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-a44582b57f4f549816bb9ed0bc8c8a47e6395aeb7dad2d338d24760c0a8f6851 | We start with four identical right triangles. In one move, we can cut one of the triangles along the height from the right angle into two triangles, resulting in a total of 5 right triangles. After performing a series of moves, let $N$ be the minimum number of moves required to ensure that there are exactly 10 triangle... | [
"6"
] | [
"We start with four identical right triangles. In one move, we can cut one of the triangles along the height from the right angle into two triangles, resulting in a total of 5 right triangles. After performing a series of moves, let $N$ be the minimum number of moves required to ensure that there are exactly 10 tri... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2734ac446da3e1496c35ec0a930db2ff09d7e09d20e7cb280eda30dfd0226a7 | We start with the first 2018 positive integers on a board. In each step, we remove some numbers and replace them with the remainder when the sum of those numbers is divided by 31. After several steps, only two numbers remain on the board, one of which is 96. Determine the other number. | [
"3"
] | [
"We start with the first 2018 positive integers on a board. In each step, we remove some numbers and replace them with the remainder when the sum of those numbers is divided by 31. After several steps, only two numbers remain on the board, one of which is 96. Determine the other number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-3ccffd331377fe719f30b773a9cb27b099d4d9791aaf214151d9c5284fa31176 | We start with the interval $[0,1]$ and construct sets $A_n$ for all $n \in \mathbb{N}$ as follows: 1. $A_1 = [0,1]$. 2. For $A_2$, divide $[0,1]$ into $4 = 2^2$ equal parts and delete the first and third parts (without endpoints), so $A_2 = [0.25, 0.5] \cup [0.75, 1]$. 3. For $A_3$, divide $[0,1]$ into $9 = 3^2$ equal ... | [
"\\( A \\) is countable."
] | [
"We start with the interval $[0,1]$ and construct sets $A_n$ for all $n \\in \\mathbb{N}$ as follows:\n1. $A_1 = [0,1]$.\n2. For $A_2$, divide $[0,1]$ into $4 = 2^2$ equal parts and delete the first and third parts (without endpoints), so $A_2 = [0.25, 0.5] \\cup [0.75, 1]$.\n3. For $A_3$, divide $[0,1]$ into $9 = ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-479e5cb7c009207b6001266c5a4299d59362d666aedf84545e0e062f5db68d46 | We start with the list of integers from 1 to 2005. We cancel the first two integers and write their sum at the end of the list, then repeat this process until only one number remains. What is the sum of all the numbers listed, including the initial list of 1 to 2005? | [
"24046868"
] | [
"We start with the list of integers from 1 to 2005. We cancel the first two integers and write their sum at the end of the list, then repeat this process until only one number remains. What is the sum of all the numbers listed, including the initial list of 1 to 2005?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-2b749b842978f18621ab46acc2c27d0019d48b3f18e4a1723ffa81d03fbe890e | We start with the numbers 1 through 20 on a blackboard. We multiply every pair of these numbers in all possible ways. We then erase all the odd numbers and all numbers that are not multiples of 3. How many numbers remain on the blackboard? (Note: Duplicate numbers are considered as one number.) | [
"43"
] | [
"We start with the numbers 1 through 20 on a blackboard. We multiply every pair of these numbers in all possible ways. We then erase all the odd numbers and all numbers that are not multiples of 3. How many numbers remain on the blackboard? (Note: Duplicate numbers are considered as one number.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-75bdbc2c50fbd642a478f1184584001ba6a7985b496f5a56ba645beb94f5609b | We start with three real numbers \(a\), \(b\), and \(c\) on a board that are not all equal. Each time, we replace these numbers with \(a - b\), \(b - c\), and \(c - a\). Let \(S_n\) be the sum of the squares of the numbers on the board after \(n\) operations. Find the smallest integer \(k\) such that \(S_k > 2020^2\). | [
"14"
] | [
"We start with three real numbers \\(a\\), \\(b\\), and \\(c\\) on a board that are not all equal. Each time, we replace these numbers with \\(a - b\\), \\(b - c\\), and \\(c - a\\). Let \\(S_n\\) be the sum of the squares of the numbers on the board after \\(n\\) operations. Find the smallest integer \\(k\\) such ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-79f8e57ad08ccf964beec64beb47c647c666dc34e27b94c3e9c47e74d449edc6 | We take 4 natural numbers. For each pair of them, we wrote their GCD (greatest common divisor). The resulting GCDs are $1, 2, 3, 4, 5, N$, where $N > 5$. Find the least possible value of $N$. For example: - $GCD(a,b) = 1$ - $GCD(a,c) = 2$ - $GCD(a,d) = 3$ - $GCD(b,c) = 4$ - $GCD(b,d) = 5$ - $GCD(c,d) = N$ | [
"14"
] | [
"We take 4 natural numbers. For each pair of them, we wrote their GCD (greatest common divisor). The resulting GCDs are $1, 2, 3, 4, 5, N$, where $N > 5$. Find the least possible value of $N$. \n\nFor example:\n- $GCD(a,b) = 1$\n- $GCD(a,c) = 2$\n- $GCD(a,d) = 3$\n- $GCD(b,c) = 4$\n- $GCD(b,d) = 5$\n- $GCD(c,d) = N... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2d33753586f8d1b35e3a11e360ce379a5f8fd205c898d40f21cda821127bfc28 | We take 5 balls from a box containing 4 white balls, 6 blue balls, and 8 red balls. In how many cases do we have at least 1 ball from every color? | [
"5,584"
] | [
"We take 5 balls from a box containing 4 white balls, 6 blue balls, and 8 red balls. In how many cases do we have at least 1 ball from every color?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-68db19df4858d6ec9afb733623e4c8f745d5ff1ea09a4d05c4f37bb6bffebd70 | We throw $n$ raisins on a circular cake randomly and independently. What is the probability that the cake can be cut into two equal halves such that all raisins lie on the same piece? | [
"\\(\\frac{n}{2^{n-1}}\\)"
] | [
"We throw $n$ raisins on a circular cake randomly and independently. What is the probability that the cake can be cut into two equal halves such that all raisins lie on the same piece?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-5a03bf03b9ebdb272c6962cfc28dae25b434dda3a76e421c72e9f4ff53ab8f1f | We throw a balanced coin until the number of heads equals double the number of tails. Find the probability that we will never stop. | [
"2 - \\sqrt{5}"
] | [
"We throw a balanced coin until the number of heads equals double the number of tails. Find the probability that we will never stop."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60537a555842bf208bd6dee0d807ab77e655c03774a1c5f1d92b498c163b17a8 | We throw a regular six-sided die $n$ times. What fraction of the possible outcomes has the sum of the outcomes divisible by $5$? | [
"\\(\\frac{1}{5} + \\frac{4}{6^n}\\) if \\(5 \\mid n\\), \\(\\frac{1}{5} - \\frac{1}{6^n}\\) if \\(5 \\nmid n\\)"
] | [
