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-8693d2c2b9b43efb724f7951935e889537f7cdb5c78622b3d098764d11cc5581 | A "white and nerdy" kid's 27 friends are to be ranked in his top 8 Myspace list, with the condition that Caelestor comes after gfour84. Let \( x \) be the number of possible combinations. What is the sum of the digits of \( x \)? | [
"45"
] | [
"A \"white and nerdy\" kid's 27 friends are to be ranked in his top 8 Myspace list, with the condition that Caelestor comes after gfour84. Let \\( x \\) be the number of possible combinations. What is the sum of the digits of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-b786d888bfffbe8f1e47cb945226a892f29fd83b3c640f02dffd4cb359d52927 | A $(2^n - 1) \times (2^n + 1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into. | [
"\\(2n\\)"
] | [
"A $(2^n - 1) \\times (2^n + 1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1652e5dc5677b509d439aa3f2f4bae1dc463aded41fc19feb883573711516f6a | A $1 \times 1$ piece is taken from the corner of a $5 \times 5$ grid. We use $k$ L-trominoes and $(8-k)$ $1 \times 3$ trominoes to tile the grid. Find the number $k$. | [
"4"
] | [
"A $1 \\times 1$ piece is taken from the corner of a $5 \\times 5$ grid. We use $k$ L-trominoes and $(8-k)$ $1 \\times 3$ trominoes to tile the grid. Find the number $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-3b68edcdbed9d69d6202d07dc4ae039c656753af2b314aaef92cfc475c7c8643 | A $1 \times 2$ rectangle is inscribed in a semicircle with its longer side on the diameter. What is the area of the semicircle? | [
"\\(\\pi\\)"
] | [
"A $1 \\times 2$ rectangle is inscribed in a semicircle with its longer side on the diameter. What is the area of the semicircle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-d5b34dcc4d3a2c7d95dc2c0c18d17a4771b9c484dc8df1f8227e3ec0ae11eba9 | A $1 \times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in increasing order. In each step, three consecutive tiles are selected, and one of the following operations is performed: - (i) The number in the middle is increased by $2$ and its neighbors are decreased by $1$. - (ii) The number in the middle is decr... | [
"678057"
] | [
"A $1 \\times 2019$ board is filled with numbers $1, 2, \\dots, 2019$ in increasing order. In each step, three consecutive tiles are selected, and one of the following operations is performed:\n- (i) The number in the middle is increased by $2$ and its neighbors are decreased by $1$.\n- (ii) The number in the middl... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3f23f9de179bf41135adc10874607a0a92bad96d4a63ca6b3698f390afb2ade6 | A $1 \times 5n$ rectangle is partitioned into tiles, where each tile is either a $1 \times 1$ square or a broken domino consisting of two $1 \times 1$ squares separated by four squares. Let $P(n)$ denote the number of such partitions of a $1 \times 5n$ rectangle. Find the value of $P(2)$. | [
"32"
] | [
"A $1 \\times 5n$ rectangle is partitioned into tiles, where each tile is either a $1 \\times 1$ square or a broken domino consisting of two $1 \\times 1$ squares separated by four squares. Let $P(n)$ denote the number of such partitions of a $1 \\times 5n$ rectangle. Find the value of $P(2)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-7e4122867bf0b4be7d131838954d6b4803894c9f9e6ed200e3444a086cd4cdc2 | A $1 \times n$ rectangle ($n \geq 1$) is divided into $n$ unit ($1 \times 1$) squares. Each square is colored red, blue, or green. Let $f(n)$ be the number of colorings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$? (The number of red squares can be z... | [
"37"
] | [
"A $1 \\times n$ rectangle ($n \\geq 1$) is divided into $n$ unit ($1 \\times 1$) squares. Each square is colored red, blue, or green. Let $f(n)$ be the number of colorings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$? (The number of red squares ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-449cc370f0f74aefbda7466e4b2bfc5eba4af85667476b421129590e7f584b68 | A $1$ and nine $0$s are written on the board. We may replace any two numbers each by their average. What is the smallest number that can eventually replace the $1$? | [
"\\(\\frac{1}{512}\\)"
] | [
"A $1$ and nine $0$s are written on the board. We may replace any two numbers each by their average. What is the smallest number that can eventually replace the $1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-3874956e1ce8a6ceb3dce2dce5613ef139f571450f981b9783fad3a49f777ed5 | A $10 \times 10 \times 10$ cube is formed of small unit cubes. A grasshopper sits in the center $O$ of one of the corner cubes. At any given moment, it can jump to the center of any adjacent cube that shares a face with its current position, as long as the jump increases the distance between point $O$ and the current p... | [
