data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A computer can apply three operations to a number: \"increase by 2,\" \"increase by 3,\" \"multiply by 2.\" The computer starts with the number 1 and is made to go through all possible combinations of 6 operations (each combination is applied to the initial number 1). After how... | math | {
"style": "rule",
"ground_truth": "486"
} | {
"index": 1400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A computer can do $10,000$ additions per second. How many additions can it do in one hour?"
}
] | math | {
"style": "rule",
"ground_truth": "36000000"
} | {
"index": 1401,
"solution": "1. **Identify the rate of additions per second**: The computer can perform $10,000$ additions per second.\n\n2. **Convert the rate to additions per hour**: \n - There are $3600$ seconds in one hour.\n - To find the total number of additions in one hour, multiply the number of addit... |
DeepScaleR | [
{
"role": "user",
"content": "A computer program evaluates expressions without parentheses in the following way:\n\n1) First, it performs multiplications and divisions from left to right one by one.\n2) Then, it performs additions and subtractions from left to right.\n\nFor example, the value of the express... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 1402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A computer program is a function that takes in 4 bits, where each bit is either a 0 or a 1, and outputs TRUE or FALSE. How many computer programs are there?"
}
] | math | {
"style": "rule",
"ground_truth": "65536"
} | {
"index": 1403,
"solution": "The function has $2^{4}$ inputs and 2 outputs for each possible input, so the answer is $2^{2^{4}}=2^{16}=65536$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A computer screen shows a $98 \\times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white ... | math | {
"style": "rule",
"ground_truth": "\\[ 98 \\]"
} | {
"index": 1404,
"solution": "Answer: $98$ .\nThere are $4\\cdot97$ adjacent pairs of squares in the border and each pair has one black and one white square. Each move can fix at most $4$ pairs, so we need at least $97$ moves. However, we start with two corners one color and two another, so at least one rectangle m... |
DeepScaleR | [
{
"role": "user",
"content": "A computer software retailer has 1200 copies of a new software package to sell. From past experience, she knows that:\n\n- Half of them will sell right away at the original price she sets,\n- Two-thirds of the remainder will sell later when their price is reduced by 40%, and\n-... | math | {
"style": "rule",
"ground_truth": "$80.90"
} | {
"index": 1405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A computer software retailer has 1200 copies of a new software package to sell. Given that half of them will sell right away at the original price, two-thirds of the remainder will sell later when the price is reduced by 40%, and the remaining copies will sell in a clearance sa... | math | {
"style": "rule",
"ground_truth": "80.90"
} | {
"index": 1406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone and a sphere are placed inside a cylinder without overlapping. All bases are aligned and rest on the same flat surface. The cylinder has a radius of 6 cm and a height of 15 cm. The cone has a radius of 6 cm and a height of 7.5 cm. The sphere has a radius of 6 cm. Find th... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{138}"
} | {
"index": 1407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]\n\nimport markers;\nsize(150);\nimport geometry;\ndraw(scale(1,.2)*arc(... | math | {
"style": "rule",
"ground_truth": "48.0"
} | {
"index": 1408,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone has a volume of $2592\\pi$ cubic inches and the vertex angle of the vertical cross section is 90 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "20.0"
} | {
"index": 1409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone is formed from a 270-degree sector of a circle of radius 18 by aligning the two straight sides. What is the result when the volume of the cone is divided by $\\pi$?"
}
] | math | {
"style": "rule",
"ground_truth": "60.75\\sqrt{141.75}"
} | {
"index": 1410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone is formed from a 300-degree sector of a circle of radius 18 by aligning the two straight sides. [asy]\nsize(110);\ndraw(Arc((0,0),1,0,300));\ndraw((1,0)--(0,0)--(.5,-.5*sqrt(3)));\nlabel(\"18\",(.5,0),S); label(\"$300^\\circ$\",(0,0),NW);\n[/asy] What is the result when ... | math | {
"style": "rule",
"ground_truth": "225\\sqrt{11}"
} | {
"index": 1411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the total surface area of the cone to the lateral surface area of the cone, given that the side length of the pyramid's base is 4 and the angle between the pyramid's height and the plane of its lateral fa... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{3}}{3}"
} | {
"index": 1412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone is inscribed in a sphere such that the slant height of the cone is equal to the diameter of the base. Find the ratio of the total surface area of the cone to the surface area of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "9/16"
} | {
"index": 1413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone is inverted and filled with water to 3/4 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth. (You should enter 10.0000 for $10\\%$ instead of 0.1000.)"
