data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A regular octahedron $A B C D E F$ is given such that $A D, B E$, and $C F$ are perpendicular. Let $G, H$, and $I$ lie on edges $A B, B C$, and $C A$ respectively such that \\frac{A G}{G B}=\\frac{B H}{H C}=\\frac{C I}{I A}=\\rho. For some choice of $\\rho>1, G H, H I$, and $I ... | math | {
"style": "rule",
"ground_truth": "(1+\\sqrt{5}) / 2"
} | {
"index": 3100,
"solution": "Let $J$ lie on edge $C E$ such that \\frac{E J}{J C}=\\rho. Then we must have that $H I J$ is another face of the icosahedron, so in particular, $H I=H J$. But since $B C$ and $C E$ are perpendicular, $H J=H C \\sqrt{2}$. By the Law of Cosines, $H I^{2}=H C^{2}+C I^{2}-2 H C \\cdot C I... |
DeepScaleR | [
{
"role": "user",
"content": "A regular octahedron has a side length of 1. What is the distance between two opposite faces?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6} / 3"
} | {
"index": 3101,
"solution": "Imagine orienting the octahedron so that the two opposite faces are horizontal. Project onto a horizontal plane; these two faces are congruent equilateral triangles which (when projected) have the same center and opposite orientations. Hence, the vertices of the octahedron project to t... |
DeepScaleR | [
{
"role": "user",
"content": "A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sph... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\\frac {a\\sqrt {b}}{c}$, where $a$, $b$, and $c$ are posi... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 3103,
"solution": "\nTo solve this problem, we need to understand the geometry of the octahedron and how the plane intersects it. An octahedron can be thought of as two square pyramids base-to-base. Each face of the octahedron is an equilateral triangle.\n\n#### Step 1: Understanding the Intersection\nTh... |
DeepScaleR | [
{
"role": "user",
"content": "A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is\n$\\mathrm{(A) \\frac{\\sqrt{3}}{12} } \\qquad \\mathrm{(B) \\frac{\\sqrt{6}}{16} } \\qquad \\mathrm{(C) \\frac{1}{6} } \\q... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 3104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular octahedron is formed by joining the midpoints of the edges of a regular tetrahedron. Calculate the ratio of the volume of this octahedron to the volume of the original tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular pentagon \\(Q_1 Q_2 \\dotsb Q_5\\) is drawn in the coordinate plane with \\(Q_1\\) at \\((1,0)\\) and \\(Q_3\\) at \\((5,0)\\). If \\(Q_n\\) is the point \\((x_n,y_n)\\), compute the numerical value of the product\n\\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\dotsm (x... | math | {
"style": "rule",
"ground_truth": "242"
} | {
"index": 3106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular pentagon has an area numerically equal to its perimeter, and a regular hexagon also has its area numerically equal to its perimeter. Compare the apothem of the pentagon with the apothem of the hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "1.06"
} | {
"index": 3107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has exterior angles each measuring 15 degrees. How many sides does the polygon have?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has interior angles of 120 degrees. How many sides does the polygon have?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 3110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has interior angles of 160 degrees, and each side is 4 units long. How many sides does the polygon have, and what is its perimeter?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 3111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has interior angles of 162 degrees. How many sides does the polygon have?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has perimeter 108 cm and each side has length 12 cm. How many sides does this polygon have?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 3113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$? \n[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3115,
"solution": "1. **Determine the interior angle of the decagon**: \n The formula for the interior angle of a regular polygon with $m$ sides is given by:\n \\[\n \\text{Interior angle} = \\frac{(m-2) \\times 180^\\circ}{m}\n \\]\n For a decagon ($m=10$), the interior angle is:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3116,
"solution": "1. **Understanding the Problem**: We are given a regular polygon with $n$ sides inscribed in a circle of radius $R$. The area of the polygon is given as $3R^2$. We need to find the value of $n$.\n\n2. **Area of a Regular Polygon**: The area $A$ of a regular polygon inscribed in a circl... |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"index": 3117,
"solution": "Imagine the shadow of the skeleton of the tetrahedron (i.e. make the entire tetrahedron translucent except for the edges). The diagonals of the square shadow must correspond to a pair of opposite edges of the tetrahedron. Both of these edges must be parallel to the plane - if they were... |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron has an edge length of $2$. Calculate the surface area of the sphere circumscribed around this tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"index": 3118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron has the numbers $1,2,3,4$ written on its four faces. Four such identical regular tetrahedrons are thrown simultaneously onto a table. What is the probability that the product of the numbers on the four faces in contact with the table is divisible by $4$?"