"$\\frac{30!}{10!10!10!}$"
] | [
"A $10 \\times 10 \\times 10$ cube is formed of small unit cubes. A grasshopper sits in the center $O$ of one of the corner cubes. At any given moment, it can jump to the center of any adjacent cube that shares a face with its current position, as long as the jump increases the distance between point $O$ and the cu... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 30
} | 36 | 0 | 0 |
omr-ws-24e85fc5c623bc1866ae336c19337543482311e3b4ab13a058f2e5cc09434977 | A $10 \times 10 \times 10$ cube is made up of $500$ white unit cubes and $500$ black unit cubes, with every two adjacent unit cubes (sharing a face) being of different colors. After removing $100$ unit cubes such that $300$ lines (each a $1 \times 1 \times 10$ portion parallel to one of the cube’s edges) have exactly o... | [
"Yes, it is possible that the number of removed black unit cubes is divisible by 4."
] | [
"A $10 \\times 10 \\times 10$ cube is made up of $500$ white unit cubes and $500$ black unit cubes, with every two adjacent unit cubes (sharing a face) being of different colors. After removing $100$ unit cubes such that $300$ lines (each a $1 \\times 1 \\times 10$ portion parallel to one of the cube’s edges) have ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-45db041ef45b494488944fda2deb76b224781f06381f60cb7700423dc828ac45 | A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (none of these segments belongs to the boundary of the square). If $P$ is the perimeter of one of these polygons, find the sum of all possible values of $P$. | [
"10"
] | [
"A $10 \\times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (none of these segments belongs to the boundary of the square). If $P$ is the perimeter of one of these polygons, find the sum of all possible values of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e82b566bceaa8b871718f27f96c0ca52df1bbb0f45273b23d14d1aacebf7ded2 | A $10 \times 10$ square table is divided into $1 \times 1$ squares. Each $1 \times 1$ square contains one of the numbers $\{1, 2, 3, \ldots, 9, 10\}$. Numbers in any two adjacent or diagonally adjacent squares are relatively prime. Let $N$ be the minimum number of times a single number must appear in the table under th... | [
"25"
] | [
"A $10 \\times 10$ square table is divided into $1 \\times 1$ squares. Each $1 \\times 1$ square contains one of the numbers $\\{1, 2, 3, \\ldots, 9, 10\\}$. Numbers in any two adjacent or diagonally adjacent squares are relatively prime. Let $N$ be the minimum number of times a single number must appear in the tab... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c969b9d1035fb3af601605d630c62784d298f4ba9d0f42c52793d8e3a9ecb7dd | A $10 \times 10$ square table is divided into $1 \times 1$ squares. In each $1 \times 1$ square, one of the numbers $\{1, 2, 3, \ldots, 9, 10\}$ is written such that numbers from any two adjacent or diagonally adjacent squares are relatively prime. Let $f(n)$ be the minimum number of times the number $n$ appears in suc... | [
"100"
] | [
"A $10 \\times 10$ square table is divided into $1 \\times 1$ squares. In each $1 \\times 1$ square, one of the numbers $\\{1, 2, 3, \\ldots, 9, 10\\}$ is written such that numbers from any two adjacent or diagonally adjacent squares are relatively prime. Let $f(n)$ be the minimum number of times the number $n$ app... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-982b5ad45f8d69c62fb5e0c15b6be1170ea38d2b5b057381ba77ec534a028ec7 | A $10 \times 10$ square table is divided into unit squares and colored such that there are 50 black and 50 white unit squares. A grid line is called a "border" if it is shared by squares of different colors. Determine the maximum and minimum number of "borders" that can be found in such a coloring. | [
"Maximum = 180, Minimum = 10"
] | [
"A $10 \\times 10$ square table is divided into unit squares and colored such that there are 50 black and 50 white unit squares. A grid line is called a \"border\" if it is shared by squares of different colors. Determine the maximum and minimum number of \"borders\" that can be found in such a coloring."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-9d9d89772dc0d06ad9c0fc36235b66105df57fa54d5c70de2676250990ce6015 | A $10 \times 10$ table consists of $100$ unit cells. A block is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e., each cell of the table is covered by some block of $C$) but no $n-1$ blocks of $C$ cover the table. Find the largest possible value of $n$. | [
