data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data 3, 4, 5, a, b with an average of 4 and a median of m, where the probability of selecting the number 4 from the set 3, 4, 5, a, b, m is $\\frac{2}{3}$, calculate the variance of the set 3, 4, 5, a, b."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 13700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data arranged in ascending order, which are -1, 0, 4, x, 7, 14, and the median is 5, find the variance of this set of data."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{74}{3}"
} | {
"index": 13701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data is multiplied by 2 and then reduced by 80 for each data point, resulting in a new set of data with an average of 1.2 and a variance of 4.4, determine the average and variance of the original data."
}
] | math | {
"style": "rule",
"ground_truth": "1.1"
} | {
"index": 13702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data pairs (3,y_{1}), (5,y_{2}), (7,y_{3}), (12,y_{4}), (13,y_{5}) corresponding to variables x and y, the linear regression equation obtained is \\hat{y} = \\frac{1}{2}x + 20. Calculate the value of \\sum\\limits_{i=1}^{5}y_{i}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 13703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data: $10.1$, $9.8$, $10$, $x$, $10.2$, the average of these data is $10$. Calculate the variance of this set of data."
}
] | math | {
"style": "rule",
"ground_truth": "0.02"
} | {
"index": 13704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data: $2$, $x$, $4$, $6$, $10$, with an average value of $5$, find the standard deviation of this data set."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 13705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of data: 10, 10, x, 8, where the median is equal to the mean, find the median of this data set."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of four-ordered real number pairs \\((a, b, c, d)\\), where \\(a, b, c, d \\in \\{0, 1, 2, 3\\}\\) and \\(a, b, c, d\\) can be the same, calculate how many such pairs exist so that \\(ad - bc\\) is odd."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 13707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of paired data $(18,24)$, $(13,34)$, $(10,38)$, $(-1,m)$, the regression equation for these data is $y=-2x+59.5$. Find the correlation coefficient $r=$______(rounded to $0.001$)."
}
] | math | {
"style": "rule",
"ground_truth": "-0.998"
} | {
"index": 13708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perf... | math | {
"style": "rule",
"ground_truth": "1025"
} | {
"index": 13709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of sample data with $8$ numbers, the average is $8$, and the variance is $12$. Two unknown numbers are added to this set of sample data to form a new set of sample data. It is known that the average of the new sample data is $9$. Find the minimum value of the varian... | math | {
"style": "rule",
"ground_truth": "13.6"
} | {
"index": 13710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 13711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a set with three elements, it can be represented as $\\{a, \\frac{b}{a}, 1\\}$ and also as $\\{a^2, a+b, 0\\}$. Find the value of $a^{2013} + b^{2013}$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a shooter who has a probability of $\\frac{3}{4}$ of hitting target A with a single shot and a probability of $\\frac{2}{3}$ of hitting target B with each of two shots, determine the probability that the shooter hits exactly one of the three shots."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{36}"
} | {
"index": 13713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a six-digit phone number, how many different seven-digit phone numbers exist such that, by crossing out one digit, you obtain the given six-digit number?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 13714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a skewed six-sided die is structured so that rolling an odd number is twice as likely as rolling an even number, calculate the probability that, after rolling the die twice, the sum of the numbers rolled is odd."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 13715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a small cube block, each face is painted with a different color. If you want to carve 1, 2, 3, 4, 5, 6 small dots on the faces of the block, and the dots 1 and 6, 2 and 5, 3 and 4 are carved on opposite faces respectively, determine the number of different carving methods... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 13716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a solid $\\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\\alpha$, and $\\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\\angle A=90^{\\circ}$. Point $P$ is a moving point on the solid $\\Omega$, a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sphere O with a radius of 2, a cone is inscribed in the sphere O. When the volume of the cone is maximized, find the radius of the sphere inscribed in the cone."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4(\\sqrt{3} - 1)}{3}"
} | {
"index": 13718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sphere resting on a flat surface and a 1.5 m tall post, the shadow of the sphere is 15 m and the shadow of the post is 3 m, determine the radius of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 13719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "3 : 2 : 1"
} | {
"index": 13720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square $ABCD$ . Let $P\\in{AB},\\ Q\\in{BC},\\ R\\in{CD}\\ S\\in{DA}$ and $PR\\Vert BC,\\ SQ\\Vert AB$ and let $Z=PR\\cap SQ$ . If $BP=7,\\ BQ=6,\\ DZ=5$ , then find the side length of the square."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square $A B C D$ on a plane, find the minimum of the ratio $\\frac{O A + O C}{O B + O D}$, where $O$ is an arbitrary point on the plane."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"index": 13722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$."
