data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 3200,
"solution": "Although I can't draw the exact picture of this problem, but it is quite easy to imagine that four vertices of the base of this pyramid is on a circle (Radius $\\frac{6}{\\sqrt{2}} = 3\\sqrt{2}$). Since all five vertices are on the sphere, the distances of the spherical center and the ... |
DeepScaleR | [
{
"role": "user",
"content": "A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters?"
}
] | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"index": 3201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is $3$ , then the minimum possible volume of the cone can... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle $ABC$ is inscribed in a circle. From the vertex $C$ of the right angle, a chord $CM$ is drawn, intersecting the hypotenuse at point $K$. Find the area of triangle $ABM$ if $BK: AB = 3:4$, $BC=2\\sqrt{2}$, $AC=4$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{36}{19} \\sqrt{2}"
} | {
"index": 3203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle $\\triangle ABC$ has sides $AC=3$, $BC=4$, and $AB=5$. When this triangle is rotated around the right-angle side $BC$, the surface area of the resulting solid is ______."
}
] | math | {
"style": "rule",
"ground_truth": "24\\pi"
} | {
"index": 3204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a fact... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle \\(ABC\\) has a perimeter of 54, with the leg \\(AC\\) being greater than 10. A circle with a radius of 6, whose center lies on leg \\(BC\\), is tangent to lines \\(AB\\) and \\(AC\\). Find the area of triangle \\(ABC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "243/2"
} | {
"index": 3206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle \\(ABC\\) is inscribed in a circle. A chord \\(CM\\) is drawn from the vertex \\(C\\) of the right angle, intersecting the hypotenuse at point \\(K\\). Find the area of triangle \\(ABM\\) if \\(AK : AB = 1 : 4\\), \\(BC = \\sqrt{2}\\), and \\(AC = 2\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{19} \\sqrt{2}"
} | {
"index": 3207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1,24 , and 3 , and the segment of length 24 is a chord of the circle. Compute the area of the triangle."
}
... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 3208,
"solution": "Let the triangle be $\\triangle A B C$, with $A C$ as the hypotenuse, and let $D, E, F, G$ be on sides $A B, B C, A C$, $A C$, respectively, such that they all lie on the circle. We have $A G=1, G F=24$, and $F C=3$. By power of a point, we have $$\\begin{aligned} & A D=\\sqrt{A G \\cd... |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "60\\text{ inches}"
} | {
"index": 3209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has area 5 and a hypotenuse of length 5. Find its perimeter."
}
] | math | {
"style": "rule",
"ground_truth": "5+3 \\sqrt{5}"
} | {
"index": 3211,
"solution": "If $x, y$ denote the legs, then $x y=10$ and $x^{2}+y^{2}=25$, so $x+y+\\sqrt{x^{2}+y^{2}}=\\sqrt{\\left(x^{2}+y^{2}\\right)+2 x y}+5=\\sqrt{45}+5=5+3 \\sqrt{5}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has integer side lengths. One of its legs is 1575 units shorter than its hypotenuse, and the other leg is less than 1991 units. Find the length of the hypotenuse of this right triangle."
}
] | math | {
"style": "rule",
"ground_truth": "1799"
} | {
"index": 3212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 3213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has legs measuring 20 inches and 21 inches. What is the length of the hypotenuse, in inches?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 3214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has legs of lengths 126 and 168 units. What is the perimeter of the triangle formed by the points where the angle bisectors intersect the opposite sides?"
}
] | math | {
"style": "rule",
"ground_truth": "230.61"
} | {
"index": 3215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{48\\pi}{5}"
} | {
"index": 3216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has one angle measuring $30^\\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\\circ$, find the area of the quad... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{3} + 50}{2}"
} | {
"index": 3217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has perimeter $2008$ , and the area of a circle inscribed in the triangle is $100\\pi^3$ . Let $A$ be the area of the triangle. Compute $\\lfloor A\\rfloor$ ."
}
] | math | {
"style": "rule",
"ground_truth": "31541"
} | {
"index": 3218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has perimeter $32$ and area $20$. What is the length of its hypotenuse?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{59}{4}"
} | {
"index": 3219,
"solution": "\nLet the legs of the triangle be $a$ and $b$, and the hypotenuse be $c$. We are given that the perimeter is 32 and the area is 20. Therefore, we have the following equations:\n\n1. \\(a + b + c = 32\\)\n2. \\(\\frac{1}{2}ab = 20 \\Rightarrow ab = 40\\)\n\nUsing the Pythagorean theorem... |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has side lengths $a, b$, and $\\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "48+\\sqrt{2016}"
} | {
"index": 3220,
"solution": "There are no integer solutions to $a^{2}+b^{2}=2016$ due to the presence of the prime 7 on the right-hand side (by Fermat's Christmas Theorem). Assuming $a<b$, the minimal solution $(a, b)=(3,45)$ which gives the answer above."