"We throw a regular six-sided die $n$ times. What fraction of the possible outcomes has the sum of the outcomes divisible by $5$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 13
} | 18 | 0 | 0 |
omr-ws-1f347512565ffb3d87bf2aa305c8b826df891a140a7c5ac60aa438df6f7bcf8a | We throw symmetric dice until we get two 6's one after another. We can make a maximum of 6 throws. What is the probability that we will throw exactly six times? | [
"\\(\\frac{1025}{46656}\\)"
] | [
"We throw symmetric dice until we get two 6's one after another. We can make a maximum of 6 throws. What is the probability that we will throw exactly six times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-c7cd67643c02ed036a63aa34c71a4036baa29a6a53f20da0f93b544d565411bb | We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \ldots, N_R$ be the lengths of the runs. Derive the asymptotics for $E(N_1^2 + \cdots + N_R^2)$ as $n \to \infty$. | [
"\\( 3n - 4 \\)"
] | [
"We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \\ldots, N_R$ be the lengths of the runs. Derive the asymptotics for $E(N_1^2 + \\cdots + N_R^2)$ as $n \\to \\infty$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-c0f4f8da568e608440abb5fbfbfb3f3d2043c68a90f8ce0897650b461b8e6d27 | We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \ldots, N_R$ be the lengths of the runs. Find $E(N_1)$. | [
"\\( 2(1 - 2^{-n}) \\)"
] | [
"We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \\ldots, N_R$ be the lengths of the runs. Find $E(N_1)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-9c5317ba1183b57a6908f55fbe60e6fffab6f70ea17e17ad504edf26f4484e1a | We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \ldots, N_R$ be the lengths of the runs. Find $E(N_1^2 + \cdots + N_R^2)$. | [
"\\( 3n - 4 \\)"
] | [
"We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \\ldots, N_R$ be the lengths of the runs. Find $E(N_1^2 + \\cdots + N_R^2)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c799dc338b340abfa5fe8ff13b915ba1b6072a361d0253cc6a84b231d957c31 | We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \ldots, N_R$ be the lengths of the runs. Find $E(N_1^2)$. | [
"\\( 2\\left(3 - \\frac{2n + 3}{2^n}\\right) \\)"
] | [
"We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \\ldots, N_R$ be the lengths of the runs. Find $E(N_1^2)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-4b7f919a5c6a9c394b6cdbe75b7108bc244fe44c0401b486dcb64a3159158a31 | We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \ldots, N_R$ be the lengths of the runs. Find the asymptotics of $\max N_i$ as $n \to \infty$. | [
"\\(\\log_2 n + O(1)\\)"
] | [
"We toss a fair coin $n$ times. Let $R$ be the number of runs and $N_1, \\ldots, N_R$ be the lengths of the runs. Find the asymptotics of $\\max N_i$ as $n \\to \\infty$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f3926911e401ffd8b2769a1b9c39297014bba10af92a81013e8e0600108633f1 | We use 15 characters, consisting of S and J, to form a sequence. For every two consecutive characters, SS must appear 5 times, SJ must appear 3 times, JS must appear 3 times, and JJ must appear 3 times. How many such sequences can be formed? | [
"70"
] | [
"We use 15 characters, consisting of S and J, to form a sequence. For every two consecutive characters, SS must appear 5 times, SJ must appear 3 times, JS must appear 3 times, and JJ must appear 3 times. How many such sequences can be formed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-6dda5ad0a1ed29d923c0d3f6cac12018d1378d5031cbc35a6440dc2f439cc638 | We want to choose telephone numbers for a city. The numbers have 10 digits, and 0 is not used in the numbers. The aim is to choose numbers such that every pair of telephone numbers differs in more than one digit or differs in a digit which is more than 1. What is the maximum number of telephone numbers that can be chos... | [
"\\(\\frac{9^{10} + 1}{2}\\)"
] | [
"We want to choose telephone numbers for a city. The numbers have 10 digits, and 0 is not used in the numbers. The aim is to choose numbers such that every pair of telephone numbers differs in more than one digit or differs in a digit which is more than 1. What is the maximum number of telephone numbers that can be... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a781a7c842130982818be244da3e8e4e854f3f91d9829254b3946c698c19f9f | We want to give 3 toys to each of 7 different children. We want to ensure that any two children have at least one toy in common. What is the minimum number of different kinds of toys we need to buy? | [
"5"
] | [
"We want to give 3 toys to each of 7 different children. We want to ensure that any two children have at least one toy in common. What is the minimum number of different kinds of toys we need to buy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-3c337ccf358829cd2e37207df3c4561c056a8253a0da33fb65d6ed3a386500ae | We want to pick $5$-person teams from a total of $m$ people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint: $m$ is determined by these conditions.) | [
"21"
] | [
"We want to pick $5$-person teams from a total of $m$ people such that:\n1. Any two teams must share exactly one member.\n2. For every pair of people, there is a team in which they are teammates.\nHow many teams are there? (Hint: $m$ is determined by these conditions.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9f697432de5b2de7d76edd77fe5d017ea24fe1ab710418b60834b80ab2b7a9f9 | We want to place 2012 pockets, each containing variously colored balls, into \( k \) boxes such that for any box, either all pockets in the box must include a ball with the same color, or all pockets in the box must include a ball having a color which is not included in any other pocket in that box. Find the smallest v... | [
"62"
] | [
"We want to place 2012 pockets, each containing variously colored balls, into \\( k \\) boxes such that for any box, either all pockets in the box must include a ball with the same color, or all pockets in the box must include a ball having a color which is not included in any other pocket in that box. Find the sma... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aca4b647717ad86582aada342e8c17906d5a0c87e859a9b83fb63e49fc7dcaba | We wish to find the sum of 40 given numbers using 40 processors. Initially, each processor has the number 0 on its screen. Each processor can add the number on its screen with a number either entered directly (only the given numbers can be entered directly) or transferred from another processor, in a unit time. Wheneve... | [
"7"
] | [
"We wish to find the sum of 40 given numbers using 40 processors. Initially, each processor has the number 0 on its screen. Each processor can add the number on its screen with a number either entered directly (only the given numbers can be entered directly) or transferred from another processor, in a unit time. Wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 7,
"tir": 1
} | 55 | 0 | 0 |
omr-ws-fde9d597e05ef5f59340dbff8af5eb2c33ba5aa9fc28054c139e64169629c0ac | We wish to tile a strip of $n$ $1$-inch by $1$-inch squares using dominos (which cover two adjacent squares) or $1$-inch square tiles (which cover one square). Each square can be covered with one or two tiles, and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part of a ... | [