}
] | math | {
"style": "rule",
"ground_truth": "42.1875"
} | {
"index": 1414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone is perfectly fitted inside a cube such that the cone's base is one face of the cube and its vertex touches the opposite face. A sphere is inscribed in the same cube. Given that one edge of the cube is 8 inches, calculate:\n1. The volume of the inscribed sphere.\n2. The v... | math | {
"style": "rule",
"ground_truth": "\\frac{128}{3}\\pi"
} | {
"index": 1415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone with a base radius of $15$ cm and a height of $30$ cm has a sphere inscribed within it. The radius of the sphere can be expressed in the form $b\\sqrt{d} - b$ cm. Determine the values of $b$ and $d$."
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"index": 1416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?"
}
] | math | {
"style": "rule",
"ground_truth": "4000"
} | {
"index": 1417,
"solution": "1. **Understanding the Problem**: We are given a cone-shaped mountain with a total height of 8000 feet. The top $\\frac{1}{8}$ of the volume of the mountain is above water. We need to find the depth of the ocean at the base of the mountain.\n\n2. **Volume of a Cone**: The volume $V$ of... |
DeepScaleR | [
{
"role": "user",
"content": "A confectionery factory received 5 rolls of ribbon, each 50 meters long, for packing cakes. How many cuts are needed to obtain pieces of ribbon that are 2 meters each?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 1418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A confectionery factory received 5 spools of ribbon, each 60 meters long, for packaging cakes. How many cuts are needed to obtain pieces of ribbon, each 1 meter 50 centimeters long?"
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"index": 1419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A conference center is setting up chairs in rows for a seminar. Each row can seat $13$ chairs, and currently, there are $169$ chairs set up. They want as few empty seats as possible but need to maintain complete rows. If $95$ attendees are expected, how many chairs should be re... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 1420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A conference hall is setting up seating for a workshop. Each row must contain $13$ chairs, and initially, there are $169$ chairs in total. The organizers expect $95$ participants to attend the workshop. To ensure all rows are completely filled with minimal empty seats, how many... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 1421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A conference lasted for 2 days. On the first day, the sessions lasted for 7 hours and 15 minutes, and on the second day, they lasted for 8 hours and 45 minutes. Calculate the total number of minutes the conference sessions lasted."
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 1422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines... | math | {
"style": "rule",
"ground_truth": "2013"
} | {
"index": 1423,
"solution": "\nTo solve this problem, we need to determine the least number of lines, \\( k \\), required to ensure that any Colombian configuration of 4027 points (where 2013 are red and 2014 are blue, with no three points collinear) can be separated such that no region contains points of both col... |
DeepScaleR | [
{
"role": "user",
"content": "A conical flask contains some water. When the flask is oriented so that its base is horizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep. When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2 inches d... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}+\\frac{\\sqrt{93}}{6}"
} | {
"index": 1424,
"solution": "$\\frac{1}{2}+\\frac{\\sqrt{93}}{6}$. Let $h$ be the height, and let $V$ be such that $V h^{3}$ equals the volume of the flask. When the base is at the bottom, the portion of the flask not occupied by water forms a cone similar to the entire flask, with a height of $h-1$; thus its volu... |
DeepScaleR | [
{
"role": "user",
"content": "A conical flask has a base radius of $15$ cm and a height of $30$ cm. The flask is filled with a liquid to a height of $10$ cm from the base, and a sphere is inscribed such that it just touches the liquid surface. What is the radius of the sphere? Express your answer in cm."