"39"
] | [
"A $10 \\times 10$ table consists of $100$ unit cells. A block is a $2 \\times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e., each cell of the table is covered by some block of $C$) but no $n-1$ blocks of $C$ cover the table. Find the largest possible value of $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5bc30740f31cbcdcd9321e8ca89f5c79de4666b4d572887e3229c1c6fcedfe2 | A $10 \times 10$ table has been covered with 39 $2 \times 2$ tiles. Calculate the minimum number of $2 \times 2$ tiles that can be removed such that the remaining tiles still cover the table without any gaps or overlaps. | [
"14"
] | [
"A $10 \\times 10$ table has been covered with 39 $2 \\times 2$ tiles. Calculate the minimum number of $2 \\times 2$ tiles that can be removed such that the remaining tiles still cover the table without any gaps or overlaps."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d577769cceb01477d1db4f3e89da67b29b549b5675a824cb5553cf499bb2b1b | A $10 \times 12$ paper rectangle is folded along the grid lines several times, forming a thick $1 \times 1$ square. How many pieces of paper can one possibly get by cutting this square along the segment connecting the midpoints of a pair of adjacent sides? | [
"2"
] | [
"A $10 \\times 12$ paper rectangle is folded along the grid lines several times, forming a thick $1 \\times 1$ square. How many pieces of paper can one possibly get by cutting this square along the segment connecting the midpoints of a pair of adjacent sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25339a89f847e5586a3e502f8fa0901488acdac4d9ccc0f11ef942c96d21b773 | A $10 \times 12$ paper rectangle is folded along the grid lines several times, forming a thick $1 \times 1$ square. How many pieces of paper can one possibly get by cutting this square along the segment connecting the midpoints of a pair of opposite sides? | [
"240"
] | [
"A $10 \\times 12$ paper rectangle is folded along the grid lines several times, forming a thick $1 \\times 1$ square. How many pieces of paper can one possibly get by cutting this square along the segment connecting the midpoints of a pair of opposite sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 7
} | 24 | 0 | 0 |
omr-ws-9708c2a9bcef1b58caff04c5c93c535ad810e6d4bb409a767872f7c963441487 | A $10 \times 14$ rectangular grid is divided into 140 unit squares and painted in a red and white chessboard pattern. We place either $0$ or $1$ in each square such that each row and each column contains an odd number of $1$s. Calculate the remainder when the number of $1$s contained in the red-painted squares is divid... | [
"0"
] | [
"A $10 \\times 14$ rectangular grid is divided into 140 unit squares and painted in a red and white chessboard pattern. We place either $0$ or $1$ in each square such that each row and each column contains an odd number of $1$s. Calculate the remainder when the number of $1$s contained in the red-painted squares is... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8544eeac30e788a695df4a2f25f7dab5d551a653afaa1061cd6384a8386688d | A $10$-digit positive integer $\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called deceptive if there exist distinct indices $i > j$ such that $\overline{a_i a_j} = 37$. Find the number of deceptive positive integers. | [
"2413851687"
] | [
"A $10$-digit positive integer $\\overline{a_9a_8a_7...a_1a_0}$ with $a_9$ nonzero is called deceptive if there exist distinct indices $i > j$ such that $\\overline{a_i a_j} = 37$. Find the number of deceptive positive integers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-1b562bc3cfe494fd94fe43596d9d3dd92f617356e45b81955801325d8bf8c32c | A $100 \times 100 \times 100$ cube is divided into a million unit cubes, each containing a light bulb. Three faces $100 \times 100$ of the large cube, sharing a common vertex, are painted red, blue, and green. Each of the $30,000$ columns (each column being a $1 \times 1 \times 100$ block) has a switch at one of its pa... | [
"10"
] | [
