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-18b29aced67a6f59ab9d0f55c22a3df13d65e6e4cb2e6200e43d48345d61c739 | We have a stack of $n$ books labeled $1, 2, \ldots, n$. In each round, we make $n$ moves where in the $i$-th move, we turn over the top $i$ books as a single unit. Determine the smallest positive integer $k$ such that after $k$ rounds, the books will return to their initial arrangement for $n=7$. | [
"4"
] | [
"We have a stack of $n$ books labeled $1, 2, \\ldots, n$. In each round, we make $n$ moves where in the $i$-th move, we turn over the top $i$ books as a single unit. Determine the smallest positive integer $k$ such that after $k$ rounds, the books will return to their initial arrangement for $n=7$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0e00ede369f67c6b72011ae0d3dabe19349e6b26329d35949d0b8a529277a1e | We have a stack of 2016 cards and a hat. We choose a card, put it in the hat, and arbitrarily divide the remaining cards into two piles. In each subsequent step, we choose one of the piles, take a card from it and put it in the hat, then divide the remaining cards in that pile into two. Is it possible, after repeating ... | [
"\\text{No}"
] | [
"We have a stack of 2016 cards and a hat. We choose a card, put it in the hat, and arbitrarily divide the remaining cards into two piles. In each subsequent step, we choose one of the piles, take a card from it and put it in the hat, then divide the remaining cards in that pile into two. Is it possible, after repea... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9cf94cfd2f27ec4d0fd1e64688a0d2bb82ef0c813066d6c263a1fcd2b365c139 | We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house? | [
"\\frac{6}{4165}"
] | [
"We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a9726fe376d32f991b6890738ad32176fb9e75a5be7d48b88a3111d4c661c62 | We have a total of 48 balls, 8 each in 6 different colors. We randomly select 35 balls. What are the probabilities that we can complete 1, 2, 3, 4, or 5 sets of 6 balls in the same color from the selected balls? | [
"1"
] | [
"We have a total of 48 balls, 8 each in 6 different colors. We randomly select 35 balls. What are the probabilities that we can complete 1, 2, 3, 4, or 5 sets of 6 balls in the same color from the selected balls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-22e29c1749e7449f39250708bf273a4ca0c3baba31e99f51a7cedb619a77569b | We have a total of 48 balls, 8 each in 6 different colors. We randomly select 35 balls. What is the probability that we cannot complete any set of 6 balls in the same color from the selected balls? | [
"0"
] | [
"We have a total of 48 balls, 8 each in 6 different colors. We randomly select 35 balls. What is the probability that we cannot complete any set of 6 balls in the same color from the selected balls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-57589985ebe508ef04cc04b2bcb96fcfa1a586fad84e590eb7950f4687593e69 | We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\triangle ABC$? | [
"52"
] | [
"We have a triangle $\\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\\triangle ABC$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16a1e01ac8d50c4eaca4aa090853a4b24a9cc8f974323618a3840abe71e8eee8 | We have a triangle $\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$? | [
"\\sqrt{39}"
] | [
"We have a triangle $\\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2cf178a1001e3892d6ce4887616959c0db9e238854b27ee5c48ad98652a6559 | We have a triangle with side lengths \( AB = 32 \), \( BC = 12 \), \( CA = 25 \). Let \( D \) be the point on \( AB \) one fourth of the way from \( A \) to \( B \), and \( E \) the midpoint of \( BF \). Moreover, let \( F \) be the midpoint of \( BC \), and let \( G, H \) be on \( CA \) such that \( CG = AH = 5 \). If... | [
"133"
] | [
"We have a triangle with side lengths \\( AB = 32 \\), \\( BC = 12 \\), \\( CA = 25 \\). Let \\( D \\) be the point on \\( AB \\) one fourth of the way from \\( A \\) to \\( B \\), and \\( E \\) the midpoint of \\( BF \\). Moreover, let \\( F \\) be the midpoint of \\( BC \\), and let \\( G, H \\) be on \\( CA \\) ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-0c20a296bf61fdaf0fc4e18bfb328033603b2b979f10f7f1848525469a1a972b | We have a triangular wooden figure with vertices \( A, B \), and \( C \) such that \( \angle C = 25^\circ \). The triangle is rotated counterclockwise around point \( A \) until \( B, A \), and the original position of vertex \( C \) are collinear. If the new position of vertex \( C \) and the original positions of \( ... | [
"\\( 50^\\circ \\)"
] | [
"We have a triangular wooden figure with vertices \\( A, B \\), and \\( C \\) such that \\( \\angle C = 25^\\circ \\). The triangle is rotated counterclockwise around point \\( A \\) until \\( B, A \\), and the original position of vertex \\( C \\) are collinear. If the new position of vertex \\( C \\) and the orig... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-794f69fcc63d6c987ded0bd640128d30dd8c0500bb475f12a9c2e5117e0b67ba | We have an $n \times n$ board where the unit square $(i,j)$ is colored black if $i=j$, red if $i<j$, and green if $i>j$. Let $a_{i,j}$ denote the color of the unit square $(i,j)$. In each move, we switch two rows and write down the $n$-tuple $(a_{1,1}, a_{2,2}, \ldots, a_{n,n})$. How many distinct $n$-tuples can be obt... | [
"\\(\\frac{3^n + 3 - 2n}{4}\\)"
] | [
"We have an $n \\times n$ board where the unit square $(i,j)$ is colored black if $i=j$, red if $i<j$, and green if $i>j$. Let $a_{i,j}$ denote the color of the unit square $(i,j)$. In each move, we switch two rows and write down the $n$-tuple $(a_{1,1}, a_{2,2}, \\ldots, a_{n,n})$. How many distinct $n$-tuples can... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-d134c769dbce47c207f0f8c53e17e939cb803751f15fd25be27d02312fd3d1e6 | We have an $n \times n$ board, where each square contains an integer, and the sum of all integers on the board is $0$. An action on a square decreases the integer in that square by the number of its neighboring squares and increases the integer in each neighboring square by 1. Determine if there exists $n \geq 2$ such ... | [
"No"
] | [
"We have an $n \\times n$ board, where each square contains an integer, and the sum of all integers on the board is $0$. An action on a square decreases the integer in that square by the number of its neighboring squares and increases the integer in each neighboring square by 1. Determine if there exists $n \\geq 2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-557eddb56c67238d95b0ee6bc2971c3a56ee0e3c1dd48d9e55a5f4d584b6810c | We have an \( m \times n \) table of squares. The bottom left and the top right squares are painted white. We can move right, up, or up-right by 1, but only to white squares. In how many ways can we color the rest of the squares white and black so that we can reach the top right square from the bottom left square? | [
"2^{(m-1)(n-1)}"
] | [
"We have an \\( m \\times n \\) table of squares. The bottom left and the top right squares are painted white. We can move right, up, or up-right by 1, but only to white squares. In how many ways can we color the rest of the squares white and black so that we can reach the top right square from the bottom left squa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-428d49b566e8c2f02312fb0dc3b86514eb87fff714011b54389f765f66ddccb2 | We have an open-ended table with two rows. Initially, the numbers $1$, $2$, …, $2005$ are written in the first 2005 squares of the first row. In each move, we write down the sum of the first two numbers of the first row as a new number which is then added to the end of this row, and drop the two numbers used in the add... | [
"24046868"
] | [
"We have an open-ended table with two rows. Initially, the numbers $1$, $2$, …, $2005$ are written in the first 2005 squares of the first row. In each move, we write down the sum of the first two numbers of the first row as a new number which is then added to the end of this row, and drop the two numbers used in th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-b524174900069a3b78aa7adb35009cfc979c397586dfbf67176932eca16cdf87 | We have an open-ended table with two rows. Initially, the numbers $1, 2, \ldots, 2005$ are written in the first $2005$ squares of the first row. In each move, we write down the sum of the first two numbers of the first row as a new number which is then added to the end of this row, and drop the two numbers used in the ... | [