"832"
] | [
"We wish to tile a strip of $n$ $1$-inch by $1$-inch squares using dominos (which cover two adjacent squares) or $1$-inch square tiles (which cover one square). Each square can be covered with one or two tiles, and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 2
} | 48 | 0 | 0 |
omr-ws-54e300e3b602a5bc9bbc166e165da174ef4a99cf3f17b341c047a51eacb9ef38 | We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of ways to write these numbers such that for every square on the board, the absolute value of the sum of numbers on the square is less than or equal to $1$. | [
"2"
] | [
"We write $1$ or $-1$ on each unit square of a $2007 \\times 2007$ board. Find the number of ways to write these numbers such that for every square on the board, the absolute value of the sum of numbers on the square is less than or equal to $1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-14235a88632eb81cd5e6cbb9c692ae5525b5a2edfca0bf52c61c2b0b0b10ccc4 | We write all natural numbers from 1 to 1,000,000 on paper in random order. Then we cut the paper into pieces, each containing two successive digits. Calculate the total number of distinct 2-digit numbers that can be formed from these pieces. | [
"100"
] | [
"We write all natural numbers from 1 to 1,000,000 on paper in random order. Then we cut the paper into pieces, each containing two successive digits. Calculate the total number of distinct 2-digit numbers that can be formed from these pieces."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-803da0dcfb990da242590f7bfda46363761ba0e5a1a26bd68402aa434f18506b | We write different positive integers into each of the squares of an $n \times n$ chessboard. In each column, we color red the square with the greatest number. We call a set of $n$ unit squares *good* if no two of them lie in the same column or row. Let $S$ be the sum of the numbers in the good set with the maximal sum.... | [
"\\dfrac{1}{n}"
] | [
"We write different positive integers into each of the squares of an $n \\times n$ chessboard. In each column, we color red the square with the greatest number. We call a set of $n$ unit squares *good* if no two of them lie in the same column or row. Let $S$ be the sum of the numbers in the good set with the maxima... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-e29a8be4631d71fe4c8ee9217410e070e3b5b1e224ddd6d5c41c655b32b058c3 | We write each number $1, 2, 3, \ldots, 17$ in the squares of a $17 \times 17$ grid such that each number appears exactly 17 times. Let $m$ be the maximum number of different numbers appearing in any single row or column. Find the minimum possible value of $m$. | [
"9"
] | [
"We write each number $1, 2, 3, \\ldots, 17$ in the squares of a $17 \\times 17$ grid such that each number appears exactly 17 times. Let $m$ be the maximum number of different numbers appearing in any single row or column. Find the minimum possible value of $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-48212c26bee854a7f41eb65155ae1a13b57a4efabdb9e8e4f9204bd3431f9fb0 | We write in each of the squares (1x1) of a chessboard the number of rectangles that contain that square. Calculate the total sum of those values. | [
"14400"
] | [
"We write in each of the squares (1x1) of a chessboard the number of rectangles that contain that square. Calculate the total sum of those values."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-31e677008f6e772c0768eeea33e0db456cdb794b1d98df937bfaacb52d054844 | We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$s can be written into the unit squares? | [
"198"
] | [
"We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$s can be written into the unit squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-620f9d19de0cae4c91ff5f03e2c92f5aace3ecc4cd4c85ea6b4b126722866d75 | We write out all the integers from 1 to 100 in a row on index cards, shuffle them, and repeatedly replace the top three cards $a_1$, $a_2$, and $a_3$ with the card $a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_1a_3 + a_1a_2a_3$. We continue this process until we are left with either one card labeled $y$ or two cards, in which... | [
"\\( 101! - 1 \\)"
] | [
"We write out all the integers from 1 to 100 in a row on index cards, shuffle them, and repeatedly replace the top three cards $a_1$, $a_2$, and $a_3$ with the card $a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_1a_3 + a_1a_2a_3$. We continue this process until we are left with either one card labeled $y$ or two cards, in ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0fc935bcd74dd71172a7d21dcc8516ef9bd78c5b5aeb75c7b1289694d494617d | We write out all the integers from 1 to 100 in a row on index cards, shuffle them, and repeatedly replace the top three cards $a_1$, $a_2$, and $a_3$ with the card $a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_1a_3 + a_1a_2a_3$. We continue this process until we are left with either one card labeled $y$ or two cards, in which... | [
"101! - 1"
] | [
"We write out all the integers from 1 to 100 in a row on index cards, shuffle them, and repeatedly replace the top three cards $a_1$, $a_2$, and $a_3$ with the card $a_1 + a_2 + a_3 + a_1a_2 + a_2a_3 + a_1a_3 + a_1a_2a_3$. We continue this process until we are left with either one card labeled $y$ or two cards, in ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f564f8595e9b5f4ad48f9e2a1fa25000138e082ce933f9854483360e8eb66eeb | We write successively all the numbers from 1 to one million. Then, we replace each number by the sum of its digits. We repeat this process until we have only single-digit numbers. Which digit appears the most frequently? | [
"1"
] | [
"We write successively all the numbers from 1 to one million. Then, we replace each number by the sum of its digits. We repeat this process until we have only single-digit numbers. Which digit appears the most frequently?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e776e627bd9740e216cf8a65e1ae96f6cb1f3cced42d721cd40889ea9720699 | We write the numbers 1, 2, 3 in the cells of a 3x3 grid and calculate the sums of the numbers in each row and each column. What is the maximal number of different sums that can be obtained? | [
"5"
] | [
"We write the numbers 1, 2, 3 in the cells of a 3x3 grid and calculate the sums of the numbers in each row and each column. What is the maximal number of different sums that can be obtained?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-dbc4d7920553128e155cb6a6001d36ef7f8af12971ec2bf54e0b9676ce6a3761 | We write the numbers \(a_1, a_2, a_3, \dots, a_{107}, a_{108}\) on a circle so that the sum of any twenty consecutive numbers on the circle is 1000. If \(a_1 = 1\), \(a_{19} = 19\), and \(a_{50} = 50\), find the value of \(a_{100}\). | [
"130"
] | [
