data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more?"
}
] | math | {
"style": "rule",
"ground_truth": "30 \\%"
} | {
"index": 34800,
"solution": "To solve this problem, we need to determine the percentage of employees who have worked at the Gauss company for 5 years or more based on the given information.\n\n1. **Assign Variables:**\n Let $x$ represent the number of employees each $\\text{X}$ on the graph represents.\n\n2. **... |
DeepScaleR | [
{
"role": "user",
"content": "The vertical coordinate of the intersection point of the new graph obtained by shifting the graph of the quadratic function $y=x^{2}+2x+1$ $2$ units to the left and then $3$ units up is ______."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \\begin{tabular}{|l|c|} \\hline\nThe Parabola & 165 feet \\\\... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 34802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertical drops of six roller coasters at Cantor Amusement Park are shown in the table. \n\\[\n\\begin{tabular}{|l|c|}\n\\hline\nCyclone & 180 feet \\\\ \\hline\nGravity Rush & 120 feet \\\\ \\hline\nScreamer & 150 feet \\\\ \\hline\nSky High & 310 feet \\\\ \\hline\nTwister... | math | {
"style": "rule",
"ground_truth": "8.\\overline{3}"
} | {
"index": 34803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertical drops of six roller coasters at Fractal Fun Park are listed in the table below:\n\\begin{tabular}{|l|c|}\n\\hline\nThe Spiral Slide & 173 feet \\\\\n\\hline\nThe Velocity Vortex & 125 feet \\\\\n\\hline\nThe Cyclone & 150 feet \\\\\n\\hline\nThe Apex Jump & 310 fee... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by \\[V=\\left\\{ \\sqrt{2}i,-\\sqrt{2}i, \\frac{1}{\\sqrt{8}}(1+i),\\frac{1}{\\sqrt{8}}(-1+i),\\frac{1}{\\sqrt{8}}(1-i),\\frac{1}{\\sqrt{8}}(-1-i) \\right\\}.\\]For each $j$, $1\\leq j\\leq 12$,... | math | {
"style": "rule",
"ground_truth": "233"
} | {
"index": 34805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.\n\nHow many triangles can be formed with vertices at the marked points?"
}
] | math | {
"style": "rule",
"ground_truth": "1130"
} | {
"index": 34806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s. Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$, and points $P_2\\,$, $P_3\\,$... | math | {
"style": "rule",
"ground_truth": "344"
} | {
"index": 34807,
"solution": "If we have points $(p,q)$ and $(r,s)$ and we want to find $(u,v)$ so $(r,s)$ is the midpoint of $(u,v)$ and $(p,q)$, then $u=2r-p$ and $v=2s-q$. So we start with the point they gave us and work backwards. We make sure all the coordinates stay within the triangle. We have: $P_7=(14,92)... |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$ . What is the angle between the diagonals $D_1D_3$ and $D_2D_5$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 34808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a $3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these circles?\n\n[asy]unitsize(1cm);\ndraw(Circle((1.8,2.4),1),linewidth(0.7));\ndraw(Circle((0,0),2),linewidth(0.7));\ndraw(C... | math | {
"style": "rule",
"ground_truth": "14\\pi"
} | {
"index": 34809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a $3 \\times 1 \\times 1$ rectangular prism are $A, B, C, D, E, F, G$, and $H$ so that $A E, B F$, $C G$, and $D H$ are edges of length 3. Point $I$ and point $J$ are on $A E$ so that $A I=I J=J E=1$. Similarly, points $K$ and $L$ are on $B F$ so that $B K=K L=L... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 34810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a cube have coordinates $(0,0,0),$ $(0,0,4),$ $(0,4,0),$ $(0,4,4),$ $(4,0,0),$ $(4,0,4),$ $(4,4,0),$ and $(4,4,4).$ A plane cuts the edges of this cube at the points $P = (0,2,0),$ $Q = (1,0,0),$ $R = (1,4,4),$ and two other points. Find the distance between t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{29}"
} | {
"index": 34811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a quadrilateral lie on the graph of $y=\\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\\ln{\\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34812,
"solution": "1. **Identify the coordinates of the vertices**: Let the $x$-coordinates of the vertices of the quadrilateral be $n, n+1, n+2, n+3$. Then, the coordinates of the vertices are $(n,\\ln(n))$, $(n+1,\\ln(n+1))$, $(n+2,\\ln(n+2))$, $(n+3,\\ln(n+3))$.\n\n2. **Use the Shoelace Theorem**: Th... |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a regular hexagon are labeled $\\cos (\\theta), \\cos (2 \\theta), \\ldots, \\cos (6 \\theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds fo... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34813,
"solution": "The existence of the Chebyshev polynomials, which express $\\cos (n \\theta)$ as a polynomial in $\\cos (\\theta)$, imply that Bob draws a blue line between $\\cos (\\theta)$ and each other vertex, and also between $\\cos (2 \\theta)$ and $\\cos (4 \\theta)$, between $\\cos (2 \\theta... |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be o... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 34814,
"solution": "Notice that there are three pairs of congruent integers mod 3 ($(1,4,7),(2,5,8),(3,6,9)$). There are $3!$ ways to order each pair individually and $3!$ ways to order the pairs as a group. Rotations are indistinguishable, so in total there are $6^4/9=\\boxed{144}$ ways."
