data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"content": "Given point $P$ is a moving point on the ellipse $\\frac{x^2}{8} + \\frac{y^2}{4} = 1$ in the first quadrant, through point $P$, two tangents $PA$ and $PB$ are drawn to the circle $x^2 + y^2 = 4$, with the points of tangency being $A$ and $B$ respectively. The line $AB$ intersects the $x$-axis and... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 29806,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that the four vertices of the quadrilateral $MNPQ$ are on the graph of the function $f(x)=\\log_{\\frac{1}{2}} \\frac{ax+1}{x+b}$, and it satisfies $\\overrightarrow{MN}= \\overrightarrow{QP}$, where $M(3,-1)$, $N\\left( \\frac{5}{3},-2\\right)$, then the area of the quadrilateral $MNPQ$ is ... | math | {
"ground_truth": "\\frac{26}{3}",
"style": "rule"
} | {
"index": 29808,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the parabola \\( C: x^{2} = 2py \\) with \\( p > 0 \\), two tangents \\( RA \\) and \\( RB \\) are drawn from the point \\( R(1, -1) \\) to the parabola \\( C \\). The points of tangency are \\( A \\) and \\( B \\). Find the minimum area of the triangle \\( \\triangle RAB \\) as \\( p \\) va... | math | {
"ground_truth": "3 \\sqrt{3}",
"style": "rule"
} | {
"index": 29810,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The base of the quadrilateral pyramid \\( S A B C D \\) is a rhombus \\( A B C D \\) with an acute angle at vertex \\( A \\). The height of the rhombus is 4, and the point of intersection of its diagonals is the orthogonal projection of vertex \\( S \\) onto the plane of the base. A sphere with ra... | math | {
"ground_truth": "8\\sqrt{2}",
"style": "rule"
} | {
"index": 29812,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f(x)=\\ln (1+x)- \\frac {x(1+λx)}{1+x}$, if $f(x)\\leqslant 0$ when $x\\geqslant 0$, calculate the minimum value of $λ$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": 29814,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the ellipse $\\frac{x^2}{16} + \\frac{y^2}{b^2} = 1$, a line passing through its left focus intersects the ellipse at points $A$ and $B$, and the maximum value of $|AF_{2}| + |BF_{2}|$ is $10$. Find the eccentricity of the ellipse.\nPlease reason step by step, and put your final answer withi... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 29818,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... | math | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": 29819,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the arithmetic sequence $\\{a_{n}\\}$, given that $a_{3}=-2, a_{n}=\\frac{3}{2}, S_{n}=-\\frac{15}{2}$, find the value of $a_{1}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{19}{6}",
"style": "rule"
} | {
"index": 29822,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Calculate:<br/>$(1)(\\sqrt{\\frac{1}{3}})^{2}+\\sqrt{0．{3}^{2}}-\\sqrt{\\frac{1}{9}}$;<br/>$(2)(\\sqrt{6}-\\sqrt{\\frac{1}{2}})-(\\sqrt{24}+2\\sqrt{\\frac{2}{3}})$;<br/>$(3)(\\frac{\\sqrt{32}}{3}-4\\sqrt{\\frac{1}{2}}+3\\sqrt{27})÷2\\sqrt{2}$;<br/>$(4)(\\sqrt{3}+\\sqrt{2}-1)(\\sqrt{3}-\\sqrt{2}+1)... | math | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": 29823,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The postal department stipulates that for letters weighing up to $100$ grams (including $100$ grams), each $20$ grams requires a postage stamp of $0.8$ yuan. If the weight is less than $20$ grams, it is rounded up to $20$ grams. For weights exceeding $100$ grams, the initial postage is $4$ yuan. F... | math | {
"ground_truth": "5.6",
"style": "rule"
} | {
"index": 29825,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that the point F(0,1) is the focus of the parabola $x^2=2py$,\n\n(1) Find the equation of the parabola C;\n(2) Points A, B, and C are three points on the parabola such that $\\overrightarrow{FA} + \\overrightarrow{FB} + \\overrightarrow{FC} = \\overrightarrow{0}$, find the maximum value of t... | math | {
"ground_truth": "\\frac{3\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": 29828,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $\\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$ . Suppose $P$ is a point inside $\\triangle ABC$ such that $\\triangle BPA\\sim \\triangle APC$ . If $AP$ intersects $BC$ at $X$ , find $\\frac{BX}{CX}$ .\n\n[i]Proposed by Nathan Ramesh\nPlease reason step by step, and put y... | math | {
"ground_truth": "25/49",
"style": "rule"
} | {