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle has sides of lengths 5 cm and 11 cm. Calculate the length of the remaining side if the side of length 5 cm is a leg of the triangle. Provide your answer as an exact value and as a decimal rounded to two decimal places."
}
] | math | {
"style": "rule",
"ground_truth": "9.80"
} | {
"index": 3221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle is inscribed in the ellipse given by the equation $x^2 + 9y^2 = 9$. One vertex of the triangle is at the point $(0,1)$, and one leg of the triangle is fully contained within the x-axis. Find the squared length of the hypotenuse of the inscribed right triangle, ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 3222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle when rotating around a large leg forms a cone with a volume of $100\\pi$ . Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 3223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle with integer leg lengths is called \"cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 3224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle with legs $10$ and $8$ is similar to another right triangle with corresponding legs $x$ and $5$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"index": 3225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangle with legs each measuring 7 cells was drawn on graph paper. Then all the lines of the grid inside the triangle were outlined. What is the maximum number of triangles that can be found in this drawing?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 3226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangular prism $ABC-A_{1}B_{1}C_{1}$ has all its vertices on the surface of a sphere. Given that $AB=3$, $AC=5$, $BC=7$, and $AA_{1}=2$, find the surface area of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{208\\pi}{3}"
} | {
"index": 3227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangular prism \\( ABC-A_{1}B_{1}C_{1} \\) has 9 edges of equal length. Point \\( P \\) is the midpoint of \\( CC_{1} \\). The dihedral angle \\( B-A_{1}P-B_{1} \\) is \\( \\alpha \\). What is \\( \\sin \\alpha \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{4}"
} | {
"index": 3228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right triangular pyramid has a base edge length of $2$, and its three side edges are pairwise perpendicular. Calculate the volume of this pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 3229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right, rectangular prism has three faces with areas of $6,8$ and $12$ square inches. What is the volume of the prism, in cubic inches?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right-angled triangle has an area of \\( 36 \\mathrm{~m}^2 \\). A square is placed inside the triangle such that two sides of the square are on two sides of the triangle, and one vertex of the square is at one-third of the longest side.\n\nDetermine the area of this square."
... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to:"
}
] | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"index": 3232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rigid board with a mass \\( m \\) and a length \\( l = 20 \\) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \\( 2m \\) was placed at its very edge. How far f... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rigid board with a mass of \\(m\\) and a length of \\(l = 24\\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \\(2m\\) is placed at its very edge. How far from the ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\\%$ to $2\\%$ is approximately:"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 3235,
"solution": "1. **Understanding the problem**: The problem states that a railroad needs to rise 600 feet to cross a mountain. The grade of the railroad, which is the ratio of the rise to the horizontal length, can be adjusted by changing the length of the track.\n\n2. **Calculating the horizontal l... |
DeepScaleR | [
{
"role": "user",
"content": "A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\\text{th}}$ number in the li... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3236,
"solution": "1. **Understanding the Problem**: We need to find the $97^{\\text{th}}$ five-digit rising number and identify which digit from the given options it does not contain.\n\n2. **Counting Rising Numbers Starting with '1'**:\n - A five-digit rising number starting with '1' can be formed by... |
DeepScaleR | [
{
"role": "user",
"content": "A river boat travels at a constant speed from point A to point B. Along the riverbank, there is a road. The boat captain observes that every 30 minutes, a bus overtakes the boat from behind, and every 10 minutes, a bus approaches from the opposite direction. Assuming that the b... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their r... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 3238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A road of 1500 meters is being repaired. In the first week, $\\frac{5}{17}$ of the total work was completed, and in the second week, $\\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{17}"
