id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-54cc4ae5ce96c6df57ba48584e4787d95125f86f3302590ef32fafd34b28b144 | Wayne, Bryson, and Young are engaged in a Nerf gun game. On a given player’s turn, he has the option of shooting once at one of his opponents. If he successfully hits them, they are removed from the game. Wayne has a 20% chance of hitting his target, Bryson has a 70% chance of hitting his target, and Young never misses... | [
"\\dfrac{1003}{4750}"
] | [
"Wayne, Bryson, and Young are engaged in a Nerf gun game. On a given player’s turn, he has the option of shooting once at one of his opponents. If he successfully hits them, they are removed from the game. Wayne has a 20% chance of hitting his target, Bryson has a 70% chance of hitting his target, and Young never m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-c78656865b6668e3be3f0943a4741269774cf18442e9d058ee67557aac3627d6 | Wayne, Bryson, and Young are engaged in a Nerf gun game. On a given player’s turn, he has the option of shooting once at one of his opponents. Wayne has a 20% chance of hitting his target, Bryson has a 70% chance of hitting his target, and Young never misses. If Wayne goes first, then Bryson, then Young, repeating that... | [
"0.1"
] | [
"Wayne, Bryson, and Young are engaged in a Nerf gun game. On a given player’s turn, he has the option of shooting once at one of his opponents. Wayne has a 20% chance of hitting his target, Bryson has a 70% chance of hitting his target, and Young never misses. If Wayne goes first, then Bryson, then Young, repeating... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-8d880fde647582960a3b8289ce38a30646b89a247dfc1309659ac210719e457b | We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 10? | [
"5"
] | [
"We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 10?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-431e31ba8cde86ffdadf7eaac39daad10a9cf1ede5a92ed04d9c5be8d9a11aaa | We are colouring numbers from 1 to 10 with red and blue colours. For every \( k = 1, 2, \ldots, a \), the subtraction of two numbers that have the same color is \( k \), and this situation is valid for both red and blue. What is the biggest value of \( a \)? | [
"4"
] | [
"We are colouring numbers from 1 to 10 with red and blue colours. For every \\( k = 1, 2, \\ldots, a \\), the subtraction of two numbers that have the same color is \\( k \\), and this situation is valid for both red and blue. What is the biggest value of \\( a \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a07aaa506289b7803cd987bf9de930388aa9c3daf727fd359552c19d2299628 | We are given $2n+1$ real numbers $a_1, a_2, \ldots, a_{2n+1}$. It is known that if we remove any one of these numbers, the remaining $2n$ numbers can be rearranged into two $n$-element groups with equal sums. Calculate the value of $a_1 + a_2 + \cdots + a_{2n+1}$ given that $n=4$. | [
"0"
] | [
"We are given $2n+1$ real numbers $a_1, a_2, \\ldots, a_{2n+1}$. It is known that if we remove any one of these numbers, the remaining $2n$ numbers can be rearranged into two $n$-element groups with equal sums. Calculate the value of $a_1 + a_2 + \\cdots + a_{2n+1}$ given that $n=4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d30e940eec428d1b446ccd5cdd16b1a3a39b871a15a7673659396b734c175d0 | We are given $b$ white balls and $n$ black balls ($b,n>0$) which are to be distributed among two urns, at least one in each. Let $s$ be the number of balls in the first urn, and $r$ the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it. Compute the probability $p$ that the d... | [
"\\dfrac{r}{2s} + \\dfrac{b - r}{2(b + n - s)}"
] | [
"We are given $b$ white balls and $n$ black balls ($b,n>0$) which are to be distributed among two urns, at least one in each. Let $s$ be the number of balls in the first urn, and $r$ the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it. Compute the probability $p$ that ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-90b43d47da41b7180486a48b3bc300d72de299355c28f24d758d3d19a4b4c187 | We are given 5040 balls in \( k \) different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest \( k \) such that, however the balls are distributed into the bags, we can arrange the bags around a circl... | [
"8"
] | [
"We are given 5040 balls in \\( k \\) different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest \\( k \\) such that, however the balls are distributed into the bags, we can arrange the bags aroun... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-e7cca4f63e59646d148bbfe6c93a6e5a9524fecbad51d44d5a93913cc9e47c93 | We are given 5771 weights weighing 1, 2, 3, ..., 5770, 5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible? | [
"2886"
] | [
"We are given 5771 weights weighing 1, 2, 3, ..., 5770, 5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 21
} | 73 | 0 | 0 |
omr-ws-16010ca3ad66b5c6a9dcce506cf3bbf1e3efabff5eb789292f5ad699cf3b8685 | We are given 6 boxes (arranged in a line from left to right) and 5 distinct balls. The balls are placed in some or all of the first 5 boxes (and hence there is a total of \(5^5\) ways of arranging them). We apply the following rule repeatedly, until none of the first five boxes has more than one ball in it: If a box ha... | [
"1296"
] | [
"We are given 6 boxes (arranged in a line from left to right) and 5 distinct balls. The balls are placed in some or all of the first 5 boxes (and hence there is a total of \\(5^5\\) ways of arranging them). We apply the following rule repeatedly, until none of the first five boxes has more than one ball in it: If a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6245faf4ac04ef5636c2cf0b0c31ed5dbc702da85e8b7bf9896d6c082f362ff2 | We are given 6 boxes arranged in a line from left to right and 5 distinct balls. The balls are placed in some or all of the first 5 boxes (resulting in a total of $5^5$ ways to arrange them). We apply the following rule repeatedly until none of the first 5 boxes has more than one ball in it: if a box has more than one ... | [
"1296"
] | [
"We are given 6 boxes arranged in a line from left to right and 5 distinct balls. The balls are placed in some or all of the first 5 boxes (resulting in a total of $5^5$ ways to arrange them). We apply the following rule repeatedly until none of the first 5 boxes has more than one ball in it: if a box has more than... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 11
} | 63 | 0 | 0 |
omr-ws-17ffa3a26fbc6fbaa821870133201593e5e4c171ed8d4024113169cbeef04260 | We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures (the figures can be moved and rotated) equals $5$. What is the greatest number of di... | [
"3"
] | [
"We are given a $4\\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures (the figures can be moved and rotated) equals $5$. What is the greatest numbe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9b6e5343a56d0f9cb6ae065d045d9b1ba68ce81d76abce4e212691f2e2125d6 | We are given a 12-pint vessel filled with a liquid and two empty vessels with capacities of 9 pints and 5 pints, with no markings on any of them. How many total possible transfers are needed to divide the liquid into two equal portions, given that no container should be empty during the process before reaching equal po... | [
"12"
] | [
"We are given a 12-pint vessel filled with a liquid and two empty vessels with capacities of 9 pints and 5 pints, with no markings on any of them. How many total possible transfers are needed to divide the liquid into two equal portions, given that no container should be empty during the process before reaching equ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-5ede40df7e6d4ddff6642f0be4392520e058ac6916bd7b69113a30131a137ccb | We are given a 12-pint vessel filled with a liquid and two empty vessels with capacities of 9 pints and 5 pints, with no markings on any of them. How many total possible transfers are needed to divide the liquid into two equal portions? | [
"8"
] | [