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$ . An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall o... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 1426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A conservatory houses five pairs of different animals, one male and one female of each type. The feeder must alternate between feeding a male and a female each time. If the feeder begins by feeding a female hippopotamus, how many ways can the feeder complete feeding all the ani... | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"index": 1427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A construction company built a cottage village consisting of three identical streets. The houses in the village are identical according to the plan and are up for sale at the same price. The business plan of the company, which includes expenses for all stages of construction, c... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 1428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A construction company in Changsha has a total of 50 slag transport vehicles, including 20 Type A and 30 Type B vehicles. Now all these slag transport vehicles are allocated to the construction of the Changsha-Zhuzhou-Xiangtan intercity light rail, with 30 vehicles sent to Site... | math | {
"style": "rule",
"ground_truth": "80000"
} | {
"index": 1429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A construction company purchased a piece of land for 80 million yuan. They plan to build a building with at least 12 floors on this land, with each floor having an area of 4000 square meters. Based on preliminary estimates, if the building is constructed with x floors (where x ... | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"index": 1430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different w... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A container holds $47\\frac{2}{3}$ cups of sugar. If one recipe requires $1\\frac{1}{2}$ cups of sugar, how many full recipes can be made with the sugar in the container? Express your answer as a mixed number."
}
] | math | {
"style": "rule",
"ground_truth": "31\\frac{7}{9}"
} | {
"index": 1432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount o... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$-inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its bas... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 1434,
"solution": "The scale factor is uniform in all dimensions, so the volume of the liquid is $\\left(\\frac{3}{4}\\right)^{3}$ of the container. The remaining section of the volume is $\\frac{1-\\left(\\frac{3}{4}\\right)^{3}}{1}$ of the volume, and therefore $\\frac{\\left(1-\\left(\\frac{3}{4}\\rig... |
DeepScaleR | [
{
"role": "user",
"content": "A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball i... | math | {
"style": "rule",
"ground_truth": "3 : 4 : 13"
} | {
"index": 1435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A container is composed of an upright hollow frustum and a hollow cylinder, each with a base radius of $12 \\,\\text{cm}$ and a height of $20 \\,\\text{cm}$. When finely granulated sand is poured into this container, it fills the frustum and partially fills the cylinder, with t... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 1436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A container with a capacity of 100 liters is filled with pure alcohol. After pouring out a portion of the alcohol, the container is filled with water. The mixture is then stirred thoroughly, and an amount of liquid equal to the first portion poured out is poured out again. The ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 1437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?"
}
] | math | {
"style": "rule",
"ground_truth": "4:40 a.m."
} | {
"index": 1438,
"solution": "1. **Convert the contest duration to hours and minutes**: The contest lasted for 1000 minutes. To convert this into hours, we divide by 60 (since there are 60 minutes in an hour):\n \\[\n \\frac{1000}{60} = 16 \\text{ hours and } 40 \\text{ minutes}\n \\]\n This means the conte... |
DeepScaleR | [
{
"role": "user",
"content": "A contest has six problems worth seven points each. On any given problem, a contestant can score either 0,1 , or 7 points. How many possible total scores can a contestant achieve over all six problems?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 1439,
"solution": "For $0 \\leq k \\leq 6$, to obtain a score that is $k(\\bmod 6)$ exactly $k$ problems must get a score of 1 . The remaining $6-k$ problems can generate any multiple of 7 from 0 to $7(6-k)$, of which there are $7-k$. So the total number of possible scores is $\\sum_{k=0}^{6}(7-k)=28$."
... |
DeepScaleR | [
{
"role": "user",
"content": "A contractor estimated that one of his two bricklayers would take $9$ hours to build a certain wall and the other $10$ hours. \nHowever, he knew from experience that when they worked together, their combined output fell by $10$ bricks per hour. \nBeing in a hurry, he put both m... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 1440,
"solution": "1. **Define the rates of the bricklayers**: Let the total number of bricks in the wall be $x$. The first bricklayer can complete the wall in $9$ hours, so his rate is $\\frac{x}{9}$ bricks per hour. The second bricklayer can complete the wall in $10$ hours, so his rate is $\\frac{x}{10... |
DeepScaleR | [
{
"role": "user",
"content": "A convex 2019-gon \\(A_{1}A_{2}\\ldots A_{2019}\\) is cut into smaller pieces along its 2019 diagonals of the form \\(A_{i}A_{i+3}\\) for \\(1 \\leq i \\leq 2019\\), where \\(A_{2020}=A_{1}, A_{2021}=A_{2}\\), and \\(A_{2022}=A_{3}\\). What is the least possible number of resul... | math | {
"style": "rule",
"ground_truth": "5049"
} | {
"index": 1441,
"solution": "Each time we draw in a diagonal, we create one new region, plus one new region for each intersection on that diagonal. So, the number of regions will be \\(1+\\text{ (number of diagonals) }+ \\text{ (number of intersections) }\\) where (number of intersections) counts an intersection o... |
DeepScaleR | [
{
"role": "user",
"content": "A convex hexagon \\( A_{1} A_{2} \\ldots A_{6} \\) is circumscribed around a circle \\( \\omega \\) with a radius of 1. Consider three segments that connect the midpoints of the opposite sides of the hexagon. What is the greatest \\( r \\) for which it can be stated that at lea... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 1442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{3\\sqrt{3}}{4}"
} | {
"index": 1443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle?"