"index": 29830,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f\\left(x\\right)=ax^{2}-bx-1$, sets $P=\\{1,2,3,4\\}$, $Q=\\{2,4,6,8\\}$, if a number $a$ and a number $b$ are randomly selected from sets $P$ and $Q$ respectively to form a pair $\\left(a,b\\right)$.<br/>$(1)$ Let event $A$ be \"the monotonically increasing interval of the fu... | math | {
"ground_truth": "\\frac{11}{16}",
"style": "rule"
} | {
"index": 29832,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that 20% of the birds are geese, 40% are swans, 10% are herons, and 20% are ducks, and the remaining are pigeons, calculate the percentage of the birds that are not herons and are ducks.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "22.22\\%",
"style": "rule"
} | {
"index": 29833,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the polar coordinate system, the equation of circle C is given by ρ = 2sinθ, and the equation of line l is given by $ρsin(θ+ \\frac {π}{3})=a$. If line l is tangent to circle C, find the value of the real number a.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"... | math | {
"ground_truth": "- \\frac {1}{2}",
"style": "rule"
} | {
"index": 29835,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that in $\\triangle ABC$, $\\sin A + 2 \\sin B \\cos C = 0$, find the maximum value of $\\tan A$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 29837,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Find the area of the triangle formed by the axis of the parabola $y^{2}=8x$ and the two asymptotes of the hyperbola $(C)$: $\\frac{x^{2}}{8}-\\frac{y^{2}}{4}=1$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": 29838,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A certain teacher received $10$, $6$, $8$, $5$, $6$ letters from Monday to Friday, then the variance of this data set is $s^{2}=$____.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3.2",
"style": "rule"
} | {
"index": 29844,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In square $ABCD$, a point $P$ is chosen at random. The probability that $\\angle APB < 90^{\\circ}$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1 - \\frac{\\pi}{8}",
"style": "rule"
} | {
"index": 29846,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 points. In a game betwee... | math | {
"ground_truth": "0.3072",
"style": "rule"
} | {
"index": 29847,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(6, 0)$ and tangent to the $y$-axis at $(0, 2)$. Find the distance between the foci of the ellipse.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": 29848,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Suppose that \\( ABCDEF \\) is a regular hexagon with sides of length 6. Each interior angle of \\( ABCDEF \\) is equal to \\( 120^{\\circ} \\).\n\n(a) A circular arc with center \\( D \\) and radius 6 is drawn from \\( C \\) to \\( E \\). Determine the area of the shaded sector.\n\n(b) A circular... | math | {
"ground_truth": "18\\pi - 27\\sqrt{3}",
"style": "rule"
} | {
"index": 29849,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system using the same unit of length. The parametric equations of line $l$ are given by $\\begin{cases}x=2+t \\\\ y=1+t\\end{cases} (t \\text{ i... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 29850,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\\cos C=3a\\cos B-c\\cos B$, $\\overrightarrow{BA}\\cdot \\overrightarrow{BC}=2$, find the area of $\\triangle ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | math | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": 29851,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given a line $y=-x+1$ and an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B.\n(1) If the eccentricity of the ellipse is $\\frac{\\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB.\n(2) (For Liberal Arts students) If segm... | math | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": 29854,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Triangle $ABC$ has $\\angle BAC=90^\\circ$ . A semicircle with diameter $XY$ is inscribed inside $\\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , $AC=4$ , and $AX=\... | math | {
"ground_truth": "39/32",
"style": "rule"
} | {
