data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "From a batch of parts, 50 are drawn, and then 40 out of these 50 are inspected. It is found that there are 38 qualified products. Calculate the pass rate of this batch of products as a percentage."
}
] | math | {
"style": "rule",
"ground_truth": "95\\%"
} | {
"index": 12300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a bottle containing 1 liter of alcohol, $\\frac{1}{3}$ liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, $\\frac{1}{3}$ liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly. Finally, 1 liter ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{27}"
} | {
"index": 12301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a certain airport, 100 planes (1 command plane and 99 supply planes) take off at the same time. When their fuel tanks are full, each plane can fly 1000 kilometers. During the flight, planes can refuel each other, and after completely transferring their fuel, planes can lan... | math | {
"style": "rule",
"ground_truth": "100000"
} | {
"index": 12302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a circular piece of paper with radius $BC$, Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 12 centimeters and of volume $432\\pi$ cubic centimeters. What is the number of ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 12303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a class, 5 students were selected to measure their heights (in cm), and the data obtained were 160, 162, 159, 160, 159. The variance $s^2$ of this group of data is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 12304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a container filled to the brim with $100\\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening r... | math | {
"style": "rule",
"ground_truth": "1.75"
} | {
"index": 12305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calcu... | math | {
"style": "rule",
"ground_truth": "1006009"
} | {
"index": 12306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a deck of cards marked with 1, 2, 3, and 4, two cards are drawn consecutively. The probability of drawing the card with the number 4 on the first draw, the probability of not drawing it on the first draw but drawing it on the second, and the probability of drawing the numb... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a deck of cards, 5 spades, 4 clubs, and 6 hearts, totaling 15 cards, are drawn. If drawing $m$ cards such that all three suits are present is a certain event, then the minimum value of $m$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a group of $3$ orthopedic surgeons, $4$ neurosurgeons, and $5$ internists, a medical disaster relief team of $5$ people is to be formed. How many different ways can the team be selected such that there is at least one person from each specialty?"
}
] | math | {
"style": "rule",
"ground_truth": "590"
} | {
"index": 12309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a group of 6 students, 4 are to be selected to participate in competitions for four subjects: mathematics, physics, chemistry, and biology. If two students, A and B, cannot participate in the biology competition, determine the number of different selection plans."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 12310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a group of boys and girls, 15 girls leave. There are then left two boys for each girl. After this 45 boys leave. There are then 5 girls for each boy. The number of girls in the beginning was:"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12311,
"solution": "1. **Define Variables:**\n Let $b$ represent the number of boys and $g$ represent the number of girls initially.\n\n2. **Translate the Problem into Equations:**\n - After 15 girls leave, the number of girls becomes $g - 15$. According to the problem, there are then two boys for ea... |
DeepScaleR | [
{
"role": "user",
"content": "From a group of six students, how many different four-student committees can be chosen?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a large bottle containing 1 liter of alcohol, 1/3 liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, 1/3 liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly again. Finally, 1/3 liter of the m... | math | {
"style": "rule",
"ground_truth": "8/27"
} | {
"index": 12313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{2}$"
} | {
"index": 12314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a plywood circle with a diameter of 30 cm, two smaller circles with diameters of 20 cm and 10 cm are cut out. What is the diameter of the largest circle that can be cut from the remaining piece of plywood?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a point \\( M \\) on the ellipse \\(\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1\\), two tangent lines are drawn to the circle with the minor axis of the ellipse as its diameter. The points of tangency are \\( A \\) and \\( B \\). The line \\( AB \\) intersects the \\(x\\)-axis and... | math | {
"style": "rule",
"ground_truth": "10/3"
} | {
"index": 12316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 12317,
"solution": "\nTo solve this problem, we need to calculate the probability that a triangle formed by choosing three vertices of a regular octagon has at least one side that is also a side of the octagon.\n\n#### Total Number of Triangles:\nThe total number of ways to choose 3 vertices from 8 verti... |
DeepScaleR | [
{
"role": "user",
