data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The digits $1,2,3,4,5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers is the digit $1$ to the left of both the digits $2$ and $3$?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 32100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\\overline{A B C}$ and $N=\\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \\cdot N$."
}
] | math | {
"style": "rule",
"ground_truth": "143745"
} | {
"index": 32101,
"solution": "By linearity of expectation and symmetry, $$\\mathbb{E}[M N]=\\mathbb{E}[(100 A+10 B+C)(100 D+10 E+F)]=111^{2} \\cdot \\mathbb{E}[A D]$$ Since $$\\mathbb{E}[A D]=\\frac{(1+2+3+4+5+6)^{2}-\\left(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}\\right)}{6 \\cdot 5}=\\frac{350}{30}$$ our answer is $1... |
DeepScaleR | [
{
"role": "user",
"content": "The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 32102,
"solution": "1. **Identify the requirement for the number to be even**: The number must end in an even digit. The available even digits are 2 and 4.\n\n2. **Determine the smallest possible even digit for the units place**: To form the smallest number, we prefer the smallest digits in the higher pl... |
DeepScaleR | [
{
"role": "user",
"content": "The digits 1, 2, 3, 4, and 5 were used, each one only once, to write a certain five-digit number \\(abcde\\) such that \\(abc\\) is divisible by 4, \\(bcd\\) is divisible by 5, and \\(cde\\) is divisible by 3. Find this number."
}
] | math | {
"style": "rule",
"ground_truth": "12453"
} | {
"index": 32103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits 1, 3, 4, and 5 are arranged randomly to form a four-digit number. What is the probability that the number is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 32104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits 2, 3, 5 and 7 are arranged randomly to form a four-digit number. What is the probability that the number is odd? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 32105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. What is the smallest possible difference between the two 2-digit numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits from 1 to 9 are each used exactly once to write three one-digit integers and three two-digit integers. The one-digit integers are equal to the length, width and height of a rectangular prism. The two-digit integers are equal to the areas of the faces of the same pris... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"index": 32108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits from 1 to 9 are randomly arranged to make a 9-digit number. What is the probability that the resulting number is divisible by 18?"
}
] | math | {
"style": "rule",
"ground_truth": "4/9"
} | {
"index": 32109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits from 1 to 9 are to be written in the nine cells of a $3 \\times 3$ grid, one digit in each cell.\n- The product of the three digits in the first row is 12.\n- The product of the three digits in the second row is 112.\n- The product of the three digits in the first co... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 32110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32111,
"solution": "Suppose that the original integer has tens digit $a$ and ones (units) digit $b$. This integer is equal to $10 a+b$. When the digits are reversed, the tens digit of the new integer is $b$ and the ones digit is $a$. This new integer is equal to $10 b+a$. Since the new two-digit integer ... |
DeepScaleR | [
{
"role": "user",
"content": "The digits of 2021 can be rearranged to form other four-digit whole numbers between 1000 and 3000. Find the largest possible difference between two such four-digit whole numbers."
}
] | math | {
"style": "rule",
"ground_truth": "1188"
} | {
"index": 32112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits of a certain three-digit number form a geometric progression. If the digits of the hundreds and units places are swapped, the new three-digit number will be 594 less than the original number. If, in the original number, the hundreds digit is removed and the remaining... | math | {
"style": "rule",
"ground_truth": "842"
} | {
"index": 32113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits of a four-digit positive integer add up to 14. The sum of the two middle digits is nine, and the thousands digit minus the units digit is one. If the integer is divisible by 11, what is the integer?"
}
] | math | {
"style": "rule",
"ground_truth": "3542"
} | {
"index": 32114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$?"
}
] | math | {
"style": "rule",
"ground_truth": "217"
} | {
"index": 32115,
"solution": "A brute-force solution to this question is fairly quick, but we'll try something slightly more clever: our numbers have the form ${\\underline{(n+3)}}\\,{\\underline{(n+2)}}\\,{\\underline{( n+1)}}\\,{\\underline {(n)}}$$= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n$, for ... |
DeepScaleR | [
{
"role": "user",
"content": "The digits of a three-digit number form a geometric progression with distinct terms. If this number is decreased by 200, the resulting three-digit number has digits that form an arithmetic progression. Find the original three-digit number."