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a regular nonagon are colored such that 1) adjacent vertices are different colors and 2) if 3 vertices form an equilateral triangle, they are all different colors. Let m be the minimum number of colors needed for a valid coloring, and n be the total number of co... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 34815,
"solution": "It's clear that m is more than 2 since it's impossible to alternate the color of the vertices without having two of the same color adjacent (since the graph is not bipartite). However, it's possible to use 3 colors. Number the vertices 1 through 9 in order and let the colors be A, B, ... |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]\nfill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray... | math | {
"style": "rule",
"ground_truth": "36 - 12\\sqrt{3} - 4\\pi"
} | {
"index": 34816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a tetrahedron in the spatial rectangular coordinate system O-xyz are located at the coordinates (1,0,1), (1,1,0), (0,1,0), and (1,1,1). Find the volume of the circumscribed sphere of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2} \\pi"
} | {
"index": 34817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of a triangle have coordinates \\(A(1 ; 3.5)\\), \\(B(13.5 ; 3.5)\\), and \\(C(11 ; 16)\\). We consider horizontal lines defined by the equations \\(y=n\\), where \\(n\\) is an integer. Find the sum of the lengths of the segments cut by these lines on the sides of ... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 34818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"index": 34819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 34820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of an equilateral triangle lie on the hyperbola \\( xy = 3 \\). The centroid of this triangle is at the origin, \\( (0,0) \\). What is the square of the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 34821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The vertices of the broken line $A B C D E F G$ have coordinates $A(-1, -7), B(2, 5), C(3, -8), D(-3, 4), E(5, -1), F(-4, -2), G(6, 4)$.\n\nFind the sum of the angles with vertices at points $B, E, C, F, D$."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 34822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns aroun... | math | {
"style": "rule",
"ground_truth": "9217"
} | {
"index": 34823,
"solution": "On the $n$th straight path, the caterpillar travels $n$ units before hitting food and his weight is $n-1$. Then his speed is $2^{1-n}$. Then right before he turns around for the $n$th time, he has traveled a total time of $\\sum_{i=1}^{n} \\frac{i}{2^{1-i}}=\\frac{1}{2} \\sum_{i=1}^{n... |
DeepScaleR | [
{
"role": "user",
"content": "The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangul... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 34824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers such that $1 < c < b < a$ with $c$ being a prime number. Determine the largest possible value of $b$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a certain rectangular solid is $8 \\text{ cm}^3$, its total surface area is $32 \\text{ cm}^2$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 34826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a certain rectangular solidis $216\\text{ cm}^3$, its total surface area is $288\\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 34827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes?"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 34828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a cylinder circumscribed around a sphere with radius $r$ is $V_{1}$, and the volume of a cone circumscribed around the same sphere is $V_{2}$. What is the minimum value of the ratio $V_{2} / V_{1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 34829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a cylinder is $54\\pi$ $\\text{cm}^3$. How many cubic centimeters are in the volume of a cone with the same radius and height as the cylinder? Express your answer in terms of $\\pi$. [asy]\nimport solids; currentprojection=orthographic(0,100,25); defaultpen(linewi... | math | {
"style": "rule",
"ground_truth": "18\\pi"
} | {
"index": 34830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 34831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a given sphere is \\( 72\\pi \\) cubic inches. Calculate the surface area of the sphere in terms of \\( \\pi \\)."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi 2^{2/3}"
} | {
"index": 34832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a given sphere is \\(72\\pi\\) cubic inches. Find the surface area of the sphere. Express your answer in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi 2^{2/3}"
} | {
"index": 34833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a hemisphere is $\\frac{500}{3}\\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi \\times 250^{2/3}"
} | {
"index": 34834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height?"