"4022030"
] | [
"We have an open-ended table with two rows. Initially, the numbers $1, 2, \\ldots, 2005$ are written in the first $2005$ squares of the first row. In each move, we write down the sum of the first two numbers of the first row as a new number which is then added to the end of this row, and drop the two numbers used i... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-55f65da32c21a2d6e1726dc2abaa3febb40e65ec67bcf80f1cd3a81394d01694 | We have an open-ended table with two rows. Initially, the numbers $1, 2, \ldots, 2005$ are written in the first 2005 squares of the first row. In each move, we write down the sum of the first two numbers of the first row as a new number which is then added to the end of this row, and drop the two numbers used in the ad... | [
"\\(4022030\\)"
] | [
"We have an open-ended table with two rows. Initially, the numbers $1, 2, \\ldots, 2005$ are written in the first 2005 squares of the first row. In each move, we write down the sum of the first two numbers of the first row as a new number which is then added to the end of this row, and drop the two numbers used in ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a5b710767b8be07062310bd8efe1e3ffc2d6ddfdc76e65cccb6781bb23f3a3a | We have chosen 38 even numbers from the set \(\{1, 2, 3, \ldots, 1000\}\). Let \(d\) be the smallest possible difference between any two of these chosen even numbers. Find the maximum possible value of \(d\). | [
"26"
] | [
"We have chosen 38 even numbers from the set \\(\\{1, 2, 3, \\ldots, 1000\\}\\). Let \\(d\\) be the smallest possible difference between any two of these chosen even numbers. Find the maximum possible value of \\(d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-873bbbc1dee1e6badaf087a13544dce7196b83acb30561a98a2fc576f65adbe2 | We have constructed a rhombus by attaching two equal equilateral triangles. By placing \( n-1 \) points on each of the three sides of each triangle, we divide the sides into \( n \) equal segments. By drawing line segments between corresponding points on each side of the triangles, we divide the rhombus into \( 2n^2 \)... | [
"\\(3n^4 - 4n^2 + 4n - 2\\)"
] | [
"We have constructed a rhombus by attaching two equal equilateral triangles. By placing \\( n-1 \\) points on each of the three sides of each triangle, we divide the sides into \\( n \\) equal segments. By drawing line segments between corresponding points on each side of the triangles, we divide the rhombus into \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93f5783382095952bb4b61068e5ae43c2ce122f3bc8b5f2d3ef790aa4d9e5454 | We have erasers, four pencils, two notebooks, and three pens, and we want to divide them between two persons so that each person receives at least one of the above stationery. In how many ways is this possible? (Note that the items are not distinct.) | [
"118"
] | [
"We have erasers, four pencils, two notebooks, and three pens, and we want to divide them between two persons so that each person receives at least one of the above stationery. In how many ways is this possible? (Note that the items are not distinct.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4c7332113e4f05d4cbda35f55e0bb89263e409bc9e4aa80cffff800185d399a | We have filled $k$ cells of an $8 \times 8$ grid with distinct numbers in such a way that the number in each square is greater than at most one filled square with a common corner. Find the maximum possible value for $k$. | [
"32"
] | [
"We have filled $k$ cells of an $8 \\times 8$ grid with distinct numbers in such a way that the number in each square is greater than at most one filled square with a common corner. Find the maximum possible value for $k$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b17283ffc5be0a4b0fb861057cd03146405e4d38e15df03fde8051476b032b77 | We have five consecutive positive integers whose sum is a cube and the sum of the three middle integers is a square. What is the smallest possible middle integer? | [
"675"
] | [
"We have five consecutive positive integers whose sum is a cube and the sum of the three middle integers is a square. What is the smallest possible middle integer?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb919f4435717bcbde25ab9bdfda7a5e7f3998272ce3f8a6c05f1c58cd43784e | We have four cards. One side of each card has either a number {1, 2} or a letter {B, C}, and the other side has the opposite. The visible sides are: 1, 2, B, and C. The claim is: "If the letter B is written on one side, then the number 1 is written on the other side." How many cards should be checked to verify or refut... | [
"2"
] | [
"We have four cards. One side of each card has either a number {1, 2} or a letter {B, C}, and the other side has the opposite. The visible sides are: 1, 2, B, and C. The claim is: \"If the letter B is written on one side, then the number 1 is written on the other side.\" How many cards should be checked to verify o... | [
"aops_c7_college_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-ef6ba6f2772dbc0643566f1f428b9c49aedd6fd53abbea9dad322d17ecfc6573 | We have four charged batteries, four uncharged batteries, and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists of putting two batteries in ... | [
"5"
] | [
"We have four charged batteries, four uncharged batteries, and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists of putting two batterie... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f79e8dc6c857491c68829ccaec851c2bef66ec7a18ec317e6bfe2dbd92d30d24 | We have four liters of acid in one concentration and six liters of acid with a different concentration. If all the acid is mixed together, a 35\% solution of acid is obtained, and if equal volumes of these solutions are taken, then a 36\% solution of acid is obtained. How much acid (in liters) is contained in each of t... | [
"1.64 liters and 1.86 liters"
] | [
"We have four liters of acid in one concentration and six liters of acid with a different concentration. If all the acid is mixed together, a 35\\% solution of acid is obtained, and if equal volumes of these solutions are taken, then a 36\\% solution of acid is obtained. How much acid (in liters) is contained in ea... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc71359d86331099794fb186dd2f30ad36ff41cd5def7b14e9ad45d162f658e9 | We have four loaded batteries, four unloaded batteries, and a radio that requires two loaded batteries to work. Determine the smallest number of attempts needed to guarantee that the radio works, where an attempt consists of placing two batteries in the radio and checking if it works. | [
"4"
] | [
"We have four loaded batteries, four unloaded batteries, and a radio that requires two loaded batteries to work. Determine the smallest number of attempts needed to guarantee that the radio works, where an attempt consists of placing two batteries in the radio and checking if it works."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac2a2fb844d3e6abb1b15e3be3c77ad63ff5481007bacaadb688bca9f1ea0d4b | We have four positive integers. If we choose three of them, take their average, and divide this average by the fourth number, we get the results $89$, $95$, $101$, and $117$. Find the four numbers. | [
"33, 42, 51, 75"
] | [
"We have four positive integers. If we choose three of them, take their average, and divide this average by the fourth number, we get the results $89$, $95$, $101$, and $117$. Find the four numbers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bf417c18f65119b12945b895be13b1d1225833f78d015c33e8f80403c21cc07 | We have four white equilateral triangles of 3 cm on each side, joined by their sides to form a triangular base pyramid. At each edge of the pyramid, two red dots are marked that divide it into three equal parts. Number the red dots so that when you traverse them in the order they were numbered, the resulting path has t... | [
"12"
] | [