"We are given a 12-pint vessel filled with a liquid and two empty vessels with capacities of 9 pints and 5 pints, with no markings on any of them. How many total possible transfers are needed to divide the liquid into two equal portions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-7bbd8ba0c6cda8a7edb46cb10b8319817904b711611bc8f6328c878fac4d80ce | We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers. Using these weights, we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all w... | [
"1"
] | [
"We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers. Using these weights, we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-06801c67af6b671c0b273afd9e7382e7b321242efeb5b1ba398c36c4c2e079b8 | We are given a combination lock consisting of 6 rotating discs. Each disc consists of digits \(0, 1, 2, \ldots, 9\) in that order (after digit 9 comes 0). The lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction, and the lock opens instantly if the current co... | [
"29"
] | [
"We are given a combination lock consisting of 6 rotating discs. Each disc consists of digits \\(0, 1, 2, \\ldots, 9\\) in that order (after digit 9 comes 0). The lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction, and the lock opens instantly if the cu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-066e4f4d850ad65f765dca364bc49a550691b508b02f676600cd5f3fec07b460 | We are given a fixed point on the circle of radius 1, and going from this point along the circumference in the positive direction on curved distances \(0, 1, 2, \ldots\) from it we obtain points with abscissas \(n = 0, 1, 2, \ldots\). How many points among them should we take to ensure that some two of them are less th... | [
"33"
] | [
"We are given a fixed point on the circle of radius 1, and going from this point along the circumference in the positive direction on curved distances \\(0, 1, 2, \\ldots\\) from it we obtain points with abscissas \\(n = 0, 1, 2, \\ldots\\). How many points among them should we take to ensure that some two of them ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-a5ec59ce0077d215bba2f64480ba73b61345725d11f05673175b32b13b441588 | We are given a map divided into $13 \times 13$ fields. A tank is stationed at one of these fields and must be hit twice to be destroyed. After the first hit, the tank moves to a neighboring field. How many shots must be fired to ensure the tank is destroyed, given that we cannot see the tank or receive any feedback reg... | [
"247"
] | [
"We are given a map divided into $13 \\times 13$ fields. A tank is stationed at one of these fields and must be hit twice to be destroyed. After the first hit, the tank moves to a neighboring field. How many shots must be fired to ensure the tank is destroyed, given that we cannot see the tank or receive any feedba... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d8c987e41ef988c28f9eb15c8be38301208c3ceecd347859ab46a3671b825bf | We are given a square of side length 1 with a lower left vertex at the origin of the Cartesian plane and sides parallel to the axes. We find the midpoint of the right side of the square and connect it to the midpoint of the opposite segment. Then we find the midpoint of the newly created segment and connect it to the m... | [
"\\(\\left(\\frac{2}{3}, \\frac{1}{3}\\right)\\)"
] | [
"We are given a square of side length 1 with a lower left vertex at the origin of the Cartesian plane and sides parallel to the axes. We find the midpoint of the right side of the square and connect it to the midpoint of the opposite segment. Then we find the midpoint of the newly created segment and connect it to ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ceda5cc9706d893f3eccd9191c4a1e127228871fbbbc6c77963f665e9b8a2043 | We are given an infinite deck of cards, each with a real number on it. For every real number \( x \), there is exactly one card in the deck that has \( x \) written on it. Two players draw disjoint sets \( A \) and \( B \) of 100 cards each from this deck. Define a rule that declares one of them a winner, satisfying th... | [
"100"
] | [
"We are given an infinite deck of cards, each with a real number on it. For every real number \\( x \\), there is exactly one card in the deck that has \\( x \\) written on it. Two players draw disjoint sets \\( A \\) and \\( B \\) of 100 cards each from this deck. Define a rule that declares one of them a winner, ... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-589d4ba7c03ed028ed94fa5acb6cfcfc2381dc8559e9d66750f2832ea757b465 | We are given an infinite sequence of $1$'s and $2$'s with the following properties: - The first element of the sequence is $1$. - There are no two consecutive $2$'s or three consecutive $1$'s. - If we replace consecutive $1$'s by a single $2$, leave the single $1$'s alone, and delete the original $2$'s, then we recover... | [
"\\(\\lfloor (\\sqrt{2} - 1) n + (1 - \\frac{\\sqrt{2}}{2}) \\rfloor\\)"
] | [
"We are given an infinite sequence of $1$'s and $2$'s with the following properties:\n- The first element of the sequence is $1$.\n- There are no two consecutive $2$'s or three consecutive $1$'s.\n- If we replace consecutive $1$'s by a single $2$, leave the single $1$'s alone, and delete the original $2$'s, then we... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84f93350e836971cc99ab11e698380a70f74a7033885f7ab6dd12767eec3b7ca | We are given some three-element subsets of $\{1,2, \dots ,n\}$ such that any two of them have at most one common element. A subset of $\{1,2, \dots ,n\}$ is called [i]nice[/i] if it does not include any of the given three-element subsets. If no matter how the three-element subsets are selected initially, we can add one... | [
"436"
] | [
"We are given some three-element subsets of $\\{1,2, \\dots ,n\\}$ such that any two of them have at most one common element. A subset of $\\{1,2, \\dots ,n\\}$ is called [i]nice[/i] if it does not include any of the given three-element subsets. If no matter how the three-element subsets are selected initially, we ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 6,
"tir": 3
} | 55 | 0 | 0 |
omr-ws-735523c7447572b7db938af5567ac93ad49e9ed23ecc56f873c75f68b64896ab | We are given sufficiently many tiles of the forms of a $2 \times 1$ rectangle and $1 \times 1$ square. Let $n > 3$ be a rational number. In how many ways can one tile a rectangle $3 \times n$ using these tiles, so that no two $2 \times 1$ rectangles have overlaps (their borders are adjacent), and each of them has the l... | [
"3^n"
] | [
"We are given sufficiently many tiles of the forms of a $2 \\times 1$ rectangle and $1 \\times 1$ square. Let $n > 3$ be a rational number. In how many ways can one tile a rectangle $3 \\times n$ using these tiles, so that no two $2 \\times 1$ rectangles have overlaps (their borders are adjacent), and each of them ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-8970bf5558b31006cc23f191eecb66ae50df31ac3a2b97d86925b6d23c3c48f5 | We are given that $$54+(98\div14)+(23\cdot 17)-200-(312\div 6)=200.$$Now, let's remove the parentheses: $$54+98\div14+23\cdot 17-200-312\div 6.$$What does this expression equal? | [
"200"
] | [
"We are given that $$54+(98\\div14)+(23\\cdot 17)-200-(312\\div 6)=200.$$Now, let's remove the parentheses: $$54+98\\div14+23\\cdot 17-200-312\\div 6.$$What does this expression equal?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89f7b5826b248237920084e98b05b6f5ea7ec013642af2f1994b47e62c282e23 | We are given two mutually tangent circles in the plane, with radii \( r_1 \) and \( r_2 \). A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of \( r_1 \) and \( r_2 \) and the condition for the solvability of the problem. | [
"\\(\\frac{2 \\sqrt{r_1 r_2}}{r_1 + r_2}\\)"
] | [
"We are given two mutually tangent circles in the plane, with radii \\( r_1 \\) and \\( r_2 \\). A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of \\( r_1 \\) and \\( r_2 \\) and the condition for the solvability of the problem."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c16b5b9478ce80a286957ef82ad1eec0b24eb2a8a9d5d37d80813402f34a5d13 | We are grouping the numbers $1, 2, 3, \ldots, 100$ into ten groups. We delete the two smallest numbers from each group and then delete the smallest ten numbers from the remaining numbers. What is the smallest possible value of the smallest number after these deletions? | [