} | {
"index": 3239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A robot colors natural numbers starting from 1 in ascending order according to the following rule: any natural number that can be expressed as the sum of two composite numbers is colored red, and those that do not meet the above criteria are colored yellow. If the numbers color... | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"index": 3240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A robot colors natural numbers starting from 1 in ascending order according to the following rule: any natural number that can be expressed as the sum of two composite numbers is colored red; those that do not meet this requirement are colored yellow. For example, 23 can be exp... | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"index": 3241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A robot has 6 modules, each needing a base and a cap attached sequentially. In how many different orders can the robot attach its bases and caps, assuming that, on each module, the base must be attached before the cap?\nA) $6! \\cdot 2^6$\nB) $(6!)^2$\nC) $12!$\nD) $\\frac{12!}... | math | {
"style": "rule",
"ground_truth": "\\frac{12!}{2^6}"
} | {
"index": 3242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A robot invented a cipher for encoding words: it replaced certain letters of the alphabet with one-digit or two-digit numbers, using only the digits 1, 2, and 3 (different letters were replaced with different numbers). Initially, it encoded itself: ROBOT = 3112131233. After enc... | math | {
"style": "rule",
"ground_truth": "2232331122323323132"
} | {
"index": 3243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 15... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 3244,
"solution": "Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. We note that $158=2 \\times 79$. Thus, after 79 groups of four j... |
DeepScaleR | [
{
"role": "user",
"content": "A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south.... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 3245,
"solution": "Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. In other words, we can consider each group of four jumps, starti... |
DeepScaleR | [
{
"role": "user",
"content": "A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $ . If the number of photograph... | math | {
"style": "rule",
"ground_truth": "$\\dfrac{1}{3}$"
} | {
"index": 3246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rod with a length of four meters has weights attached as follows: $20 \\mathrm{~kg}$ at one end, and at distances of one, two, and three meters from that end, weights of $30, 40, 50$ $\\mathrm{kg}$ respectively. Additionally, a weight of $60 \\mathrm{~kg}$ is attached at the ... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 3247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A roll of quarters contains $40$ quarters and a roll of dimes contains $50$ dimes. James has a jar that contains $83$ quarters and $159$ dimes. Lindsay has a jar that contains $129$ quarters and $266$ dimes. James and Lindsay pool these quarters and dimes and make complete rol... | math | {
"style": "rule",
"ground_truth": "\\$5.50"
} | {
"index": 3248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room has a floor with dimensions \\(7 \\times 8\\) square meters, and the ceiling height is 4 meters. A fly named Masha is sitting in one corner of the ceiling, while a spider named Petya is in the opposite corner of the ceiling. Masha decides to travel to visit Petya by the ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{265}"
} | {
"index": 3249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room is 19 feet long and 11 feet wide. Find the ratio of the length of the room to its perimeter. Express your answer in the form $a:b$."
}
] | math | {
"style": "rule",
"ground_truth": "19:60"
} | {
"index": 3250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$."
}
] | math | {
"style": "rule",
"ground_truth": "7:38"
} | {
"index": 3251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room is 25 feet long and 15 feet wide. Find the ratio of the length of the room to its perimeter and the ratio of the width of the room to its perimeter. Express both your answers in the form $a:b$."
}
] | math | {
"style": "rule",
"ground_truth": "3:16"
} | {
"index": 3252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is $12$ m. What is the area (in $m^2$ ) of the room?\n\n![Image](https://cdn.artofproblemsolving... | math | {
"style": "rule",
"ground_truth": "18\\pi"
} | {
"index": 3253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter."
}
] | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"index": 3254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A room measures 16 feet by 12 feet and includes a column with a square base of 2 feet on each side. Find the area in square inches of the floor that remains uncovered by the column."