"We have four white equilateral triangles of 3 cm on each side, joined by their sides to form a triangular base pyramid. At each edge of the pyramid, two red dots are marked that divide it into three equal parts. Number the red dots so that when you traverse them in the order they were numbered, the resulting path ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-a6af8c04af111c25538c5f4a7284abe2ff7ddbd43067ebaf4f39d619bb00d9bc | We have many three-element subsets of a 1000-element set. We know that the union of every 5 of them has at least 12 elements. Find the maximum possible number of these subsets. | [
"444"
] | [
"We have many three-element subsets of a 1000-element set. We know that the union of every 5 of them has at least 12 elements. Find the maximum possible number of these subsets."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ecd14a13bc940c8d0c34fec440d276a435631e80f3e68efc17f7aa612020242 | We have placed $n > 3$ cards around a circle, facing downwards. In one step, we may perform the following operation with three consecutive cards: calling the one in the center $B$, and the two on the ends $A$ and $C$, we put card $C$ in the place of $A$, then move $A$ and $B$ to the places originally occupied by $B$ an... | [
"28"
] | [
"We have placed $n > 3$ cards around a circle, facing downwards. In one step, we may perform the following operation with three consecutive cards: calling the one in the center $B$, and the two on the ends $A$ and $C$, we put card $C$ in the place of $A$, then move $A$ and $B$ to the places originally occupied by $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-24fe21b08e65feeb46acc3580a40aa5ff5bc7769404ab559e86cdb09db6f21b3 | We have placed some balls in three boxes. In each round, we can choose two boxes that contain \(a\) and \(b\) balls, where \(a \geq b\). We then move \(b\) balls from the first box to the second box, resulting in the boxes containing \(a - b\) and \(2b\) balls, respectively. Suppose the initial number of balls in the t... | [
"50"
] | [
"We have placed some balls in three boxes. In each round, we can choose two boxes that contain \\(a\\) and \\(b\\) balls, where \\(a \\geq b\\). We then move \\(b\\) balls from the first box to the second box, resulting in the boxes containing \\(a - b\\) and \\(2b\\) balls, respectively. Suppose the initial number... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc33e0ff2b95370945f5fc3316665cd06c8245fd0efd008894585f9cdee0c64d | We have positive integers \(a < b < c < d\) such that \(a, b,\) and \(c\) form an arithmetic progression while \(b, c,\) and \(d\) form a geometric progression. If \(d - a = 30\), evaluate \(a + b + c + d\). | [
"129"
] | [
"We have positive integers \\(a < b < c < d\\) such that \\(a, b,\\) and \\(c\\) form an arithmetic progression while \\(b, c,\\) and \\(d\\) form a geometric progression. If \\(d - a = 30\\), evaluate \\(a + b + c + d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-0b8a7d167c1eb7ecd3aafe2cb4e4cb618fbf82b0a87b70219041b3e57623cabe | We have ten red marbles, nine green marbles, and eleven blue marbles. Marbles of the same color are indistinguishable. Suppose we arrange these thirty marbles in a line. A marble is called isolated if there is no marble with the same color adjacent to it. Find the number of arrangements that contain exactly two isolate... | [
"0"
] | [
"We have ten red marbles, nine green marbles, and eleven blue marbles. Marbles of the same color are indistinguishable. Suppose we arrange these thirty marbles in a line. A marble is called isolated if there is no marble with the same color adjacent to it. Find the number of arrangements that contain exactly two is... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-02d2f95d7484de4d2e79c760a065e9b397d507f6ba765a109f0af4dd919f23e8 | We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$? | [
"1"
] | [
"We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-011a3d322e9dc4e501dbc61dd258d1318b8c54ccd74f8465068d3b30f2309739 | We have that $3 \cdot f(x) + 4 \cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is $9$, then what is the minimum possible degree of $g(x)$? | [
"9"
] | [
"We have that $3 \\cdot f(x) + 4 \\cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is $9$, then what is the minimum possible degree of $g(x)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c9078d7f78ab3dbf34530ab69bef7283d0ff46572face63d3521e3d0ac170d9 | We have that $\lceil x \rceil - \lfloor x \rfloor = 0.$ Then, what is $\lceil x \rceil - x$? | [
"0"
] | [
"We have that $\\lceil x \\rceil - \\lfloor x \\rfloor = 0.$ Then, what is $\\lceil x \\rceil - x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30cb714cbcf888eb6a5e2265a70ae5736a14b988d0995a6a9e814ec650edd917 | We have that $p, q, r$ are consecutive real values in an arithmetic sequence in some order. Given that $(p + q - r)^2 + (p + r - q)^2 + (q + r - p)^2 = (p + q + r)^4$, the minimum possible value of $(p + q + r)^2$ can be expressed as $\frac{a}{b}$. Find $a + b$. | [
"4"
] | [
"We have that $p, q, r$ are consecutive real values in an arithmetic sequence in some order. Given that $(p + q - r)^2 + (p + r - q)^2 + (q + r - p)^2 = (p + q + r)^4$, the minimum possible value of $(p + q + r)^2$ can be expressed as $\\frac{a}{b}$. Find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-8cd8e8bb1c7a744bdbe5b4405349445a022dda081d936f1594d35497a5ed10e7 | We have the coins \( C_1, C_2, \cdots, C_n \). For each \( k \), \( C_k \) is biased so that when tossed, it has probability \( \frac{1}{2k+1} \) of showing heads. If \( n \) coins are tossed, what is the probability that the number of heads is odd? (Putnam, as found in the Intermediate Counting and Probability book) | [
"\\(\\frac{n}{2n+1}\\)"
] | [
"We have the coins \\( C_1, C_2, \\cdots, C_n \\). For each \\( k \\), \\( C_k \\) is biased so that when tossed, it has probability \\( \\frac{1}{2k+1} \\) of showing heads. If \\( n \\) coins are tossed, what is the probability that the number of heads is odd? (Putnam, as found in the Intermediate Counting and Pr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-21a1089b66d401426cb76206d7d8580abf2054721297c771db690071506a93a3 | We have two coins, one fair with a probability of heads equal to $\frac{1}{2}$ and the other biased with a probability of heads equal to $\frac{1}{3}$. We choose one of the coins at random and toss it 10 times. Find the conditional probability that the chosen coin is the fair one, given that we got exactly 4 heads. | [
"\\dfrac{59049}{124585}"
] | [
"We have two coins, one fair with a probability of heads equal to $\\frac{1}{2}$ and the other biased with a probability of heads equal to $\\frac{1}{3}$. We choose one of the coins at random and toss it 10 times. Find the conditional probability that the chosen coin is the fair one, given that we got exactly 4 hea... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33c8ce2d25cbce37f224aa341ba1e82ff2ddd25ab28d09168ee3df8ecaaa731d | We have two decks of 36 cards each, numbered 1 through 36. Each card in the first deck is matched with its pair in the second deck (i.e., 1-1, 2-2, etc.). After shuffling one deck randomly and placing it on top of the original (unshuffled) deck, find the sum of the distances between each matched pair of cards. The dist... | [
"1260"
] | [
"We have two decks of 36 cards each, numbered 1 through 36. Each card in the first deck is matched with its pair in the second deck (i.e., 1-1, 2-2, etc.). After shuffling one deck randomly and placing it on top of the original (unshuffled) deck, find the sum of the distances between each matched pair of cards. The... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-b5334a5c2fa1456b795ebc46dd58005f1cb0a24cdf48aeffa178a3805c030e3d | We have two decks of cards, each with $n$ cards. There is a matching between the decks (a bijection). One deck is shuffled and placed on top of the other non-shuffled deck. For each card on the top shuffled deck, we count the number of cards between it and its corresponding card in the second deck. What is the sum of t... | [
"\\( n(n-1) \\)"
] | [
"We have two decks of cards, each with $n$ cards. There is a matching between the decks (a bijection). One deck is shuffled and placed on top of the other non-shuffled deck. For each card on the top shuffled deck, we count the number of cards between it and its corresponding card in the second deck. What is the sum... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 6
} | 51 | 0 | 0 |