"13"
] | [
"We are grouping the numbers $1, 2, 3, \\ldots, 100$ into ten groups. We delete the two smallest numbers from each group and then delete the smallest ten numbers from the remaining numbers. What is the smallest possible value of the smallest number after these deletions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-a6f9f95067d12d67dfb06d388c158e6ef5defd4af4a6af74ddbf317a0bdf7e59 | We are looking for a treasure hidden under one of the tiles covering a room. Each time a tile is picked up and the treasure is not found, the direction to the treasure appears on the tile. Given that the treasure can be found by picking up $n$ tiles, determine the maximum number of tiles that can cover the room. | [
"\\(2^n\\)"
] | [
"We are looking for a treasure hidden under one of the tiles covering a room. Each time a tile is picked up and the treasure is not found, the direction to the treasure appears on the tile. Given that the treasure can be found by picking up $n$ tiles, determine the maximum number of tiles that can cover the room."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1191730da1c95f673c64e4c19e702f06949c1c81a448a539360a4ada5c016089 | We are numbering the rows and columns of a $29 \times 29$ chessboard with numbers $1, 2, \ldots, 29$ (top row numbered $1$ and first column numbered $1$). We select some squares such that for each selected square, there is at most one other selected square with both row and column numbers greater than or equal to those... | [
"43"
] | [
"We are numbering the rows and columns of a $29 \\times 29$ chessboard with numbers $1, 2, \\ldots, 29$ (top row numbered $1$ and first column numbered $1$). We select some squares such that for each selected square, there is at most one other selected square with both row and column numbers greater than or equal t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9638de52ef94629d9a5931781cda65b048a1afa7808a8b748c1a7d6a4632f926 | We are placing \( n \) integers whose sum is 94 over a circle such that each number is equal to the absolute value of the difference of the next two numbers (clockwise). What is the largest \( n \) that makes such placement possible? | [
"141"
] | [
"We are placing \\( n \\) integers whose sum is 94 over a circle such that each number is equal to the absolute value of the difference of the next two numbers (clockwise). What is the largest \\( n \\) that makes such placement possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e2a58141850b7eb71161447d96ca1e826dbdc1eb754f94ec24a96d5838d51f32 | We are splitting a giant cake for 2020 persons as follows: The first person takes $\frac{1}{2020}$ of the cake, the second person takes $\frac{2}{2020}$ of the remaining cake, the third person takes $\frac{3}{2020}$ of the remaining cake, and so on, until the $2019^{\text{th}}$ person takes $\frac{2019}{2020}$ of the r... | [
"45th guest"
] | [
"We are splitting a giant cake for 2020 persons as follows: The first person takes $\\frac{1}{2020}$ of the cake, the second person takes $\\frac{2}{2020}$ of the remaining cake, the third person takes $\\frac{3}{2020}$ of the remaining cake, and so on, until the $2019^{\\text{th}}$ person takes $\\frac{2019}{2020}... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-c1b5c1ae8051cc5c5b9375e4aba86b03f47fa499b82efecad64078c19e5b6a3d | We are writing either $0$ or $1$ to unit squares of an $8 \times 8$ chessboard. If the sum of numbers is even vertically, horizontally, or diagonally, what is the greatest possible value of the sum of all numbers on the board? | [
"48"
] | [
"We are writing either $0$ or $1$ to unit squares of an $8 \\times 8$ chessboard. If the sum of numbers is even vertically, horizontally, or diagonally, what is the greatest possible value of the sum of all numbers on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-8b50c24b8a28ba9bc9c4b4a7f86589a2ce53ab7b77eb6ac689ff11a4967cb7ec | We arrange \( n \) boys and \( n \) girls in one row. Let \( A \) be the number of arrangements that cannot be split into two parts such that each part has an equal number of boys and girls. Let \( B \) be the number of arrangements that can be split into two parts with an equal number of boys and girls in exactly one ... | [
"2"
] | [
"We arrange \\( n \\) boys and \\( n \\) girls in one row. Let \\( A \\) be the number of arrangements that cannot be split into two parts such that each part has an equal number of boys and girls. Let \\( B \\) be the number of arrangements that can be split into two parts with an equal number of boys and girls in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-0d9babd848d927a74b9e531f3131f25dba12edb539b3f37c53113dd323c42ca1 | We arrange the numbers $1, 2, 3, \ldots, 16$ in a $4 \times 4$ board such that the numbers $1, 2, \ldots, 4$ appear in the first row in increasing order, $5, 6, \ldots, 8$ in the second row in increasing order, and so on. Every minute, we can take a number in a square, subtract 2 from it, and add 1 to each of two neigh... | [
"\\dfrac{16!}{2}"
] | [
"We arrange the numbers $1, 2, 3, \\ldots, 16$ in a $4 \\times 4$ board such that the numbers $1, 2, \\ldots, 4$ appear in the first row in increasing order, $5, 6, \\ldots, 8$ in the second row in increasing order, and so on. Every minute, we can take a number in a square, subtract 2 from it, and add 1 to each of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-fed7b5df035b3bed0484b0c4872dfa445b4d47ab2dda00a7e2554d377b389050 | We attach the numbers $1$, $0$, $0$, $0$, $0$, $0$ to the vertices of a regular hexagon. We are allowed to transform the numbers using the following rules: - (a) Add an arbitrary integer to the numbers at two opposite vertices. - (b) Add an arbitrary integer to the numbers at three vertices forming an equilateral trian... | [
"2"
] | [
"We attach the numbers $1$, $0$, $0$, $0$, $0$, $0$ to the vertices of a regular hexagon. We are allowed to transform the numbers using the following rules:\n- (a) Add an arbitrary integer to the numbers at two opposite vertices.\n- (b) Add an arbitrary integer to the numbers at three vertices forming an equilatera... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-55b0f7bfdd75cd7043fb123bc278b4c4a2aed8bd16c8ff64c35619eb0d63f22a | We begin with an equilateral triangle with side length 1. We divide each side into three segments of equal length, and add an equilateral triangle to each side using the middle third as a base. We then repeat this process indefinitely. What is the area of the resulting figure? | [
"\\(\\frac{9\\sqrt{3}}{20}\\)"
] | [
"We begin with an equilateral triangle with side length 1. We divide each side into three segments of equal length, and add an equilateral triangle to each side using the middle third as a base. We then repeat this process indefinitely. What is the area of the resulting figure?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc270a4ea83855000bb1606f8262f3768d485ad40acddfb51a05ca5aef07ec49 | We build a tower of $2 \times 1$ dominoes such that each story consists of $55$ dominoes covering a $10 \times 11$ rectangle. The tower is called [i]stable[/i] if for every non-lattice point of the $10 \times 11$ rectangle, there is a domino that has an inner point above it. How many stories is the lowest [i]stable[/i]... | [
"5"
] | [
"We build a tower of $2 \\times 1$ dominoes such that each story consists of $55$ dominoes covering a $10 \\times 11$ rectangle. The tower is called [i]stable[/i] if for every non-lattice point of the $10 \\times 11$ rectangle, there is a domino that has an inner point above it. How many stories is the lowest [i]st... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49e19be45f573c38d78a9dda510b3f61ecd10fdfa101d73988306a62e0830632 | We call $A_{1}, A_{2}, A_{3}$ *mangool* if there is a permutation $\pi$ such that $A_{\pi(2)} \not\subset A_{\pi(1)}$ and $A_{\pi(3)} \not\subset A_{\pi(1)} \cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb{N}$ like $X, A_{1}, A_{2}, \dots, A_{n}$. To each good family, we correspond a graph with... | [
"All graphs can be represented by a good family."