}
] | math | {
"style": "rule",
"ground_truth": "45^\\circ"
} | {
"index": 13723,
"solution": "\nGiven a square \\(ABCD\\) with side length \\(1\\), points \\(P\\) and \\(Q\\) are on sides \\(AB\\) and \\(AD\\) respectively. We are to find the angle \\( \\angle PCQ \\) given that the perimeter of \\( \\triangle APQ \\) is \\(2\\).\n\nLet \\( AP = x \\) and \\( AQ = y \\). Then,... |
DeepScaleR | [
{
"role": "user",
"content": "Given a square $R_1$ with an area of 25, where each side is trisected to form a smaller square $R_2$, and this process is repeated to form $R_3$ using the same trisection points strategy, calculate the area of $R_3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{400}{81}"
} | {
"index": 13724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square ABCD with a side length of 2, points M and N are the midpoints of sides BC and CD, respectively. If vector $\\overrightarrow {MN}$ = x $\\overrightarrow {AB}$ + y $\\overrightarrow {AD}$, find the values of xy and $\\overrightarrow {AM}$ • $\\overrightarrow {MN}$... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square \\( ABCD \\). Point \\( N \\) lies on side \\( AD \\) such that \\( AN : ND = 2 : 3 \\), point \\( F \\) lies on side \\( CD \\) such that \\( DF : FC = 1 : 4 \\), and point \\( K \\) lies on side \\( AB \\) such that \\( AK : KB = 1 : 4 \\). Find the angle \\( \... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 13726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square \\( PQRS \\) with an area of \\( 120 \\, \\text{cm}^2 \\). Point \\( T \\) is the midpoint of \\( PQ \\). The ratios are given as \\( QU: UR = 2:1 \\), \\( RV: VS = 3:1 \\), and \\( SW: WP = 4:1 \\).\n\nFind the area, in \\(\\text{cm}^2\\), of quadrilateral \\( T... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 13727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square \\(ABCD\\) on the plane, find the minimum of the ratio \\(\\frac{OA+OC}{OB+OD}\\), where \\(O\\) is an arbitrary point on the plane."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"index": 13728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square \\(ABCD\\) with side length 1, determine the maximum value of \\(PA \\cdot PB \\cdot PC \\cdot PD\\) where point \\(P\\) lies inside or on the boundary of the square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 13729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square \\(ABCD\\) with side length 2, \\(E\\) is the midpoint of \\(AB\\). The square is folded along lines \\(EC\\) and \\(ED\\) so that \\(AE\\) coincides with \\(BE\\), and point \\(A\\) coincides with point \\(B\\), named point \\(O\\). Calculate the volume of the t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square grid of size $2023 \\times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six... | math | {
"style": "rule",
"ground_truth": "338"
} | {
"index": 13731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square grid with the letters AMC9 arranged as described, starting at the 'A' in the middle, determine the number of different paths that allow one to spell AMC9 without revisiting any cells."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 13732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square initially painted black, with $\\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{729}{8192}"
} | {
"index": 13733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square piece of paper with side length $s$ folded in half diagonally, then cut along a line perpendicular to the fold from the midpoint of the hypotenuse to the opposite side, forming a large rectangle and two smaller, identical triangles, find the ratio of the perimete... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 13734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square pyramid \\(M-ABCD\\) with a square base such that \\(MA = MD\\), \\(MA \\perp AB\\), and the area of \\(\\triangle AMD\\) is 1, find the radius of the largest sphere that can fit into this square pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"index": 13735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square region with a side length of 1 meter, and a total of 5120 beans within the square with 4009 beans within the inscribed circle, determine the approximate value of pi rounded to three decimal places."