"We write the numbers \\(a_1, a_2, a_3, \\dots, a_{107}, a_{108}\\) on a circle so that the sum of any twenty consecutive numbers on the circle is 1000. If \\(a_1 = 1\\), \\(a_{19} = 19\\), and \\(a_{50} = 50\\), find the value of \\(a_{100}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-753bff33410ad93840a0ea4515b7cafcdb2687a91172c8c9faf791fe50bc497d | We write the sum of the reciprocals of the numbers from $1$ to $2013$ as the irreducible fraction $\frac{a}{b}$. That is: $$1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2013} = \frac{a}{b}$$ What is the largest integer value of $n$ such that $b$ is a multiple of $3^n$? | [
"5"
] | [
"We write the sum of the reciprocals of the numbers from $1$ to $2013$ as the irreducible fraction $\\frac{a}{b}$. That is:\n$$1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{2013} = \\frac{a}{b}$$\nWhat is the largest integer value of $n$ such that $b$ is a multiple of $3^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 7
} | 20 | 0 | 0 |
omr-ws-61e4eb731fdb569cb47f58dda15a673160f04989c61f02bd69fe6e7c374e3ca4 | We wrote numbers $1, 2, 3, \ldots, 2012$ on the whiteboard. Every minute, one student chooses two numbers $x$ and $y$, erases them, and writes the number $2x + 2y$ back on the whiteboard. This activity continues until one number $N$ remains. Find $\frac{N}{1000}$. | [
"\\frac{2012 \\times 2013 \\times 2^{2010}}{1000}"
] | [
"We wrote numbers $1, 2, 3, \\ldots, 2012$ on the whiteboard. Every minute, one student chooses two numbers $x$ and $y$, erases them, and writes the number $2x + 2y$ back on the whiteboard. This activity continues until one number $N$ remains. Find $\\frac{N}{1000}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c7d30fd93b96c63abe82637a84e80024c5bded92714d10af985f520766a6804 | Weightlifter Ruslan has just finished the exercise with a weight, which has $n$ small weights on one side and $n$ on the other. At each stage, he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $k$. What is the minimal number of sta... | [
"\\( 2\\lfloor n/k \\rfloor + 2 \\)"
] | [
"Weightlifter Ruslan has just finished the exercise with a weight, which has $n$ small weights on one side and $n$ on the other. At each stage, he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $k$. What is the minimal number o... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-ff996ce28c85eb954183f63ab99e03901a5e24dbb4c74b2d3e5e9f4b5a0dc5f2 | Wendy and Lucy play a game on an $m \times n$ chessboard, where $m$ and $n$ are integers $\ge 4$. Lucy places a rider on the chessboard first. Wendy then places a turn on an empty square that is one block horizontally and two blocks vertically, or two blocks horizontally and one block vertically, away from Lucy's squar... | [
"(m, n) \\text{ with both } m \\text{ and } n \\text{ even}"
] | [
"Wendy and Lucy play a game on an $m \\times n$ chessboard, where $m$ and $n$ are integers $\\ge 4$. Lucy places a rider on the chessboard first. Wendy then places a turn on an empty square that is one block horizontally and two blocks vertically, or two blocks horizontally and one block vertically, away from Lucy'... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-fb272c3b5e52d045f4b9b21c46cb994be6652a7c56c1622441b322f9b3990fee | Wendy dyes a 3x3 checkered board with 7 different colors (including white) under the following rules: 1. One grid at a time is dyed with one color. 2. Two adjacent times cannot be dyed in the same grid or with the same color. 3. A square can be dyed by multiple colors, and the final mixed colors are revealed. How many ... | [
"5105212"
] | [
"Wendy dyes a 3x3 checkered board with 7 different colors (including white) under the following rules:\n1. One grid at a time is dyed with one color.\n2. Two adjacent times cannot be dyed in the same grid or with the same color.\n3. A square can be dyed by multiple colors, and the final mixed colors are revealed.\n... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fc64b1bc076ad826f3354df2e0a5c58d2ca53773c641fa795d6f6ecbd09904eb | Wendy has 180 feet of fencing. She needs to enclose a rectangular space with an area that is ten times its perimeter. If she uses up all her fencing material, how many feet is the largest side of the enclosure? | [
"60"
] | [
"Wendy has 180 feet of fencing. She needs to enclose a rectangular space with an area that is ten times its perimeter. If she uses up all her fencing material, how many feet is the largest side of the enclosure?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54ca701c15d3bd29e5be02e1f11cec5e41a6dc07dceedf8f3a3ed04da061b4f4 | Wendy requires 6000 dollars per year in extra income. She has 50,000 dollars to invest and can invest in B-rated bonds paying 15 percent per year or in a Certificate of Deposit paying 7 percent per year. How much money should be invested in each to realize exactly 6000 dollars in interest per year? | [
"\\( x = 31250 \\), \\( y = 18750 \\)"
] | [
"Wendy requires 6000 dollars per year in extra income. She has 50,000 dollars to invest and can invest in B-rated bonds paying 15 percent per year or in a Certificate of Deposit paying 7 percent per year. How much money should be invested in each to realize exactly 6000 dollars in interest per year?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 31
} | 34 | 0 | 0 |
omr-ws-65915de712bb41e09b6c7e13cc1499934e11e1a3ab9f40efba274e74ccf3e3d9 | Wesley starts with the numbers $1, 2, 3, \ldots, 10$ on a board. Using the operations $A$ (delete $n$ and $n + 1$ and write $n + 2$) and $B$ (delete $n$ and $n + 4$ and write $n - 1$), what is the smallest number he can obtain on the board? | [
"-3"
] | [
"Wesley starts with the numbers $1, 2, 3, \\ldots, 10$ on a board. Using the operations $A$ (delete $n$ and $n + 1$ and write $n + 2$) and $B$ (delete $n$ and $n + 4$ and write $n - 1$), what is the smallest number he can obtain on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9803e49875af9c7eebb22b1f70446d4e9745701bc5155b851fef9aa16730bed1 | Western music is based on 12 notes. If we keep going up by $n$ semitones, let $J(n)$ be the number of jumps it takes to get back to the original note. If $M$ is the maximum value of $J$, how many values of $n$ from 1 to 11 are there such that $J(n) = M$? | [
"3"
] | [
"Western music is based on 12 notes. If we keep going up by $n$ semitones, let $J(n)$ be the number of jumps it takes to get back to the original note. If $M$ is the maximum value of $J$, how many values of $n$ from 1 to 11 are there such that $J(n) = M$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51dacafb27934a428ae183684ae1a5e0aabf176509d42460643ec5d14c22a298 | What $m \times n$ board can be tiled with $1 \times k$ dominoes? Please use complex numbers. | [
"\\( k \\mid n \\)"
] | [
"What $m \\times n$ board can be tiled with $1 \\times k$ dominoes? Please use complex numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-400055f2b7ecc313ae729d5d416a4281ebbdbf2525714bf0bc28d2feea91f43d | What about a function \( f \) that satisfies the property \( |f(x) - f(y)| \ge k|x - y| \) where \( k > 0 \)? Is it necessarily monotonous? Continuous? | [
"No, the function is not necessarily monotonous or continuous."