}
] | math | {
"style": "rule",
"ground_truth": "179"
} | {
"index": 1444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $... | math | {
"style": "rule",
"ground_truth": "241"
} | {
"index": 1445,
"solution": "Every pair of vertices of the polyhedron determines either an edge, a face diagonal or a space diagonal. We have ${26 \\choose 2} = \\frac{26\\cdot25}2 = 325$ total line segments determined by the vertices. Of these, $60$ are edges. Each triangular face has $0$ face diagonals and each ... |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron $Q$ has vertices $V_1,V_2,\\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts pro... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 1446,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Understanding the Problem:**\n - A convex polyhedron $Q$ has $n$ vertices and $100$ edges.\n - Each vertex $V_k$ is associated with a plane $P_k$ that cuts all edges meeting at $V_k$.\n - The cuts do not intersect each other inside or on $Q$.... |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron S has vertices U1, U2, …, Um, and 120 edges. This polyhedron is intersected by planes Q1, Q2, …, Qm, where each plane Qk intersects only those edges that are connected to vertex Uk. No two planes intersect within the volume or on the surface of S. As a resul... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 1447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron \\( P \\) has 2021 edges. By cutting off a pyramid at each vertex, which uses one edge of \\( P \\) as a base edge, a new convex polyhedron \\( Q \\) is obtained. The planes of the bases of the pyramids do not intersect each other on or inside \\( P \\). How... | math | {
"style": "rule",
"ground_truth": "6063"
} | {
"index": 1448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron \\(Q\\) has \\(30\\) vertices, \\(70\\) edges, and \\(40\\) faces, of which \\(30\\) are triangular and \\(10\\) are pentagonal. Compute the number of space diagonals in polyhedron \\(Q\\)."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 1449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\\circ}, 72^{\\circ}$, and $72^{\\circ}$. Determine, with proof, the maximum possible value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ 36 \\]"
} | {
"index": 1450,
"solution": "Answer: 36 Solution: Consider such a polyhedron with $V$ vertices, $E$ edges, and $F=n$ faces. By Euler's formula we have $V+F=E+2$. Next, note that the number of pairs of incident faces and edges is both $2E$ and $3F$, so $2E=3F$. Now, since our polyhedron is convex, the sum of the de... |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. For the \\( V \\) vertices, each vertex has \\( T \\) triangular faces and \\( P \\) pentagonal faces intersecting. Find the value of \\( P + T + V \\)."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 1451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than a... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 1452,
"solution": "In the same ways as above, we find that there are 48 vertices. Now, notice that there are $\\binom{48}{2}$ total possible ways to choose two vertices. However, we must remove the cases where the segments do not lie in the interior of the polyhedron. We get\n\\[\\binom{48}{2}-12\\binom{... |
DeepScaleR | [
{
"role": "user",
"content": "A convex polyhedron is bounded by 4 regular hexagonal faces and 4 regular triangular faces. At each vertex of the polyhedron, 2 hexagons and 1 triangle meet. What is the volume of the polyhedron if the length of its edges is one unit?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23\\sqrt{2}}{12}"
} | {
"index": 1453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "4(36 + \\sqrt{113})"
} | {
"index": 1454,
"solution": "1. **Area and Diagonal Relationship**: \n Given that the area of convex quadrilateral $ABCD$ is $2002$, we can use the inequality for the area of a convex quadrilateral split by its diagonals:\n \\[\n [ABCD] \\leq \\frac{1}{2} (AC \\cdot BD)\n \\]\n This inequality holds bec... |
DeepScaleR | [
{
"role": "user",
"content": "A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relat... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 1455,
"solution": "By Bretschneider's Formula, \\[30=\\tfrac{1}{4}\\sqrt{4u^2v^2-(b^2+d^2-a^2-c^2)^2}=\\tfrac{1}{4}\\sqrt{4u^2v^2-441}.\\] Thus, $uv=3\\sqrt{1649}$. Also, \\[[ABCD]=\\tfrac 12 \\cdot uv\\sin{\\theta};\\] solving for $\\sin{\\theta}$ yields $\\sin{\\theta}=\\tfrac{40}{\\sqrt{1649}}$. Since... |
DeepScaleR | [
{
"role": "user",
"content": "A convex quadrilateral is determined by the points of intersection of the curves \\( x^{4}+y^{4}=100 \\) and \\( x y=4 \\); determine its area."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{17}"
} | {
"index": 1456,
"solution": "By symmetry, the quadrilateral is a rectangle having \\( x=y \\) and \\( x=-y \\) as axes of symmetry. Let \\( (a, b) \\) with \\( a>b>0 \\) be one of the vertices. Then the desired area is \\( (\\sqrt{2}(a-b)) \\cdot(\\sqrt{2}(a+b))=2\\left(a^{2}-b^{2}\\right)=2 \\sqrt{a^{4}-2 a^{2} b... |