] | [
"We call $A_{1}, A_{2}, A_{3}$ *mangool* if there is a permutation $\\pi$ such that $A_{\\pi(2)} \\not\\subset A_{\\pi(1)}$ and $A_{\\pi(3)} \\not\\subset A_{\\pi(1)} \\cup A_{\\pi(2)}$. A good family is a family of finite subsets of $\\mathbb{N}$ like $X, A_{1}, A_{2}, \\dots, A_{n}$. To each good family, we corre... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-dcb6f8ec974f27648df534c31926f2b4353cbe1da8e6a5065759134f942fe5f1 | We call a $6 \times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwis... | [
"518400"
] | [
"We call a $6 \\times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-b603a812361b05055563a0f8e07fa86087126d6c52357522235d5ca1e460a358 | We call a coloring of an \(m \times n\) table (\(m, n \geq 5\)) in three colors a good coloring if the following two conditions are satisfied: (i) Each cell has the same number of neighboring cells of two other colors. (ii) Each corner cell has no neighboring cells of its color. Find all pairs \((m, n)\) for which ther... | [
"(m, n) \\text{ with both } m \\text{ and } n \\text{ even}"
] | [
"We call a coloring of an \\(m \\times n\\) table (\\(m, n \\geq 5\\)) in three colors a good coloring if the following two conditions are satisfied:\n(i) Each cell has the same number of neighboring cells of two other colors.\n(ii) Each corner cell has no neighboring cells of its color.\nFind all pairs \\((m, n)\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-fe85ae229756b6b48853efb08f614f59b89a1b20a9fcf1eb99dfa95b26448b2f | We call a family of vectors \( v_1, \ldots, v_k \) "nice" if every vector in \( \mathbb{R}^n \) can be written as a linear combination of \( v_i \) with strictly positive coefficients. Find the best \( k_n \) such that any nice family with \( k_n \) vectors contains a proper subfamily which is also nice. | [
"\\(2n\\)"
] | [
"We call a family of vectors \\( v_1, \\ldots, v_k \\) \"nice\" if every vector in \\( \\mathbb{R}^n \\) can be written as a linear combination of \\( v_i \\) with strictly positive coefficients. Find the best \\( k_n \\) such that any nice family with \\( k_n \\) vectors contains a proper subfamily which is also n... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a02d9da738a8bb6222bc0621fdef77589f0dda6d0f95f34aaa9281aa61a943ab | We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there? | [
"240"
] | [
"We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0b043e0466f32ba5d193ba9027f1fc5014cd66342191f808cb6c9321f8d38bc | We call a number delicious if it has exactly 4 factors and is divisible by 2. How many numbers less than 99 are delicious? | [
"15"
] | [
"We call a number delicious if it has exactly 4 factors and is divisible by 2. How many numbers less than 99 are delicious?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-56371ce4787c56d5c13ed0a322f00c35e0493ba65bb2dce9d9e3ca31512a8b0d | We call a pair \((a, b)\) of positive integers "friendly" if their difference is a perfect square of an integer. The numbers \(1, 2, 3, \ldots, n\) are colored with three distinct colors. It is known that there are two "friendly" positive integers with the same color. Find the least possible value of \(n\). | [
"14"
] | [
"We call a pair \\((a, b)\\) of positive integers \"friendly\" if their difference is a perfect square of an integer. The numbers \\(1, 2, 3, \\ldots, n\\) are colored with three distinct colors. It is known that there are two \"friendly\" positive integers with the same color. Find the least possible value of \\(n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32103b674a6ddfc6bc09db01139e23997b58d95fa6f085ff3e622bce6b511cf7 | We call a pair of positive integers quare if their product is a perfect square. A subset $S$ of $\{1, 2, \ldots, 100\}$ is chosen, such that no two distinct elements of $S$ form a quare pair. Find the maximum possible number of elements in such a set $S$. | [
"61"
] | [
"We call a pair of positive integers quare if their product is a perfect square. A subset $S$ of $\\{1, 2, \\ldots, 100\\}$ is chosen, such that no two distinct elements of $S$ form a quare pair. Find the maximum possible number of elements in such a set $S$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f2b653f015b91af83806ace28544240413a6bf9d90b233016bb92f3d235c620 | We call a permutation $(a_1, a_2, \cdots, a_n)$ of the set $\{1, 2, \cdots, n\}$ "good" if for any three natural numbers $i < j < k$, $n \nmid a_i + a_k - 2a_j$. Find all natural numbers $n \ge 3$ such that there exists a "good" permutation of the set $\{1, 2, \cdots, n\}$. | [
"\\( n = 2^k \\)"
] | [
"We call a permutation $(a_1, a_2, \\cdots, a_n)$ of the set $\\{1, 2, \\cdots, n\\}$ \"good\" if for any three natural numbers $i < j < k$, $n \\nmid a_i + a_k - 2a_j$. Find all natural numbers $n \\ge 3$ such that there exists a \"good\" permutation of the set $\\{1, 2, \\cdots, n\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-bb27bcbc7976eb805f379195532d0da2a8157f140a10c1fe35e9744827858fe9 | We call a positive integer alternative if its decimal digits are alternatively odd and even. Find all positive integers \(n\) such that \(n\) has an alternative multiple. | [
"\\text{All positive integers not divisible by }20"
] | [
"We call a positive integer alternative if its decimal digits are alternatively odd and even. Find all positive integers \\(n\\) such that \\(n\\) has an alternative multiple."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-329ad55a47e582da7985501520fe1891ad6815268a96771b9c871f525bf7fdcf | We call a positive integer basically prime if it is not divisible by a prime less than $15$. I pick all the integers from $1$ to $n$, and take the ratio of the number of basically prime integers to the number of integers that are not basically prime. As $n$ becomes large, to what number does this ratio approach? | [
"\\dfrac{192}{809}"
] | [
"We call a positive integer basically prime if it is not divisible by a prime less than $15$. I pick all the integers from $1$ to $n$, and take the ratio of the number of basically prime integers to the number of integers that are not basically prime. As $n$ becomes large, to what number does this ratio approach?"
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ce2b28160941a33a3d997059462ff21e948a0f104404cbf6d6649a64584ed29 | We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in a corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles, and some $S$-rectangles, one forms a re... | [
"994000"
] | [
"We call a rectangle of size $2 \\times 3$ (or $3 \\times 2$) without one cell in a corner a $P$-rectangle. We call a rectangle of size $2 \\times 3$ (or $3 \\times 2$) without two cells in opposite corners a $S$-rectangle. Using some squares of size $2 \\times 2$, some $P$-rectangles, and some $S$-rectangles, one ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-4b1c659cb215f504778a39d9ef57991fbe1e156166d9ba424763bfa785f4d75d | We call a rectangle of size \( 1 \times 2 \) a domino. A rectangle of size \( 2 \times 3 \) with two opposite corners removed (under the center of the rectangle) is called a tetramino. These figures can be rotated. Determine the minimal number of dominoes required to tile a rectangle of size \( 2008 \times 2010 \) usin... | [
"672680"
] | [
"We call a rectangle of size \\( 1 \\times 2 \\) a domino. A rectangle of size \\( 2 \\times 3 \\) with two opposite corners removed (under the center of the rectangle) is called a tetramino. These figures can be rotated. Determine the minimal number of dominoes required to tile a rectangle of size \\( 2008 \\times... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-097b16d273e112ec42748e02ae90d98529fa3d75c93c3cdd24b21e5fc086eb54 | We call a sequence of \(n\) digits one or zero a code. A subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. a) What is the least number of palindromes in a code? b) What is the least number of nice ... | [
"n"
] | [
"We call a sequence of \\(n\\) digits one or zero a code. A subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code.\na) What is the least number of palindromes in a code?\nb) What is the least number... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-248dd42b368af4007eb0acd80f681d968d3dd0b9890f38164a757637502ff1cd | We call a sequence of letters "red" if it consists only of the letters \(C, H, I, A\) (some of these letters may not appear in the sequence) and in which \(C\) is never immediately followed by \(H\), \(H\) is never immediately followed by \(I\), \(I\) is never immediately followed by \(A\), and \(A\) is never immediate... | [