"content": "From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows: \n75, 80, 85, 65, 95, 100, 70, 55, 65, 75, 85, 110, 120, 80, 85, 80, 75, 90, 90, 95, 70, 60, 60, 75, 90, 95, 65, ... | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 12318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a set consisting of three blue cards labeled $X$, $Y$, $Z$ and three orange cards labeled $X$, $Y$, $Z$, two cards are randomly drawn. A winning pair is defined as either two cards with the same label or two cards of the same color. What is the probability of drawing a win... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 12319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a set of integers $\\{1, 2, 3, \\ldots, 12\\}$, eight distinct integers are chosen at random. What is the probability that, among those selected, the third smallest number is $4$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{165}"
} | {
"index": 12320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 12321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a square with a side length of $6 \\text{ cm}$, identical isosceles right triangles are cut off from each corner so that the area of the square is reduced by $32\\%$. What is the length of the legs of these triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"index": 12322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a standard deck of 52 cards, what is the probability that the top four cards include only cards from the black suits ($\\spadesuit$ and $\\clubsuit$)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{276}{4998}"
} | {
"index": 12323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are"
}
] | math | {
"style": "rule",
"ground_truth": "$3^{1/3},\\ 2^{1/2}$"
} | {
"index": 12324,
"solution": "To find the greatest and the next to the greatest values among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$, we can compare these values by considering their logarithmic forms or by direct calculation.\n\n1. **Convert to comparable forms:**\n We can compare $a^{1/b}$ by considering $b \\cdo... |
DeepScaleR | [
{
"role": "user",
"content": "From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 12325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From an arbitrary tetrahedron, four smaller tetrahedra are separated by four planes passing through the midpoints of the edges emanating from each vertex. Calculate the ratio of the volume of the remaining body to the volume of the original tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 12326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From an external point \\(A\\), a tangent \\(AB\\) and a secant \\(ACD\\) are drawn to a circle. Find the area of triangle \\(CBD\\), given that the ratio \\(AC : AB = 2 : 3\\) and the area of triangle \\(ABC\\) is 20."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 12327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From milk with a fat content of $5\\%$, cottage cheese with a fat content of $15.5\\%$ is produced, while there remains whey with a fat content of $0.5\\%$. How much cottage cheese is obtained from 1 ton of milk?"
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 12328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From noon till midnight, Clever Cat sleeps under the oak tree and from midnight till noon he is awake telling stories. A poster on the tree above him says \"Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time\". For how many hours a day doe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 12329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From point $A$ outside a circle, a tangent and a secant are drawn to the circle. The distance from point $A$ to the point of tangency is 16, and the distance from point $A$ to one of the intersection points of the secant with the circle is 32. Find the radius of the circle if t... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From point $A$ to point $B$, three cars depart at equal time intervals. They all arrive at $B$ simultaneously, and then they proceed to point $C$, which is 120 km away from $B$. The first car arrives at point $C$ one hour after the second car. After reaching point $C$, the thir... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From point $A$, Leigh walked 40 yards south, 60 yards west, 10 yards north, and 20 yards east to point $B$. What is the length, in yards, of $\\overline{AB}$?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 12333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From point \\( A \\), two rays are drawn intersecting a given circle: one at points \\( B \\) and \\( C \\), and the other at points \\( D \\) and \\( E \\). It is known that \\( AB = 7 \\), \\( BC = 7 \\), and \\( AD = 10 \\). Determine \\( DE \\)."
}
] | math | {
"style": "rule",
"ground_truth": "0.2"
} | {
"index": 12334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From points A and B, a motorcyclist and a cyclist respectively set off towards each other simultaneously and met at a distance of 4 km from B. At the moment the motorcyclist arrived in B, the cyclist was at a distance of 15 km from A. Find the distance AB."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From post office $A$, a car leaves heading towards post office $B$. After 20 minutes, a motorcyclist departs in pursuit of the car, traveling at a speed of 60 km/h. Upon catching up with the car, the motorcyclist delivers a package to the driver's cab and immediately turns back... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 12336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From six balls numbered respectively $1$, $2$, $3$, $4$, $5$, $6$, which are all of the same size, three balls are randomly drawn. Find the probability that exactly two of the balls have consecutive numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{5}"
} | {
"index": 12337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $10$ numbers $0-9$, select $3$ numbers. Find:<br/>\n$(1)$ How many unique three-digit numbers can be formed without repeating any digits?<br/>\n$(2)$ How many unique three-digit odd numbers can be formed without repeating any digits?"