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 34835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a prism is equal to the area of its base times its depth. If the prism has identical bases with area $400 \\mathrm{~cm}^{2}$ and depth 8 cm, what is its volume?"
}
] | math | {
"style": "rule",
"ground_truth": "3200 \\mathrm{~cm}^{3}"
} | {
"index": 34836,
"solution": "The volume of a prism is equal to the area of its base times its depth. Here, the prism has identical bases with area $400 \\mathrm{~cm}^{2}$ and depth 8 cm, and so its volume is $400 \\mathrm{~cm}^{2} \\times 8 \\mathrm{~cm} = 3200 \\mathrm{~cm}^{3}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a rectangular prism is 360 cubic units where \\(a\\), \\(b\\), and \\(c\\) are integers with \\(1 < c < b < a\\). Determine the largest possible value of \\(b\\)."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a rectangular solid each of whose side, front, and bottom faces are $12\\text{ in}^{2}$, $8\\text{ in}^{2}$, and $6\\text{ in}^{2}$ respectively is:"
}
] | math | {
"style": "rule",
"ground_truth": "$24\\text{ in}^{3}$"
} | {
"index": 34838,
"solution": "1. **Identify the equations based on the given areas of the faces:**\n Let the side lengths of the rectangular solid be $a$, $b$, and $c$. The areas of the side, front, and bottom faces are given as $12\\text{ in}^2$, $8\\text{ in}^2$, and $6\\text{ in}^2$ respectively. Therefore, w... |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a regular octagonal prism is $8 \\, \\mathrm{m}^{3}$, and its height is $2.2 \\, \\mathrm{m}$. Find the lateral surface area of the prism."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt{2.2 \\cdot (\\sqrt{2}-1)}"
} | {
"index": 34839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"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."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt{2.2 (\\sqrt{2} - 1)}"
} | {
"index": 34840,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a regular triangular prism is $8$, the base edge length that minimizes the surface area of the prism is __________."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt[3]{4}"
} | {
"index": 34841,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\\circ}$ to the base, is $9 \\mathrm{~cm}^{3}$. Find the total surface area of the pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\sqrt{3} (1 + \\sqrt{2})"
} | {
"index": 34842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a sphere is $72\\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bott... | math | {
"style": "rule",
"ground_truth": "90\\pi \\sqrt[3]{4}"
} | {
"index": 34843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of a sphere is increased to $72\\pi$ cubic inches. What is the new surface area of the sphere? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi \\cdot 2^{2/3}"
} | {
"index": 34844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the box is 240 cubic units where $a,b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$? [asy]\n\ndraw((0,0)--(0,3)--(6,3)--(6,0)--cycle);\ndraw((2,1.5)--(2,4.5)--(8,4.5)--(8,1.5)--cycle);\ndraw((0,0)--(2,1.5)--(2,4.5)--(0,3)--cycle)... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the box is 360 cubic units where $a, b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the geometric body formed by points whose distance to line segment AB is no greater than three units is $216 \\pi$. Calculate the length of the line segment AB."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the parallelepiped determined by the three-dimensional vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ is 4. Find the volume of the parallelepiped determined by the vectors $\\mathbf{a} + \\mathbf{b},$ $\\mathbf{b} + 3 \\mathbf{c},$ and $\\mathbf{c} - 7 \... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 34848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the parallelepiped determined by vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\\mathbf{a} + 2\\mathbf{b},$ $\\mathbf{b} + \\mathbf{c},$ and $2\\mathbf{c} - 5\\mathbf{a}.$"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 34849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the parallelepiped generated by $\\begin{pmatrix} 2 \\\\ 3 \\\\ 4 \\end{pmatrix},$ $\\begin{pmatrix} 1 \\\\ k \\\\ 2 \\end{pmatrix},$ and $\\begin{pmatrix} 1 \\\\ 2 \\\\ k \\end{pmatrix}$ is 15. Find $k,$ where $k > 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 34850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the parallelepiped generated by the vectors $\\begin{pmatrix} 3 \\\\ 4 \\\\ 5 \\end{pmatrix}$, $\\begin{pmatrix} 2 \\\\ m \\\\ 3 \\end{pmatrix}$, and $\\begin{pmatrix} 2 \\\\ 3 \\\\ m \\end{pmatrix}$ is 20. Find $m$, where $m > 0$."