"2916"
] | [
"We call a sequence of letters \"red\" if it consists only of the letters \\(C, H, I, A\\) (some of these letters may not appear in the sequence) and in which \\(C\\) is never immediately followed by \\(H\\), \\(H\\) is never immediately followed by \\(I\\), \\(I\\) is never immediately followed by \\(A\\), and \\(... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecc93b8d7fdcf386e49a4817bda4264da2ced8f5b2e9c83173684e38e67020a1 | We call a set of positive integers *good* if the greatest common divisor of all elements in the set is $1$. Let $a_n$ be the number of *good* subsets of $\{1, 2, \ldots, n\}$. Find all integers $n \ge 2019$ such that for all $0 < k < n$, $k \in \mathbb{Z}$, the inequality $\frac{a_{n+k}}{a_n} < \frac{a_n}{a_{n-k}}$ hol... | [
"all odd $n \\geq 2019$"
] | [
"We call a set of positive integers *good* if the greatest common divisor of all elements in the set is $1$. Let $a_n$ be the number of *good* subsets of $\\{1, 2, \\ldots, n\\}$. Find all integers $n \\ge 2019$ such that for all $0 < k < n$, $k \\in \\mathbb{Z}$, the inequality $\\frac{a_{n+k}}{a_n} < \\frac{a_n}{... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-83c843d8297ee496fe90d64c2c587d6076f110f6e89fb38d8d9f772d2d42e225 | We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly have two units. Determine the number of regular size tables $n \times n$ ($n > 1$ - given a fixed positive integer). (We can assume that the rows and columns of t... | [
"\\frac{(n!)^2}{2^n} \\sum_{k=0}^n \\frac{(-1)^k (2n - 2k)!}{k! (n - k)! 2^{n - k}}"
] | [
"We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly have two units. Determine the number of regular size tables $n \\times n$ ($n > 1$ - given a fixed positive integer). (We can assume that the rows and column... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-c304e16fb7d62d41614d2dbc0a24e962a2b75030b0e499bbcd06c7486acf5eed | We call an angle of a convex $n$-gon "good" if its sine is equal to the product of the sines of all other angles of the $n$-gon. What is the maximum number of good angles that an $n$-gon can have for each natural $n \ge 3$? | [
"1"
] | [
"We call an angle of a convex $n$-gon \"good\" if its sine is equal to the product of the sines of all other angles of the $n$-gon. What is the maximum number of good angles that an $n$-gon can have for each natural $n \\ge 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-851187e8bff4e108068f350766cca4787ce19cf5c6e48fa4935ee0d2376c6f1b | We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positive reals $a_1, a_2, \cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0 = x$, and $x_k = \sqrt{x_{k-1} a_k}$ for all $1 \leq k \leq n$. Finally, let $y = x_n$. A number $M > 0$ is "tremendous" if for any $x \in \mathbb{... | [
"(4, \\infty)"
] | [
"We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positive reals $a_1, a_2, \\cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0 = x$, and $x_k = \\sqrt{x_{k-1} a_k}$ for all $1 \\leq k \\leq n$. Finally, let $y = x_n$. A number $M > 0$ is \"tremendous\" if for any $x \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-778d7404de25e37d546fc2ce5e571d09c3cf68eefdf24e5b1d5c80a27aef4d0f | We can conduct the following moves to a real number \( x \): choose a positive integer \( n \), and positive reals \( a_1, a_2, \ldots, a_n \) whose reciprocals sum up to 1. Let \( x_0 = x \), and \( x_k = \sqrt{x_{k-1} a_k} \) for all \( 1 \leq k \leq n \). Finally, let \( y = x_n \). A number \( M > 0 \) is called "t... | [
"\\( M \\geq 4 \\)"
] | [
"We can conduct the following moves to a real number \\( x \\): choose a positive integer \\( n \\), and positive reals \\( a_1, a_2, \\ldots, a_n \\) whose reciprocals sum up to 1. Let \\( x_0 = x \\), and \\( x_k = \\sqrt{x_{k-1} a_k} \\) for all \\( 1 \\leq k \\leq n \\). Finally, let \\( y = x_n \\). A number \... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-6328e67fab7092c722aa111a9b0e3b9e96c059497644c3511e8fa25ccaee812a | We can label the squares of an $8 \times 8$ chess board from $1$ to $64$ in $64!$ different ways. For each way, we find $D$, the largest difference between the labels of two squares which are adjacent (orthogonally or diagonally). What is the smallest possible value of $D$? | [
"9"
] | [
"We can label the squares of an $8 \\times 8$ chess board from $1$ to $64$ in $64!$ different ways. For each way, we find $D$, the largest difference between the labels of two squares which are adjacent (orthogonally or diagonally). What is the smallest possible value of $D$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-d460c31aa2afb5c0649c055e0d95a51e7582584276687a71d2c8c6056dd2ae8a | We can write $(x + 35)^n = \sum_{i=0}^n c_i x^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$? | [
"50"
] | [
"We can write $(x + 35)^n = \\sum_{i=0}^n c_i x^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22eaa043a641080c57c59fb470e39b3e6e79b343be496f944242725441e89b38 | We can write $\sqrt{20 + 2\sqrt{91}} + \sqrt{20 - 2\sqrt{91}}$ in the form $x + \sqrt{y}$, where $x$ and $y$ are integers. Determine $x + y$. | [
"52"
] | [
"We can write $\\sqrt{20 + 2\\sqrt{91}} + \\sqrt{20 - 2\\sqrt{91}}$ in the form $x + \\sqrt{y}$, where $x$ and $y$ are integers. Determine $x + y$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f49a41158f6c3e75a0b59308f1ba45bdfe18cd52da80833a1386deb178b4e9fb | We can write \[\sum_{k = 1}^{100} (-1)^k \cdot \frac{k^2 + k + 1}{k!} = \frac{a}{b!} - c,\]where $a,$ $b,$ and $c$ are positive integers. Find the smallest possible value of $a + b + c.$ | [
"202"
] | [
"We can write\n\\[\\sum_{k = 1}^{100} (-1)^k \\cdot \\frac{k^2 + k + 1}{k!} = \\frac{a}{b!} - c,\\]where $a,$ $b,$ and $c$ are positive integers. Find the smallest possible value of $a + b + c.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-129067c1a313a6212a5f24491ffde23f82b2785242577125cb3e3ae77f3e21a2 | We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points. | [
"8100"
] | [
"We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-384530ad9461226afd8807a8cf5a3b0b61e38677115297ce5eaaec8297bcb5bc | We choose \( n > 3 \) points on a circle and number them \( 1 \) to \( n \) in some order. We say that two non-adjacent points \( A \) and \( B \) are related if, in one of the arcs \( AB \), all the points are marked with numbers less than those at \( A \) and \( B \). If \( n = 10 \), calculate the sum of all possibl... | [
""
] | [
"We choose \\( n > 3 \\) points on a circle and number them \\( 1 \\) to \\( n \\) in some order. We say that two non-adjacent points \\( A \\) and \\( B \\) are related if, in one of the arcs \\( AB \\), all the points are marked with numbers less than those at \\( A \\) and \\( B \\). If \\( n = 10 \\), calculate... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-ef745127867d984afcff50519f2646c80c76d512fc46c8b0d7b4da1f610633c7 | We choose \( n \) numbers from \(-1, 1, 0\) and write them in a line such that their sum is 0. If the number of \(-1\)s is odd, the line is considered "bad"; if it is even, the line is considered "good". How many different good and bad lines can we get? | [
"\\frac{1}{2} \\left(3^{n} - (-1)^{n} \\right)"
] | [
"We choose \\( n \\) numbers from \\(-1, 1, 0\\) and write them in a line such that their sum is 0. If the number of \\(-1\\)s is odd, the line is considered \"bad\"; if it is even, the line is considered \"good\". How many different good and bad lines can we get?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3aff01789c66922fd739c54e3dbd440c273a3bd7b28cad160cc20136da5b45c2 | We choose \( n+1 \) points \( x_0 = 0, x_1, x_2, \ldots, x_n = 1 \) in the interval \([0,1]\). Assume the sequence of the distances between these points such that each distance occurs at least twice, except for the distance 1. Let \( S \) be the set of all distinct distances between these points. Calculate the sum of a... | [
"3"
] | [