omr-ws-1809edb35afd08563c7f8a7782bf8872552d8d7925fabacd13cd3dd85d292e26 | We have two dice, each with numbers from 1 to 6 on each side. We can manipulate the probabilities of each number appearing on each die such that the sum of the probabilities for each die is 1 and each probability is positive. After manipulating both dice, we throw them simultaneously. Let \( P(C_k) \) be the probabilit... | [
"no_valid_answer_found"
] | [
"We have two dice, each with numbers from 1 to 6 on each side. We can manipulate the probabilities of each number appearing on each die such that the sum of the probabilities for each die is 1 and each probability is positive. After manipulating both dice, we throw them simultaneously. Let \\( P(C_k) \\) be the pro... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62af257c1c7aa3bbf370384d5666cf5d36a3dca80cafefac91757cb57c3a8ab4 | We have two distinct positive integers $a$ and $b$ with $a|b$. Each of $a$ and $b$ consists of $2n$ decimal digits. The first $n$ digits of $a$ are identical to the last $n$ digits of $b$, and the last $n$ digits of $a$ are identical to the first $n$ digits of $b$. Determine $a$ and $b$. | [
"\\( a = 142857 \\), \\( b = 857142 \\)"
] | [
"We have two distinct positive integers $a$ and $b$ with $a|b$. Each of $a$ and $b$ consists of $2n$ decimal digits. The first $n$ digits of $a$ are identical to the last $n$ digits of $b$, and the last $n$ digits of $a$ are identical to the first $n$ digits of $b$. Determine $a$ and $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 23
} | 24 | 0 | 0 |
omr-ws-c58cce250e8ca92dd0964f0a3d81ae407c6108be4d19a5f44e0c9e6fd466e692 | We have two distinct positive integers \( a \) and \( b \) with \( a \mid b \). Each of \( a \) and \( b \) consists of \( 2n \) decimal digits. The first \( n \) digits of \( a \) are identical to the last \( n \) digits of \( b \), and vice versa. Determine \( a \) and \( b \). | [
"857142"
] | [
"We have two distinct positive integers \\( a \\) and \\( b \\) with \\( a \\mid b \\). Each of \\( a \\) and \\( b \\) consists of \\( 2n \\) decimal digits. The first \\( n \\) digits of \\( a \\) are identical to the last \\( n \\) digits of \\( b \\), and vice versa. Determine \\( a \\) and \\( b \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-1ae7b6ec53914ff35039a1a2f6b343a816b472a70c036ca3ef22f1cf1b9d5767 | We have two lines \(r\) and \(s\), and determine three points on each of them: \(A_{1}\), \(A_{2}\), \(A_{3}\) on \(r\); \(B_{1}\), \(B_{2}\), \(B_{3}\) on \(s\). Then we connect with lines any two points with different subscriptions: \(A_{1}B_{2}\), \(A_{1}B_{3}\); \(A_{2}B_{1}\), \(A_{2}B_{3}\); \(A_{3}B_{1}\), \(A_{... | [
"1"
] | [
"We have two lines \\(r\\) and \\(s\\), and determine three points on each of them: \\(A_{1}\\), \\(A_{2}\\), \\(A_{3}\\) on \\(r\\); \\(B_{1}\\), \\(B_{2}\\), \\(B_{3}\\) on \\(s\\). Then we connect with lines any two points with different subscriptions: \\(A_{1}B_{2}\\), \\(A_{1}B_{3}\\); \\(A_{2}B_{1}\\), \\(A_{... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65e56e982a26fdd09accf6fbf9dc6c9a5060313096e6b62aa3ceef465b705b3e | We have two positive integers both less than 1000. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers. | [
"120"
] | [
"We have two positive integers both less than 1000. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-087e2b689f78db6c26afc2514ba511760c30773f6a0d46b4b413bc99359d87f3 | We have two similar dice, one red and one blue, each with dimensions 1*1*1 and vertexes on integer points in the coordinate system. In each step, we can move the blue die one unit along one axis (x, y, or z) without rotating it, and simultaneously rotate the red die 90 degrees about that axis without moving its center.... | [
"\\text{All initial conditions}"
] | [
"We have two similar dice, one red and one blue, each with dimensions 1*1*1 and vertexes on integer points in the coordinate system. In each step, we can move the blue die one unit along one axis (x, y, or z) without rotating it, and simultaneously rotate the red die 90 degrees about that axis without moving its ce... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-4e7fb6d414fc595e654e9ac9b1be16892c2db851719a4f32fe6b1c6c7000304c | We have two similar solids. The volume of the larger one is \(16\sqrt{2}\) times the volume of the smaller one. The sum of their surface areas is 216 square inches. What is the surface area of the smaller solid? | [
"24",
"24\n\nExplanation:\nThe forum discussion for problem #12 provides a detailed solution. The key points are:\n- The volume ratio of the larger solid to the smaller solid is \\(16\\sqrt{2}\\), which implies the linear scale factor \\(y\\) between the two solids is \\(2\\sqrt{2}\\).\n- Since surface area scale... | [
"We have two similar solids. The volume of the larger one is \\(16\\sqrt{2}\\) times the volume of the smaller one. The sum of their surface areas is 216 square inches. What is the surface area of the smaller solid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 26
} | 57 | 0 | 0 |
omr-ws-856a36ea0afa28093892b0dcf29e7a94b1f178e98dad5986d6788b68fb18a55d | We have two solutions of a salt in water. To obtain a mixture containing 10 grams of salt and 90 grams of water, one takes twice as much (by weight) of the first solution as the second. One week elapses and 200 grams of water has evaporated from each kilogram of the first and second solution. Now to obtain the same mix... | [
"5 grams and 20 grams"
] | [
"We have two solutions of a salt in water. To obtain a mixture containing 10 grams of salt and 90 grams of water, one takes twice as much (by weight) of the first solution as the second. One week elapses and 200 grams of water has evaporated from each kilogram of the first and second solution. Now to obtain the sam... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8654f9947145c0b0fde79ebf58b9e4a16d7d63b8f25caa3c4946c6e0cbe21448 | We have two tanks, one filled with pure glycerin, and the other with water. Using two three-liter scoops, one for ladling glycerin from the first tank, the other for ladling water from the second tank, glycerin was transferred from the first tank to the second tank, and a scoop-full of the contents of the second was tr... | [
"10 liters"
] | [
"We have two tanks, one filled with pure glycerin, and the other with water. Using two three-liter scoops, one for ladling glycerin from the first tank, the other for ladling water from the second tank, glycerin was transferred from the first tank to the second tank, and a scoop-full of the contents of the second w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b68f6814ee9b29e23d33c22241af5aebf7e6611e7edbca6b6f3b23fee988dab2 | We have two urns. The first urn contains 3 white balls and 4 black balls, and the second urn contains 5 white balls and 6 black balls. We flip an unbiased coin with a probability of $\frac{4}{10}$ for heads. If the coin lands on heads, we move one ball from the first urn to the second urn. If it does not land on heads,... | [
"\\(\\frac{4312}{9555}\\)"
] | [
"We have two urns. The first urn contains 3 white balls and 4 black balls, and the second urn contains 5 white balls and 6 black balls. We flip an unbiased coin with a probability of $\\frac{4}{10}$ for heads. If the coin lands on heads, we move one ball from the first urn to the second urn. If it does not land on ... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-cb934d86c304b0606f90c012b5b86730da309e303a89b830636ae6926003bdd8 | We have written numbers $1, 2, \ldots, n$ in the unit squares of an $m \times n$ rectangle such that each number $i$ appears exactly $m$ times. We are allowed to swap the numbers in any two unit squares within the same column. After rearranging the numbers so that each row contains all the numbers $1, 2, \ldots, n$, ca... | [
"no_valid_answer_found"
] | [
"We have written numbers $1, 2, \\ldots, n$ in the unit squares of an $m \\times n$ rectangle such that each number $i$ appears exactly $m$ times. We are allowed to swap the numbers in any two unit squares within the same column. After rearranging the numbers so that each row contains all the numbers $1, 2, \\ldots... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df900ea7a6a03bb5f6e268839cae186242657be2e5fd5e5106c9cc5e9003b5ea | We know that $$2^{22000} = \underbrace{4569878...229376}_{6623\,\,\, digits}.$$ For how many positive integers $n < 22000$ is it also true that the first digit of $2^n$ is 4? | [
