data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.\nThese checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside dow... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"index": 34900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points?"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 34901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$ , such that for any initial configuration of p... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 34902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2020\\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.\n・For any square, there are at least two pieces that are on the d... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"index": 34903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2022$ grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the ... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"index": 34904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of t... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"index": 34905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \\ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly $1$ . W... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"index": 34906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\\times 3 \\times 3$ cube with these... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 34907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $4$ distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code $A$ is used in the first week, find the probability that code $A$ is used in the seventh week."
}... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"index": 34908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neith... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 34909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \\text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine ... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 34910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move si... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 34911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $n$ pawns on $n$ distinct squares of a $19\\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in... | math | {
"style": "rule",
"ground_truth": "361"
} | {
"index": 34912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $n\\geq 3$ cities in a country and between any two cities $A$ and $B$ , there is either a one way road from $A$ to $B$ , or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any oth... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"index": 34913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $n\\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same ti... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 34914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34915,
"solution": "\nGiven that there are 10 birds on the ground and for any 5 of them, there are at least 4 birds on a circle, we need to determine the least possible number of birds on the circle with the most birds.\n\nTo solve this, consider the following steps:\n\n1. **Initial Assumption**: Let \\(... |
DeepScaleR | [
{
"role": "user",
"content": "There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting po... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34916,
"solution": "To solve this problem, we need to find the least time $T > 0$ such that at least $5$ of the horses are again at the starting point. Each horse $k$ returns to the starting point at multiples of $k$ minutes. Therefore, we are looking for the smallest time $T$ that is a common multiple o... |
DeepScaleR | [
{
"role": "user",
"content": "There are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point o... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34917,
"solution": "1. **Understanding the problem**: We need to find the smallest time $T > 0$ such that at least 5 of the horses are at the starting point. Each Horse $k$ returns to the starting point every $k$ minutes.\n\n2. **Divisibility condition**: For Horse $k$ to be at the starting point at time... |
DeepScaleR | [
{
"role": "user",
"content": "There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ p... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34918,
"solution": "To solve this problem, we need to consider the relationships each person has with others in the circle. Each person knows exactly three others: the two adjacent people and the one directly across the circle. We need to find how many ways we can pair up these 10 people such that each p... |
DeepScaleR | [
{
"role": "user",
"content": "There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What... | math | {
"style": "rule",
"ground_truth": "3920000"
} | {
"index": 34919,
"solution": "To solve this problem, we need to find the greatest total score Pete can get by removing all stones. Initially, we have 100 piles, each containing 400 stones.\n\n### Strategy\nTo maximize the total score, Pete should aim to keep the piles as balanced as possible until they are empty. ... |
DeepScaleR | [
{
"role": "user",
"content": "There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2001}"
} | {
"index": 34920,
"solution": "The problem involves calculating the probabilities $P_s$ and $P_d$ and finding their absolute difference.\n\n1. **Total number of marbles**:\n There are $1001$ red marbles and $1001$ black marbles, making a total of $1001 + 1001 = 2002$ marbles.\n\n2. **Calculating $P_s$ (probabilit... |
DeepScaleR | [
{
"role": "user",
"content": "There are $12$ small balls in a bag, which are red, black, and yellow respectively (these balls are the same in other aspects except for color). The probability of getting a red ball when randomly drawing one ball is $\\frac{1}{3}$, and the probability of getting a black ball i... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 34921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 34922,
"solution": "To solve this problem, we need to find the minimum number of people that must be seated such that any additional person must sit next to someone already seated. We aim to maximize the number of empty seats between seated people under this constraint.\n\n1. **Understanding the Pattern*... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys?"