}
] | math | {
"style": "rule",
"ground_truth": "842"
} | {
"index": 32116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits of a two-digit number $AB$ are reversed to form a second two-digit number, and the lesser of the two-digit numbers is subtracted from the greater. What prime number must be a factor of the difference if $A\\neq B$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The digits of the positive integer $N$ consist only of 1s and 0s, and $225$ divides $N$. What is the minimum value of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "111,111,100"
} | {
"index": 32118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number?"
}
] | math | {
"style": "rule",
"ground_truth": "-3 + 8i"
} | {
"index": 32119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The dilation, centered at $2 + 3i,$ with scale factor 3, takes $-1 - i$ to which complex number?"
}
] | math | {
"style": "rule",
"ground_truth": "-7 - 9i"
} | {
"index": 32120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The dimensions of a part on a drawing are $7{}_{-0.02}^{+0.05}$ (unit: $mm$), indicating that the maximum requirement for processing this part should not exceed ______, and the minimum should not be less than ______."
}
] | math | {
"style": "rule",
"ground_truth": "6.98"
} | {
"index": 32121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a, y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is:"
... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 32122,
"solution": "1. **Given Information and Equations**:\n - The dimensions of rectangle $R$ are $a$ and $b$ with $a < b$.\n - We need to find a rectangle with dimensions $x$ and $y$ such that $x < a$, $y < a$, the perimeter is one-third that of $R$, and the area is one-third that of $R$.\n\n2. **... |
DeepScaleR | [
{
"role": "user",
"content": "The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002$ in$^3$. Find the minimum possible sum of the three dimensions."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 32123,
"solution": "1. **Understanding the Problem**: We need to find the minimum possible sum of the three dimensions of a rectangular box with a volume of $2002$ cubic inches, where all dimensions are positive integers.\n\n2. **Prime Factorization of the Volume**: The volume of the box is given as $200... |
DeepScaleR | [
{
"role": "user",
"content": "The dimensions of a triangle are tripled to form a new triangle. If the area of the new triangle is 54 square feet, how many square feet were in the area of the original triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation wi... | math | {
"style": "rule",
"ground_truth": "294"
} | {
"index": 32125,
"solution": "Let there be $m$ members and $n$ members for the square and $c$ for the number of columns of the other formation. We have $n^2 +5 = c(c+7) \\implies n^2+5 = \\left(c+\\frac{7}{2}\\right)^2 -\\frac{49}{4} \\implies n^2 - \\left(c+\\frac{7}{2}\\right)^2 = -\\frac{69}{4} \\implies \\left... |
DeepScaleR | [
{
"role": "user",
"content": "The distance \\( AB \\) is 100 km. Cyclists depart simultaneously from \\( A \\) and \\( B \\) towards each other with speeds of 20 km/h and 30 km/h respectively. Along with the first cyclist from \\( A \\), a fly departs with a speed of 50 km/h. The fly travels until it meets ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 32126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance a dog covers in 3 steps is the same as the distance a fox covers in 4 steps and the distance a rabbit covers in 12 steps. In the time it takes the rabbit to run 10 steps, the dog runs 4 steps and the fox runs 5 steps. Initially, the distances between the dog, fox, ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 32127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between A and C is the absolute value of (k-7) plus the distance between B and C is the square root of ((k-4)^2 + (-1)^2). Find the value of k that minimizes the sum of these two distances."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"index": 32128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between Ivan's house and his grandmother's house is 12 km. Exactly at 12:00, Ivan left his house and walked along the straight road to his grandmother's house at a speed of 1 m/s. At 12:30, Ivan's parents called his grandmother, informed her that Ivan was coming to... | math | {
"style": "rule",
"ground_truth": "12:47"
} | {
"index": 32129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between Luga and Volkhov is 194 km, between Volkhov and Lodeynoe Pole is 116 km, between Lodeynoe Pole and Pskov is 451 km, and between Pskov and Luga is 141 km. What is the distance between Pskov and Volkhov?"