"index": 29855,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $ABC$, it is known that $\\sqrt{3}\\sin{2B} = 1 - \\cos{2B}$.\n\n(1) Find the value of angle $B$; \n(2) If $BC = 2$ and $A = \\frac{\\pi}{4}$, find the area of triangle $ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3 + \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": 29856,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given any point $P$ on the line $l: x-y+4=0$, two tangent lines $AB$ are drawn to the circle $O: x^{2}+y^{2}=4$ with tangent points $A$ and $B$. The line $AB$ passes through a fixed point ______; let the midpoint of segment $AB$ be $Q$. The minimum distance from point $Q$ to the line $l$ is ______... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 29857,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that \\( x \\) and \\( y \\) are non-zero real numbers and they satisfy \\(\\frac{x \\sin \\frac{\\pi}{5} + y \\cos \\frac{\\pi}{5}}{x \\cos \\frac{\\pi}{5} - y \\sin \\frac{\\pi}{5}} = \\tan \\frac{9 \\pi}{20}\\),\n1. Find the value of \\(\\frac{y}{x}\\).\n2. In \\(\\triangle ABC\\), if \\(... | math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": 29859,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $\\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles.\n\nSuppose that the edges of $WXYZ$ have lengths of either $24$ or $49$, and no face of $WXYZ$ is equilateral. Determine the sur... | math | {
"ground_truth": "48 \\sqrt{2257}",
"style": "rule"
} | {
"index": 29861,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Rectangle $ABCD$ has $AB = CD = 3$ and $BC = DA = 5$. The rectangle is first rotated $90^\\circ$ clockwise around vertex $D$, then it is rotated $90^\\circ$ clockwise around the new position of vertex $C$ (after the first rotation). What is the length of the path traveled by point $A$?\nA) $\\frac... | math | {
"ground_truth": "\\frac{\\pi(\\sqrt{34} + 5)}{2}",
"style": "rule"
} | {
"index": 29863,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \\frac{a^{2}+1}{ab} - 2) \\cdot c + \\frac{\\sqrt{2}}{c - 1}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "4 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": 29864,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-10, 15]$. Find the probability that the roots of the polynomial\n\\[ x^4 + 3ax^3 + (3a - 3)x^2 + (-5a + 4)x - 3 \\]\nare all real.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "\\frac{23}{25}",
"style": "rule"
} | {
"index": 29865,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Juan rolls a fair regular dodecahedral die marked with the numbers 1 through 12. Then Amal rolls a fair eight-sided die. What is the probability that the product of the two rolls is a multiple of 4?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": 29866,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $p, q, r,$ and $s$ be the roots of the polynomial $3x^4 - 8x^3 - 15x^2 + 10x - 2 = 0$. Find $pqrs$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 29871,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that \\( O \\) is the circumcenter of \\( \\triangle ABC \\), and the equation\n\\[\n\\overrightarrow{A O} \\cdot \\overrightarrow{B C} + 2 \\overrightarrow{B O} \\cdot \\overrightarrow{C A} + 3 \\overrightarrow{C O} \\cdot \\overrightarrow{A B} = 0,\n\\]\nfind the minimum value of \\( \\fra... | math | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 29875,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Evaluate $|5 - e|$ where $e$ is the base of the natural logarithm.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2.28172",
"style": "rule"
} | {
"index": 29878,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows:\n\n0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot:\n\n(1) Hits either the 10 or 9 ring; (2) Scores less than 7 rings.\nPlease reason ... | math | {
"ground_truth": "0.03",
"style": "rule"
} | {
"index": 29880,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let point $P$ be a point on the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of the ellipse, and let $I$ be the incenter of $\\triangle PF_1F_2$. If $S_{\\triangle IPF_1} + S_{\\triangle IPF_2} = 2S_{\\triangle IF_1F_2}$... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 29882,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0.294",
"style": "rule"
} | {
"index": 29887,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In triangle $\\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and the vectors $\\overrightarrow{m}=({\\cos C, \\cos({\\frac{\\pi}{2}-B})})$, $\\overrightarrow{n}=({\\cos({-4\\pi+B}), -\\sin C})$, and $\\overrightarrow{m} \\cdot \\overrightarrow{n}=-\\frac{\\sqrt{2... | math | {
"ground_truth": "5 + 2\\sqrt{2} + \\sqrt{5}",
"style": "rule"
} | {