}
] | math | {
"style": "rule",
"ground_truth": "320"
} | {
"index": 12338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers, and calculate the probability that these $2$ numbers are coprime."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers. The probability that these $2$ numbers are coprime is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, choose any $4$ vertices, the probability that these $4$ points lie in the same plane is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"index": 12341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, choose any $4$ vertices. The probability that these $4$ points lie in the same plane is ______ (express the result as a simplified fraction)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"index": 12342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"index": 12343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, select any $4$ vertices. The probability that these $4$ points lie in the same plane is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"index": 12344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 10 numbers $0, 1, 2, \\cdots, 9$, select 3 such that their sum is an even number not less than 10. How many different ways are there to make such a selection?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 12345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 1000 natural numbers ranging from 1 to 1000, a certain number of them are drawn. If the sum of any two numbers drawn is not a multiple of 5, then the maximum number of numbers that can be drawn from these 1000 natural numbers is ."
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 12346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 16 vertices of a $3 \\times 3$ grid comprised of 9 smaller unit squares, what is the probability that any three chosen vertices form a right triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "9/35"
} | {
"index": 12347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 20 natural numbers 1, 2, 3, ..., 20, if three numbers are randomly selected and their sum is an even number greater than 10, then there are $\\boxed{\\text{answer}}$ such sets of numbers."
}
] | math | {
"style": "rule",
"ground_truth": "563"
} | {
"index": 12348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers at a minimum must be taken to ensure that among the selected numbers, there will definitely be a pair whose sum is a multiple of ten?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers must be chosen to ensure that there are at least two numbers whose sum is a multiple of 10?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 200 natural numbers from 1 to 200, how many numbers must be selected to ensure that there are at least 2 numbers whose sum is a multiple of 5?"
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"index": 12351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 2015 natural numbers between 1 and 2015, what is the maximum number of numbers that can be found such that their product multiplied by 240 is a perfect square?"
}
] | math | {
"style": "rule",
"ground_truth": "134"
} | {
"index": 12352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 4 digits 0, 1, 2, 3, select 3 digits to form a three-digit number without repetition. How many of these three-digit numbers are divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\\frac{m}{n}$? Provide the answer in simplest fraction form."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{2020}{2017}"
} | {
"index": 12354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 12355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the center \\( O \\) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \\( O \\) appears at a right angle. What is the ratio of the sides of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "3 : 4 : 5"
} | {
"index": 12356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the consecutive natural numbers 1, 2, 3, …, 2014, select $n$ numbers such that for any two numbers chosen, one is not five times the other. Find the maximum value of $n$ and explain the reason."