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"index": 32130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between cities $A$ and $B$ is 435 km. A train left city $A$ at a speed of 45 km/h. After 40 minutes, another train left city $B$ heading towards the first train at a speed of 55 km/h. What distance will be between them one hour before they meet?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 32131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between every two utility poles along the road is 50 meters. Xiao Wang travels at a constant speed in a car, and sees 41 utility poles in 2 minutes after seeing the first pole. How many meters does the car travel per hour?"
}
] | math | {
"style": "rule",
"ground_truth": "60000"
} | {
"index": 32132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between location A and location B originally required a utility pole to be installed every 45m, including the two poles at both ends, making a total of 53 poles. Now, the plan has been changed to install a pole every 60m. Excluding the two poles at both ends, how m... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between locations A and B is 135 kilometers. Two cars, a large one and a small one, travel from A to B. The large car departs 4 hours earlier than the small car, but the small car arrives 30 minutes earlier than the large car. The speed ratio of the small car to th... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 32134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between locations A and B is 291 kilometers. Persons A and B depart simultaneously from location A and travel to location B at a constant speed, while person C departs from location B and heads towards location A at a constant speed. When person B has traveled \\( ... | math | {
"style": "rule",
"ground_truth": "221"
} | {
"index": 32135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between points A and B is 1200 meters. Dachen starts from point A, and 6 minutes later, Xiaogong starts from point B. After another 12 minutes, they meet. Dachen walks 20 meters more per minute than Xiaogong. How many meters does Xiaogong walk per minute?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 32136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between the two intersections of $x=y^4$ and $x+y^2=1$ is $\\sqrt{u+v\\sqrt5}$. Find the ordered pair, $(u,v)$."
}
] | math | {
"style": "rule",
"ground_truth": "(-2,2)"
} | {
"index": 32137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between two cities on a map is 15 inches. If the scale is 0.25 inches = 3 miles, how many miles apart are the actual cities?"
}
] | math | {
"style": "rule",
"ground_truth": "180\\text{ miles}"
} | {
"index": 32138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between two parallel lines $l_1: 3x + 4y - 2 = 0$ and $l_2: ax + 6y = 5$ is _______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 32139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between two parallel lines $x-y=1$ and $2x-2y+3=0$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{2}}{4}"
} | {
"index": 32140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance between two vectors is the magnitude of their difference. Find the value of $t$ for which the vector\n\\[\\bold{v} = \\begin{pmatrix} 2 \\\\ -3 \\\\ -3 \\end{pmatrix} + t \\begin{pmatrix} 7 \\\\ 5 \\\\ -1 \\end{pmatrix}\\]is closest to\n\\[\\bold{a} = \\begin{pmat... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{75}"
} | {
"index": 32141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from $A$ to $B$ is covered 3 hours and 12 minutes faster by a passenger train compared to a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train covers 288 km more. If the speed of each train is increased by $10 \\mat... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 32142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from Goteborg to Jonkiping on a map is 88 cm. The scale on the map is 1 cm: 15 km. How far is it between the two city centers, in kilometers?"
}
] | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"index": 32143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from Stockholm to Malmö on a map is 120 cm. The scale on the map is 1 cm: 20 km. If there is a stop in between at Lund, which is 30 cm away from Malmö on the same map, how far is it from Stockholm to Malmö passing through Lund, in kilometers?"