] | [
"What about a function \\( f \\) that satisfies the property \\( |f(x) - f(y)| \\ge k|x - y| \\) where \\( k > 0 \\)? Is it necessarily monotonous? Continuous?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-f4d1afd7c73975cc26fcc1f00747ead33a0d9b37da3cfccee69eb56753061717 | What advanced methods can be used to solve the equation \( x^2 = e^x \) algebraically, besides graphing and numerical methods like the Intermediate Value Theorem or Newton's method? Would the Lambert W function be applicable here? | [
"\\( x = e^{-W \\left(-\\frac{1}{2}\\right)} \\)"
] | [
"What advanced methods can be used to solve the equation \\( x^2 = e^x \\) algebraically, besides graphing and numerical methods like the Intermediate Value Theorem or Newton's method? Would the Lambert W function be applicable here?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 3,
"tir": 18
} | 48 | 0 | 0 |
omr-ws-28a7701ff288821175e8be3ffb1d4fe574633187f6e65aaca99a132611dea493 | What angle do the vectors ${\bf a} = (2, 3)$, ${\bf b} = (-2, 5)$, ${\bf c} =(-5, 1)$, and ${\bf d} = (-1, 1)$ make with the unit vector ${\bf i}$? | [
"$\\arccos \\frac{2}{\\sqrt {13}},\\ \\arccos \\left(-\\frac{2}{\\sqrt {29}}\\right),\\ \\arccos \\left(-\\frac{5}{\\sqrt {26}}\\right),\\ 135^\\circ$"
] | [
"What angle do the vectors ${\\bf a} = (2, 3)$, ${\\bf b} = (-2, 5)$, ${\\bf c} =(-5, 1)$, and ${\\bf d} = (-1, 1)$ make with the unit vector ${\\bf i}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37d42d63f3a29ad601ae7c77ee7bc157929ca42bf889bd6f4541ddc059398fbe | What angle is formed by the abscissa axis and the tangent to the parabola $y=x^2+4x-17$ drawn at the point $M (5/2, -3/4)$? Write the equation of the tangent. | [
"$\\arctan 9,\\ y=9x-23\\frac{1}{4}$"
] | [
"What angle is formed by the abscissa axis and the tangent to the parabola $y=x^2+4x-17$ drawn at the point $M (5/2, -3/4)$? Write the equation of the tangent."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e55eae1fee09bededadfd4456d86c2a4f9c1f9b5a5e1f60f83e39d9a74b6297b | What are all automorphisms of the symmetric group \( S_n \)? | [
"For \\( n \\leq 2 \\), \\( \\text{Aut}(S_n) = 1 \\).\nFor \\( 3 \\leq n \\neq 6 \\), \\( \\text{Aut}(S_n) \\cong S_n \\).\nFor \\( n = 6 \\), \\( \\text{Out}(S_6) = \\mathbb{Z}/2 \\)."
] | [
"What are all automorphisms of the symmetric group \\( S_n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 14
} | 28 | 0 | 0 |
omr-ws-e8618d726f3a89adc809ff79bb9c8333ea88545893908c7633f6660b27ceecb4 | What are all functions \( f \) such that \( f(a)f(b)f(c) = f(a) + f(b) + f(c) \)? | [
"\\( f(x) = 0 \\), \\( f(x) = \\sqrt{3} \\), or \\( f(x) = -\\sqrt{3} \\)"
] | [
"What are all functions \\( f \\) such that \\( f(a)f(b)f(c) = f(a) + f(b) + f(c) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-5a229ef5d116492ccf4b3d9eb28707ba9ce11a83a2ffc4b9fab507493dff763b | What are all idempotent elements in the ring $\mathbb{Z}_n$? | [
"$2^m$, where $m$ is the number of distinct prime factors of $n$."
] | [
"What are all idempotent elements in the ring $\\mathbb{Z}_n$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-040c26838b8dc49fe6741cea33f87240a3cd3e942f7e2a6f8cd6eff54a3084ae | What are all integers \(k\) for which the quadratic equation \(x^2 + kx + k + 17 = 0\) has integer solutions? | [
"\\( k = -17, -9, -7, 21, 13, 11 \\)"
] | [
"What are all integers \\(k\\) for which the quadratic equation \\(x^2 + kx + k + 17 = 0\\) has integer solutions?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-23144ec4252b38a695eb3adcaafd92784576377be63d00dc3dda2d472d7897f1 | What are all of the possible numbers of distinct regions of the plane that could be separated by three distinct lines contained within the plane? | [
"4, 6, 7"
] | [
"What are all of the possible numbers of distinct regions of the plane that could be separated by three distinct lines contained within the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-afcc6914cbd66f3eb3acf2466ce5d1bd9f420633c0a2892c59541521fec4c960 | What are all possible integer lengths for the unknown side of a triangle if the other two sides have lengths of 3 and 5? List all the possible values. | [
"3, 4, 5, 6, 7"
] | [
"What are all possible integer lengths for the unknown side of a triangle if the other two sides have lengths of 3 and 5? List all the possible values."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 22
} | 24 | 0 | 0 |
omr-ws-cab5b91e2f539232195c98c6bb3c3a144722e851e0ffa55a8ece4d9ad0998763 | What are all possible values of $\gcd(m, 3n)$ where $n$ is odd? | [
"\\text{All positive odd integers}"
] | [
"What are all possible values of $\\gcd(m, 3n)$ where $n$ is odd?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f837f44cdea46198152e2bed8c0d996146b0aff9495a6707398739653c7df00 | What are all possible values of \(a\) and \(b\) if \(|z - 1| + |z + 1| = 4\), where \(z = a + bi\) is a complex number? | [
"\\(\\frac{a^2}{4} + \\frac{b^2}{3} = 1\\)"
] | [
"What are all possible values of \\(a\\) and \\(b\\) if \\(|z - 1| + |z + 1| = 4\\), where \\(z = a + bi\\) is a complex number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-e74d3ccffa78e93ea9e7d5adc0d851492eeaccb39f2db1a0077f67b70f52b8f4 | What are all real number solutions to the equation $x^{y} = y^{x}$? | [
"\\( (t, t) \\) for any real \\( t \\) and \\( \\left( q^{\\frac{1}{q-1}}, q^{\\frac{q}{q-1}} \\right) \\) for any real \\( q \\neq 1 \\)."
] | [
"What are all real number solutions to the equation $x^{y} = y^{x}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 6
} | 17 | 0 | 0 |
omr-ws-46d9a2962c4ffcc6f81719de1e08a34f368b10c0db39bc492776fbbe28439808 | What are all real values of \( x \) that satisfy the equation \((x-3)(x^2+4x+4) = x-3\)? | [
"-1"
] | [
"What are all real values of \\( x \\) that satisfy the equation \\((x-3)(x^2+4x+4) = x-3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1dd844b83b650f26b933ed1f6f3c8d06081da19b6c81806b75fe1d80ffa7cf70 | What are all solutions to the equation $f(2x) = f(x)$ if $f(x)$ can be any function? | [
"\\( f(x) = g(\\{ \\log_2 |x| \\}) \\)"
] | [
"What are all solutions to the equation $f(2x) = f(x)$ if $f(x)$ can be any function?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 20
} | 43 | 0 | 0 |
omr-ws-3609801b64b314287f7d9428ac43615702427506741fb4b55d630ab13d05d630 | What are all solutions to the equation $f(2x) = f(x)$ if $f(x)$ is a polynomial? | [
"\\( f(x) = 0 \\)"
] | [
"What are all solutions to the equation $f(2x) = f(x)$ if $f(x)$ is a polynomial?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54ca331ec747106184a3f974b17dcfbe607cdc3d9fc87f113dc10802fd705370 | What are all the automorphisms of the group \((\mathbb{Q}^*, \cdot)\)? | [
"All automorphisms of the group \\((\\mathbb{Q}^*, \\cdot)\\) are given by functions that permute the primes and possibly flip the sign."