}
] | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"index": 34923,
"solution": "1. **Understanding the Ratio**: The problem states that there are 2 boys for every 3 girls in the class. This sets up a ratio of boys to total students. For every 5 students (2 boys + 3 girls), 2 are boys.\n\n2. **Calculating the Fraction of Boys**: The fraction of the class that are ... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2$ teachers and $4$ students. They need to be divided into two groups and sent to two locations, A and B, for social practice activities. Each group consists of $1$ teacher and $2$ students. How many different arrangements are there in total?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking pa... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34925,
"solution": "Let's define the variables for the Venn Diagram:\n- $a$: Number of students taking exactly Bridge and Yoga.\n- $b$: Number of students taking exactly Bridge and Painting.\n- $c$: Number of students taking all three classes (Yoga, Bridge, and Painting).\n- $d$: Number of students takin... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. ... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 34926,
"solution": "\nConsider \\(2018\\) players sitting around a round table, and a deck of \\(K\\) cards distributed among them. The rules of the game allow a player to draw one card from each of their two neighbors, provided both neighbors have at least one card. The game ends when no player can make... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2022$ equally spaced points on a circular track $\\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \\ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\\gamma$ from $A_1$ to $A_2$, the... | math | {
"style": "rule",
"ground_truth": "2042222"
} | {
"index": 34927,
"solution": "\nThere are \\(2022\\) equally spaced points on a circular track \\(\\gamma\\) of circumference \\(2022\\). The points are labeled \\(A_1, A_2, \\ldots, A_{2022}\\) in some order, each label used once. Initially, Bunbun the Bunny begins at \\(A_1\\). She hops along \\(\\gamma\\) from ... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) \n\nStarting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/... | math | {
"style": "rule",
"ground_truth": "3031"
} | {
"index": 34928,
"solution": "\nLet the number of users on Mathbook be \\( n = 2022 \\). We are tasked with finding the minimum number of friendships that must exist initially so that eventually every user can become friends with every other user, given the condition that a new friendship can only form between two... |
DeepScaleR | [
{
"role": "user",
"content": "There are $20n$ members in the Trumpington marching band, and when they line up in rows of 26, there are 4 band members left over. If $n$ is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching ban... | math | {
"style": "rule",
"ground_truth": "940"
} | {
"index": 34929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction ... | math | {
"style": "rule",
"ground_truth": "\\frac{22}{45}"
} | {
"index": 34931,
"solution": "1. **Calculate the number of girls at Colfax Middle School:**\n - The total number of students at Colfax Middle School is $270$.\n - The ratio of boys to girls at Colfax Middle School is $5:4$. Let the number of boys be $5x$ and the number of girls be $4x$.\n - The sum of boys a... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical."
}
] | math | {
"style": "rule",
"ground_truth": "548"
} | {
"index": 34932,
"solution": "Let $a_{n}$ be the number of ways to form $n$-letter strings made up of As and Bs such that no more than $3$ adjacent letters are identical.\n Note that, at the end of each $n$-letter string, there are $3$ possibilities for the last letter chain: it must be either $1$, $2$, or $3$ let... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\\mid z \\mid = 1$. These numbers have the form $z_{m} = \\cos\\theta_{m} + i\\sin\\theta_{m}$, where $0\\leq\\theta_{1} < \\theta_{2} < \\ldots < \\theta_{2n} < 360$ and angles are measured in degre... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 34933,
"solution": "$z$ can be written in the form $\\text{cis\\,}\\theta$. Rearranging, we find that $\\text{cis\\,}{28}\\theta = \\text{cis\\,}{8}\\theta+1$\n Since the real part of $\\text{cis\\,}{28}\\theta$ is one more than the real part of $\\text{cis\\,} {8}\\theta$ and their imaginary parts are e... |
DeepScaleR | [
{
"role": "user",
"content": "There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1$. These numbers have the form $z_{m} = \\cos\\theta_{m} + i\\sin\\theta_{m}$, where $0\\leq\\theta_{1} < \\theta_{2} < \\dots < \\theta_{2n} < 360$ and angles are measured in degrees. Find th... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 34934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $2n$ students in a school $(n \\in \\mathbb{N}, n \\geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to h... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34935,
"solution": "We prove that for any $n \\geq 2$ the answer is 6. First we show that less than 6 trips is not sufficient. In that case the total quantity of students in all trips would not exceed $5n$. A student meets $n-1$ other students in each trip, so he or she takes part on at least 3 excursion... |
DeepScaleR | [
{
"role": "user",