}
] | math | {
"style": "rule",
"ground_truth": "1665"
} | {
"index": 12357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the diagram below, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 15.$ [asy]\nimport cse5; import olympiad;\nsize(120);\nadd(grid(8,8));\ndraw((0,0)--(8,0),linewidth(1.2));\ndraw((0,0)--(0,8),linewidth(1.2));\nlabel(\"$x$\",(8,0),E);\... | math | {
"style": "rule",
"ground_truth": "39.5"
} | {
"index": 12358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the digits $1$, $2$, $3$, $4$, form a four-digit number with the first digit being $1$, and having exactly two identical digits in the number. How many such four-digit numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the digits 0, 1, 2, 3, 4, 5, 6, select 2 even numbers and 1 odd number to form a three-digit number without repeating digits. The number of such three-digit numbers that are divisible by 5 is ____. (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 12360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the digits 0, 1, 2, 3, 4, 5, select three different digits to form a three-digit even number. How many such numbers can be formed? (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 12361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the first 539 positive integers, we select some such that their sum is at least one-third of the sum of the original numbers. What is the minimum number of integers we need to select for this condition to be satisfied?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 12362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the five numbers \\\\(1, 2, 3, 4, 5\\\\), select any \\\\(3\\\\) to form a three-digit number without repeating digits. When the three digits include both \\\\(2\\\\) and \\\\(3\\\\), \\\\(2\\\\) must be placed before \\\\(3\\\\) (not necessarily adjacent). How many such t... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 12363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the five points consisting of the four vertices and the center of a square, any two points are chosen. The probability that the distance between these two points is not less than the side length of the square is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 12364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the following infinite list of numbers, how many are integers? $$\\sqrt{4096},\\sqrt[3]{4096},\\sqrt[4]{4096},\\sqrt[5]{4096},\\sqrt[6]{4096},\\ldots$$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 12365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the four numbers $0,1,2,3$, we want to select $3$ digits to form a three-digit number with no repeating digits. What is the probability that this three-digit number is divisible by $3$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{9}"
} | {
"index": 12366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the integers 1 to 2020, there are a total of 1616 integers that are not multiples of 5. These 1616 numbers need to be divided into groups (each group may have a different number of elements), such that the difference (larger number minus smaller number) between any two num... | math | {
"style": "rule",
"ground_truth": "404"
} | {
"index": 12367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the natural numbers 1 to 100, each time we take out two different numbers so that their sum is greater than 100, how many different ways are there to do this?"
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 12368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the natural numbers 1 to 2008, the maximum number of numbers that can be selected such that the sum of any two selected numbers is not divisible by 3 is ____."
}
] | math | {
"style": "rule",
"ground_truth": "671"
} | {
"index": 12369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the probability of randomly selecting two different numbers such that both numbers are odd is $\\_\\_\\_\\_\\_\\_\\_\\_\\_$, and the probability that the product of the two numbers is even is $\\_\\_\\_\\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{18}"
} | {
"index": 12370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, four different numbers are selected, denoted as $a$, $b$, $c$, $d$ respectively. If the parity of $a+b$ is the same as the parity of $c+d$, then the total number of ways to select $a$, $b$, $c$, $d$ is ______ (provide the... | math | {
"style": "rule",
"ground_truth": "912"
} | {
"index": 12371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1, 2, 3, 4, 5$, 3 numbers are randomly drawn (with replacement) to form a three-digit number. What is the probability that the sum of its digits equals 9?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{19}{125}$"
} | {
"index": 12372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 10$, a number $a$ is randomly selected, and from the numbers $-1, -2, \\cdots, -10$, a number $b$ is randomly selected. What is the probability that $a^{2} + b$ is divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 12373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 20$, calculate the probability that 3 numbers randomly selected form an arithmetic sequence."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{38}"
} | {
"index": 12375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 2005$, choose $n$ different numbers. If it is always possible to find three numbers among these $n$ numbers that can form the side lengths of a triangle, determine the minimum value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 12376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers $1,2,3, \\cdots, 2014$, select 315 different numbers (order does not matter) to form an arithmetic sequence. Among these, the number of ways to form an arithmetic sequence that includes the number 1 is ___. The total number of ways to form an arithmetic sequenc... | math | {
"style": "rule",
"ground_truth": "5490"
} | {
"index": 12377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 12378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 0, 1, 2, 3, and 4, select three different digits to form a three-digit number and calculate the total number of such numbers that are odd."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 12379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 4, 5, a five-digit number is formed with digits not repeating. What is the probability of randomly selecting a five-digit number $\\overline{abcde}$ that satisfies the condition \"$a < b > c < d > e$\"?"
}
] | math | {
"style": "rule",
"ground_truth": "2/15"
} | {
"index": 12381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 4, 5, and 6, two distinct numbers are taken out each time and denoted as $a$ and $b$. The total number of distinct values obtained for $3^{\\frac{a}{b}}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 12382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 4, 5, two numbers are randomly selected to be the base and the true number (antilogarithm) of a logarithm, respectively. The total number of different logarithmic values that can be obtained is ___."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"index": 12384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers 2, 3, 4, 5, 6, 7, 8, 9, two different numbers are selected to be the base and the exponent of a logarithm, respectively. How many different logarithmic values can be formed?"