}
] | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"index": 32144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from \\(A\\) to \\(B\\) is 999 km. Along the road, there are kilometer markers indicating the distances to \\(A\\) and \\(B\\): 0।999, 1।998, \\(\\ldots, 999।0. How many of these markers have only two different digits?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 32145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from city $A$ to city $B$ is $999$ km. Along the highway leading from $A$ to $B$, there are kilometer markers indicating the distances from the marker to $A$ and $B$ as shown:\n\n\n\nHow many of these markers use only two... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 32146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ de... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 32148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from point $\\left(1,0\\right)$ to the line $3x+4y-2+\\lambda \\left(2x+y+2\\right)=0$, where $\\lambda \\in R$, needs to be determined."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 32149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 32150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from the center \\( O \\) of a sphere with radius 12, which is circumscribed around a regular quadrangular pyramid, to a lateral edge is \\( 4 \\sqrt{2} \\). Find: \n1) the height of the pyramid;\n2) the distance from point \\( O \\) to the lateral face of the pyra... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}\\left(2 \\sqrt{2} - 1\\right)"
} | {
"index": 32151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.\n\n$$\nAB = 18, EO = 7, AE = 2 BE, R = ?\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from the origin to the line $3x+2y-13=0$ is $\\sqrt {13}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 32153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from the point $(3,0)$ to one of the asymptotes of the hyperbola $\\frac{{x}^{2}}{16}-\\frac{{y}^{2}}{9}=1$ is $\\frac{9}{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{5}"
} | {
"index": 32154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from the point of intersection of a circle's diameter with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.\n\n$$\nAB = 18, EO = 7, AE = 2BE, R = ?\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance from the point where a diameter of a circle intersects a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.\n\nGiven:\n\\[ AB = 18 \\, \\text{cm}, \\, EO = 7 \\, \\text{cm}, \\, AE = 2 \\, BE \... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map?"
}
] | math | {
"style": "rule",
"ground_truth": "30:1"
} | {
"index": 32157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distance traveled by the center \\( P \\) of a circle with radius 1 as it rolls inside a triangle with side lengths 6, 8, and 10, returning to its initial position."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distances between the points are given as $A B = 30$, $B C = 80$, $C D = 236$, $D E = 86$, $E A = 40$. What is the distance $E C$?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 32159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distances between the points on a line are given as $2, 4, 5, 7, 8, k, 13, 15, 17, 19$. Determine the value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 32161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distances from one end of the diameter of a circle to the ends of a chord parallel to this diameter are 5 and 12. Find the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 32162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distances from point \\( P \\), which lies inside an equilateral triangle, to its vertices are 3, 4, and 5. Find the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "9 + \\frac{25\\sqrt{3}}{4}"
} | {
"index": 32163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from the base to the top) at certain angles, with the sum of these three angles being ... | math | {
"style": "rule",
"ground_truth": "374"
} | {
"index": 32164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The distribution of populations in a group of counties is shown in this pie chart. What percent of the counties have fewer than 100,000 residents?\n\n[asy]\ndraw(Circle((0,0),25),linewidth(1));\ndraw((-24,-7)--(0,0)--(24,-7),linewidth(1));\ndraw((0,0)--(7,-24),linewidth(1));\nl... | math | {
"style": "rule",
"ground_truth": "84\\%"
} | {
"index": 32165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The divisors of a natural number \\( n \\) (including \\( n \\) and 1) which has more than three divisors, are written in ascending order: \\( 1 = d_{1} < d_{2} < \\ldots < d_{k} = n \\). The differences \\( u_{1} = d_{2} - d_{1}, u_{2} = d_{3} - d_{2}, \\ldots, u_{k-1} = d_{k}... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The dollar is now worth $\\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\\frac{1}{2{}^2{}^{n-1}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\\frac{1}{b... | math | {
"style": "rule",
"ground_truth": "506"
} | {
"index": 32167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the equation $y = \\frac{3x}{2x^2 - 6x + 4}$ is the set of all real numbers with the exception of the values $x = A$ and $x = B$. What is $A + B$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the function $f(x) = \\arcsin(\\log_{m}(nx))$ is a closed interval of length $\\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000."
}
] | math | {
"style": "rule",
"ground_truth": "371"
} | {
"index": 32169,
"solution": "We start with the same method as above. The domain of the arcsin function is $[-1, 1]$, so $-1 \\le \\log_{m}(nx) \\le 1$.\n\\[\\frac{1}{m} \\le nx \\le m\\] \\[\\frac{1}{mn} \\le x \\le \\frac{m}{n}\\] \\[\\frac{m}{n} - \\frac{1}{mn} = \\frac{1}{2013}\\] \\[n = 2013m - \\frac{2013}{m... |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the function $f(x) = \\arcsin(\\log_{m}(nx))$ is a closed interval of length $\\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "5371"
} | {
"index": 32170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the function $f(x)=\\log_{\\frac{1}{2}}(\\log_4(\\log_{\\frac{1}{4}}(\\log_{16}(\\log_{\\frac{1}{16}}x))))$ is an interval of length $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?"