}
] | math | {
"style": "rule",
"ground_truth": "3 + \\frac{2\\sqrt{15}}{3}"
} | {
"index": 34851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the solid formed by rotating an isosceles right triangle with legs of length 1 around its hypotenuse is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{6}\\pi"
} | {
"index": 34852,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}\\pi"
} | {
"index": 34853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \\sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{128\\pi}{3}"
} | {
"index": 34854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \\frac{4}{3}\\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:\n\n$\\frac{4}{3}\\pi r^3 =... | math | {
"style": "rule",
"ground_truth": "0.95"
} | {
"index": 34855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The volumes of a regular tetrahedron and a regular octahedron have equal edge lengths. Find the ratio of their volumes without calculating the volume of each polyhedron."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 34856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The walls of a room are in the shape of a triangle $A B C$ with $\\angle A B C=90^{\\circ}, \\angle B A C=60^{\\circ}$, and $A B=6$. Chong stands at the midpoint of $B C$ and rolls a ball toward $A B$. Suppose that the ball bounces off $A B$, then $A C$, then returns exactly to... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{21}"
} | {
"index": 34857,
"solution": "Let $C^{\\prime}$ be the reflection of $C$ across $A B$ and $B^{\\prime}$ be the reflection of $B$ across $A C^{\\prime}$; note that $B^{\\prime}, A, C$ are collinear by angle chasing. The image of the path under these reflections is just the line segment $M M^{\\prime}$, where $M$ is... |
DeepScaleR | [
{
"role": "user",
"content": "The water storage volume of two reservoirs, A and B, changes with time. The relationship between the water storage volume of reservoir A, $f(t)$ (in hundreds of tons), and time, $t$ (in hours), is given by $f(t)=2+\\sin t$, where $t\\in[0,12]$. The relationship between the wate... | math | {
"style": "rule",
"ground_truth": "6.721"
} | {
"index": 34858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 16 feet, and its height is 96 feet. The water in the tank is $25\\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\\sqr... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 34859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 8 feet, and its height is 64 feet. The water in the tank is $40\\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\\sqrt[... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 34860,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The water tank in the diagram is in the shape of an inverted right circular cone. The radius of its base is 20 feet, and its height is 100 feet. The water in the tank fills 50% of the tank's total capacity. Find the height of the water in the tank, which can be expressed in the... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 34861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The wavelength of red light that the human eye can see is $0.000077$ cm. Please round the data $0.000077$ to $0.00001$ and express it in scientific notation as ______."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\times 10^{-5}"
} | {
"index": 34862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The weight of grain in a sample of 256 grains is 18 grains, and the total weight of rice is 1536 dan. Calculate the amount of mixed grain in the total batch of rice."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 34863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34864,
"solution": "Let an $O$ signal a move that ends in the outer circle and $I$ signal a move that ends in the inner circle. Now notice that for a string of $15$ moves to end at $A$, the difference between $O$'s and $I$'s in the string must be a multiple of $5$.\n$15$ $I$'s: Trivially $1$ case.\n$5$ $... |
DeepScaleR | [
{
"role": "user",
"content": "The whole numbers from 1 to 1000 are written. How many of these numbers have at least two 7's appearing side-by-side?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 34865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The whole numbers from 1 to 2016 inclusive are written on a blackboard. Moritz underlines all the multiples of two in red, all the multiples of three in blue, and all the multiples of four in green. How many numbers does Moritz underline exactly twice?"