"index": 29888,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role":... | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 29889,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the ellipse $C: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a>b>0)$, its foci are equal to the minor axis length of the ellipse $Ω:x^{2}+ \\frac{y^{2}}{4}=1$, and the major axis lengths of C and Ω are equal.\n(1) Find the equation of ellipse C;\n(2) Let $F_1$, $F_2$ be the left and right fo... | math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 29890,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability of drawing exactly one red ball.<br/>$(2)$ If the random variable $X$ represents the number of red balls drawn, find the di... | math | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": 29894,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Choose any $2$ numbers from $-5$, $-3$, $-1$, $2$, and $4$. Let the maximum product obtained be denoted as $a$, and the minimum quotient obtained be denoted as $b$. Then the value of $\\frac{a}{b}$ is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "u... | math | {
"ground_truth": "-\\frac{15}{4}",
"style": "rule"
} | {
"index": 29895,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given vectors $\\overrightarrow{a}=(\\cos x,\\sin x)$ and $\\overrightarrow{b}=(3,-\\sqrt{3})$, with $x\\in[0,\\pi]$.\n$(1)$ If $\\overrightarrow{a}\\parallel\\overrightarrow{b}$, find the value of $x$; $(2)$ Let $f(x)=\\overrightarrow{a}\\cdot \\overrightarrow{b}$, find the maximum and minimum va... | math | {
"ground_truth": "-2\\sqrt{3}",
"style": "rule"
} | {
"index": 29901,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Consider a sequence of complex numbers $\\left\\{z_{n}\\right\\}$ defined as \"interesting\" if $\\left|z_{1}\\right|=1$ and for every positive integer $n$, the following holds:\n$$\n4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.\n$$\n\nFind the largest constant $C$ such that for any \"interesting\" s... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 29906,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "**Problem Statement**: Let $r$ and $k$ be integers such that $-5 < r < 8$ and $0 < k < 10$. What is the probability that the division $r \\div k$ results in an integer value? Express your answer as a common fraction.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"r... | math | {
"ground_truth": "\\frac{33}{108}",
"style": "rule"
} | {
"index": 29909,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given three positive numbers \\( a, b, \\mathrm{and} c \\) satisfying \\( a \\leq b+c \\leq 3a \\) and \\( 3b^2 \\leq a(a+c) \\leq 5b^2 \\), what is the minimum value of \\(\\frac{b-2c}{a}\\)?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{18}{5}",
"style": "rule"
} | {
"index": 29911,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Five people can paint a house in 10 hours. How many hours would it take four people to paint the same house and mow the lawn if mowing the lawn takes an additional 3 hours per person, assuming that each person works at the same rate for painting and different rate for mowing?\nPlease reason step b... | math | {
"ground_truth": "15.5",
"style": "rule"
} | {
"index": 29912,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f(x)=2\\sin(2x-\\frac{\\pi}{3})-1$, find the probability that a real number $a$ randomly selected from the interval $\\left[0,\\frac{\\pi}{2}\\right]$ satisfies $f(a) > 0$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 29916,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the parabola $y^{2}=2x$ with focus $F$, a line passing through $F$ intersects the parabola at points $A$ and $B$. If $|AB|= \\frac{25}{12}$ and $|AF| < |BF|$, determine the value of $|AF|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": 29917,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let us call a number \\( \\mathrm{X} \\) \"50-podpyirayushchim\" if for any 50 real numbers \\( a_{1}, \\ldots, a_{50} \\) whose sum is an integer, there exists at least one \\( a_i \\) such that \\( \\left|a_{i}-\\frac{1}{2}\\right| \\geq X \\).\n\nFind the greatest 50-podpyirayushchee \\( X \\),... | math | {
"ground_truth": "0.01",
"style": "rule"
} | {
"index": 29918,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Consider the ellipse (C) with the equation \\\\(\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)\\\\) and an eccentricity of \\\\(\\frac{\\sqrt{3}}{3}\\\\). A line passing through the right focus (F2(c,0)) perpendicular to the x-axis intersects the ellipse at points A and B, such that |AB|... | math | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": 29919,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given $m$ and $n∈\\{lg2+lg5,lo{g}_{4}3,{(\\frac{1}{3})}^{-\\frac{3}{5}},tan1\\}$, the probability that the function $f\\left(x\\right)=x^{2}+2mx+n^{2}$ has two distinct zeros is ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": 29920,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3, \\dots, 2020\\}$. What is the probability that $abc + ab + a$ is divisible by $4$?\nA) $\\frac{1}{4}$\nB) $\\frac{1}{32}$\nC) $\\frac{8}{32}$\nD) $\\frac{9}{32}$\nE) $\\frac{1}{16... | math | {