"2131"
] | [
"We know that $$2^{22000} = \\underbrace{4569878...229376}_{6623\\,\\,\\, digits}.$$ For how many positive integers $n < 22000$ is it also true that the first digit of $2^n$ is 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-768e2860d088a81419c7dcafd6fdb3b054dd539d31e7815ce445f21e226dcd96 | We know the following to be true: $\bullet$ 1. $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$ $\bullet$ 2. $Z$ = $K \times K^2.$ What is the value of $K$ for which $Z$ is a perfect square? | [
"9"
] | [
"We know the following to be true:\n\n$\\bullet$ 1. $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$\n\n$\\bullet$ 2. $Z$ = $K \\times K^2.$\n\nWhat is the value of $K$ for which $Z$ is a perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6408682cc5a6b7deec49b4c15425c4735e581c3edb5d5e735586f52c5835605a | We list the integers from $1$ to $n$. From this list, we delete the integer $m$. The average of the remaining $n-1$ numbers is $\frac{134}{11}$. Determine $n$ and $m$. | [
"\\( n = 23 \\), \\( m = 8 \\)"
] | [
"We list the integers from $1$ to $n$. From this list, we delete the integer $m$. The average of the remaining $n-1$ numbers is $\\frac{134}{11}$. Determine $n$ and $m$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 39
} | 42 | 0 | 0 |
omr-ws-2f92ec9f1592c484a0e3fd904519172c36d1783688996cd25fed8ad6b021a163 | We make $n$ triangles with wire. For any two triangles, they are not adjacent by an edge or by part of an edge (i.e., they can be adjacent only by a vertex). For each vertex, there are exactly three triangles adjacent to that vertex. Find the maximum value of $n$. | [
"7"
] | [
"We make $n$ triangles with wire. For any two triangles, they are not adjacent by an edge or by part of an edge (i.e., they can be adjacent only by a vertex). For each vertex, there are exactly three triangles adjacent to that vertex. Find the maximum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f28d0700752b42b591b78bf0ced1773d0355e9c1c5c50170389b3d8b2907d8e | We measure the angles of inclination of a tower, emerging on level ground, at distances of 50 m and 100 m from its base, respectively. The two measures sum up to 45°. Determine the height of the tower. | [
"\\(25\\sqrt{17} - 75 \\approx 28.08 \\text{ m}\\)"
] | [
"We measure the angles of inclination of a tower, emerging on level ground, at distances of 50 m and 100 m from its base, respectively. The two measures sum up to 45°. Determine the height of the tower."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-dd96b16f77ed19550289916b680c8e396a465159cdaab9b5669e2cecd0922bb6 | We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack, at the $2$nd step t... | [
"10000"
] | [
"We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack, at the $2$nd s... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-5f0aa59cda39a985feb7fcd0e06c5f667ee0c69ea4d311a2367c3461bfc48059 | We paint every point of a circumference with one of ten colors. Let $N$ be the smallest number such that in any coloring of the circumference using ten colors, there will always exist an isosceles trapezium or a rectangle with all their vertices of the same color. Calculate $N$. | [
"22"
] | [
"We paint every point of a circumference with one of ten colors. Let $N$ be the smallest number such that in any coloring of the circumference using ten colors, there will always exist an isosceles trapezium or a rectangle with all their vertices of the same color. Calculate $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-6ee39f8db12697e7261431c40a7d6d26a0d6eb77d866903a53f9c2e3873138f1 | We paint the numbers $1, 2, 3, 4, 5$ with either red or blue. Let $S$ be the set of all triples $(x, y, z)$ such that $x + y = z$ and $x, y, z$ are all the same color. Calculate the minimum possible number of elements in $S$. | [
"1"
] | [
"We paint the numbers $1, 2, 3, 4, 5$ with either red or blue. Let $S$ be the set of all triples $(x, y, z)$ such that $x + y = z$ and $x, y, z$ are all the same color. Calculate the minimum possible number of elements in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-9655482ba17b70849f6f953f3252b33c2926aa2f87af51e03c7763ef74967324 | We perform 1000 independent trials, in each of which we simultaneously throw 100 symmetric dice. A trial series is considered successful if at most one die shows a 6. Find the approximate probability that exactly five out of six trial series are successful (Hint: Use one of the limit theorems). | [
"0"
] | [
"We perform 1000 independent trials, in each of which we simultaneously throw 100 symmetric dice. A trial series is considered successful if at most one die shows a 6. Find the approximate probability that exactly five out of six trial series are successful (Hint: Use one of the limit theorems)."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 30
} | 30 | 0 | 0 |
omr-ws-74066679ecdc2735c98cc285d71e485c5bb083fb9e7b4ac55d3cbc3e5709d000 | We pick a random point $(x, y, z)$ with integer coordinates from the cube $S = \{(x, y, z) : |x|, |y|, |z| \le 10\}$. What is the probability of the event that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{x + y + z}$? | [
"\\dfrac{380}{3087}"
] | [
"We pick a random point $(x, y, z)$ with integer coordinates from the cube $S = \\{(x, y, z) : |x|, |y|, |z| \\le 10\\}$. What is the probability of the event that $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = \\frac{1}{x + y + z}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-de996aa52d94a6da578ea6071bb5b1da5a1d3e0bb1b026cfbc622a89a739f43e | We pick three distinct digits \(a\), \(b\), and \(c\), all different from zero. We form the six possible three-digit numbers using \(a\), \(b\), and \(c\). The sum of these six numbers is 2886. Find all possible values of \(a\), \(b\), and \(c\). | [
"(1, 3, 9), (1, 4, 8), (1, 5, 7), (2, 3, 8), (2, 4, 7), (2, 5, 6), (3, 4, 6)"
] | [
"We pick three distinct digits \\(a\\), \\(b\\), and \\(c\\), all different from zero. We form the six possible three-digit numbers using \\(a\\), \\(b\\), and \\(c\\). The sum of these six numbers is 2886. Find all possible values of \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 30
} | 34 | 0 | 0 |
omr-ws-a675cc7a6d9d7f1469d7670a0816b340b859e1d5ebdcef119be63351546e1e4b | We place 2023 integers in a circle such that each pair of adjacent integers differ by 1 or 2. Define sets $A$ and $B$ where an integer belongs to $A$ if it is greater than both its neighbors and to $B$ if it is smaller than both its neighbors. Given that the sum of all elements in $A$ exceeds the sum of all elements in... | [
"1011"
] | [
"We place 2023 integers in a circle such that each pair of adjacent integers differ by 1 or 2. Define sets $A$ and $B$ where an integer belongs to $A$ if it is greater than both its neighbors and to $B$ if it is smaller than both its neighbors. Given that the sum of all elements in $A$ exceeds the sum of all elemen... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-ee5d6bbbde3d4ba8f8f2899d86ac8053d29c311b646aa6347655b2d97df522cd | We place 8 distinct integers in the vertices of a cube and write the greatest common divisor (GCD) of each pair of adjacent vertices on the edge connecting them. Let \( E \) be the sum of the numbers on the edges and \( V \) the sum of the numbers on the vertices. If it is possible for \( E = V \), find the smallest po... | [
"-1"
] | [
"We place 8 distinct integers in the vertices of a cube and write the greatest common divisor (GCD) of each pair of adjacent vertices on the edge connecting them. Let \\( E \\) be the sum of the numbers on the edges and \\( V \\) the sum of the numbers on the vertices. If it is possible for \\( E = V \\), find the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c8e528ef7f547cded0922238b381bd63767c1b1cb34f887ddaf935a6892e4766 | We place 8 distinct integers in the vertices of a cube and write the greatest common divisor (GCD) of each pair of adjacent vertices on the edge connecting them. Let \( E \) be the sum of the numbers on the edges and \( V \) the sum of the numbers on the vertices. If the integers placed on the vertices are \( 2, 3, 5, ... | [
"69"
] | [