... | math | {
"style": "rule",
"ground_truth": "$\\frac{13}{16}$"
} | {
"index": 3119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron has two spheres, one inscribed within it and the other circumscribed around it. Between each face of the tetrahedron and the circumscribed sphere, there are four smaller spheres centered on each face. Given a point \\( P \\) within the circumscribed sphere... | math | {
"style": "rule",
"ground_truth": "0.2"
} | {
"index": 3120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron is formed by joining the centers of four non-adjacent faces of a cube. Determine the ratio of the volume of the tetrahedron to the volume of the cube."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron is inscribed in a cylinder such that two opposite edges of the tetrahedron are the diameters of the cylinder's bases. Find the ratio of the volume of the cylinder to the volume of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\pi}{2}"
} | {
"index": 3122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now t... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 3123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangle $EFG$ with a side length of $a$ covers a square $ABCD$ with a side length of 1. Find the minimum value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{2}{\\sqrt{3}}"
} | {
"index": 3124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangle is inscribed in a circle with a diameter of $\\sqrt{12}$. Another regular triangle is constructed on its height as a side, and a new circle is inscribed in this triangle. Find the radius of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 3125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangular prism $ABC A_{1} B_{1} C_{1}$ with the base $ABC$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere of radius 3. Segment $CD$ is a diameter of this sphere. Find the volume of the prism if $A D = 2 \\sqrt{6}$."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{15}"
} | {
"index": 3126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangular prism \\( A B C A_{1} B_{1} C_{1} \\) is inscribed in a sphere, where \\( A B C \\) is the base and \\( A A_{1}, B B_{1}, C C_{1} \\) are the lateral edges. The segment \\( C D \\) is the diameter of this sphere, and point \\( K \\) is the midpoint of the e... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}"
} | {
"index": 3127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangular prism \\(A B C A_{1} B_{1} C_{1}\\) with base \\(A B C\\) and lateral edges \\(A A_{1}, B B_{1}, C C_{1}\\) is inscribed in a sphere. The segment \\(C D\\) is the diameter of this sphere, and point \\(K\\) is the midpoint of edge \\(A A_{1}\\). Find the vol... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}"
} | {
"index": 3128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangular prism has a triangle $ABC$ with side $a$ as its base. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges and are located at distances of $a / 2, a, 3a / 2$ from the base plane, respectively. Find the angle between the planes $ABC$ and $A_{1}B... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 3129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A regular triangular pyramid \\(SABC\\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \\(A\\) and \\(B\\) of the base \\(ABC\\), and these medians do not intersect. It is known that the edges of a certain cube lie on th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 3130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June 17 is a relatively prime date because the greatest common factor of 6 and 17 is 1. How many relatively prime dates are in the month with the few... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A relatively prime date is defined as a date where the day and the month number are coprime. Determine how many relatively prime dates are in the month with 31 days and the highest number of non-relatively prime dates?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 3132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A relay race preparatory team is composed of members from 8 school basketball teams and 2 school football teams, totaling 10 people. If 2 people are randomly selected, find the probability that, given one of them is a football team member, the other one is also a football team ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{17}"
} | {
"index": 3133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per mo... | math | {
"style": "rule",
"ground_truth": "307050"
} | {
"index": 3134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per mo... | math | {
"style": "rule",
"ground_truth": "303000"
} | {
"index": 3135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan pe... | math | {
"style": "rule",
"ground_truth": "4050"
} | {
"index": 3136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$."