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 12385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers \\(1,2, \\cdots, 14\\), select \\(a_{1}, a_{2}, a_{3}\\) in ascending order such that \\(a_{2} - a_{1} \\geq 3\\) and \\(a_{3} - a_{2} \\geq 3\\). How many different ways are there to select the numbers satisfying these conditions?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 12386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers \\\\(1, 2, \\ldots, 100\\\\) totaling \\\\(100\\\\) numbers, three numbers \\\\(x, y, z\\\\) are chosen in sequence. The probability that these three numbers satisfy \\\\(x+z=2y\\\\) is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{198}"
} | {
"index": 12387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the numbers \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(4\\\\), \\\\(5\\\\), \\\\(6\\\\), two numbers are selected to form an ordered pair of real numbers \\\\((x, y)\\\\). The probability that \\\\(\\dfrac{x}{y+1}\\\\) is an integer is equal to \\_\\_\\_\\_\\_\\_"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{15}"
} | {
"index": 12388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the point $(x, y)$, a legal move is a move to $\\left(\\frac{x}{3}+u, \\frac{y}{3}+v\\right)$, where $u$ and $v$ are real numbers such that $u^{2}+v^{2} \\leq 1$. What is the area of the set of points that can be reached from $(0,0)$ in a finite number of legal moves?"
}... | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\pi}{4}"
} | {
"index": 12389,
"solution": "We claim that the set of points is the disc with radius $\\frac{3}{2}$ centered at the origin, which clearly has area $\\frac{9 \\pi}{4}$. First, we show that the set is contained in this disc. This is because if we are currently at a distance of $r$ from the origin, then we can't end... |
DeepScaleR | [
{
"role": "user",
"content": "From the sequence of natural numbers $1, 2, 3, 4, \\ldots$, erase every multiple of 3 and 4, but keep every multiple of 5 (for example, 15 and 20 are not erased). After removing the specified numbers, write the remaining numbers in a sequence: $A_{1}=1, A_{2}=2, A_{3}=5, A_{4}=... | math | {
"style": "rule",
"ground_truth": "3314"
} | {
"index": 12390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $S={1,2,3,...,100}$, three numbers are randomly selected and arranged in ascending order. Find the probability that $50$ is exactly in the middle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{66}"
} | {
"index": 12391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\left\\{ \\frac{1}{3}, \\frac{1}{2}, 2, 3 \\right\\}$, select a number and denote it as $a$. From the set $\\{-2, -1, 1, 2\\}$, select another number and denote it as $b$. Then, the probability that the graph of the function $y=a^{x}+b$ passes through the third q... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 12392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{ -3, 0, 0, 4, 7, 8\\}$, find the probability that the product of two randomly selected numbers is $0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 12393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{1, 2, 3, 4, \\ldots, 20\\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "525"
} | {
"index": 12394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{1, 2, 3, \\ldots, 10\\}$, select 3 different elements such that the sum of these three numbers is a multiple of 3, and the three numbers cannot form an arithmetic sequence. Calculate the number of ways to do this."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 12395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{1, 2, \\cdots, 20\\}$, choose 5 numbers such that the difference between any two numbers is at least 4. How many different ways can this be done?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 12396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{1,2,3, \\cdots, 10\\}$, six distinct integers are chosen at random. What is the probability that the second smallest number among the chosen integers is 3?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{3}$"
} | {
"index": 12397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{1,2,3, \\cdots, 20\\}$, choose 4 different numbers such that these 4 numbers form an arithmetic sequence. Determine the number of such arithmetic sequences."
}
] | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 12398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set $\\{10, 11, 12, \\ldots, 19\\}$, 5 different numbers were chosen, and from the set $\\{90, 91, 92, \\ldots, 99\\}$, 5 different numbers were also chosen. It turned out that the difference of any two numbers from the ten chosen numbers is not divisible by 10. Find t... | math | {
"style": "rule",
"ground_truth": "545"
} | {
"index": 12399,
"solution": ""
} |
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