] | [
"What are all the automorphisms of the group \\((\\mathbb{Q}^*, \\cdot)\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 10
} | 31 | 0 | 0 |
omr-ws-4afcba2d99282e78c5d9445c9b99fbc9bd8872afb97492706abd1baae33ce52d | What are all the factors of a polynomial if two of the zeros of the polynomial are $-17$ and $4+5i$? | [
"$(x+17)$, $(x-4-5i)$, $(x-4+5i)$"
] | [
"What are all the factors of a polynomial if two of the zeros of the polynomial are $-17$ and $4+5i$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 6
} | 16 | 0 | 0 |
omr-ws-428137bbd9d224e1f8ab31b05ad5e24981963239eca5a8b0805e236788090521 | What are all the possible areas of a hexagon that has all its angles equal and whose sides measure $1, 2, 3, 4, 5,$ and $6$, in some order? | [
"\\dfrac{65\\sqrt{3}}{4}"
] | [
"What are all the possible areas of a hexagon that has all its angles equal and whose sides measure $1, 2, 3, 4, 5,$ and $6$, in some order?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-fbb128e97fa0c7d5f89475b33d356816fc22c47825e1c01349bf617df9bd0933 | What are all the possible rational zeros for the function \( f(x) = 6x^3 + 6x^2 - 15x - 2 \)? | [
"\\(\\pm 1, \\pm \\frac{1}{2}, \\pm \\frac{1}{3}, \\pm \\frac{1}{6}, \\pm 2, \\pm \\frac{2}{3}\\)"
] | [
"What are all the possible rational zeros for the function \\( f(x) = 6x^3 + 6x^2 - 15x - 2 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-97aef3d11cbecea689624f0077d7c4ad15bcf344e54df7da1aa9c48d8eeabca9 | What are all the possible rational zeros for the function \( f(x) = x^3 + 8x + 6 \)? | [
"\\(\\pm 1, \\pm 2, \\pm 3, \\pm 6\\)"
] | [
"What are all the possible rational zeros for the function \\( f(x) = x^3 + 8x + 6 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-ab7fd6ecbde34c8b97f3bb53c7e19455cdf0858ea8e1c34c143f3458972a753a | What are all the possible values of \( x \) in the equation \( \sin 2x - \cos 2x = \frac{\sqrt{6}}{2} \)? | [
"\\( x = \\frac{(-1)^k \\pi}{6} + \\frac{k\\pi}{2} + \\frac{\\pi}{8} \\)"
] | [
"What are all the possible values of \\( x \\) in the equation \\( \\sin 2x - \\cos 2x = \\frac{\\sqrt{6}}{2} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 11
} | 15 | 0 | 0 |
omr-ws-ce329ab6cf1cd6ee93c444666396e3a12e819c0cd94affefb78488bc9062b0e8 | What are all the possible values of the 4 by 4 determinant of $I + P_{even}$, where $P_{even}$ is any of the 12 even permutations of the identity matrix $I$? | [
"\\{0, 4, 16\\}"
] | [
"What are all the possible values of the 4 by 4 determinant of $I + P_{even}$, where $P_{even}$ is any of the 12 even permutations of the identity matrix $I$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 22
} | 25 | 0 | 0 |
omr-ws-a495b1296e102edc3d6ab61e54f2855af81543efeab3eea994158cf84479eb33 | What are all the possible values that the measure of the angle $\angle FCH$ can take in the rectangular parallelepiped $ABCDEFGH$? | [
"The measure of the angle $\\angle FCH$ can take a range of values."
] | [
"What are all the possible values that the measure of the angle $\\angle FCH$ can take in the rectangular parallelepiped $ABCDEFGH$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-b8a8fd898c22659a4d0c7c2769f1a673f65a69f94fde298dc4e3f7df993e136f | What are all the rectangle sizes that can be made using tiles made by combining $3$ unit squares, allowing rotations of the tiles by multiples of $90$ degrees? | [
"m \\times n \\text{ where } 3 \\mid m \\text{ or } 3 \\mid n"
] | [
"What are all the rectangle sizes that can be made using tiles made by combining $3$ unit squares, allowing rotations of the tiles by multiples of $90$ degrees?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-11ed928e27983427a81b8d7ddd9e98d2abc8436cb395fba0f042094df8c5a456 | What are all the solutions to the equation \( x^{\log_{10}(5x)} = 2 \)? | [
"\\( x = \\frac{1}{10}, 2 \\)"
] | [
"What are all the solutions to the equation \\( x^{\\log_{10}(5x)} = 2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 30
} | 34 | 0 | 0 |
omr-ws-fe380543e8210468c811cc1850ff9810461f1d1132a76746c9521695ae2d9fc6 | What are all the subgroups of the group \( G \), where \( G \) is the group of six permutations on the set \(\{1, 2, 3\}\)? | [
"\\(\\{e\\}, \\{(1), (12)\\}, \\{(1), (13)\\}, \\{(1), (23)\\}, \\{(1), (123), (132)\\}, S_3\\)"
] | [
"What are all the subgroups of the group \\( G \\), where \\( G \\) is the group of six permutations on the set \\(\\{1, 2, 3\\}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 25
} | 28 | 0 | 0 |
omr-ws-e68fa76e8ed1a60199a4f8f6f3f4a93ea09e4d86474ccd7b447f6fb837af75fa | What are all values of $k$ for which at least one pair of real numbers $(x,y)$ satisfies $\sin{x}+\cos{y}-\sin{x}\cos{y}=k$? | [
"\\([-3,1]\\)"
] | [
"What are all values of $k$ for which at least one pair of real numbers $(x,y)$ satisfies $\\sin{x}+\\cos{y}-\\sin{x}\\cos{y}=k$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-9082122f7d4fb20909ece186818d2413d9c07f06aa058277d53ba4d488f9fccb | What are all values of $p$ such that for every $q>0$, we have $$\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. | [
"[0,3)"
] | [
"What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form."