}
] | math | {
"style": "rule",
"ground_truth": "1223456"
} | {
"index": 3137,
"solution": "Let $\\left\\{r_{n}\\right\\}_{n \\geq 0}$ be the repunits (so $r_{0}=1, r_{1}=11$, and so on). We see that for any $n$, there is $$r_{n-1}+r_{n-2}+\\cdots+r_{0}<\\frac{r_{n}}{10}+\\frac{r_{n}}{100}+\\cdots<\\frac{r_{n}}{9}<r_{n}$$ so $r_{n}$ is only needed when all possible combinatio... |
DeepScaleR | [
{
"role": "user",
"content": "A research study group is investigating the traffic volume at a certain intersection near the school during the rush hour from 8:00 to 10:00. After long-term observation and statistics, they have established a simple function model between traffic volume and average vehicle spe... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{3}"
} | {
"index": 3138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A reservoir is equipped with an inlet tap A and outlet taps B and C. It is known that opening taps A and B together for 8 hours or opening taps A, B, and C together for a whole day can fill an empty reservoir, and opening taps B and C together for 8 hours can empty a full reser... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment?"
}
] | math | {
"style": "rule",
"ground_truth": "1.14"
} | {
"index": 3140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A residential building has a construction investment of 250 yuan per square meter. Considering a service life of 50 years and an annual interest rate of 5%, calculate the monthly rent per square meter in yuan required to recoup the total investment."
}
] | math | {
"style": "rule",
"ground_truth": "1.14"
} | {
"index": 3141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A restricted path of length $n$ is a path of length $n$ such that for all $i$ between 1 and $n-2$ inclusive, if the $i$th step is upward, the $i+1$st step must be rightward. Find the number of restricted paths that start at $(0,0)$ and end at $(7,3)$."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 3142,
"solution": "This is equal to the number of lattice paths from $(0,0)$ to $(7,3)$ that use only rightward and diagonal (upward+rightward) steps plus the number of lattice paths from $(0,0)$ to $(7,2)$ that use only rightward and diagonal steps, which is equal to the number of paths (as defined abov... |
DeepScaleR | [
{
"role": "user",
"content": "A retailer purchases a gadget at $50$ minus $10%$. He aims to sell the gadget at a gain of $25%$ on his cost price after offering a $15%$ discount on the marked price. Determine the gadget's marked price in dollars."
}
] | math | {
"style": "rule",
"ground_truth": "66.18"
} | {
"index": 3143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rhombus $ABCD$ is given with $\\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$ , $DP = 2$ , $CP = 3$ . Determine the length of the segment $AP$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"index": 3144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rhombus has an area of 108 square units. The lengths of its diagonals have a ratio of 3 to 2. What is the length of the longest diagonal, in units?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 3145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 3146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rich emir was admiring a new jewel, a small golden plate in the shape of an equilateral triangle, decorated with diamonds. He noticed that the shadow of the plate forms a right triangle, with the hypotenuse being the true length of each side of the plate. \n\nWhat is the angl... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 3147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone and a sphere possess the same radius, $r$. If the volume of the cone is one-third of the volume of the sphere, determine the ratio of the height of the cone to the radius $r$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 3148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone has a base with radius $600$ and height $200\\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose... | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 3149,
"solution": "The easiest way is to unwrap the cone into a circular sector. Center the sector at the origin with one radius on the positive $x$-axis and the angle $\\theta$ going counterclockwise. The circumference of the base is $C=1200\\pi$. The sector's radius (cone's sweep) is $R=\\sqrt{r^2+h^2}... |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$?"
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"index": 3150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{2 + 3\\sqrt{2}}"
} | {
"index": 3151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touc... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 3152,
"solution": "The path is a circle with radius equal to the slant height of the cone, which is $\\sqrt {r^{2} + h^{2}}$. Thus, the length of the path is $2\\pi\\sqrt {r^{2} + h^{2}}$.\n Also, the length of the path is 17 times the circumference of the base, which is $34r\\pi$. Setting these equal gi... |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone has base radius 1. The vertex is K. P is a point on the circumference of the base. The distance KP is 3. A particle travels from P around the cone and back by the shortest route. What is its minimum distance from K?"