}
] | math | {
"style": "rule",
"ground_truth": "3.13"
} | {
"index": 13736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square side of length $s$. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is:"
}
] | math | {
"style": "rule",
"ground_truth": "$s\\sqrt{2}$"
} | {
"index": 13737,
"solution": "1. **Calculate the area of the square**: \n The area of a square with side length $s$ is given by:\n \\[\n \\text{Area of square} = s^2\n \\]\n\n2. **Determine the length of the diagonal of the square**: \n The diagonal of a square divides it into two 45-45-90 right triang... |
DeepScaleR | [
{
"role": "user",
"content": "Given a square with a side length of $1$, if two points are randomly selected with equal probability from the center and the vertices of the square, what is the probability that the distance between these two points is $\\frac{\\sqrt{2}}{2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 13738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 13739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a stalk of bamboo with nine sections, with three sections from the bottom holding 3.9 liters, and the four sections from the top holding three liters, determine the combined volume of the middle two sections."
}
] | math | {
"style": "rule",
"ground_truth": "2.1"
} | {
"index": 13740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a structure formed by joining eight unit cubes where one cube is at the center, and each face of the central cube is shared with one additional cube, calculate the ratio of the volume to the surface area in cubic units to square units."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 13741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a student measures his steps from one sidewalk to another on a number line where each marking represents 3 meters, calculate the position marking $z$ after taking 5 steps from the starting point, given that the student counts 8 steps and the total distance covered is 48 m... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 13742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a sufficiently large positive integer \\( n \\), which can be divided by all the integers from 1 to 250 except for two consecutive integers \\( k \\) and \\( k+1 \\), find \\( k \\)."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 13743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a supermarket with 100 customers, the table shows the number of customers in each age group and payment method category. Determine the number of customers aged between 40 and 60 years old who do not use mobile payment."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 13744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tesseract (4-dimensional hypercube), calculate the sum of the number of edges, vertices, and faces."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 13745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron $ABCD$, each edge is colored red or blue with equal probability. What is the probability that point $A$ can reach point $B$ through red edges within the tetrahedron $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron $ABCD$, with $AD$ perpendicular to plane $BCD$, $BC$ perpendicular to $CD$, $AD=2$, $BD=4$, calculate the surface area of the circumscribed sphere of tetrahedron $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"index": 13747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron $P-ABC$ with its circumscribed sphere's center $O$ on $AB$, and $PO \\perp$ plane $ABC$, $2AC = \\sqrt{3}AB$. If the volume of the tetrahedron $P-ABC$ is $\\frac{3}{2}$, find the volume of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}\\pi"
} | {
"index": 13748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron $P-ABC$, in the base $\\triangle ABC$, $\\angle BAC=60^{\\circ}$, $BC=\\sqrt{3}$, $PA\\perp$ plane $ABC$, $PA=2$, then the surface area of the circumscribed sphere of this tetrahedron is ______."
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"index": 13749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron O-ABC, where $\\angle BOC=90^\\circ$, $OA \\perpendicular$ plane BOC, and $AB= \\sqrt{10}$, $BC= \\sqrt{13}$, $AC= \\sqrt{5}$. Points O, A, B, and C are all on the surface of sphere S. Find the surface area of sphere S."
}
] | math | {
"style": "rule",
"ground_truth": "14\\pi"
} | {
"index": 13750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron P-ABC, if PA, PB, and PC are mutually perpendicular, and PA=2, PB=PC=1, then the radius of the inscribed sphere of the tetrahedron P-ABC is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}"
} | {