"We choose \\( n+1 \\) points \\( x_0 = 0, x_1, x_2, \\ldots, x_n = 1 \\) in the interval \\([0,1]\\). Assume the sequence of the distances between these points such that each distance occurs at least twice, except for the distance 1. Let \\( S \\) be the set of all distinct distances between these points. Calculat... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ab0b97cc738408b642d7fa2736a799d0a840701c39c653940ea8fe0071985dd | We choose a subset $S$ from the set $A = \{1, 2, 3, \ldots, 1001\}$ with the condition that for any two elements $x$ and $y$ in $S$, their sum $x + y$ is not in $S$. What is the largest possible size of the set $S$? | [
"501"
] | [
"We choose a subset $S$ from the set $A = \\{1, 2, 3, \\ldots, 1001\\}$ with the condition that for any two elements $x$ and $y$ in $S$, their sum $x + y$ is not in $S$. What is the largest possible size of the set $S$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f7cff043ad0786f5edeb557c9fee568de76677ce525346b998ec5631a557931 | We choose randomly $k$ positive natural numbers, $X_{1}, X_{2}, \ldots, X_{k}$, which are independent and identically distributed (iid) with $P(X_{i}=n)=2^{-n}$. Let $N$ be the number of non-chosen numbers in the set $\{1, 2, 3, \ldots, \max\{X_{i}\}\}$. Find the distribution of $N$. | [
"P(N = m) = \\frac{1}{2^{m+1}} \\text{ for } m \\geq 0"
] | [
"We choose randomly $k$ positive natural numbers, $X_{1}, X_{2}, \\ldots, X_{k}$, which are independent and identically distributed (iid) with $P(X_{i}=n)=2^{-n}$. Let $N$ be the number of non-chosen numbers in the set $\\{1, 2, 3, \\ldots, \\max\\{X_{i}\\}\\}$. Find the distribution of $N$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-3a5a41be51129b6923465ba6fb258c603748a2c85d364a8b900b18c7b597b0ab | We color each number in the set \( S = \{1, 2, \ldots, 61\} \) with one of 25 given colors, where it is not necessary that every color gets used. Let \( m \) be the number of non-empty subsets of \( S \) such that every number in the subset has the same color. What is the minimum possible value of \( m \)? | [
"119"
] | [
"We color each number in the set \\( S = \\{1, 2, \\ldots, 61\\} \\) with one of 25 given colors, where it is not necessary that every color gets used. Let \\( m \\) be the number of non-empty subsets of \\( S \\) such that every number in the subset has the same color. What is the minimum possible value of \\( m \... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-1832c2e944a1eed02318f7c26349eab81d8d32caf96b4d425b8c4128b0fe8d13 | We color each unit of an $8 \times 8$ table either green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ blue unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. | [
"(0, 8)"
] | [
"We color each unit of an $8 \\times 8$ table either green or blue such that there are $a$ green unit squares in each $3 \\times 3$ square and $b$ blue unit squares in each $2 \\times 4$ rectangle. Find all possible values of $(a, b)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-32c8ae86681390644a12fe52fd174c0f482d58f2f4a49bb0c8e2b14fe1df8995 | We color one of the numbers $1, \ldots, 8$ with white or black according to the following rules: i) Number $4$ gets colored white and at least one of the following numbers gets colored black. ii) If two numbers $a, b$ are colored in different colors and $a + b \leq 8$, then number $a + b$ gets colored black. iii) If tw... | [
"BBBWBBBW, BWBWBWBW, BWBWBWBB"
] | [
"We color one of the numbers $1, \\ldots, 8$ with white or black according to the following rules:\ni) Number $4$ gets colored white and at least one of the following numbers gets colored black.\nii) If two numbers $a, b$ are colored in different colors and $a + b \\leq 8$, then number $a + b$ gets colored black.\n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-f4828fd428b6a9d2d343e6c2f42ebaa47bf7be04ad4cb81e5e0a91d7206d0655 | We color some unit squares in a $99 \times 99$ square grid with one of 5 given distinct colors, such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares. | [
"1900"
] | [
"We color some unit squares in a $99 \\times 99$ square grid with one of 5 given distinct colors, such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 8,
"tir": 6
} | 51 | 0 | 0 |
omr-ws-69aa2a706e06b685f4de6c19ee50593a4fe111e406330fe3890140051872f95f | We color some unit squares in a $99 \times 99$ square grid with one of five given distinct colors such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares and give an example in this cas... | [
"1900"
] | [
"We color some unit squares in a $99 \\times 99$ square grid with one of five given distinct colors such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares and give an example in th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-e7951534cefc622da43634c842b15860f8b8c00786ada9b4fd033547909a2f75 | We color the cells of an $n \times n$ board either red or blue. A red queen can move in any direction but only between red cells, and a blue queen can move in any direction but only between blue cells. Two queens attack each other if one can move to the other's position in a single move. Find the greatest integer $k$ s... | [
"\\( k = \\frac{n^2}{2} \\) for even \\( n \\) and \\( k = \\frac{n(n+1)}{2} \\) for odd \\( n \\)"
] | [
"We color the cells of an $n \\times n$ board either red or blue. A red queen can move in any direction but only between red cells, and a blue queen can move in any direction but only between blue cells. Two queens attack each other if one can move to the other's position in a single move. Find the greatest integer... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-852cb00ac7874d9af0e5b0f55cb376e9d4d1a063275240682fb864be07049a07 | We color the numbers $2, 3, 4, \ldots, 1024$ with $k$ different colors such that if $a$ is a multiple of $b$, then $a$ and $b$ have different colors. Find the minimum value of $k$. | [
"10"
] | [
"We color the numbers $2, 3, 4, \\ldots, 1024$ with $k$ different colors such that if $a$ is a multiple of $b$, then $a$ and $b$ have different colors. Find the minimum value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4bd1ae487dbebd68dd3d94a38a3dc0e1c6461494225a8fde5c67d7da217708f | We color the numbers from $1$ to $n$ with three distinct colors: white, black, and red, such that each color appears more than $n/4$ times. Let $S$ be the set of all triples $(a, b, c)$ where $a, b, c$ are distinct numbers from $1$ to $n$, $a + b = c$, and $a, b, c$ are of different colors. Calculate the minimum possib... | [
"351"
] | [
"We color the numbers from $1$ to $n$ with three distinct colors: white, black, and red, such that each color appears more than $n/4$ times. Let $S$ be the set of all triples $(a, b, c)$ where $a, b, c$ are distinct numbers from $1$ to $n$, $a + b = c$, and $a, b, c$ are of different colors. Calculate the minimum p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-7f1e2fdb7f7187ddd23cc30e6ed28aea5de0b148d96ee9d42e8c2dd147c33d71 | We color the vertices of a regular polygon with $n$ sides using two colors. A diagonal is considered good if it connects two vertices colored with different colors. A coloring is perfect if it is possible to divide the polygon internally into triangles by tracing only good diagonals, with no two diagonals sharing any p... | [
"2"
] | [
"We color the vertices of a regular polygon with $n$ sides using two colors. A diagonal is considered good if it connects two vertices colored with different colors. A coloring is perfect if it is possible to divide the polygon internally into triangles by tracing only good diagonals, with no two diagonals sharing ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-206f96e72cf7bf6efd6d6b064f7aacbca38c5b5ca3f1a8416e7fac2431f21dcf | We colour $8N^3$ lattice points red. On each red point $P$, we write the total number of red points on the vertical and horizontal lines through $P$. Let $M(N)$ be the smallest number that is written at least $N+1$ times. Calculate $M(3)$. | [
"12"
] | [
"We colour $8N^3$ lattice points red. On each red point $P$, we write the total number of red points on the vertical and horizontal lines through $P$. Let $M(N)$ be the smallest number that is written at least $N+1$ times. Calculate $M(3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-52491da4b13cd5f44a04b8f759511461ffe3cc6a5e6d4df082257fdf5e505cb5 | We colour every square of a $2009 \times 2009$ board with one of $n$ colours. A colour is called connected if either there is only one square of that colour or any two squares of the same colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares of another colo... | [