"ground_truth": "\\frac{9}{32}",
"style": "rule"
} | {
"index": 29921,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\\frac {(a+b)^{2}-c^{2}}{3ab}=1$. \n$(1)$ Find $\\angle C$; \n$(2)$ If $c= \\sqrt {3}$ and $b= \\sqrt {2}$, find $\\angle B$ and the area of $\\triangle ABC$.\... | math | {
"ground_truth": "\\frac {3+ \\sqrt {3}}{4}",
"style": "rule"
} | {
"index": 29922,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest squa... | math | {
"ground_truth": "\\frac{500}{3}",
"style": "rule"
} | {
"index": 29923,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the Cartesian coordinate system, there are points $P_{1}, P_{2}, \\ldots, P_{n-1}, P_{n}, \\ldots (n \\in \\mathbb{N}^{*})$. Let the coordinates of point $P_{n}$ be $(n, a_{n})$, where $a_{n}= \\frac {2}{n} (n \\in \\mathbb{N}^{*})$. The line passing through points $P_{n}$ and $P_{n+1}$ forms a... | math | {
"ground_truth": "\\frac {125}{6}",
"style": "rule"
} | {
"index": 29924,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that a smaller circle is entirely inside a larger circle, such that the larger circle has a radius $R = 2$, and the areas of the two circles form an arithmetic progression, with the largest area being that of the larger circle, find the radius of the smaller circle.\nPlease reason step by st... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 29927,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Suppose convex hexagon $ \\text{HEXAGN}$ has $ 120^\\circ$ -rotational symmetry about a point $ P$ —that is, if you rotate it $ 120^\\circ$ about $ P$ , it doesn't change. If $ PX\\equal{}1$ , find the area of triangle $ \\triangle{GHX}$ .\nPlease reason step by step, and put your final a... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 29928,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let \\(\\theta\\) be an angle in the second quadrant, and if \\(\\tan (\\theta+ \\frac {\\pi}{3})= \\frac {1}{2}\\), calculate the value of \\(\\sin \\theta+ \\sqrt {3}\\cos \\theta\\).\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": 29929,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Calculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.\n\n$A_{1}(-2, 0, -4)$\n\n$A_{2}(-1, 7, 1)$\n\n$A_{3}(4, -8, -4)$\n\n$A_{4}(1, -4, 6)$\nPlease reason step by step, and put your fi... | math | {
"ground_truth": "5\\sqrt{2}",
"style": "rule"
} | {
"index": 29933,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In $\\triangle ABC$, $2\\sin ^{2} \\frac{A}{2}= \\sqrt{3}\\sin A$, $\\sin (B-C)=2\\cos B\\sin C$, find the value of $\\frac{AC}{AB}$ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1+\\sqrt{13}}{2}",
"style": "rule"
} | {
"index": 29934,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the function $f(x)= \\sqrt {3}\\sin x\\cdot\\cos x- \\frac {1}{2}\\cos 2x$ $(x\\in\\mathbb{R})$. \n$(1)$ Find the minimum value and the smallest positive period of the function $f(x)$. \n$(2)$ Let $\\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectiv... | math | {
"ground_truth": "2 \\sqrt {3}",
"style": "rule"
} | {
"index": 29940,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In an isosceles triangle \\(ABC\\) with base \\(AC\\) equal to 37, the exterior angle at vertex \\(B\\) is \\(60^\\circ\\). Find the distance from vertex \\(C\\) to line \\(AB\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "18.5",
"style": "rule"
} | {
"index": 29941,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given that \\( p \\) and \\( q \\) are positive integers such that \\( p + q > 2017 \\), \\( 0 < p < q \\leq 2017 \\), and \\((p, q) = 1\\), find the sum of all fractions of the form \\(\\frac{1}{pq}\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": 29942,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The average age of 40 sixth-graders is 12. The average age of 30 of their teachers is 45. What is the average age of all these sixth-graders and their teachers?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "26.14",
"style": "rule"
} | {
"index": 29944,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given the equation $x^3 - 12x^2 + 27x - 18 = 0$ with roots $a$, $b$, $c$, find the value of $\\frac{1}{a^3} + \\frac{1}{b^3} + \\frac{1}{c^3}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{13}{24}",
"style": "rule"
} | {