DeepScaleR | [
{
"role": "user",
"content": "A convex quadrilateral is divided by its diagonals into four triangles; the areas of three of them are \\(10 \\, \\text{cm}^2\\), \\(20 \\, \\text{cm}^2\\), and \\(30 \\, \\text{cm}^2\\), and each is less than the area of the fourth triangle. Find the area of the given quadrila... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 1457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convex quadrilateral is drawn in the coordinate plane such that each of its vertices $(x, y)$ satisfies the equations $x^{2}+y^{2}=73$ and $x y=24$. What is the area of this quadrilateral?"
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 1458,
"solution": "The vertices all satisfy $(x+y)^{2}=x^{2}+y^{2}+2 x y=73+2 \\cdot 24=121$, so $x+y= \\pm 11$. Similarly, $(x-y)^{2}=x^{2}+y^{2}-2 x y=73-2 \\cdot 24=25$, so $x-y= \\pm 5$. Thus, there are four solutions: $(x, y)=(8,3),(3,8),(-3,-8),(-8,-3)$. All four of these solutions satisfy the orig... |
DeepScaleR | [
{
"role": "user",
"content": "A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?"
}
] | math | {
"style": "rule",
"ground_truth": "11475"
} | {
"index": 1459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A convoy of cars is moving on a highway at a speed of 80 km/h with a distance of 10 meters between the cars. Upon passing a speed limit sign, all cars reduce their speed to 60 km/h. What will be the distance between the cars in the convoy after passing the sign?"
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 1460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cooperative farm can purchase two types of feed mixtures from a neighboring farm to feed its animals. The Type I feed costs $30 per sack and contains 10 kg of component A and 10 kg of component B. The Type II feed costs $50 per sack and contains 10 kg of component A, 20 kg of... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 1461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cooperative receives apple and grape juice in identical containers and produces an apple-grape drink in identical cans. One container of apple juice is enough for exactly 6 cans of the drink, and one container of grape juice is enough for exactly 10 cans. When the recipe of t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 1462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coordinate paper is folded once such that the point \\((0,2)\\) overlaps with the point \\((4,0)\\). If the point \\((7,3)\\) overlaps with the point \\((m, n)\\), what is the value of \\(m+n\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "6.8"
} | {
"index": 1463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\\_1$ are $\\begin{cases} x=\\sqrt{2}\\sin(\\alpha + \\frac{\\pi}{4}) \\\\ y=\\sin(2\\alpha) + 1 \\end{cases}$, where $\\alpha$ is th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2} - 1"
} | {
"index": 1464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coordinate system and parametric equations problem (4-4):\n\nIn the rectangular coordinate system $x0y$, the parametric equations of line $l$ are given by $\\begin{cases} x = \\frac{1}{2}t \\ y = \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{3}}{2}t \\end{cases}$, where $t$ is the par... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"index": 1465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coordinate system and parametric equations problem (4-4):\n\nIn the rectangular coordinate system $xoy$, a curve $C_1$ is defined by the parametric equations $\\begin{cases} x = 1 + \\cos \\alpha \\\\ y = \\sin^2 \\alpha - \\frac{9}{4} \\end{cases}$, where $\\alpha$ is the pa... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"index": 1466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\\sqrt{2}, \\frac{\\pi}{4})$, and the polar coordinate equation of line $l$ is $\\rho... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 1467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A copper cube with an edge length of $l = 5 \\text{ cm}$ is heated to a temperature of $t_{1} = 100^{\\circ} \\text{C}$. Then, it is placed on ice, which has a temperature of $t_{2} = 0^{\\circ} \\text{C}$. Determine the maximum depth the cube can sink into the ice. The specifi... | math | {
"style": "rule",
"ground_truth": "0.06"
} | {
"index": 1468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 1469,
"solution": "1. **Identify the repeating pattern**: The problem states that the entire floor is tiled in the same way as the corner shown, and each of the four corners looks the same. This suggests a repeating pattern across the floor.\n\n2. **Determine the size of the repeating unit**: The solutio... |
DeepScaleR | [
{
"role": "user",