"2"
] | [
"We colour every square of a $2009 \\times 2009$ board with one of $n$ colours. A colour is called connected if either there is only one square of that colour or any two squares of the same colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares of anothe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-859a31c95c9bc3364a1564d2aed520837416a97bbf5ac4d556cac8ae5e6ae8d1 | We consider $5 \times 5$ tables containing a real number in each of the 25 cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is balanced if the number in the middle cell of every row and column is the average of the numbers in that row or column. A cell ... | [
"8"
] | [
"We consider $5 \\times 5$ tables containing a real number in each of the 25 cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is balanced if the number in the middle cell of every row and column is the average of the numbers in that row or column. A... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-10aa76847520867f479666e6e058ef7905ad38c5b730d1bb37e11c5c84358e0e | We consider a square of side $8$. By joining the midpoints of each pair of adjacent sides, we get a second square, and if we continue like this, joining the midpoints of the adjacent sides of each drawn square, we obtain a new square. What is the area of the sixth square obtained? | [
"1"
] | [
"We consider a square of side $8$. By joining the midpoints of each pair of adjacent sides, we get a second square, and if we continue like this, joining the midpoints of the adjacent sides of each drawn square, we obtain a new square. What is the area of the sixth square obtained?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e5f3a16ec954b071fa77950e32d70ed82a15a09c5748918d159a07d3cda6065b | We consider all possible pairs of 10-gons \( A \) and \( B \) (possibly non-convex) in the plane. The contour of polygon \( B \) divides polygon \( A \) into \( k \) parts. Find the largest possible value of \( k \). | [
"51"
] | [
"We consider all possible pairs of 10-gons \\( A \\) and \\( B \\) (possibly non-convex) in the plane. The contour of polygon \\( B \\) divides polygon \\( A \\) into \\( k \\) parts. Find the largest possible value of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-bea3d21d0dde00a243153d44e8ce17cd2125d570f9ea78b61b88e42977e01b7a | We consider maps on two spheres, say the earth and the moon. Each region on the earth is a country, and has a corresponding region or colony on the moon. We desire to color the countries and colonies so that: 1) Each country receives the same color as its colony, and 2) If two countries or colonies share a common borde... | [
"8"
] | [
"We consider maps on two spheres, say the earth and the moon. Each region on the earth is a country, and has a corresponding region or colony on the moon. We desire to color the countries and colonies so that:\n1) Each country receives the same color as its colony, and\n2) If two countries or colonies share a commo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-cc41702bb454ff275d5380a1251463fe16af8282b89e9d7c9ef8aa4a95f0b714 | We consider security codes consisting of four digits. We say that one code dominates another code if each digit of the first code is at least as large as the corresponding digit in the second code. For example, 4961 dominates 0761, because 4 ≥ 0, 9 ≥ 7, 6 ≥ 6, and 1 ≥ 1. We would like to assign a colour to each securit... | [
"37"
] | [
"We consider security codes consisting of four digits. We say that one code dominates another code if each digit of the first code is at least as large as the corresponding digit in the second code. For example, 4961 dominates 0761, because 4 ≥ 0, 9 ≥ 7, 6 ≥ 6, and 1 ≥ 1. We would like to assign a colour to each se... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-dff0b214d104f306ed810370ec9670b903123fdc72c2e222610d342f84d12aeb | We construct a base 2 number by flipping a coin repeatedly. If we get tails, we write a $0$ and stop. If we get heads, we write a $1$ and continue flipping. Each subsequent flip follows the same rule. What is the expected base 10 value of the constructed number? | [
"\\(\\frac{4}{3}\\)"
] | [
"We construct a base 2 number by flipping a coin repeatedly. If we get tails, we write a $0$ and stop. If we get heads, we write a $1$ and continue flipping. Each subsequent flip follows the same rule. What is the expected base 10 value of the constructed number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d50bcfbdeba8a1ee715c71a1b73cfb242c950fbc2bd2baf47a2d23d61448091 | We construct a solid figure through the following process: 1. Start with a solid clay cube of side length 8. 2. Remove all clay within a smaller cube formed by connecting the centers of each face of the solid cube. 3. Add back the clay in the shape of an even smaller cube, formed by connecting the centers of the faces ... | [
"\\dfrac{512(2\\sqrt{2} - 1)}{7}"
] | [
"We construct a solid figure through the following process:\n1. Start with a solid clay cube of side length 8.\n2. Remove all clay within a smaller cube formed by connecting the centers of each face of the solid cube.\n3. Add back the clay in the shape of an even smaller cube, formed by connecting the centers of th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-c0e6fd55ba6002dee587645046b1a68120157af9d030b8cbad3824e4f06b6dde | We construct right triangles $ABC, BDC, DEC, EZC$ with $\widehat{A} = \widehat{CBD} = \widehat{CDE} = \widehat{CEZ} = 90^\circ$. Given $CA=25$, $AB=36$, $BD=9$, $DE=4$, $EZ=1$, and $ZC=x$, find the area of the square with side length $x$. | [
"2019"
] | [
"We construct right triangles $ABC, BDC, DEC, EZC$ with $\\widehat{A} = \\widehat{CBD} = \\widehat{CDE} = \\widehat{CEZ} = 90^\\circ$. Given $CA=25$, $AB=36$, $BD=9$, $DE=4$, $EZ=1$, and $ZC=x$, find the area of the square with side length $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-298ebeb6c31db03bc1e613b51025ccf8a8be52fc173b9c658f13e9c27c6bf91b | We construct the Determinant-1 Triangle as follows: - In the \( n \)th row, the first and last entries equal 1. - Let \( d \) be an entry of the \( n \)th row other than a first or last entry. Let \( b \) and \( c \) be the two entries in the previous row directly above \( d \) and let \( a \) be the entry that is dire... | [
"50"
] | [
"We construct the Determinant-1 Triangle as follows:\n- In the \\( n \\)th row, the first and last entries equal 1.\n- Let \\( d \\) be an entry of the \\( n \\)th row other than a first or last entry. Let \\( b \\) and \\( c \\) be the two entries in the previous row directly above \\( d \\) and let \\( a \\) be t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-1abbe0f7f6c2f3b185a1f292e8d4a0512101eef6d9440feb0e718d2ed1ea29e9 | We continuously sample from the uniform distribution U(0,1) until the sum of the samples exceeds 1. What is the expected number of samples needed? | [
"e"
] | [
"We continuously sample from the uniform distribution U(0,1) until the sum of the samples exceeds 1. What is the expected number of samples needed?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f66df63c4b645560d56c7be5ec8ea69b7897943757aaccb67f1b784710da44f | We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Let $T(n)$ be the maximum number of moves required to reach a state where everyone has at most one card. Find $T(10)$. | [
"20"
] | [
"We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Let $T(n)$ be the maximum number of moves required to reach a state where everyone has at most one card. Find $T(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-95d841d8903af4054e5896f3927239ceee7cdf30ff0ddcd8e83d385f7d456d20 | We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon. To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. Initially, some hexagons are infected by a virus. If a hexagon has been infected, it will always be infecte... | [
"\\( f(n) = 3n^2 - 3n - \\left\\lfloor \\frac{3}{2}(n-1)^2 \\right\\rfloor = 3n^2 - \\left\\lfloor \\frac{3}{2}n^2 + \\frac{3}{2} \\right\\rfloor \\)"
] | [