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-e0295dfcda5ae09c99df2a4edd76951aab39cbbeea54d1ec94a3ec65456bac01 | What are all values of \( n \) such that \( \phi(n) = 38 \)? | [
"No solutions."
] | [
"What are all values of \\( n \\) such that \\( \\phi(n) = 38 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-bc0c889ee63290f885fa8a2b2de21c206dee73a8b9c3a292ea3432bfaace8c6f | What are examples of a $5$-Sylow subgroup of $GL_3(\mathbb{F}_5)$? | [
"\\( H=\\left\\{ \\begin{pmatrix} 1& *&*\\\\0&1&*\\\\0&0&1\\end{pmatrix}\\right\\} \\)"
] | [
"What are examples of a $5$-Sylow subgroup of $GL_3(\\mathbb{F}_5)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-80c11b00df250d18efa92a36b5c5be2fc0232233b0d905b05b8d5346462af6eb | What are good approaches to finding the indefinite integral of $\int \frac{dx}{1+\sin x+2\cos x}$? | [
"\\frac{1}{2} \\ln\\left| \\frac{\\tan\\left(\\frac{x}{2}\\right) + 1}{\\tan\\left(\\frac{x}{2}\\right) - 3} \\right| + C"
] | [
"What are good approaches to finding the indefinite integral of $\\int \\frac{dx}{1+\\sin x+2\\cos x}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-93aa262f68c59a81439f41d96af007e866d969e4ac762141df1f637512fc76da | What are the 2-dimensional subspaces of $\mathbb{C}^{3}$ that are invariant under multiplication by the matrix \[A= \begin{pmatrix}\lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}\] i.e., subspaces $V$ where $A(V) \subseteq V$? | [
"\\(\\{\\left(\\begin{matrix}p\\\\q\\\\0\\end{matrix}\\right)|p,q\\in \\mathbb{C}\\}\\)"
] | [
"What are the 2-dimensional subspaces of $\\mathbb{C}^{3}$ that are invariant under multiplication by the matrix \\[A= \\begin{pmatrix}\\lambda & 1 & 0 \\\\ 0 & \\lambda & 1 \\\\ 0 & 0 & \\lambda \\end{pmatrix}\\] i.e., subspaces $V$ where $A(V) \\subseteq V$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-cc70dd0118064d809815de9f91acc326942b6d0698cc08f01e5885a19daf4944 | What are the 2007th, 2008th, and 2009th digits after the decimal point in the decimal expansion of \( \frac{1}{2008} \)? | [
"007"
] | [
"What are the 2007th, 2008th, and 2009th digits after the decimal point in the decimal expansion of \\( \\frac{1}{2008} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-33adbacfcec36e54001b976bfb4aad222e9d73b851cc960da7a222c9e86551b3 | What are the 2016th, 2017th, and 2018th natural numbers whose square ends with the digits 2001? | [
"2519751, 2520249, 2521001"
] | [
"What are the 2016th, 2017th, and 2018th natural numbers whose square ends with the digits 2001?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-4217401c35b79384c7829be5012dbd5d65fa445b2e425191a1e6ffe486f6f94f | What are the 2x2 matrices \( A \) with integer entries such that for all \( n > 1 \), there exists a 2x2 matrix \( X(n) \) with integer entries such that \( (X(n))^n = A \)? | [
"The matrices are \\( 0_2 \\), \\( I_2 \\), and the specific forms listed above."
] | [
"What are the 2x2 matrices \\( A \\) with integer entries such that for all \\( n > 1 \\), there exists a 2x2 matrix \\( X(n) \\) with integer entries such that \\( (X(n))^n = A \\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-85cf1ba3e34d2efefcf26d2e6a50c9795c1d048b15bce3f0bc7a41e473f6e649 | What are the 99th, 100th, and 101st elements in the colex order on $\mathbb{N}^{(4)}$? | [
"3579, 4579, 1679"
] | [
"What are the 99th, 100th, and 101st elements in the colex order on $\\mathbb{N}^{(4)}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-d3f94ea1eb8adeaaea70e45bab20e6bb2b9f402c1e1e73ad93d6e8f5b6100456 | What are the Cauchy-Riemann Equations in polar coordinates for the modulus and the argument? | [
"\\(\\frac{\\partial R}{\\partial r} = \\frac{1}{r} \\frac{\\partial \\phi}{\\partial \\theta}\\) and \\(\\frac{1}{r} \\frac{\\partial R}{\\partial \\theta} = -\\frac{\\partial \\phi}{\\partial r}\\)"
] | [
"What are the Cauchy-Riemann Equations in polar coordinates for the modulus and the argument?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-436fef725c2c5c113f63f6ace0a3ae9ff0ff0098fa6b254ead97908fdd4a9802 | What are the Fourier coefficients of the function \( f(x) = \sum_{n \geq 2} \frac{\cos(nx)}{n \ln n} \)? | [
"a_n = \\begin{cases} \n0 & \\text{if } n = 0 \\text{ or } n = 1, \\\\\n\\frac{1}{n \\ln n} & \\text{if } n \\geq 2,\n\\end{cases} \\quad \\text{and} \\quad b_n = 0 \\text{ for all } n \\geq 1"
] | [
"What are the Fourier coefficients of the function \\( f(x) = \\sum_{n \\geq 2} \\frac{\\cos(nx)}{n \\ln n} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-882f6402ca5740faac90bea5fcd28d0e11bf57920d8be1c669af4a2da20081e9 | What are the Fourier transforms of the functions \( f(x) = e^{-x^2} \) and \( g(x) = x e^{-(x^2 + 2x)} \)? | [
" -\\sqrt{\\pi} e^{1 + ik - k^2/4} \\left(1 + \\frac{ik}{2}\\right) "
] | [
"What are the Fourier transforms of the functions \\( f(x) = e^{-x^2} \\) and \\( g(x) = x e^{-(x^2 + 2x)} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-e152e54096aaa54816b2607b07d82de73506ca95389adb27b8ef4fa2618e3b46 | What are the Representative Fractions of Map A and Map B, given that Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m? | [
"Map A: \\(\\frac{1}{80}\\), Map B: \\(\\frac{1}{220}\\)"
] | [
"What are the Representative Fractions of Map A and Map B, given that Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4e57bd5681cd7d9320f0e1ebd5ee7556a4cfcfe78fefd609d1dfd2540fc14546 | What are the angles of the triangle formed by joining the three points of contact of a circle inscribed in a triangle? | [
"\\(\\textbf{(D)}\\ \\text{are always acute angles}\\)"
] | [
"What are the angles of the triangle formed by joining the three points of contact of a circle inscribed in a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a40a632a0d0f3cd3119b4db5e14e31dcbad1e8d0efe1602958fed5034cb35fd7 | What are the barycentric coordinates of the orthocenter of $\triangle BIC$, where $I$ is the incenter of $\triangle ABC$, with $\triangle ABC$ taken as the reference triangle? | [
"\\((a : c-a : b-a)\\)"
] | [