"content": "A corporation plans to expand its sustainability team to include specialists in three areas: energy efficiency, waste management, and water conservation. The company needs 95 employees to specialize in energy efficiency, 80 in waste management, and 110 in water conservation. It... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 1470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A couch cost 62,500 rubles. Its price was adjusted by 20% each month, either increasing or decreasing. Over six months, the price increased three times and decreased three times (the order of these changes is unknown). Can you uniquely determine the price of the couch after six... | math | {
"style": "rule",
"ground_truth": "55296"
} | {
"index": 1471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A council consists of nine women and three men. During their meetings, they sit around a round table with the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways can the nine chairs and three stools be arranged around the ro... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 1472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A counter begins at 0 . Then, every second, the counter either increases by 1 or resets back to 0 with equal probability. The expected value of the counter after ten seconds can be written as $\\frac{m}{n}$, where $m, n$ are positive integers and $\\operatorname{gcd}(m, n)=1$. ... | math | {
"style": "rule",
"ground_truth": "103324"
} | {
"index": 1473,
"solution": "The probability that the counter is equal to $k$ corresponds to the last $k$ seconds all being increases by 1 and the second before that being a reset to 0 , which happens with probability $2^{-k-1}$. The only contradiction to this is when $k=10$ and the counter gets there by only coun... |
DeepScaleR | [
{
"role": "user",
"content": "A courier set out on a moped from city $A$ to city $B$, which are 120 km apart. One hour later, a second courier on a motorcycle left from $A$. The second courier caught up to the first, delivered a message, and immediately returned to $A$ at the same speed, arriving back in $A... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 1474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight i... | math | {
"style": "rule",
"ground_truth": "27.1"
} | {
"index": 1475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determin... | math | {
"style": "rule",
"ground_truth": "27.1"
} | {
"index": 1476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 1477,
"solution": "1. **Identify the positions**: Let's denote the cowboy's initial position as point $C$, the stream as a horizontal line to the north of $C$, and the cabin as point $B$. According to the problem, $C$ is 4 miles south of the stream and 8 miles west and 7 miles north of $B$. \n\n2. **Calc... |
DeepScaleR | [
{
"role": "user",
"content": "A cowboy is 5 miles north of a stream which flows due west. He is also 10 miles east and 6 miles south of his cabin. He wishes to water his horse at the stream and then return home. Determine the shortest distance he can travel to accomplish this.\nA) $5 + \\sqrt{256}$ miles\nB... | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{356}"
} | {
"index": 1478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cowboy is 6 miles south of a stream which flows due east. He is also 12 miles west and 10 miles north of his cabin. Before returning to his cabin, he wishes to fill his water barrel from the stream and also collect firewood 5 miles downstream from the point directly opposite ... | math | {
"style": "rule",
"ground_truth": "11 + \\sqrt{305}"
} | {
"index": 1479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cowboy is initially 6 miles south and 2 miles west of a stream that flows due northeast. His cabin is located 12 miles east and 9 miles south of his initial position. He wants to water his horse at the stream and then return to his cabin. What is the shortest distance he can ... | math | {
"style": "rule",
"ground_truth": "8 + \\sqrt{545}"
} | {
"index": 1480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A crew of workers was tasked with pouring ice rinks on a large and a small field, where the area of the large field is twice the area of the small field. The part of the crew working on the large field had 4 more workers than the part of the crew working on the small field. Whe... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \\mathrm{~cm}^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters, given that the area of the cross is $810 \\mathrm{~cm}^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cross, consisting of two identical large squares and two identical small squares, was placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is 810 cm²."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cryptographer designed the following method to encode natural numbers: first, represent the natural number in base 5, then map the digits in the base 5 representation to the elements of the set $\\{V, W, X, Y, Z\\}$ in a one-to-one correspondence. Using this correspondence, h... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 1485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. \nSecond, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set \n$\\{V, W, X, Y,... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 1486,