"index": 13751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\( A B C D \\) with side lengths \\( A B = 41 \\), \\( A C = 7 \\), \\( A D = 18 \\), \\( B C = 36 \\), \\( B D = 27 \\), and \\( C D = 13 \\), let \\( d \\) be the distance between the midpoints of edges \\( A B \\) and \\( C D \\). Find the value of \\( d... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 13752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\( P-ABC \\) with its four vertices on the surface of sphere \\( O \\), where \\( PA = PB = PC \\) and \\( \\triangle ABC \\) is an equilateral triangle with side length 2. \\( E \\) and \\( F \\) are the midpoints of \\( AC \\) and \\( BC \\) respectively,... | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"index": 13753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\( P-ABCD \\) where the edges \\( AB \\) and \\( BC \\) each have a length of \\(\\sqrt{2}\\), and all other edges have a length of 1, find the volume of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{6}"
} | {
"index": 13754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\( S-ABC \\), point \\( A_1 \\) is the centroid of \\( \\triangle SBC \\). Point \\( G \\) is on the segment \\( AA_1 \\) such that \\(\\frac{|AG|}{|GA_1|}=3\\). Line \\( SG \\) intersects the plane of \\( \\triangle ABC \\) at point \\( M \\). Determine \\... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 13755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\(A B C D\\) with the following edge lengths: \\(A B = 2\\), \\(C D = 2\\sqrt{5}\\), \\(A C = B D = 3\\), and \\(A D = B C = \\sqrt{5}\\), find the volume of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 13756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\(A B C D\\), where \\(B D = D C = C B = \\sqrt{2}\\), \\(A C = \\sqrt{3}\\), and \\(A B = A D = 1\\), find the cosine of the angle between line \\(B M\\) and line \\(A C\\), where \\(M\\) is the midpoint of \\(C D\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"index": 13757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\(ABCD\\), in what ratio does the plane passing through the intersection points of the medians of the faces \\(ABC\\), \\(ABD\\), and \\(BCD\\) divide the edge \\(BD\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 13758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tetrahedron \\(ABCD\\). Points \\(M\\), \\(N\\), and \\(K\\) lie on edges \\(AD\\), \\(BC\\), and \\(DC\\) respectively, such that \\(AM:MD = 1:3\\), \\(BN:NC = 1:1\\), and \\(CK:KD = 1:2\\). Construct the section of the tetrahedron with the plane \\(MNK\\). In what rat... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 13759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a three-digit natural number \\(abc\\) with digits \\(a\\), \\(b\\), and \\(c\\) in the hundreds, tens, and units places respectively, and where \\(a > b\\) and \\(c > b\\), with \\(a, b, c \\in \\{0, 2, 3, 4, 5\\}\\) and \\(a, b, c\\) distinct, calculate the number of co... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a tower with a height of $8$ cubes, where a blue cube must always be at the top, determine the number of different towers the child can build using $2$ red cubes, $4$ blue cubes, and $3$ green cubes."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 13761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a trapezoid \\( ABCD \\) with \\( AD \\parallel BC \\). It turns out that \\( \\angle ABD = \\angle BCD \\). Find the length of segment \\( BD \\) if \\( BC = 36 \\) and \\( AD = 64 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 13762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a trapezoid \\( MNPQ \\) with bases \\( MQ \\) and \\( NP \\). A line parallel to the bases intersects the lateral side \\( MN \\) at point \\( A \\), and the lateral side \\( PQ \\) at point \\( B \\). The ratio of the areas of the trapezoids \\( ANPB \\) and \\( MABQ \\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{46}}{3}"
} | {
"index": 13763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a trapezoid \\(ABCD\\) with bases \\(AB\\) and \\(CD\\), and angles \\(\\angle C = 30^\\circ\\) and \\(\\angle D = 80^\\circ\\). Find \\(\\angle ACB\\), given that \\(DB\\) is the bisector of \\(\\angle D\\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a trapezoid \\(ABCD\\). A point \\(M\\) is chosen on its lateral side \\(CD\\) such that \\( \\frac{CM}{MD} = \\frac{4}{3} \\). It turns out that segment \\( BM \\) divides the diagonal \\( AC \\) into two segments, the ratio of the lengths of which is also \\( \\frac{4}{... | math | {
"style": "rule",
"ground_truth": "7/12"
} | {
"index": 13765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a trapezoid with labeled points as shown in the diagram, the point of intersection of the extended non-parallel sides is labeled as \\(E\\), and the point of intersection of the diagonals is labeled as \\(F\\). Similar right triangles \\(BFC\\) and \\(DFA\\) have correspo... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\sqrt{3}}"
} | {
"index": 13766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with angles $\\angle A = 60^{\\circ}, \\angle B = 75^{\\circ}, \\angle C = 45^{\\circ}$ , let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$ , and $Q$ be the foot of the perpendicular from $B$ onto ... | math | {