"We place 8 distinct integers in the vertices of a cube and write the greatest common divisor (GCD) of each pair of adjacent vertices on the edge connecting them. Let \\( E \\) be the sum of the numbers on the edges and \\( V \\) the sum of the numbers on the vertices. If the integers placed on the vertices are \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f0f64678f456ebf835ab526aa14f4fcd6d2ee9e58327419e37e49fd2b8f0e6e | We place N cars on a 10 × 10 grid. Each car occupies a single cell and points up, down, left, or right. In a move, we may choose a car and shift it one cell forward to a vacant cell or remove it from the grid if it has reached the edge. Suppose that it is not possible to remove all of the cars from the grid. Find the s... | [
"2"
] | [
"We place N cars on a 10 × 10 grid. Each car occupies a single cell and points up, down, left, or right. In a move, we may choose a car and shift it one cell forward to a vacant cell or remove it from the grid if it has reached the edge. Suppose that it is not possible to remove all of the cars from the grid. Find ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-8236fa93f1a63900dca869cb56a46bd2de0cc869954799b95b94d12b23e8b0bb | We place \( 2n \) points in the unit square independently, according to a uniform distribution. These points are the vertices of a graph \( G_{2n} \). Two points are connected by an edge if the slope of the segment connecting them is nonnegative. Let \( f(n) \) be the probability that the graph \( G_{2n} \) has a 1-fac... | [
"1"
] | [
"We place \\( 2n \\) points in the unit square independently, according to a uniform distribution. These points are the vertices of a graph \\( G_{2n} \\). Two points are connected by an edge if the slope of the segment connecting them is nonnegative. Let \\( f(n) \\) be the probability that the graph \\( G_{2n} \\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bc66fdf3ac0b5fe1c14a536659ea31958e8797eea144d5d08b07a4c2c83ecc87 | We place red and blue marbles on a circle, starting with two red marbles. It is allowed to: i) add one red marble and change the color of each of its neighboring marbles (red to blue and vice versa); ii) remove one red marble and change the color of each of its neighboring marbles. After a finite number of operations, ... | [
"2"
] | [
"We place red and blue marbles on a circle, starting with two red marbles. It is allowed to:\ni) add one red marble and change the color of each of its neighboring marbles (red to blue and vice versa);\nii) remove one red marble and change the color of each of its neighboring marbles.\nAfter a finite number of oper... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-3d4ae75516b19b3df0ecc6cd4e86de566b58b9f81a831e61194420828fe1ec32 | We place the numbers $1, 2, \dots, 2006$ randomly around a circle. A move consists of swapping two neighboring numbers. After a series of moves, all numbers are diametrically opposite to their starting positions. Let $S$ be the sum of all numbers that were involved in a swap with another number such that their sum is $... | [
"0"
] | [
"We place the numbers $1, 2, \\dots, 2006$ randomly around a circle. A move consists of swapping two neighboring numbers. After a series of moves, all numbers are diametrically opposite to their starting positions. Let $S$ be the sum of all numbers that were involved in a swap with another number such that their su... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-bcd9f064ab65c8722dbed8dbf1008c55f3c9249d1b4dd4f254fda2fdeda530ba | We place the numbers from $1$ to $81$ in a $9 \times 9$ board. Let $P_k$ denote the product of the numbers in the $k$-th column and $Q_k$ denote the product of the numbers in the $k$-th row. Find the minimum possible value of $\sum_{k=1}^{9} |P_k - Q_k|$. | [
"0"
] | [
"We place the numbers from $1$ to $81$ in a $9 \\times 9$ board. Let $P_k$ denote the product of the numbers in the $k$-th column and $Q_k$ denote the product of the numbers in the $k$-th row. Find the minimum possible value of $\\sum_{k=1}^{9} |P_k - Q_k|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6d7eff49d53ae948f9510b41d8eb24f9664fc9fd1e7d132082fd8d2eb60e405 | We place three kings on a 3 × 3 grid such that no two kings occupy the same square. What is the probability that no two kings attack each other? (Two kings attack each other if the squares they occupy share at least one vertex). | [
"\\(\\frac{2}{21}\\)"
] | [
"We place three kings on a 3 × 3 grid such that no two kings occupy the same square. What is the probability that no two kings attack each other? (Two kings attack each other if the squares they occupy share at least one vertex)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-15f942bd7eabfe26842ad95a3a8559c670dfbc466050fe541f754a28c8039c24 | We play the following game with an equilateral triangle of $\frac{n(n+1)}{2}$ coins (with $n > 1$ coins on each side). Initially, all of the coins are turned heads up. On each turn, we may turn over three coins that are mutually adjacent; the goal is to make all of the coins eventually turned tails up. What is the $7^{... | [
"11"
] | [
"We play the following game with an equilateral triangle of $\\frac{n(n+1)}{2}$ coins (with $n > 1$ coins on each side). Initially, all of the coins are turned heads up. On each turn, we may turn over three coins that are mutually adjacent; the goal is to make all of the coins eventually turned tails up. What is th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-391376d23169e7efd316e876401626836210778a8f27137a5129a7b9c9c504c0 | We possess $5$ white marbles and $10$ black marbles. How many arrangements can we create when we place them in a sequence from left to right, ensuring that there is at least one black marble positioned immediately after every white one? | [
"252"
] | [
"We possess $5$ white marbles and $10$ black marbles. How many arrangements can we create when we place them in a sequence from left to right, ensuring that there is at least one black marble positioned immediately after every white one?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-9d19ffab597590f67eabad4aa2bccc4d843a2cdf54976cfe2a9c06686bcdd0d7 | We put 0 or 1 in every cell of an \( m \times n \) chessboard. If the number of 0s and the number of 1s is equal, we call this an "equal putting." Let \( a \) be a real number. If we can find \( m \) and \( n \) such that in each of the \( m \) rows and \( n \) columns, the percentage of 1s is not less than \( a \) and... | [
"50"
] | [
"We put 0 or 1 in every cell of an \\( m \\times n \\) chessboard. If the number of 0s and the number of 1s is equal, we call this an \"equal putting.\" Let \\( a \\) be a real number. If we can find \\( m \\) and \\( n \\) such that in each of the \\( m \\) rows and \\( n \\) columns, the percentage of 1s is not l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-ff8df2c06305400c4dd7a80c4da02a6e416e1af76d119067b44b8e329676d151 | We put the numbers $1, 2, \ldots, 3m$ evenly into 3 drawers (so that each drawer contains $m$ numbers). For a given $m$, let $S(m)$ be the number of ways to pick one number from each drawer such that one of them is the sum of the rest two. Find $S(5)$. | [
"15"
] | [
"We put the numbers $1, 2, \\ldots, 3m$ evenly into 3 drawers (so that each drawer contains $m$ numbers). For a given $m$, let $S(m)$ be the number of ways to pick one number from each drawer such that one of them is the sum of the rest two. Find $S(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-7a6832ef39642bdeff64c6744e9ebc01955759b9ecf97364399b8f2cf8b702a1 | We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number? | [
"606"
] | [
"We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 16
} | 21 | 0 | 0 |
omr-ws-5b1cdee2fe3883d0c230413e4f79efc3bf37d01b3686564784d5549ab25f486e | We randomly place numbers from the set \( S = \{1, 2, 3, \ldots, n^2\} \) in each box of an \( n \times n \) board. Find the best constant \( \alpha \) such that in every random placement, there will exist a \( 3 \times 3 \) square whose sum of numbers is greater than \( \alpha \). | [
"\\frac{9(n^2 + 1)}{2}"
] | [