"content": "There are $4$ cards, marked with $0$, $1$, $2$, $3$ respectively. If two cards are randomly drawn from these $4$ cards to form a two-digit number, what is the probability that this number is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 34936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $4$ male athletes and $3$ female athletes.<br/>$(1)$ Now $7$ athletes are lined up. If all female athletes are together, how many ways are there to arrange them?<br/>$(2)$ Now the male athletes are sent to two different venues for training, with at least one athlete i... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $5$ different books to be arranged in a row. Book A must be adjacent to Book B, while Book C cannot be adjacent to Book D. The number of different arrangement methods is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 34938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $5$ different books to be distributed among three students, with each student receiving at least $1$ book and at most $2$ books. The number of different distribution methods is $\\_\\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 34939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $5$ people arranged in a row. Among them, persons A and B must be adjacent, and neither of them can be adjacent to person D. How many different arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 34940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $5$ people participating in a lottery, each drawing a ticket from a box containing $5$ tickets ($3$ of which are winning tickets) without replacement until all $3$ winning tickets have been drawn, ending the activity. The probability that the activity ends exactly aft... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 34941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $52$ people in a room. what is the largest value of $n$ such that the statement \"At least $n$ people in this room have birthdays falling in the same month\" is always true?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34942,
"solution": "1. **Understanding the Problem:**\n We need to determine the largest number \\( n \\) such that in any group of 52 people, at least \\( n \\) people will have their birthdays in the same month.\n\n2. **Applying the Pigeonhole Principle:**\n The Pigeonhole Principle states that if ... |
DeepScaleR | [
{
"role": "user",
"content": "There are $7$ different books to be distributed among three people, A, B, and C.<br/>$(1)$ If one person gets $1$ book, another gets $2 books, and the third gets $4 books, how many different ways can the books be distributed?<br/>$(2)$ If one person gets $3 books, and the other... | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 34943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $7$ students, among whom $A_{1}$, $A_{2}$, $A_{3}$ excel in mathematics, $B_{1}$, $B_{2}$ excel in physics, and $C_{1}$, $C_{2}$ excel in chemistry. One student with excellent performance in mathematics, one in physics, and one in chemistry will be selected to form a ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 34944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.<br/>$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find th... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"index": 34945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ i... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 34946,
"solution": "1. **Identify the total number of points and the position of $P$:**\n The square contains $81$ grid points, arranged in a $9 \\times 9$ grid. Point $P$, being at the center, is located at the center grid point.\n\n2. **Determine the number of points excluding $P$:**\n Excluding po... |
DeepScaleR | [
{
"role": "user",
"content": "There are $F$ fractions $\\frac{m}{n}$ with the properties: $m$ and $n$ are positive integers with $m<n$, $\\frac{m}{n}$ is in lowest terms, $n$ is not divisible by the square of any integer larger than 1, and the shortest sequence of consecutive digits that repeats consecutive... | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 34947,
"solution": "We start with 1111109 fractions, as above, and want to remove all of the fractions in $U, V$ and $W$. Since each fraction in $W$ is in $U$ and $V$, it is enough to remove those $U$ and $V$ only. The total number of fractions in $U$ and $V$ (that is, in $U \\cup V$) equals the number o... |
DeepScaleR | [
{
"role": "user",
"content": "There are $N$ lockers, labeled from 1 to $N$, placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker $n$ and there are more than $n$ open lockers, she keeps locker... | math | {
"style": "rule",
"ground_truth": "2046"
} | {
"index": 34948,
"solution": "Note that in the first run-through, we will leave all lockers $2^{n}-1$ open. This is because after having locker $2^{n}-1$ open, we will close the next $2^{n}-1$ lockers and then start at locker $2^{n}-1+2^{n}-1+1=2^{n+1}-1$. Now we want 1 to be the last locker that is open. We know ... |
DeepScaleR | [
{
"role": "user",
"content": "There are $N$ natural numbers written on a board, where $N \\geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 34949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$. Find the remainder when $N$ is divided by $1000$."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"index": 34950,
"solution": "Note that $30=2\\cdot 3\\cdot 5$. Since $\\gcd(2, 3, 5)=1$, by CRT, for each value $k=0\\ldots 29$ modulo $30$ there exists a unique ordered triple of values $(a, b, c)$ such that $k\\equiv a\\pmod{2}$, $k\\equiv b\\pmod{3}$, and $k\\equiv c\\pmod{5}$. Therefore, we can independently ... |
DeepScaleR | [
{
"role": "user",
"content": "There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible?"