"style": "rule",
"ground_truth": "1132"
} | {
"index": 13767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ \\triangle ABC $ with sides $ a $, $ b $, and $ c $ opposite angles $ A $, $ B $, and $ C $, respectively, consider vectors $ \\overrightarrow{m} = (a, \\sqrt{3}b) $ and $ \\overrightarrow{n} = (\\cos A, \\sin B) $ being parallel.\n(I) Find the angle $ A $.\n... | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{7}"
} | {
"index": 13768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with $\\angle B = \\frac{\\pi}{3}$, \n(I) if $AB=8\\sqrt{3}$ and $AC=12$, find the area of $\\triangle ABC$; \n(II) if $AB=4$ and $\\vec{BM} = \\vec{MN} = \\vec{NC}$ with $AN=2\\sqrt{3}BM$, find the length of $AM$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 13769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $A=2C$.\n\n(Ⅰ) If $\\triangle ABC$ is an acute triangle, find the range of $\\frac{a}{c}$.\n\n(Ⅱ) If $b=1, c=3$, find the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\\cos B=b\\cos C$, and the dot product $\\vec{AB}\\cdot \\vec{BC} = -3$,\n1. Find the area of $\\triangle ABC$;\n2. Find the minimum value of side $AC$.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 13771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and centroid $G$. If $2\\sin A\\overrightarrow{GA}+\\sqrt{3}\\sin B\\overrightarrow{GB}+3\\sin C\\cdot \\overrightarrow{GC}=\\vec{0}$, then $\\cos B=$_______."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{12}"
} | {
"index": 13772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and it is known that $$2\\sin^{2}(B+C)= \\sqrt {3}\\sin2A.$$ \n(Ⅰ) Find the degree measure of $A$; \n(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\sqrt {3}"
} | {
"index": 13773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with sides $a$, $b$, $c$, and area $S$ satisfying $S=a^{2}-(b-c)^{2}$, and $b+c=8$. \n$(1)$ Find $\\cos A$; \n$(2)$ Find the maximum value of $S$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{17}"
} | {
"index": 13774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with sides opposite angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, and with $b = 3$, $c = 2$, and the area $S_{\\triangle ABC} = \\frac{3\\sqrt{3}}{2}$:\n1. Determine the value of angle $A$;\n2. When angle $A$ is obtuse, find the height from ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{57}}{19}"
} | {
"index": 13775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively, and it is given that $(3b-c)\\cos A = a\\cos C$.\n(1) Find the value of $\\cos A$;\n(2) If the area of $\\triangle ABC$ is $S=2\\sqrt{2}$, find the minimum value of the... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}+2\\sqrt{2}"
} | {
"index": 13776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, let vectors $\\overrightarrow{m}=(1-\\cos(A+B), \\cos \\frac{A-B}{2})$ and $\\overrightarrow{n}=(\\frac{5}{8}, \\cos \\frac{A-B}{2})$, and it's known that $\\overrighta... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"index": 13777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ whose side lengths form an arithmetic sequence with a common difference of $2$, and the sine of its largest angle is $\\frac{\\sqrt{3}}{2}$, find the perimeter of this triangle."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 13778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with an area of $S$, and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = S$.\n\n(I) Find the value of $\\tan 2A$;\n\n(II) If $\\cos C = \\frac{3}{5}$, and $|\\overrightarrow{AC} - \\overrightarrow{AB}| = 2$, find the area $S$ of $\\triangle... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 13779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. It is known that $2a\\sin (C+ \\frac{\\pi}{6})=b+c$.\n\n1. Find the value of angle $A$.\n2. If $B= \\frac{\\pi}{4}$ and $b-a= \\sqrt{2}- \\sqrt{3}$, find the area o... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{3}}{4}"
} | {
"index": 13780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $\\frac{a}{b}=\\frac{1+\\cos A}{\\cos C}$.\n(1) Find angle $A$;\n(2) If $a=1$, find the maximum area $S$ of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"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 $b = 2\\sqrt{3}$, fin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"index": 13782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with area $S$ and sides $a$, $b$, $c$ that satisfy the equations: $S=a^{2}-(b-c)^{2}$, $b+c=8$, find the maximum value of the area $S$ of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {64}{17}"
} | {
"index": 13783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with internal angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. $B$ is an acute angle. Vector $m=(2\\sin B, -\\sqrt{3})$ and vector $n=(\\cos 2B, 2\\cos^2 \\frac{B}{2} - 1)$ are parallel.\n\n(1) Find the value of angle ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with its three interior angles $A$, $B$, and $C$ satisfying: $$A+C=2B, \\frac {1}{\\cos A}+ \\frac {1}{\\cos C}=- \\frac { \\sqrt {2}}{\\cos B}$$, find the value of $$\\cos \\frac {A-C}{2}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}}{2}"