"We randomly place numbers from the set \\( S = \\{1, 2, 3, \\ldots, n^2\\} \\) in each box of an \\( n \\times n \\) board. Find the best constant \\( \\alpha \\) such that in every random placement, there will exist a \\( 3 \\times 3 \\) square whose sum of numbers is greater than \\( \\alpha \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d17e57b89dee50e015639928cb6293d0721d74d42ae47eba7e82258392a10930 | We randomly place numbers from the set \( S = \{1, 2, 3, \ldots, n^2\} \) in each box of an \( n \times n \) board. Find the best constant \(\alpha\) such that in every random placement, there will exist a \(3 \times 3\) square whose sum of numbers is greater than \(\alpha\). | [
"\\dfrac{9n^4 - 48n^3 + 121n^2 - 216n + 264}{2(n - 2)^2}"
] | [
"We randomly place numbers from the set \\( S = \\{1, 2, 3, \\ldots, n^2\\} \\) in each box of an \\( n \\times n \\) board. Find the best constant \\(\\alpha\\) such that in every random placement, there will exist a \\(3 \\times 3\\) square whose sum of numbers is greater than \\(\\alpha\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8434a771daa58185a4d1de349408252b51558e1b4d9a18751c41ae58daaf19db | We randomly place numbers from the set \( S = \{1, 2, 3, \ldots, n^2\} \) into each box of an \( n \times n \) board. Find the best constant \( \alpha \) such that in every random placement, there will exist a \( 3 \times 3 \) square whose sum of numbers is greater than \( \alpha \). | [
"\\frac{9(n^2 + 1)}{2}"
] | [
"We randomly place numbers from the set \\( S = \\{1, 2, 3, \\ldots, n^2\\} \\) into each box of an \\( n \\times n \\) board. Find the best constant \\( \\alpha \\) such that in every random placement, there will exist a \\( 3 \\times 3 \\) square whose sum of numbers is greater than \\( \\alpha \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2ca728d9946443e5be235f3de2cecaa741026bac8811c934c973290249142ec6 | We repeatedly roll a tetrahedral die with sides $1, 2, 3, 4$ until the sum of the last two rolls is prime. What is the probability that the last number rolled is a $1$? | [
"\\(\\frac{15}{44}\\)"
] | [
"We repeatedly roll a tetrahedral die with sides $1, 2, 3, 4$ until the sum of the last two rolls is prime. What is the probability that the last number rolled is a $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-1c7dd1b20d4846dd29e7a755438218c7657e74551605bead26f1a07b67a43542 | We roll $6n$ dice. Let $p_n$ be the probability of having at least $n$ times the number 6. Calculate the value of $p_3 - p_4$. | [
"0.0124"
] | [
"We roll $6n$ dice. Let $p_n$ be the probability of having at least $n$ times the number 6. Calculate the value of $p_3 - p_4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-f0358dcc479250910b24bef31ad98a2fadd30c3021435f5a0d0830d62f671c63 | We roll a dice continuously until the first time that all numbers $1, 2, \dots, 6$ appear. Find the expected value of the numbers that appear exactly one time. | [
"\\dfrac{7}{2}"
] | [
"We roll a dice continuously until the first time that all numbers $1, 2, \\dots, 6$ appear. Find the expected value of the numbers that appear exactly one time."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-bd5a1ec53d2a7c1699cb1afd3d1b963ae12f92cdf14cd50aedd46505175b3ffe | We roll a dice until we get a $6$ two times. Let $Y$ be the number of all rolls and $X$ be the number of "$1$". Find $E(Y - X|X = 4)$. | [
"10"
] | [
"We roll a dice until we get a $6$ two times. Let $Y$ be the number of all rolls and $X$ be the number of \"$1$\". Find $E(Y - X|X = 4)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-a74cbfe648da6dbd119dd5f3f859cd50ba71a2563ad65d1341b9aa96c748cc3c | We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls? | [
"\\dfrac{5}{32}"
] | [
"We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6c2f0085db866ef3460b18da7f0cd91b5bf50cdee85093084e40bbda08e76a0 | We roll two fair dice to get \( x \) and \( y \), respectively. What is the probability that \( x \), \( y \), and \( 3 \) can form a triangle? | [
"\\(\\frac{7}{12}\\)"
] | [
"We roll two fair dice to get \\( x \\) and \\( y \\), respectively. What is the probability that \\( x \\), \\( y \\), and \\( 3 \\) can form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 61
} | 92 | 0 | 0 |
omr-ws-e00c9468cf663292cf8905dce7f563026eb89c14b753d381dd6aa80eba0fc031 | We say a sequence of natural numbers $(a_1, \ldots, a_k)$ is "cyclic" if the last digit of $a_i$ is the first digit of $a_{i+1}$ for each $i = 1, 2, \ldots, k - 1$, and the last digit of $a_k$ is the first digit of $a_1$. For instance, $(8118)$, $(23, 32)$, and $(12, 24, 48, 801)$ are cyclic. At least how many integers... | [
"450"
] | [
"We say a sequence of natural numbers $(a_1, \\ldots, a_k)$ is \"cyclic\" if the last digit of $a_i$ is the first digit of $a_{i+1}$ for each $i = 1, 2, \\ldots, k - 1$, and the last digit of $a_k$ is the first digit of $a_1$. For instance, $(8118)$, $(23, 32)$, and $(12, 24, 48, 801)$ are cyclic. At least how many... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f2a881aa077b1ce8ad63bd1bb0311df3d211afeebdcf0da6f49341d7a43d5f1b | We say that $k$ is the distance between numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ if $a_k \neq b_k$. All numbers from $44444$ to $60000$ are written in a random order. Find the minimum value of the sum of distances between neighboring numbers. | [
"76050"
] | [
"We say that $k$ is the distance between numbers $\\overline{a_1a_2a_3a_4a_5}$ and $\\overline{b_1b_2b_3b_4b_5}$ if $a_k \\neq b_k$. All numbers from $44444$ to $60000$ are written in a random order. Find the minimum value of the sum of distances between neighboring numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-51e1dcd63be1930d0321094116776b4d55903e51d4ba9e2ebabe53be5845f45a | We say that a number is ascending if its digits are, from left-to-right, in nondecreasing order. We say that a number is descending if the digits are, from left-to-right, in nonincreasing order. Let $a_n$ be the number of $n$-digit positive integers which are ascending, and $b_n$ be the number of $n$-digit positive int... | [
"\\left( \\dfrac{1}{9}, 1 \\right)"
] | [
"We say that a number is ascending if its digits are, from left-to-right, in nondecreasing order. We say that a number is descending if the digits are, from left-to-right, in nonincreasing order. Let $a_n$ be the number of $n$-digit positive integers which are ascending, and $b_n$ be the number of $n$-digit positiv... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b40f8497f2287c008419211ea783450e0ba390faa6f7ee124a5c53d076f0d19d | We say that a permutation $\sigma$ of the set $\{1, 2, \ldots, n\}$ preserves divisibility if $\sigma (a)$ divides $\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\{1, 2, \ldots, 40\}$ preserve divisibility? | [
"96"
] | [
"We say that a permutation $\\sigma$ of the set $\\{1, 2, \\ldots, n\\}$ preserves divisibility if $\\sigma (a)$ divides $\\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\\{1, 2, \\ldots, 40\\}$ preserve divisibility?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-94bddc5f73bf934b43b124b18a858e60c98c5ba01c2d30cef94e6f99abef5006 | We say that a quadruple of nonnegative real numbers \((a, b, c, d)\) is balanced if \[ a + b + c + d = a^2 + b^2 + c^2 + d^2. \] Find all positive real numbers \( x \) such that \[ (x - a)(x - b)(x - c)(x - d) \geq 0 \] for every balanced quadruple \((a, b, c, d)\). | [
"\\( x \\geq \\frac{3}{2} \\)"
] | [
"We say that a quadruple of nonnegative real numbers \\((a, b, c, d)\\) is balanced if \n\\[ a + b + c + d = a^2 + b^2 + c^2 + d^2. \\]\nFind all positive real numbers \\( x \\) such that \n\\[ (x - a)(x - b)(x - c)(x - d) \\geq 0 \\]\nfor every balanced quadruple \\((a, b, c, d)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 10
} | 19 | 0 | 0 |
omr-ws-db23dd641afede45999ce687334a374bb18a33f38481c6979b06837ed8e7b47b | We say that a real number $V$ is good if there exist two closed convex subsets $X$ and $Y$ of the unit cube in $\mathbb{R}^3$, each with volume $V$, such that for each of the three coordinate planes, the projections of $X$ and $Y$ onto that plane are disjoint. Find $\sup \{V \mid V \text{ is good}\}$. | [
"\\dfrac{1}{4}"
] | [
"We say that a real number $V$ is good if there exist two closed convex subsets $X$ and $Y$ of the unit cube in $\\mathbb{R}^3$, each with volume $V$, such that for each of the three coordinate planes, the projections of $X$ and $Y$ onto that plane are disjoint. Find $\\sup \\{V \\mid V \\text{ is good}\\}$."
