
} | {
"index": 29774,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given an equilateral triangle $ABC$ with a circle of radius $3$ tangent to line $AB$ at $B$ and to line $AC$ at $C$, find the area of the circle that passes through vertices $A$, $B$, and $C$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "36\\pi",
"style": "rule"
} | {
"index": 29776,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The midpoints of the sides of a parallelogram with area $P$ are joined to form a smaller parallelogram inside it. What is the ratio of the area of the smaller parallelogram to the area of the original parallelogram? Express your answer as a common fraction.\nPlease reason step by step, and put you... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": 29779,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that $\\frac{3\\pi}{4} < \\alpha < \\pi$, find the values for the following:\n1. $\\tan\\alpha$ given $\\tan\\alpha + \\frac{1}{\\tan\\alpha} = -\\frac{10}{3}$.\n2. The value of $$\\frac{5\\sin^2\\left(\\frac{\\alpha}{2}\\right) + 8\\sin\\left(\\frac{\\alpha}{2}\\right)\\cos\\left(\\frac{\\a... | math | {
"ground_truth": "-\\frac{5}{4}",
"style": "rule"
} | {
"index": 29780,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The solutions to the equation $x(5x+2) = 6(5x+2)$ are ___.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{2}{5}",
"style": "rule"
} | {
"index": 29781,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that $α \\in ( \\frac{5}{4}π, \\frac{3}{2}π)$, and it satisfies $\\tan α + \\frac{1}{\\tan α} = 8$, find the value of $\\sin α \\cos α$ and $\\sin α - \\cos α$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "- \\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": 29782,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that vertex $E$ of right $\\triangle ABE$ (with $\\angle ABE = 90^\\circ$) is inside square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and hypotenuse $AE$, find the area of $\\triangle ABF$ where the length of $AB$ is $\\sqrt{2}$.\nPlease reason step by step, and put your ... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 29783,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A projectile is launched with an initial velocity $u$ at an angle $\\phi$ from the ground. The trajectory can be described by the parametric equations:\n\\[\nx = ut \\cos \\phi, \\quad y = ut \\sin \\phi - \\frac{1}{2} g t^2,\n\\]\nwhere $t$ is the time, $g$ is the acceleration due to gravity, and... | math | {
"ground_truth": "\\frac{\\pi}{8}",
"style": "rule"
} | {
"index": 29786,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Compute $\\arccos (\\cos 3).$ All functions are in radians.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3 - 2\\pi",
"style": "rule"
} | {
"index": 29788,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "1. Given that the terminal side of angle $\\alpha$ passes through point $P(4, -3)$, find the value of $2\\sin\\alpha + \\cos\\alpha$.\n2. Given that the terminal side of angle $\\alpha$ passes through point $P(4a, -3a)$ ($a \\neq 0$), find the value of $2\\sin\\alpha + \\cos\\alpha$.\n3. Given tha... | math | {
"ground_truth": "-\\frac{2}{5}",
"style": "rule"
} | {
"index": 29791,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The solutions to the equations $z^2 = 9 + 9\\sqrt{7}i$ and $z^2 = 5 + 5\\sqrt{2}i$, where $i = \\sqrt{-1}$, form the vertices of a parallelogram in the complex plane. Determine the area of this parallelogram, which can be written in the form $p\\sqrt{q} - r\\sqrt{s}$, where $p, q, r, s$ are positi... | math | {
"ground_truth": "\\frac{3}{4}\\sqrt{35} - \\frac{3}{4}\\sqrt{15}",
"style": "rule"
} | {
"index": 29793,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\\mu = 1$ and the variance $\\sigma^2 = 4$. Given that $P(X \\geq 2) = 0.2$, calculate the probability $P(0 \\leq X \\leq 1)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "us... | math | {
"ground_truth": "0.3",
"style": "rule"
} | {
"index": 29794,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter.\nPlease reason step by step, and put y... | math | {
"ground_truth": "1:4",
"style": "rule"
} | {
"index": 29795,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given complex numbers $u$ and $v$ such that $|u+v| = 3$ and $|u^2 + v^2| = 10,$ determine the largest possible value of $|u^3 + v^3|.$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "31.5",
"style": "rule"
} | {
"index": 29798,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, \\pi]$ and $[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$, respectively. Define $P(\\alpha)$ as the probability that\n\\[\\cos^2{x} + \\cos^2{y} < \\alpha\\]\nwhere $\\alpha$ is a constant with $1 < \\alpha \\leq 2$. Find the... | math | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": 29799,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "$O$ is the origin, and $F$ is the focus of the parabola $C:y^{2}=4x$. A line passing through $F$ intersects $C$ at points $A$ and $B$, and $\\overrightarrow{FA}=2\\overrightarrow{BF}$. Find the area of $\\triangle OAB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "\\dfrac{3\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": 29800,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A spinner has four sections labeled 1, 2, 3, and 4, each section being equally likely to be selected. If you spin the spinner three times to form a three-digit number, with the first outcome as the hundreds digit, the second as the tens digit, and the third as the unit digit, what is the probabili... | math | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": 29802,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Consider the infinite series defined by the following progression:\n\\[2 + \\frac{1}{3} + \\frac{1}{9} + \\frac{1}{3^2} + \\frac{1}{9^2} + \\frac{1}{3^3} + \\frac{1}{9^3} + \\cdots\\]\nDetermine the limit of this series as it extends to infinity.\nA) $\\frac{1}{3}$\nB) $3$\nC) $\\frac{21}{8}$\nD) ... | math | {
"ground_truth": "\\frac{21}{8}",
"style": "rule"
} | {
"index": 29805,
"split": "train",
"acc@8": null
} |
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