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 34951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $n \\geq 2$ coins, each with a different positive integer value. Call an integer $m$ sticky if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the mini... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 34952,
"solution": "Sort a stick by increasing value. Note that all sticks must contain 1 by necessity, or the largest and second largest sticky values would not be consecutive. So, let's say a stick's highest coin value is $a$, and all the other terms have a value of $S$. If $a \\geq S+2$, we cannot bui... |
DeepScaleR | [
{
"role": "user",
"content": "There are $n$ balls that look identical, among which one ball is lighter than the others (all other balls have equal weight). If using an unweighted balance scale as a tool, it takes at least 5 weighings to find the lighter ball, then the maximum value of $n$ is ___."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"index": 34953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $n$ mathematicians attending a conference. Each mathematician has exactly 3 friends (friendship is mutual). If they are seated around a circular table such that each person has their friends sitting next to them on both sides, the number of people at the table is at l... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 34954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that\n ($1$) for each $k,$ the math... | math | {
"style": "rule",
"ground_truth": "332"
} | {
"index": 34955,
"solution": "It is a well-known fact that the set $0, a, 2a, ... (n-1)a$ forms a complete set of residues if and only if $a$ is relatively prime to $n$.\n Thus, we have $a$ is relatively prime to $n$. In addition, for any seats $p$ and $q$, we must have $ap - aq$ not be equivalent to either $p - q... |
DeepScaleR | [
{
"role": "user",
"content": "There are $n$ students in the math club. When grouped in 4s, there is one incomplete group. When grouped in 3s, there are 3 more complete groups than with 4s, and one incomplete group. When grouped in 2s, there are 5 more complete groups than with 3s, and one incomplete group. ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34956,
"solution": "Suppose that, when the $n$ students are put in groups of 2, there are $g$ complete groups and 1 incomplete group. Since the students are being put in groups of 2, an incomplete group must have exactly 1 student in it. Therefore, $n=2g+1$. Since the number of complete groups of 2 is 5 ... |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 boys, each with a unique weight and height. For any two boys, $\\mathbf{A}$ and $\\mathbf{B}$, if $\\mathbf{A}$ is heavier than $\\mathbf{B}$, or if $\\mathbf{A}$ is taller than $\\mathbf{B}$, then $\\mathbf{A}$ is not considered worse than $\\mathbf{B}$. A boy who... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 boys, each with different weights and heights. For any two boys $\\mathbf{A}$ and $\\mathbf{B}$, if $\\mathbf{A}$ is heavier than $\\mathbf{B}$, or if $\\mathbf{A}$ is taller than $\\mathbf{B}$, we say that \" $\\mathrm{A}$ is not inferior to B\". If a boy is not i... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 cards, each card is written with two different numbers from $1, 2, 3, 4, 5$, and the numbers on any two cards are not completely the same. These 10 cards are to be placed into five boxes labeled $1, 2, 3, 4, 5$, with the rule that a card with numbers $i, j$ can onl... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 34959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\\ b,\\ c\\ (a > b > c)$ are drawn from the cards at the same time.\r\nFind the probability such that $ \\int_0^a (x^2 \\minus{} 2bx \\plus{} 3c)\\ dx \\equal{} 0$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1/30"
} | {
"index": 34960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 cities in a state, and some pairs of cities are connected by roads. There are 40 roads altogether. A city is called a \"hub\" if it is directly connected to every other city. What is the largest possible number of hubs?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34961,
"solution": "If there are $h$ hubs, then $\\binom{h}{2}$ roads connect the hubs to each other, and each hub is connected to the other $10-h$ cities; we thus get $\\binom{h}{2}+h(10-h)$ distinct roads. So, $40 \\geq\\binom{ h}{2}+h(10-h)=-h^{2} / 2+19 h / 2$, or $80 \\geq h(19-h)$. The largest $h \... |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 different mathematics books, 9 different Chinese language books, and 8 different English books. In how many different ways can you take two books of different subjects?"