] | [
"aops_c7_college_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-de8474e20baa20fc941971dc34f6fda2249dbaa03ca7847837d429705513139a | We say that the positive integer $q$ is the $k$-successor of the positive integer $n$ if there exists a positive integer $p$ such that $n + p^2 = q^2$. Let $A$ be the set of all positive integers $n$ such that they have at least one $k$-successor but their $k$-successors do not have $k$-successors. Define the set $A$. | [
"A = \\{ p \\text{ prime} \\mid p \\equiv 3 \\pmod{4} \\} \\cup \\{ 4p \\mid p \\text{ prime} \\}"
] | [
"We say that the positive integer $q$ is the $k$-successor of the positive integer $n$ if there exists a positive integer $p$ such that $n + p^2 = q^2$. Let $A$ be the set of all positive integers $n$ such that they have at least one $k$-successor but their $k$-successors do not have $k$-successors. Define the set ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5a1f85a5dd5ba8807100d45de764aaf637eb2023e07abefeae0cf70a19d0f0b | We say that the sets $A$, $B$, and $C$ form a "sunflower" if $A \cap B = A \cap C = B \cap C$. ($A \cap B$ denotes the intersection of the sets $A$ and $B$.) If $A$, $B$, and $C$ are independently randomly chosen $4$-element subsets of the set $\{1, 2, 3, 4, 5, 6\}$, what is the probability that $A$, $B$, and $C$ form ... | [
"\\(\\frac{1}{25}\\)"
] | [
"We say that the sets $A$, $B$, and $C$ form a \"sunflower\" if $A \\cap B = A \\cap C = B \\cap C$. ($A \\cap B$ denotes the intersection of the sets $A$ and $B$.) If $A$, $B$, and $C$ are independently randomly chosen $4$-element subsets of the set $\\{1, 2, 3, 4, 5, 6\\}$, what is the probability that $A$, $B$, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-aba012d031a91dd69237902dc7dac94f2eca44595591b1d40ae2986b5121837b | We say that two lattice points are neighboring if the distance between them is $1$. We say that a point lies at distance $d$ from a line segment if $d$ is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point $A$ is nearby a line segment if the distance between $... | [
"5"
] | [
"We say that two lattice points are neighboring if the distance between them is $1$. We say that a point lies at distance $d$ from a line segment if $d$ is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point $A$ is nearby a line segment if the distance betw... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-96da657dd49aee1ad82448e9f3ba0c4c1f376935d1e8ed8246c3efc582d1ee24 | We say that we extend a finite sequence of positive integers $(a_1, \dotsc, a_n)$ if we replace it by \[ (1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n) \] i.e., each element $k$ of the original sequence is replaced by $1,2,\dotsc,k$. Géza takes the sequence $(1,2,\dotsc,9)$... | [
"\\(\\frac{2018}{2026}\\)"
] | [
"We say that we extend a finite sequence of positive integers $(a_1, \\dotsc, a_n)$ if we replace it by\n\\[\n(1,2,\\dotsc,a_1-1,a_1,1,2,\\dotsc,a_2-1,a_2,1,2,\\dotsc,a_3-1,a_3,\\dotsc,1,2,\\dotsc,a_n-1,a_n)\n\\]\ni.e., each element $k$ of the original sequence is replaced by $1,2,\\dotsc,k$. Géza takes the sequenc... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 8
} | 10 | 0 | 0 |
omr-ws-317e461290b0da8f2467ba18a390af829a1032b458c6a9bb2111bc975279dcda | We select 2024 real numbers, all lying between 1 and 2 (inclusive), such that their sum is 3000. Compute the maximum possible value of the sum of their squares. | [
"4952"
] | [
"We select 2024 real numbers, all lying between 1 and 2 (inclusive), such that their sum is 3000. Compute the maximum possible value of the sum of their squares."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-afeb4c7f737e1723f8948c1c3885d35f9c67066fab13f92a1d439d1ecf75553d | We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \frac{m+\alpha}{n} \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$. | [
"1000"
] | [
"We select a real number $\\alpha$ uniformly and at random from the interval $(0,500)$. Define \\[ S = \\frac{1}{\\alpha} \\sum_{m=1}^{1000} \\sum_{n=m}^{1000} \\frac{m+\\alpha}{n} \\] Let $p$ denote the probability that $S \\ge 1200$. Compute $1000p$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5858cc569887fb0052c52e948b90f1ec51a84fe4e7e5826b67bd5a588bafcce | We select two random nonnegative real numbers \( x \) and \( y \) such that \( y < 2 \) and \( x < 4 \). What is the probability that \( y \) is greater than or equal to the square root of \( x \)? | [
"\\(\\frac{1}{3}\\)"
] | [
"We select two random nonnegative real numbers \\( x \\) and \\( y \\) such that \\( y < 2 \\) and \\( x < 4 \\). What is the probability that \\( y \\) is greater than or equal to the square root of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-962f4a8451bb6570651480a3583a12f27a754ba593cbde23ac57c4cd36e567bc | We split the numbers in the set $\{1, 2, \dots, 2017\}$ into 6 pairwise disjoint sets. Let $S$ be the smallest possible value of $c$ such that there exist three numbers $a, b, c$ (not necessarily distinct) in the same set satisfying $a + b = c$. Find $S$. | [
"366"
] | [
"We split the numbers in the set $\\{1, 2, \\dots, 2017\\}$ into 6 pairwise disjoint sets. Let $S$ be the smallest possible value of $c$ such that there exist three numbers $a, b, c$ (not necessarily distinct) in the same set satisfying $a + b = c$. Find $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-91797e1ec25ec211ef07c36335a5f1b388ac27de211f5e6272dfc223e4a553fe | We stack cubes with side lengths 1, 2, 3, 4, 5, 6, and 7 on top of each other. What is the surface area of the resulting figure? | [
"658"
] | [
"We stack cubes with side lengths 1, 2, 3, 4, 5, 6, and 7 on top of each other. What is the surface area of the resulting figure?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 35,
"tir": 5
} | 40 | 0 | 0 |
omr-ws-bd6ef172ee22c792aa2f8c195770a663caaff1a8775709deb77c2c8ef7bcfb12 | We start a game with all the integers from $2$ to $2015$ written on a blackboard, in this order. A "change" consists of selecting two numbers, say $a$ and $b$, and replacing them with $\frac{4a+3b}{5}$ and $\frac{24a-7b}{25}$, respectively. After performing a series of such "changes," we aim to obtain all the integers ... | [
"0"
] | [
"We start a game with all the integers from $2$ to $2015$ written on a blackboard, in this order. A \"change\" consists of selecting two numbers, say $a$ and $b$, and replacing them with $\\frac{4a+3b}{5}$ and $\\frac{24a-7b}{25}$, respectively. After performing a series of such \"changes,\" we aim to obtain all th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
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