}
] | math | {
"style": "rule",
"ground_truth": "271"
} | {
"index": 32171,
"solution": "To find the domain of the function \n\\[ f(x) = \\log_{\\frac{1}{2}}\\left(\\log_4\\left(\\log_{\\frac{1}{4}}\\left(\\log_{16}\\left(\\log_{\\frac{1}{16}}x\\right)\\right)\\right)\\right), \\]\nwe need to ensure that each logarithm in the nested expression is defined and positive, as ... |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the function $y=\\sin x$ is $[a,b]$, and its range is $\\left[-1, \\frac{1}{2}\\right]$. Calculate the maximum value of $b-a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"index": 32172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the function \\( f(x) \\) is \\( (0,1) \\), and the function is defined as follows:\n\\[ \nf(x)=\\begin{cases}\nx, & \\text{if } x \\text{ is an irrational number}, \\\\\n\\frac{p+1}{q}, & \\text{if } x=\\frac{p}{q}, \\; p, q \\in \\mathbf{N}^{*}, \\; (p, q) = 1, ... | math | {
"style": "rule",
"ground_truth": "16/17"
} | {
"index": 32173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The domain of the function \\( f(x) \\) is \\( D \\). If for any \\( x_{1}, x_{2} \\in D \\), when \\( x_{1} < x_{2} \\), it holds that \\( f(x_{1}) \\leq f(x_{2}) \\), then \\( f(x) \\) is called a non-decreasing function on \\( D \\). Suppose that the function \\( f(x) \\) is... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 32174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The eccentricity of the ellipse $\\frac {x^{2}}{9}+ \\frac {y^{2}}{4+k}=1$ is $\\frac {4}{5}$. Find the value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 32175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The eccentricity of the ellipse given by the parametric equations $\\begin{cases} x=3\\cos\\theta \\\\ y=4\\sin\\theta\\end{cases}$ is $\\frac{\\sqrt{7}}{\\sqrt{3^2+4^2}}$, calculate this value."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {7}}{4}"
} | {
"index": 32176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The eccentricity of the ellipse given that the slope of line $l$ is $2$, and it intersects the ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ at two different points, where the projections of these two intersection points on the $x$-axis are exactly the two f... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"index": 32177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The eccentricity of the hyperbola defined by the equation $\\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1$ given that a line with a slope of -1 passes through its right vertex A and intersects the two asymptotes of the hyperbola at points B and C, and if $\\overrightarrow {A... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 32178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ecology club at a school has 30 members: 12 boys and 18 girls. A 4-person committee is to be chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{530}{609}"
} | {
"index": 32179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The edge of a regular tetrahedron is equal to \\(\\sqrt{2}\\). Find the radius of the sphere whose surface touches all the edges of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 32180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 32181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The edge of cube \\( ABCD A_1 B_1 C_1 D_1 \\) is 12. Point \\( K \\) lies on the extension of edge \\( BC \\) at a distance of 9 from vertex \\( C \\). Point \\( L \\) on edge \\( AB \\) is at a distance of 5 from \\( A \\). Point \\( M \\) divides segment \\( A_1 C_1 \\) in a ... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 32182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The edge of the cube \\(E F G H E_1 F_1 G_1 H_1\\) is equal to 2. Points \\(A\\) and \\(B\\) are taken on the edges \\(E H\\) and \\(H H_1\\) such that \\(\\frac{E A}{A H} = 2\\) and \\(\\frac{B H}{B H_1} = \\frac{1}{2}\\). A plane is drawn through the points \\(A\\), \\(B\\), ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{\\frac{2}{11}}"
} | {
"index": 32183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The edges meeting at one vertex of a rectangular parallelepiped are in the ratio of $1: 2: 3$. What is the ratio of the lateral surface areas of the cylinders that can be circumscribed around the parallelepiped?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13} : 2\\sqrt{10} : 3\\sqrt{5}"
} | {
"index": 32184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The edges of a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 32185,
"solution": "1. **Identify the Key Points and Their Coordinates:**\n - Let $P$ be a point on edge $AB$ and $Q$ be a point on edge $CD$ of a regular tetrahedron with vertices $A, B, C, D$ and each edge of length 1.\n - Assume $P$ and $Q$ are the midpoints of $AB$ and $CD$, respectively, since s... |
DeepScaleR | [
{
"role": "user",
"content": "The eight digits in Jane's phone number and the four digits in her office number have the same sum. The digits in her office number are distinct, and her phone number is 539-8271. What is the largest possible value of Jane's office number?"