"index": 29945,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\\angle ASB = 30^\\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$.\nPlease reason step by s... | math | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": 29946,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Calculate the volume of the tetrahedron with vertices at points \\( A_{1}, A_{2}, A_{3}, A_{4} \\) and its height dropped from the vertex \\( A_{4} \\) to the face \\( A_{1} A_{2} A_{3} \\).\n\n\\( A_{1}(-2, -1, -1) \\)\n\n\\( A_{2}(0, 3, 2) \\)\n\n\\( A_{3}(3, 1, -4) \\)\n\n\\( A_{4}(-4, 7, 3) \\... | math | {
"ground_truth": "\\frac{140}{\\sqrt{1021}}",
"style": "rule"
} | {
"index": 29950,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Let $ABCD$ be a cyclic quadrilateral with sides $AB$, $BC$, $CD$, and $DA$. The side lengths are distinct integers less than $10$ and satisfy $BC + CD = AB + DA$. Find the largest possible value of the diagonal $BD$.\nA) $\\sqrt{93}$ \nB) $\\sqrt{\\frac{187}{2}}$\nC) $\\sqrt{\\frac{191}{2}}$ \nD) ... | math | {
"ground_truth": "\\sqrt{\\frac{191}{2}}",
"style": "rule"
} | {
"index": 29952,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given two non-collinear vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{a}|=|\\overrightarrow{b}|$, and $\\overrightarrow{a}\\perp(\\overrightarrow{a}-2\\overrightarrow{b})$, the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ______.\nPlease ... | math | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": 29953,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ Determine the area of $\\triangle ABC.$\n\n[asy]\ndraw((0,0)--(2,5)--(11,5)--(18,0)--cycle);\ndraw((11,5)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(2,5),NW);\nlabel(\"$B$\",(11,5),NE);\nlabe... | math | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": 29955,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In a 200-meter race, Sonic is 16 meters ahead of Dash when Sonic finishes the race. The next time they race, Sonic starts 2.5 times this lead distance behind Dash, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Determine the distance Sonic i... | math | {
"ground_truth": "19.2",
"style": "rule"
} | {
"index": 29958,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and a die with 8 faces numbered 1 to 8 is rolled. Determine the probability that the product of the numbers on the tile and the die will be a square.\nPlease reason step by step, and put your final answer... | math | {
"ground_truth": "\\frac{7}{48}",
"style": "rule"
} | {
"index": 29959,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Point $A$ lies on the line $y=\\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1334/255",
"style": "rule"
} | {
"index": 29961,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Given positive real numbers $a$, $b$, $c$, $d$ satisfying $a^{2}-ab+1=0$ and $c^{2}+d^{2}=1$, find the value of $ab$ when $\\left(a-c\\right)^{2}+\\left(b-d\\right)^{2}$ reaches its minimum.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\sqrt{2}}{2} + 1",
"style": "rule"
} | {
"index": 29964,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "The area of triangle \\(ABC\\) is 1. Let \\(A_1\\), \\(B_1\\), and \\(C_1\\) be the midpoints of the sides \\(BC\\), \\(CA\\), and \\(AB\\) respectively. Points \\(K\\), \\(L\\), and \\(M\\) are taken on segments \\(AB_1\\), \\(CA_1\\), and \\(BC_1\\) respectively. What is the minimum area of the ... | math | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": 29965,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^{2}+c^{2}-b^{2}=ac$, $c=2$, and point $G$ satisfies $| \\overrightarrow{BG}|= \\frac { \\sqrt {19}}{3}$ and $\\overrightarrow{BG}= \\frac {1}{3}( \\overrightarrow{BA}+... | math | {
"ground_truth": "\\frac {3 \\sqrt {21}}{14}",
"style": "rule"
} | {
"index": 29966,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "In parallelogram $ABCD$ , $AB = 10$ , and $AB = 2BC$ . Let $M$ be the midpoint of $CD$ , and suppose that $BM = 2AM$ . Compute $AM$ .\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": 29968,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"content": "Point $(x, y)$ is randomly selected from the rectangular region defined by vertices $(0, 0), (3013, 0), (3013, 3014),$ and $(0, 3014)$. What is the probability that $x > 8y$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3013}{48224}",
"style": "rule"
} | {
"index": 29969,
"split": "train",
"acc@8": null
} |
DeepScaleR | [
{
"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.\nPlease reason step by step, and p... | math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 29971,
"split": "train",
"acc@8": null
} |
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