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 3153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone has for its base a circle having the same radius as a given sphere.\nThe volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:"
}
] | math | {
"style": "rule",
"ground_truth": "2/1"
} | {
"index": 3154,
"solution": "1. **Identify the given information:**\n - The base of the cone and the sphere have the same radius, denoted as $r$.\n - The volume of the cone is half that of the sphere.\n\n2. **Write down the formulas for the volumes:**\n - Volume of the sphere, $V_{\\text{sphere}} = \\frac{4}... |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is cut into five pieces by four planes parallel to its base, each piece having equal height. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"index": 3155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is inscribed in a right prism as shown. What is the ratio of the volume of the cone to the volume of the prism? Express your answer as a common fraction in terms of $\\pi$. [asy]\nimport three; import graph3;\ndefaultpen(linewidth(0.8));\nsize(200);\ndraw(... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"index": 3156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is inscribed in a right rectangular prism as shown. The base of the prism has dimensions, where one side is exactly twice the length of the other ($a$ and $2a$). The cone's base fits perfectly into the base of the prism making one side of the rectangle the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{24}"
} | {
"index": 3157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume and surface area (not including the base) are filled with water and exposed to air, respectively? Express your answer as a decimal to the nearest ten-thousandth."
}... | math | {
"style": "rule",
"ground_truth": "55.5556\\%"
} | {
"index": 3158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is placed on a table, pointing upwards. The vertical cross-section triangle, perpendicular to the base, has a vertex angle of 90 degrees. The diameter of the cone's base is 16 inches. A sphere is placed inside the cone so that it touches the sides of the c... | math | {
"style": "rule",
"ground_truth": "\\frac{256}{3}\\pi"
} | {
"index": 3159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is sliced into five equal-height sections by planes parallel to its base. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"index": 3160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{61}"
} | {
"index": 3161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is sliced into five pieces by planes parallel to its base. Each slice has the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"index": 3162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is sliced into five pieces of equal height by planes parallel to its base. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"index": 3163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is sliced into four pieces by planes parallel to its base, as shown in the figure. All of these pieces have the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Express your answer as a common fra... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{37}"
} | {
"index": 3164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest p... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{27}"
} | {
"index": 3165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is $12\\sqrt{3}$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the con... | math | {
"style": "rule",
"ground_truth": "288\\pi"
} | {
"index": 3166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The co... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 3167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 14 and an altitude of 20, and the axes of the cylinder and cone coincide. The height of the cylinder is three times its radius. Find the radius of the cylinder."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{140}{41}"
} | {
"index": 3168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30}{11}"
} | {
"index": 3169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"index": 3170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"index": 3171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 such that its bases are parallel to the base of the hemisphere and the top of the cylinder touches the top of the hemisphere. What is the height of the cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"index": 3172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"index": 3173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cone has a base with a circumference of $16\\pi$ inches and a height of 30 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $192\\pi$ cubic inches. What is the ratio of the shorter height to the origina... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 3174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cone has a base with a circumference of $20\\pi$ inches and a height of 40 inches. The height of the cone is reduced while the circumference stays the same. After reduction, the volume of the cone is $400\\pi$ cubic inches. What is the ratio of the new height to the ori... | math | {
"style": "rule",
"ground_truth": "400\\pi"
} | {
"index": 3175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\\pi$. Find $W$, as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{284}{3}"
} | {
"index": 3176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"index": 3177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cylinder with a height of 5 inches has a radius of 3 inches. A cone with a height of 3 inches and the same radius is placed on top of the cylinder. What is the total surface area of the combined object, in square inches? Express your answer in terms of \\(\\pi\\)."