}
] | math | {
"style": "rule",
"ground_truth": "9876"
} | {
"index": 32186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take o... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 32187,
"solution": "1. **Identify the total number of students and the students in each category**: \n - Total number of students = 93\n - Students taking math = 70\n - Students taking foreign language = 54\n\n2. **Use the principle of inclusion-exclusion**:\n - The principle states that for any ... |
DeepScaleR | [
{
"role": "user",
"content": "The electronic clock on the International Space Station displayed time in the format HH:MM. Due to an electromagnetic storm, the device malfunctioned, and each digit on the display either increased by 1 or decreased by 1. What was the actual time of the storm if the clock displ... | math | {
"style": "rule",
"ground_truth": "11:18"
} | {
"index": 32188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The elevator buttons in Harvard's Science Center form a $3 \\times 2$ grid of identical buttons, and each button lights up when pressed. One day, a student is in the elevator when all the other lights in the elevator malfunction, so that only the buttons which are lit can be se... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 32189,
"solution": "We first note that there are $2^{6}-1=63$ possibilities for lights in total. We now count the number of duplicates we need to subtract by casework on the number of buttons lit. To do this, we do casework on the size of the minimal \"bounding box\" of the lights: - If the bounding box ... |
DeepScaleR | [
{
"role": "user",
"content": "The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ (where $a > b > 0$) has an eccentricity of $e = \\frac{2}{3}$. Points A and B lie on the ellipse and are not symmetrical with respect to the x-axis or the y-axis. The perpendicular bisector of segment AB intersects the x-axi... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 32191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y+3)^2 = 1$ are tangent. Compute $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 32192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ellipse $x^2+4y^2=4$ and the hyperbola $x^2-m(y+2)^2 = 1$ are tangent. Compute $m.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13}"
} | {
"index": 32193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ellipse \\(\\frac{x^{2}}{2002^{2}}+\\frac{y^{2}}{1949^{2}}=1\\) intersects with two lines passing through the origin at points \\(A, B, C,\\) and \\(D\\). The slope of line \\(AB\\) is \\(k\\), and the slope of line \\(CD\\) is \\(-\\frac{1949^{2}}{2002^{2} \\cdot k}\\). Gi... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 32194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ellipse whose equation is\n\\[\\frac{x^2}{25} + \\frac{y^2}{9} = 1\\]is graphed below. The chord $\\overline{AB}$ passes through a focus $F$ of the ellipse. If $AF = \\frac{3}{2},$ then find $BF.$\n\n[asy]\nunitsize (0.6 cm);\n\npair A, B, F;\n\nF = (4,0);\nA = (35/8,3*sq... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 32195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The endpoints of a diameter of circle $M$ are $(-1,-4)$ and $(-7,6)$. What are the coordinates of the center of circle $M$? Express your answer as an ordered pair."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,1)"
} | {
"index": 32196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The endpoints of a line segment $MN$ with a fixed length of $4$ move on the parabola $y^{2}=x$. Let $P$ be the midpoint of the line segment $MN$. The minimum distance from point $P$ to the $y$-axis is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{4}"
} | {
"index": 32197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 32198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The endpoints of a line segment \\\\(AB\\\\) with a fixed length of \\\\(3\\\\) move on the parabola \\\\(y^{2}=x\\\\). Let \\\\(M\\\\) be the midpoint of the line segment \\\\(AB\\\\). The minimum distance from \\\\(M\\\\) to the \\\\(y\\\\)-axis is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{4}"
} | {
"index": 32199,
"solution": ""
} |
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