} | {
"index": 13785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $a=2$, $A= \\frac{\\pi}{3}$, and $\\frac{\\sqrt{3}}{2} - \\sin(B-C) = \\sin 2B$, find the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 13786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $\\cos A= \\frac {3}{4}$, $\\cos C= \\frac {1}{8}$,\n(I) find the ratio $a:b:c$;\n(II) if $| \\overrightarrow{AC}+ \\overrightarrow{BC}|= \\sqrt {46}$, find the ... | math | {
"style": "rule",
"ground_truth": "\\frac {15 \\sqrt {7}}{4}"
} | {
"index": 13787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is given by $S= \\frac{a^{2}+b^{2}-c^{2}}{4}$ and $\\sin A= \\frac{3}{5}$.\n\n1. Find $\\sin B$.\n2. If side $c=5$, find the area of $\\tri... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{2}"
} | {
"index": 13788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$, where $2 \\sqrt {2}(\\sin ^{2}A-\\sin ^{2}C)=(a-b)\\sin B$, and the radius of the circumcircle is $\\sqrt {2}$.\n(1) Find $\\angle C$;\n(2) Find the maximum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}}{2}"
} | {
"index": 13789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$, where the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that vector $\\overrightarrow{m} = (\\sin A + \\sin C, \\sin B - \\sin A)$ and vector $\\overrightarrow{n} = (\\sin A - \\sin C... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"index": 13790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle ABC, let the lengths of the sides opposite to angles A, B, C be a, b, c, respectively. If a, b, c satisfy $a^2 + c^2 - b^2 = \\sqrt{3}ac$,\n(1) find angle B;\n(2) if b = 2, c = $2\\sqrt{3}$, find the area of triangle ABC."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 13791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( A B C \\) with sides \\( A B = \\sqrt{17} \\), \\( B C = 5 \\), and \\( A C = 4 \\). Point \\( D \\) is taken on the side \\( A C \\) such that \\( B D \\) is the altitude of triangle \\( A B C \\). Find the radius of the circle passing through points \\( A... | math | {
"style": "rule",
"ground_truth": "5/6"
} | {
"index": 13792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\) where \\( |AB| = |AC| \\) and \\( \\angle BAC = 80^\\circ \\). Inside the triangle, there is a point \\( M \\) such that \\( \\angle MBC = 30^\\circ \\) and \\( \\angle MCB = 10^\\circ \\). Find \\( \\angle AMC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 13793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\) with sides \\( AB=13 \\), \\( BC=20 \\), and \\( AC=21 \\). Point \\( K \\) is on side \\( AB \\), point \\( L \\) is on side \\( AC \\), and point \\( N \\) is on side \\( BC \\). It is known that \\( AK=4 \\), \\( CN=1 \\), and \\( CL=\\frac{20}{2... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 13794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\) with the condition:\n\n\\[ \\cos (\\angle A - \\angle B) + \\sin (\\angle A + \\angle B) = 2 \\]\n\nFind the side \\( BC \\) if \\( AB = 4 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 13795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\), \\( X \\) and \\( Y \\) are points on side \\( AB \\), with \\( X \\) closer to \\( A \\) than \\( Y \\), and \\( Z \\) is a point on side \\( AC \\) such that \\( XZ \\) is parallel to \\( YC \\) and \\( YZ \\) is parallel to \\( BC \\). Suppose \... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 13796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( ACE \\) with a point \\( B \\) on segment \\( AC \\) and a point \\( D \\) on segment \\( CE \\) such that \\( BD \\) is parallel to \\( AE \\). A point \\( Y \\) is chosen on segment \\( AE \\), and segment \\( CY \\) is drawn, intersecting \\( BD \\) at p... | math | {
"style": "rule",
"ground_truth": "39/25"
} | {
"index": 13797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( \\triangle ABC \\) and a point \\( D \\) on its side \\( BC \\) such that \\( AB = 2BD \\) and \\( AC = 3CD \\), and an ellipse \\( \\Gamma \\) can be constructed with \\( A \\) and \\( D \\) as foci passing through points \\( B \\) and \\( C \\), find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{21}}{7}"
} | {
"index": 13798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a triangle \\( \\triangle ABC \\) with \\(\\angle B = 90^\\circ\\). The incircle touches sides \\(BC\\), \\(CA\\), and \\(AB\\) at points \\(D\\), \\(E\\), and \\(F\\) respectively. Line \\(AD\\) intersects the incircle at another point \\(P\\), and \\(PF \\perp PC\\). Fi... | math | {
"style": "rule",
"ground_truth": "3:4:5"
} | {
"index": 13799,
"solution": ""
} |
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