}
] | math | {
"style": "rule",
"ground_truth": "27,072"
} | {
"index": 3255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A root of unity is a complex number that is a solution to $z^{n}=1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^{2}+a z+b=0$ for some integers $a$ and $b$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 3257,
"solution": "The only real roots of unity are 1 and -1. If $\\zeta$ is a complex root of unity that is also a root of the equation $z^{2}+a z+b$, then its conjugate $\\bar{\\zeta}$ must also be a root. In this case, $|a|=|\\zeta+\\bar{\\zeta}| \\leq|\\zeta|+|\\bar{\\zeta}|=2$ and $b=\\zeta \\bar{\\... |
DeepScaleR | [
{
"role": "user",
"content": "A rotating disc is divided into five equal sectors labeled $A$, $B$, $C$, $D$, and $E$. The probability of the marker stopping on sector $A$ is $\\frac{1}{5}$, the probability of it stopping in $B$ is $\\frac{1}{5}$, and the probability of it stopping in sector $C$ is equal to ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 3258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A round cake is cut into \\( n \\) pieces with 3 cuts. Find the product of all possible values of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 3259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A round pizza is $\\frac13$ of an inch thick and has a diameter of 12 inches. It is cut into 12 congruent pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 3260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $... | math | {
"style": "rule",
"ground_truth": "$\\frac{3\\sqrt{7}-\\sqrt{3}}{2}$"
} | {
"index": 3261,
"solution": "1. **Understanding the Geometry**: The problem describes a round table with radius $4$ and six rectangular place mats each with width $1$ and length $x$. Each place mat has two corners on the edge of the table, and the inner corners of each mat touch the inner corner of an adjacent mat... |
DeepScaleR | [
{
"role": "user",
"content": "A round-robin tennis tournament consists of each player playing every other player exactly once. How many matches will be held during an 8-person round-robin tennis tournament?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 3262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament wa... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A row consists of 10 chairs, but chair #5 is broken and cannot be used. Mary and James each sit in one of the available chairs, choosing their seats at random from the remaining chairs. What is the probability that they don't sit next to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 3264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rubber ball is released from a height of 120 feet and rebounds to three-quarters of the height it falls each time it bounces. How far has the ball traveled when it strikes the ground for the fifth time?"
}
] | math | {
"style": "rule",
"ground_truth": "612.1875"
} | {
"index": 3265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rubber band is 4 inches long. An ant begins at the left end. Every minute, the ant walks one inch along rightwards along the rubber band, but then the band is stretched (uniformly) by one inch. For what value of $n$ will the ant reach the right end during the $n$th minute?"
... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 3266,
"solution": "7 The ant traverses $1 / 4$ of the band's length in the first minute, $1 / 5$ of the length in the second minute (the stretching does not affect its position as a fraction of the band's length), $1 / 6$ of the length in the third minute, and so on. Since $$1 / 4+1 / 5+\\cdots+1 / 9<0.2... |
DeepScaleR | [
{
"role": "user",
"content": "A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \\(\\frac{1}{4}\\) inch."
}
] | math | {
"style": "rule",
"ground_truth": "2\\%"
} | {
"index": 3267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 3268,
"solution": "1. **Define the dimensions and areas:**\n - Let the length of the inner rectangle be $x$ feet. Since the width is given as 1 foot, the area of the inner rectangle is $x \\cdot 1 = x$ square feet.\n - The second largest rectangle, which includes the inner rectangle and a 1-foot wide... |
DeepScaleR | [
{
"role": "user",
"content": "A rugby team scored 24 points, 17 points, and 25 points in the seventh, eighth, and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A running competition on an unpredictable distance is conducted as follows. On a circular track with a length of 1 kilometer, two points \\( A \\) and \\( B \\) are randomly selected (using a spinning arrow). The athletes then run from \\( A \\) to \\( B \\) along the shorter a... | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"index": 3270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by $10\\pi $ feet, how wide is the track in feet?\n\n[asy]size(100); path g=scale(2)*unitcircle;\nfilldraw(unitcircle^^g,evenodd+grey,black);\n[/asy]"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A said: \"I am 10 years old, 2 years younger than B, and 1 year older than C.\"\nB said: \"I am not the youngest, C and I have a 3-year difference, and C is 13 years old.\"\nC said: \"I am younger than A, A is 11 years old, and B is 3 years older than A.\"\nAmong the three stat... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 3272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3273,
"solution": "To find the smallest possible range of a set of five observations with an arithmetic mean of $10$ and a median of $12$, we need to consider the properties of the mean and median, and how they affect the arrangement of the numbers in the set.\n\n1. **Understanding the Mean and Median**:... |
DeepScaleR | [
{
"role": "user",
"content": "A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sample is divided into 5 groups, with a total of 160 data points in the first, second, and third groups, and a total of 260 data points in the third, fourth, and fifth groups, and the frequency of the third group is 0.20. Calculate the frequency of the third group."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 3275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\\lfloor ... | math | {
"style": "rule",
"ground_truth": "947"
} | {
"index": 3276,
"solution": "With the same reasoning as Solution 1, in order to get largest possible value of D, we can construct that our set of numbers as $\\underbrace{1,1,1...1,}_\\text{n times}\\underbrace{2,2,2...2,}_\\text{n times}\\underbrace{3,3,3...3,}_\\text{n times}........\\underbrace{1000,1000,1000..... |
DeepScaleR | [
{
"role": "user",
"content": "A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 3277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A satellite is launched vertically from the Earth's pole with the first cosmic velocity. What is the maximum distance the satellite will reach from the Earth's surface? (The gravitational acceleration at the Earth's surface is $g = 10 \\, \\mathrm{m/s^2}$, and the Earth's radiu... | math | {
"style": "rule",
"ground_truth": "6400"
} | {
"index": 3278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A scale drawing of a park shows that one inch represents 800 feet. A line segment in the drawing that is 4.75 inches long represents how many feet?"