"We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon. To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. Initially, some hexagons are infected by a virus. If a hexagon has been infected, it will always be in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 6
} | 19 | 0 | 0 |
omr-ws-9b8635aeaf20f24bced7d0c44b6eeaf5dbaccf829f6496f627dd8b11dece76d9 | We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and $f(b)=3b+1$ if $b$ is odd $f(b)=\frac{b}{2}$ if $b$ is even. What is the smallest possible number of integers in the domain of $f$? | [
"15"
] | [
"We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and\n\n$f(b)=3b+1$ if $b$ is odd\n\n$f(b)=\\frac{b}{2}$ if $b$ is even.\n\nWhat is the smallest possible number of integers in the domain of $f$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff46e9411f5ec97df018920c8cc213ae33458964079488a511722680ef17514f | We define a sequence \( x_1 = \sqrt{3}, x_2 = -1, x_3 = 2 - \sqrt{3} \), and for all \( n \geq 4 \), \[ (x_n + x_{n-3})(1 - x^2_{n-1}x^2_{n-2}) = 2x_{n-1}(1 + x^2_{n-2}). \] Suppose \( m \) is the smallest positive integer for which \( x_m \) is undefined. Compute \( m \). | [
"10"
] | [
"We define a sequence \\( x_1 = \\sqrt{3}, x_2 = -1, x_3 = 2 - \\sqrt{3} \\), and for all \\( n \\geq 4 \\),\n\\[\n(x_n + x_{n-3})(1 - x^2_{n-1}x^2_{n-2}) = 2x_{n-1}(1 + x^2_{n-2}).\n\\]\nSuppose \\( m \\) is the smallest positive integer for which \\( x_m \\) is undefined. Compute \\( m \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-1b09b3a4cab28586a656be9f5dcf8e6d40c38ab3b0a9bb47ce114cb1e72c78c1 | We define a sequence of natural numbers by the initial values \(a_0 = a_1 = a_2 = 1\) and the recursion \[ a_n = \left\lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \right\rfloor \] for all \(n \ge 3\). Find the value of \(a_{2022}\). | [
"674"
] | [
"We define a sequence of natural numbers by the initial values \\(a_0 = a_1 = a_2 = 1\\) and the recursion \n\\[ a_n = \\left\\lfloor \\frac{n}{a_{n-1}a_{n-2}a_{n-3}} \\right\\rfloor \\]\nfor all \\(n \\ge 3\\). Find the value of \\(a_{2022}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-9c81e031bd794a5cc2911091f46b49d38d307584ebbd8487f4032c7281f25024 | We define a sequence of numbers \(a_n\) such that \(a_0 = 1\) and for all \(n \geq 0\): \[2a_{n+1}^3 + 2a_n^3 = 3a_{n+1}^2 a_n + 3a_{n+1} a_n^2.\] Find the sum of all possible values of \(a_{2023}\). | [
"\\(\\frac{2^{2024} + 2^{-2023}}{3}\\)"
] | [
"We define a sequence of numbers \\(a_n\\) such that \\(a_0 = 1\\) and for all \\(n \\geq 0\\):\n\\[2a_{n+1}^3 + 2a_n^3 = 3a_{n+1}^2 a_n + 3a_{n+1} a_n^2.\\]\nFind the sum of all possible values of \\(a_{2023}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-472ba1faff298f0d4d60d8e0a79254eeeb1f3b0c1aafcb34ce96472fdf15b44c | We define the function \( f \) as follows: \[ f(x) = \begin{cases} \frac{x}{2} & \text{if } x \text{ is even} \\ x-1 & \text{if } x \text{ is odd} \end{cases} \] The path of \( x \), denoted as \( \text{path}(x) \), is defined such that: 1. \( x \in \text{path}(x) \) 2. If \( y \in \text{path}(x) \), then \( f(y) \in \... | [
"a"
] | [
"We define the function \\( f \\) as follows:\n\\[ f(x) = \\begin{cases}\n\\frac{x}{2} & \\text{if } x \\text{ is even} \\\\\nx-1 & \\text{if } x \\text{ is odd}\n\\end{cases} \\]\n\nThe path of \\( x \\), denoted as \\( \\text{path}(x) \\), is defined such that:\n1. \\( x \\in \\text{path}(x) \\)\n2. If \\( y \\in... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-c7f83e151f5794412a8d3ff83251b40ae30c815fa56af7ce1530bbe92f953c7a | We define the sets \( A_1, A_2, \ldots, A_{160} \) such that \( |A_i| = i \) for all \( i = 1, 2, \ldots, 160 \). By removing the same number of elements from each of these sets in each step, we create new sets \( M_1, M_2, \ldots, M_n \). Find the minimum value of \( n \) such that all elements from \( A_1, A_2, \ldot... | [
"8"
] | [
"We define the sets \\( A_1, A_2, \\ldots, A_{160} \\) such that \\( |A_i| = i \\) for all \\( i = 1, 2, \\ldots, 160 \\). By removing the same number of elements from each of these sets in each step, we create new sets \\( M_1, M_2, \\ldots, M_n \\). Find the minimum value of \\( n \\) such that all elements from ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-3c77676612f8dc65815230e6d4e58d46290558776b35cb74e3f78aaa3ea8e84d | We define two sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ as follows: First, $\mathbf{v}_0 = \begin{pmatrix} 1 \\ 3 \end{pmatrix},$ $\mathbf{w}_0 = \begin{pmatrix} 4 \\ 0 \end{pmatrix}.$ Then for all $n \ge 1,$ $\mathbf{v}_n$ is the projection of $\mathbf{w}_{n - 1}$ onto $\mathbf{v}_0,$ and $\mathbf{w}_... | [
"\\begin{pmatrix} 4/9 \\\\ 4/3 \\end{pmatrix}"
] | [
"We define two sequences of vectors $(\\mathbf{v}_n)$ and $(\\mathbf{w}_n)$ as follows: First, $\\mathbf{v}_0 = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix},$ $\\mathbf{w}_0 = \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}.$ Then for all $n \\ge 1,$ $\\mathbf{v}_n$ is the projection of $\\mathbf{w}_{n - 1}$ onto $\\mathbf{v... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3221bfcb89ec544767f99c21875610334539d566ef801d26680b04bf20221ee5 | We denote with \(m_a\), \(m_g\) the arithmetic mean, respectively geometric mean, of the positive numbers \(x, y\). a) If \(m_a + m_g = y - x\), determine the value of \(\frac{x}{y}\). b) Prove that there exists exactly one pair of different positive integers \((x, y)\) for which \(m_a + m_g = 40\). | [
"(5, 45)"
] | [
"We denote with \\(m_a\\), \\(m_g\\) the arithmetic mean, respectively geometric mean, of the positive numbers \\(x, y\\).\n\na) If \\(m_a + m_g = y - x\\), determine the value of \\(\\frac{x}{y}\\).\n\nb) Prove that there exists exactly one pair of different positive integers \\((x, y)\\) for which \\(m_a + m_g = ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-18d0e7c77f1bf8429ca15431758a48886c790f8619d05ce19c4e3d6d2d7a48c8 | We distribute 65 balls of different shapes and colors into two boxes. The colors of the balls are either white, yellow, red, or black. Suppose that when we choose 5 balls of the same size, we find at least two balls with the same color. Determine the minimum number of balls in one of the boxes that must have the same s... | [
"9"
] | [
"We distribute 65 balls of different shapes and colors into two boxes. The colors of the balls are either white, yellow, red, or black. Suppose that when we choose 5 balls of the same size, we find at least two balls with the same color. Determine the minimum number of balls in one of the boxes that must have the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-f2efb4cc0a9e528559acffa935523685fe6f8512dad185dd8c38993c2acd065e | We divide 25 arbitrary numbers into \( k \) subsets such that: 1) Every subset contains 5 numbers; 2) Each pair of subsets shares at most 1 number. Find the maximum value of \( k \). | [
"30"
] | [
"We divide 25 arbitrary numbers into \\( k \\) subsets such that:\n1) Every subset contains 5 numbers;\n2) Each pair of subsets shares at most 1 number.\nFind the maximum value of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a63d3d6c5a4d5493a2df39acff288788b4ca379c44cb530cdd04310fa02af428 | We divide a square into $16^2$ smaller squares, all of the same size. Each smaller square is painted with a certain color, with the restriction that in each column and each row at most $4$ distinct colors appear. Determine the maximum number of colors that can be used. | [
"16"
] | [
"We divide a square into $16^2$ smaller squares, all of the same size. Each smaller square is painted with a certain color, with the restriction that in each column and each row at most $4$ distinct colors appear. Determine the maximum number of colors that can be used."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-41a34fa91750e59ba72af41f08e0e0d30cd5786f1bb6d18b9645681b7be6bf39 | We divide the entire \(\mathbb{Z}\) into \(n\) subsets such that the difference of any two elements in a subset will not be a prime number. \(n\) is at least? | [
"4"
] | [
"We divide the entire \\(\\mathbb{Z}\\) into \\(n\\) subsets such that the difference of any two elements in a subset will not be a prime number. \\(n\\) is at least?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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