}
... | math | {
"style": "rule",
"ground_truth": "30\\pi + 9\\sqrt{2}\\pi"
} | {
"index": 3178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinde... | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"index": 3179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is $3$ feet deep. How deep, in feet, would the oil have been if the tank had been sta... | math | {
"style": "rule",
"ground_truth": "12.1"
} | {
"index": 3180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles)."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 3181,
"solution": "If there are two edges on a single diameter, there would be six diameters. There are four ways to put the third number, and four equilateral triangles. There are $4+ 2 \\cdot 2\\cdot 6 = 28$ ways. Then, if one length was $\\sqrt{3}$ but no side on the diameter, there would be twelve wa... |
DeepScaleR | [
{
"role": "user",
"content": "A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right octagonal pyramid has two cross sections obtained by slicing the pyramid with planes parallel to the octagonal base. The area of the smaller cross section is $256\\sqrt{2}$ square feet and the area of the larger cross section is $576\\sqrt{2}$ square feet. The distance ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 3183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyram... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 3184,
"solution": "Let $B$ and $C$ be the vertices adjacent to $A$ on the same base as $A$, and let $D$ be the last vertex of the triangular pyramid. Notice that we can already find some lengths. We have $AB=AC=12$ (given) and $BC=BD=\\sqrt{144+h^2}$ by the Pythagorean Theorem. Let $M$ be the midpoint of... |
DeepScaleR | [
{
"role": "user",
"content": "A right pyramid has a square base that measures 10 cm on each side. Its peak is 12 cm above the center of its base. What is the sum of the lengths of the pyramid's eight edges? Express your answer to the nearest whole number.\n\n[asy]\nsize(150);\ndraw((0,0)--(3,3)--(13,3)--(10... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 3185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right pyramid has a square base where each side measures 15 cm. The height of the pyramid, measured from the center of the base to the peak, is 15 cm. Calculate the total length of all edges of the pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "60 + 4\\sqrt{337.5}"
} | {
"index": 3186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches?"
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{7}"
} | {
"index": 3187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right pyramid has a square base with side length 10 cm. Its peak is 12 cm above the center of its base. What is the total surface area of the pyramid, in square centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 3188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units?"
}
] | math | {
"style": "rule",
"ground_truth": "288\\sqrt{3}"
} | {
"index": 3189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given tha... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volum... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3191,
"solution": "Let $P'$ be the prism similar to $P$, and let the sides of $P'$ be of length $x,y,z$, such that $x \\le y \\le z$. Then\n\\[\\frac{x}{a} = \\frac{y}{b} = \\frac zc < 1.\\]\n Note that if the ratio of similarity was equal to $1$, we would have a prism with zero volume. As one face of $P... |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism $Q$ has integral side lengths $a, b, c$ with $a \\le b \\le c$. A plane parallel to one of the faces of $Q$ cuts $Q$ into two prisms, one of which is similar to $Q$, with both having nonzero volumes. The middle side length $b = 3969$. Determine the num... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid).... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 3193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism has edge lengths $\\log_{5}x, \\log_{8}x,$ and $\\log_{10}x.$ Given that the sum of its surface area and volume is twice its volume, find the value of $x$.\nA) $1,000,000$\nB) $10,000,000$\nC) $100,000,000$\nD) $1,000,000,000$\nE) $10,000,000,000$"
}... | math | {
"style": "rule",
"ground_truth": "100,000,000"
} | {
"index": 3194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism has integer side lengths $a$ , $b$ , and $c$ . If $\\text{lcm}(a,b)=72$ , $\\text{lcm}(a,c)=24$ , and $\\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?\n\n*Proposed by Deyuan Li and Andrew Milas*"
... | math | {
"style": "rule",
"ground_truth": "3024"
} | {
"index": 3195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\\log_{2}x, \\log_{3}x,$ and $\\log_{4}x.$ What is $x?$"
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 3196,
"solution": "1. **Identify the expressions for surface area and volume:**\n - The surface area (SA) of a right rectangular prism is given by:\n \\[\n SA = 2(ab + bc + ca)\n \\]\n where \\(a\\), \\(b\\), and \\(c\\) are the lengths of the edges. For this problem, \\(a = \\log_{2}x\... |
DeepScaleR | [
{
"role": "user",
"content": "A right rectangular prism, with edge lengths $\\log_{5}x, \\log_{6}x,$ and $\\log_{8}x,$ must satisfy the condition that the sum of the squares of its face diagonals is numerically equal to 8 times the volume. What is $x?$\nA) $24$\nB) $36$\nC) $120$\nD) $\\sqrt{240}$\nE) $240$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{240}"
} | {
"index": 3197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right square pyramid with base edges of length $12$ units each and slant edges of length $15$ units each is cut by a plane that is parallel to its base and $4$ units above its base. What is the volume, in cubic units, of the top pyramid section that is cut off by this plane?"... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3} \\times \\left(\\frac{(144 \\cdot (153 - 8\\sqrt{153}))}{153}\\right) \\times (\\sqrt{153} - 4)"
} | {
"index": 3198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right square pyramid with base edges of length $8\\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 3199,
"solution": ""
} |
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