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 34866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The whole numbers from 1 to \\( 2k \\) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \\( k \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 34867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Det... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 34868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minu... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 34869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The wizard revealed the secret of wisdom. For those who wanted to know precisely where the secret is hidden, he left a clue in his magic book:\n\n\\[ 5 \\cdot \\text{BANK} = 6 \\cdot \\text{SAD} \\]\n\nEach letter in this clue represents a certain digit. Find these digits and s... | math | {
"style": "rule",
"ground_truth": "55.543065317"
} | {
"index": 34870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work wit... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 34871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ worker... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 34872,
"solution": "Suppose that it takes $x$ hours for one worker to create one widget, and $y$ hours for one worker to create one whoosit.\n Therefore, we can write that (note that two hours is similar to having twice the number of workers, and so on):\n$100 = 300x + 200y$\n$2(60) = 240x + 300y$\n$3(50... |
DeepScaleR | [
{
"role": "user",
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can pr... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 34873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The workers laid a floor of size $n\\times n$ ($10 <n <20$) with two types of tiles: $2 \\times 2$ and $5\\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’... | math | {
"style": "rule",
"ground_truth": "12, 15, 18"
} | {
"index": 34874,
"solution": "\nTo solve this problem, we aim to find all integer values of \\( n \\) (where \\( 10 < n < 20 \\)) for which an \\( n \\times n \\) floor can be completely covered using the same number of \\( 2 \\times 2 \\) and \\( 5 \\times 1 \\) tiles. We cannot cut the tiles and they should not ... |
DeepScaleR | [
{
"role": "user",
"content": "The world is currently undergoing a major transformation that has not been seen in a century. China is facing new challenges. In order to enhance students' patriotism and cohesion, a certain high school organized a knowledge competition on \"China's national conditions and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{3}"
} | {
"index": 34875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 34876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The writer Arthur has $n \\ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having writt... | math | {
"style": "rule",
"ground_truth": "$n\\le6$"
} | {
"index": 34877,
"solution": "\nTo solve the problem, we need to determine the values of \\( n \\) for which it is possible that each co-author accurately remembers the number of books written with Arthur, and these numbers correspond to the first \\( n \\) Fibonacci numbers. \n\nThe Fibonacci sequence is defined ... |
DeepScaleR | [
{
"role": "user",
"content": "The year 2000 is a leap year. The year 2100 is not a leap year. The following are the complete rules for determining a leap year:\n\n(i) Year \\(Y\\) is not a leap year if \\(Y\\) is not divisible by 4.\n\n(ii) Year \\(Y\\) is a leap year if \\(Y\\) is divisible by 4 but not by... | math | {
"style": "rule",
"ground_truth": "244"
} | {
"index": 34878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34879,
"solution": "To find the next palindromic year after 2002, we need to identify a year that reads the same forwards and backwards. We start by checking the years sequentially from 2003 onwards.\n\n1. **Check year 2003**: This is not a palindrome.\n2. **Check year 2004**: This is not a palindrome.\n... |
DeepScaleR | [
{
"role": "user",
"content": "The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again?"
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 34880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The year 2009 has the property that rearranging its digits never results in a smaller four-digit number (numbers do not start with zero). In which year will this property first repeat?"
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 34881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The year 2013 has arrived, and Xiao Ming's older brother sighed and said, \"This is the first year in my life that has no repeated digits.\" It is known that Xiao Ming's older brother was born in a year that is a multiple of 19. How old is the older brother in 2013?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 34882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent."
}
] | math | {
"style": "rule",
"ground_truth": "-19\\%"
} | {
"index": 34883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is:"
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 34884,
"solution": "1. **Understanding the percentage changes**: \n - A 25% increase in population means the population becomes 125% of its original, which can be expressed as a multiplication by $\\frac{5}{4}$.\n - A 25% decrease means the population becomes 75% of its previous amount, which can be ... |
DeepScaleR | [
{
"role": "user",
"content": "The young man paid with $100 for a gift that cost $18 and received $79 in change from Mr. Wang. Mr. Wang then returned the counterfeit $100 bill to the neighbor."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"index": 34885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 17\\qquad\\textbf{(E)}\\ 18$"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The zookeeper is distributing a pile of peaches among several monkeys. If each monkey receives 6 peaches, there are 57 peaches left. If each monkey should receive 9 peaches, 5 monkeys get nothing, and one monkey gets only 3 peaches. How many peaches are there in total?"
}
] | math | {
"style": "rule",
"ground_truth": "273"
} | {
"index": 34888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. What time does Leo think it is?\n\nA) 11:30 \nB) 11:45... | math | {
"style": "rule",
"ground_truth": "12:30"
} | {
"index": 34889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?"
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 34890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{55}"
} | {
"index": 34891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are... | math | {
"style": "rule",
"ground_truth": "19/165"
} | {
"index": 34892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump."
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 34893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $13$ positive integers greater than $\\sqrt{15}$ and less than $\\sqrt[3]{B}$ . What is the smallest integer value of $B$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "4097"
} | {
"index": 34894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 34895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$ , and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group."
}
] | math | {
"style": "rule",
"ground_truth": "3/455"
} | {
"index": 34896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $168$ primes below $1000$ . Then sum of all primes below $1000$ is,"
}
] | math | {
"style": "rule",
"ground_truth": "76127"
} | {
"index": 34897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?"
}
] | math | {
"style": "rule",
"ground_truth": "100!"
} | {
"index": 34898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to ... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 34899,
"solution": ""
} |
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