}
] | math | {
"style": "rule",
"ground_truth": "242"
} | {
"index": 34962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 different natural numbers, their sum is 604, and these 10 numbers have the same sum of digits. What is the largest number among these 10 numbers? $\\qquad"
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"index": 34963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 different televisions, including 3 type A, 3 type B, and 4 type C. Now, 3 televisions are randomly selected from them. If at least two different types are included, calculate the total number of different ways to select them."
}
] | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 34964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 horizontal roads and 10 vertical roads in a city, and they intersect at 100 crossings. Bob drives from one crossing, passes every crossing exactly once, and return to the original crossing. At every crossing, there is no wait to turn right, 1 minute wait to go stra... | math | {
"style": "rule",
"ground_truth": "90 \\leq S<100"
} | {
"index": 34965,
"solution": "Obviously, the route of driving is a non-self-intersecting closed polyline. Regard each crossing as a vertex, then the route is regarded as a 100-gon.An interior angle may be greater than or equal to a straight angle.. By the formula of the sum of the angles of the polygon, the sum of... |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 34966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 numbers written in a circle, and their sum is 100. It is known that the sum of any three consecutive numbers is not less than 29.\n\nDetermine the smallest number \\( A \\) such that in any such set of numbers, each number does not exceed \\( A \\)."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.\n\nWhat is the smallest number \\( A \\) such that in any such set of numbers, each number does not exceed \\( A \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 people who want to choose a committee of 5 people among them. They do this by first electing a set of $1,2,3$, or 4 committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assum... | math | {
"style": "rule",
"ground_truth": "7560"
} | {
"index": 34969,
"solution": "There are $\\binom{10}{5}$ ways to choose the 5-person committee. After choosing the committee, there are $2^{5}-2=30$ ways to choose the leaders. So the answer is $30 \\cdot\\binom{10}{5}=7560$."
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.\n\n(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.\n\n(2) Given that among the 3 selected q... | math | {
"style": "rule",
"ground_truth": "\\dfrac{93}{125}"
} | {
"index": 34970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 sheikhs each with a harem of 100 wives standing on the bank of a river along with a yacht that can hold $n$ passengers. According to the law, a woman must not be on the same bank, on the yacht, or at any stopover point with a man unless her husband is present. What... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals?"
}
] | math | {
"style": "rule",
"ground_truth": "696"
} | {
"index": 34972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 steps, and one can take 1, 2, or 3 steps at a time to complete them in 7 moves. Calculate the total number of different ways to do this."
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 34973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10 street lamps, and to save electricity, three of them are turned off. However, the two at the ends must not be turned off, and no two consecutive lamps should be off. Calculate the number of ways to do this."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \\( k \\) societies. Suppo... | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"index": 34975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls?"