}
] | math | {
"style": "rule",
"ground_truth": "3800"
} | {
"index": 3279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A scalene triangle has side lengths which are prime numbers and the length of its perimeter is also prime. What is its smallest possible perimeter?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 3280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A scanning code consists of a $7 \\times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\\textit{symmetric}$ if its look does not cha... | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"index": 3281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school arranges five classes every morning from Monday to Friday, each lasting 40 minutes. The first class starts from 7:50 to 8:30, with a 10-minute break between classes. A student returns to school after taking leave. If he arrives at the classroom randomly between 8:50 an... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{4}"
} | {
"index": 3282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school arranges for five people, \\( A \\), \\( B \\), \\( C \\), \\( D \\), and \\( E \\), to enter into three classes, with each class having at least one person, and \\( A \\) and \\( B \\) cannot be in the same class. Calculate the total number of different arrangements."... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 3283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band?"
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 3284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars."
}
] | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"index": 3285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school club designed an unconventional rectangular checkerboard that measures 10 by 6. Calculate the probability that a randomly chosen unit square does not touch any edge of the board or is not adjacent to any edge."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 3286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and t... | math | {
"style": "rule",
"ground_truth": "-13.5"
} | {
"index": 3287,
"solution": "To find $t$ and $s$, we use the expected value formula, which is given by:\n\\[\n\\text{Expected Value} = \\sum (\\text{Outcome} \\cdot \\text{Probability})\n\\]\n\n1. **Calculating $t$:** \n $t$ is the average number of students per class if a teacher is picked at random. Since eac... |
DeepScaleR | [
{
"role": "user",
"content": "A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let be the average value obtained if a teacher is picked at random and the number o... | math | {
"style": "rule",
"ground_truth": "-13.5"
} | {
"index": 3288,
"solution": "1. **Calculate $t$ (average number of students per teacher):**\n The average number of students per teacher, $t$, is calculated by taking the sum of the number of students in each class and dividing by the number of teachers. Since each teacher teaches one class, the probability of p... |
DeepScaleR | [
{
"role": "user",
"content": "A school has 1200 students, and each student participates in exactly \\( k \\) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \\( k \\)."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 3289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school has between 130 and 210 students enrolled. Every afternoon, all the students gather to participate in a singing session. The students are divided into eight distinct groups. If two students are absent from school, the groups can all have the same number of students. Wh... | math | {
"style": "rule",
"ground_truth": "1870"
} | {
"index": 3290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school has between 150 and 200 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into six distinct sections of students. If one student is absent from school, the sections can all have the same number of... | math | {
"style": "rule",
"ground_truth": "1575"
} | {
"index": 3291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school has between 150 and 250 students enrolled. Each day, all the students split into eight different sections for a special workshop. If two students are absent, each section can contain an equal number of students. Find the sum of all possible values of student enrollment... | math | {
"style": "rule",
"ground_truth": "2626"
} | {
"index": 3292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school has eight identical copies of a particular book. At any given time, some of these copies are in the school library and some are with students. How many different ways are there for some of the books to be in the library and the rest to be with students if at least one ... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"index": 3293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how ma... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 3294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is __... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 3295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. R... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 3296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school library purchased 17 identical books. How much do they cost if they paid more than 11 rubles 30 kopecks, but less than 11 rubles 40 kopecks for 9 of these books?"
}
] | math | {
"style": "rule",
"ground_truth": "2142"
} | {
"index": 3297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school offers 7 courses for students to choose from, among which courses A, B, and C cannot be taken together due to scheduling conflicts, allowing at most one of them to be chosen. The school requires each student to choose 3 courses. How many different combinations of cours... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 3298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school organized a trip to the Expo Park for all third-grade students and rented some large buses. Initially, the plan was to have 28 people on each bus. After all the students boarded, it was found that 13 students could not get on the buses. So, they decided to have 32 peop... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 3299,
"solution": ""
} |
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