"solution": "1. **Identify the pattern and the base-5 system**: The problem states that three consecutive integers are coded as $VYZ, VYX, VVW$. We know that in base-5, each digit represents powers of 5, starting from the rightmost digit (units place) to the leftmost digit.\n\n2. **Analyze the chan... |
DeepScaleR | [
{
"role": "user",
"content": "A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (asumming that the letter $A$ is one place to the right of the letter $Z$). The second time this same lett... | math | {
"style": "rule",
"ground_truth": "s"
} | {
"index": 1487,
"solution": "1. **Identify the pattern of replacement:** The problem states that each time a letter appears, it is replaced by the letter that is $1, 1+2, 1+2+3, \\ldots$ places to the right in the alphabet. This sequence represents the sum of the first $n$ natural numbers, where $n$ is the occurre... |
DeepScaleR | [
{
"role": "user",
"content": "A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"index": 1488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube \\(ABCDA_1B_1C_1D_1\\) has edge length 1. Point \\(M\\) is taken on the side diagonal \\(A_1D\\), and point \\(N\\) is taken on \\(CD_1\\), such that the line segment \\(MN\\) is parallel to the diagonal plane \\(A_1ACC_1\\). Find the minimum length of \\(MN\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 1489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has a square pyramid placed on one of its faces. Determine the sum of the combined number of edges, corners, and faces of this new shape."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 1490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has a surface area of 216 square centimeters. What is the volume of the cube, in cubic centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 1491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has an edge length of 30. A rectangular solid has edge lengths 20, 30, and $L$. If the cube and the rectangular solid have equal surface areas, what is the value of $L$?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 1492,
"solution": "The cube has six identical square faces, each of which is 30 by 30. Therefore, the surface area of the cube is $6(30^{2})=5400$. The rectangular solid has two faces that are 20 by 30, two faces that are 20 by $L$, and two faces that are 30 by $L$. Thus, the surface area of the rectangu... |
DeepScaleR | [
{
"role": "user",
"content": "A cube has edge length 4 m. One end of a rope of length 5 m is anchored to the centre of the top face of the cube. What is the integer formed by the rightmost two digits of the integer closest to 100 times the area of the surface of the cube that can be reached by the other end... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 1493,
"solution": "The top face of the cube is a square, which we label $ABCD$, and we call its centre $O$. Since the cube has edge length 4, then the side length of square $ABCD$ is 4. This means that $O$ is a perpendicular distance of 2 from each of the sides of square $ABCD$, and thus is a distance of... |
DeepScaleR | [
{
"role": "user",
"content": "A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled u... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1+\\sqrt{5}}{2}"
} | {
"index": 1494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]\n/* AMC8 1998 #17 Problem */\npair A=(0,48), B=(0,0), C=... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 1495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has side length $6$. Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)\n\n[asy]\nimport three;\nreal t=-0.05... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 1496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has side length 1. Find the product of the lengths of the diagonals of this cube (a diagonal is a line between two vertices that is not an edge)."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 1497,
"solution": "There are 12 diagonals that go along a face and 4 that go through the center of the cube, so the answer is $\\sqrt{2}^{12} \\cdot \\sqrt{3}^{4}=576$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\\circ$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 1498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{27}"
} | {
"index": 1499,
"solution": "When a dot is removed from a face with an even number of dots, that face then has an odd number of dots. When a dot is removed from a face with an odd number of dots, that face then has an even number of dots. Initially, there are 3 faces with an even number of dots and 3 faces with an... |
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