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 34976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 chairs arranged in a circle. If \\( n \\) people are sitting on these chairs, such that any new person sitting down will always sit on a chair adjacent to one of the \\( n \\) people, what is the minimum value of \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 34977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest va... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 34978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 34979,
"solution": "House $n$ ends up red if and only if the largest odd divisor of $n$ is of the form $4 k+1$. We have 25 values of $n=4 k+1 ; 13$ values of $n=2(4 k+1)$ (given by $k=0,1,2, \\ldots, 12$ ); 7 values of $n=4(4 k+1)(k=0,1, \\ldots, 6) ; 3$ values of $n=8(4 k+1)(k=0,1,2) ; 2$ of the form $n... |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 34980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 people in a room with ages $1,2, \\ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 34981,
"solution": "For a cute pair $(a, b)$ we would have $$a \\geq \\frac{b}{2}+7, b \\geq \\frac{a}{2}+7$$ Solving the system, we get that $a$ and $b$ must both be at least 14. However 14 could only be paired with itself or a smaller number; therefore, only people with age 15 or above can be paired wi... |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result?"
}
] | math | {
"style": "rule",
"ground_truth": "197"
} | {
"index": 34982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 points marked on a circle, painted either red or blue. Some points are connected by segments, with each segment having one blue end and one red end. It is known that no two red points are connected to the same number of segments. What is the maximum possible numbe... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 34983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 points on a coordinate plane. Let \\( N \\) be the number of triplets \\((A, B, C)\\) that satisfy the following conditions: the vertices are chosen from these 100 points, \\( A \\) and \\( B \\) have the same y-coordinate, and \\( B \\) and \\( C \\) have the sam... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"index": 34984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denomin... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 34985,
"solution": "\nTo solve this problem, we aim to find the smallest positive integer \\( n \\) such that after \\( n \\) steps, the 100 numbers on the board are all pairwise coprime regardless of their initial values.\n\n### Key Observations\n\n1. **Irreducible Fractions**: At each step, Alex forms ... |
DeepScaleR | [
{
"role": "user",
"content": "There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the righ... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"index": 34986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switches numbered 2, 3, and 5 are flipped, how many lamps remain on?"
}
] | math | {
"style": "rule",
"ground_truth": "499"
} | {
"index": 34987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on?"
}
] | math | {
"style": "rule",
"ground_truth": "499"
} | {
"index": 34988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 34989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 1000 toy bears in a row. The bears come in three colors: white, brown, and black. Among any three consecutive bears, there is at least one bear of each color. Iskander tries to guess the colors of the bears. He made five guesses:\n\n- The 2nd bear from the left is whi... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 1001 people sitting around a round table, each of whom is either a knight (always tells the truth) or a liar (always lies). It turned out that next to each knight there is exactly one liar, and next to each liar there is exactly one knight. What is the minimum number ... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 34991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 101 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 34992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 101 people participating in a Secret Santa gift exchange. As usual each person is randomly assigned another person for whom (s)he has to get a gift, such that each person gives and receives exactly one gift and no one gives a gift to themself. What is the probability ... | math | {
"style": "rule",
"ground_truth": "0.96039"
} | {
"index": 34993,
"solution": "Let $D_{k}$ denote the number of derangements of $\\{1,2, \\ldots, k\\}$. (A derangement is a permutation in which no element appears in its original position.) Call the first three people $A, B$, and $C$. Let $X \\rightarrow Y$ denote that $X$ gives a gift to $Y$ and let $X \\nrighta... |
DeepScaleR | [
{
"role": "user",
"content": "There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that ha... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 11 seats, and now we need to arrange for 2 people to sit down. It is stipulated that the middle seat (the 6th seat) cannot be occupied, and the two people must not sit next to each other. How many different seating arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 34997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 111 balls in a box, each being red, green, blue, or white. It is known that if 100 balls are drawn, it ensures getting balls of all four colors. Find the smallest integer $N$ such that if $N$ balls are drawn, it can ensure getting balls of at least three different col... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 34998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "There are 12 different-colored crayons in a box. How many ways can Karl select four crayons if the order in which he draws them out does not matter?"
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"index": 34999,
"solution": ""
} |
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