"What are the barycentric coordinates of the orthocenter of $\\triangle BIC$, where $I$ is the incenter of $\\triangle ABC$, with $\\triangle ABC$ taken as the reference triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-e66be725ab0462af038a3782ddcfed05955c7eadebff764ed244b72cc24b2d5d | What are the barycentric coordinates of the point \( P(x:y:z) \) after being inverted with respect to the circumcircle? | [
"\\left( \\dfrac{a^2}{x} : \\dfrac{b^2}{y} : \\dfrac{c^2}{z} \\right)"
] | [
"What are the barycentric coordinates of the point \\( P(x:y:z) \\) after being inverted with respect to the circumcircle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ac2ee320b3e076d13da4eb74f2ea60eaac8a454deae27c2a2c5f261557e75e1a | What are the barycentric coordinates of the point which is the reflection of the point \((0,1,1)\) in the external angular bisector of angle \(A\) of triangle \(ABC\)? | [
"(4, -1, -1)"
] | [
"What are the barycentric coordinates of the point which is the reflection of the point \\((0,1,1)\\) in the external angular bisector of angle \\(A\\) of triangle \\(ABC\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-d7836243efee73965b871ac9e608e4dcdb1e018933b90039881f01f809778f9e | What are the barycentric coordinates of the reflection of the point \((0,1,1)\) in the external angular bisector of vertex \(A\) of triangle \(ABC\)? | [
"(4, -1, -1)"
] | [
"What are the barycentric coordinates of the reflection of the point \\((0,1,1)\\) in the external angular bisector of vertex \\(A\\) of triangle \\(ABC\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-03e442036b867fd3c1c5feeffadf3d079b58847f5b91b1ac2ce5fd311d6d0538 | What are the best values of \( m \) and \( k \) (both greater than 0) for which the inequality \[ \sqrt{1 + m \sum{(a-b)^2}} + k \left( \frac{abcde + abcef + abdef + acdef + bcdef + abcdf}{6} \right) \geq (1 + k) \left( \frac{a + b + c + d + e + f}{6} \right) \] holds for all non-negative real numbers \( a, b, c, d, e,... | [
"\\dfrac{1}{4}"
] | [
"What are the best values of \\( m \\) and \\( k \\) (both greater than 0) for which the inequality\n\\[\n\\sqrt{1 + m \\sum{(a-b)^2}} + k \\left( \\frac{abcde + abcef + abdef + acdef + bcdef + abcdf}{6} \\right) \\geq (1 + k) \\left( \\frac{a + b + c + d + e + f}{6} \\right)\n\\]\nholds for all non-negative real n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-877af833f6437d38374a191804e8457b5d59bf752a8668d683619f971420d77c | What are the bounds on \(a\) and \(b\) if the polynomial equation \(x^4 + ax^3 + bx^2 + ax + 1 = 0\) has exactly 1, 2, 3, or 4 real solutions? | [
"(a, b) = (-4, 6) \\text{ or } (4, 6)"
] | [
"What are the bounds on \\(a\\) and \\(b\\) if the polynomial equation \\(x^4 + ax^3 + bx^2 + ax + 1 = 0\\) has exactly 1, 2, 3, or 4 real solutions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e82b6f0e9ac4b21cf1edef8b09e6dd7ec42d73a88f172a29188e2ba3f0b257b | What are the chances that at least one student has their birthday on an exam day, given there are 50 students and 5 exams on different days out of 365 days in the year? | [
"1 - \\left( \\frac{360}{365} \\right)^{50}"
] | [
"What are the chances that at least one student has their birthday on an exam day, given there are 50 students and 5 exams on different days out of 365 days in the year?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-f15ce018350279dfbf503e90b3758d1b8f5601d070ed8f426abd1153840e0d73 | What are the chances that two people in a room have the same birthday, given that only the month and day are considered? | [
"\\( 1 - \\left( \\frac{365}{365} \\times \\frac{364}{365} \\times \\frac{363}{365} \\times \\ldots \\times \\frac{345}{365} \\right) \\)"
] | [
"What are the chances that two people in a room have the same birthday, given that only the month and day are considered?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-1d625b696a3d30ca2cec3b285d3b48babdf225ad6e1c9a8822426e0b2f67c5fb | What are the closed subgroups of $\mathbb{R}^n$ for $n > 1$? | [
"\\mathbb{R}^k \\times \\mathbb{Z}^m"
] | [
"What are the closed subgroups of $\\mathbb{R}^n$ for $n > 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-b89dfa85227b65866aa63ac7e9b0e11a52afa97929673e4484237abd680ee49f | What are the coefficients in the Taylor series of \( f(z) = \frac{1}{(z-a)(z-b)} \) centered at \( z_0 \)? | [
"\\( c_n = \\frac{1}{a-b}\\left(\\frac{1}{(b-z_0)^{n+1}} - \\frac{1}{(a-z_0)^{n+1}}\\right) \\) when \\( a \\ne b \\) and \\( c_n = -(n+1) \\left(\\frac{1}{z_0-a}\\right)^{n} \\) when \\( a = b \\)"
] | [
"What are the coefficients in the Taylor series of \\( f(z) = \\frac{1}{(z-a)(z-b)} \\) centered at \\( z_0 \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a755013c2d7a0f2748b79a6d4d61b41994f3af568de9756ab58d9749bb94902 | What are the coefficients of the polynomial $(x - 1)^{n-1}$ using a subtraction-based Pascal's Trapezoid? | [
"The coefficients of \\((x - 1)^{n-1}\\) using a subtraction-based Pascal's Trapezoid are derived from the pattern where each entry is the right number minus the left number."
] | [
"What are the coefficients of the polynomial $(x - 1)^{n-1}$ using a subtraction-based Pascal's Trapezoid?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-3bef92e71bee4206c144dcb93239749b77c531b50d94112117f82304ad02fe9a | What are the coefficients of the polynomial $(x^3 + 2x^2 + 3x + 1)(x + 1)^{n-1}$ using Pascal's Trapezoids? | [
"\\binom{n-1}{m} + 3\\binom{n-1}{m-1} + 2\\binom{n-1}{m-2} + \\binom{n-1}{m-3}"
] | [
"What are the coefficients of the polynomial $(x^3 + 2x^2 + 3x + 1)(x + 1)^{n-1}$ using Pascal's Trapezoids?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-f9ad60a1b252363843fa223959d14c2c15fb12fbde2d318b1af5e3f6fb2fc9f1 | What are the complex matrices \( A \) of size \( n \) such that \( A^* A A = I_n \)? | [
"I_n"
] | [
"What are the complex matrices \\( A \\) of size \\( n \\) such that \\( A^* A A = I_n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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