"A $100 \\times 100 \\times 100$ cube is divided into a million unit cubes, each containing a light bulb. Three faces $100 \\times 100$ of the large cube, sharing a common vertex, are painted red, blue, and green. Each of the $30,000$ columns (each column being a $1 \\times 1 \\times 100$ block) has a switch at one... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d443d2ff85b1666ce9c58159bb3c8af38be52f20b81af214b997c4bc03d8fe3 | A $100 \times 100$ board is divided into unit squares and colored with 4 colors such that every row and every column has exactly 25 squares of each color. Let $N$ be the number of ways to choose 2 rows and 2 columns such that their 4 intersections are painted with different colors. Calculate the remainder when $N$ is d... | [
"200"
] | [
"A $100 \\times 100$ board is divided into unit squares and colored with 4 colors such that every row and every column has exactly 25 squares of each color. Let $N$ be the number of ways to choose 2 rows and 2 columns such that their 4 intersections are painted with different colors. Calculate the remainder when $N... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-847f8636800f06492404d3a29d7ee358abc39abd82b2eefdc464a2973b95c3e2 | A $100 \times 100$ chessboard is given with each square filled with either $-1$ or $1$. The product of all numbers in each row and in each column must equal $-1$. Determine the number of ways to fill this chessboard. | [
"2^{9801}",
"\\(2^{99 \\cdot 99}\\)"
] | [
"A $100 \\times 100$ chessboard is given with each square filled with either $-1$ or $1$. The product of all numbers in each row and in each column must equal $-1$. Determine the number of ways to fill this chessboard."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-ff9e56cef2f1ad64157978a7bce9a3afd7785dcfa005d2f645b4a5c928eea8b3 | A $100 \times 100$ closed square napkin is thrown on a plane and covers some lattice points (points with integral coordinates). Find the minimum value of $k$ such that for every position of the napkin, there are $k$ lines that contain all these lattice points. | [
"2"
] | [
"A $100 \\times 100$ closed square napkin is thrown on a plane and covers some lattice points (points with integral coordinates). Find the minimum value of $k$ such that for every position of the napkin, there are $k$ lines that contain all these lattice points."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-503a533ccba44748419562c9a59c8cbd46b042cefca602632661d93754e66544 | A $100 \times 100$ square floor consisting of $10000$ squares has a $2 \times 2$ square removed from its center. Determine the number of ways the remaining floor can be completely tiled using $1 \times 3$ rectangular tiles. | [
"0"
] | [
"A $100 \\times 100$ square floor consisting of $10000$ squares has a $2 \\times 2$ square removed from its center. Determine the number of ways the remaining floor can be completely tiled using $1 \\times 3$ rectangular tiles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2de31cbd15836b12eb399acaec9ac773c08071b04ae08671708c41f5d9266e2e | A $100 \times 100$ square floor consisting of $10000$ squares has a $2 \times 2$ square removed from one of its corners. If the remaining floor is to be tiled using $1 \times 3$ rectangular tiles, calculate the remainder when the total number of squares that cannot be tiled is divided by $4$. | [
"3"
] | [
"A $100 \\times 100$ square floor consisting of $10000$ squares has a $2 \\times 2$ square removed from one of its corners. If the remaining floor is to be tiled using $1 \\times 3$ rectangular tiles, calculate the remainder when the total number of squares that cannot be tiled is divided by $4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32b572f575b97027cb0e50f09be01cb447bf6e59cd71e053ac913d7466a44f04 | A $100 \times 100$ square is divided into $9$ rectangles by two vertical and two horizontal lines. The inner rectangle has dimensions $45 \times 30$ centimeters, and the sides of the remaining rectangles are not integers. Find the sum of the areas of the four corner rectangles. | [
"3550"
] | [
"A $100 \\times 100$ square is divided into $9$ rectangles by two vertical and two horizontal lines. The inner rectangle has dimensions $45 \\times 30$ centimeters, and the sides of the remaining rectangles are not integers. Find the sum of the areas of the four corner rectangles."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-242a075d630d7e9ef40e7ad55ebd0944e997e028355d81f3ca72076d99748a33 | A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. After permuting the numbers in the table, let $S$ be the set of all possible sums of the rows and columns. Find the minimum possible value of $|S|$. | [
"100"
] | [
"A $100 \\times 100$-table is filled with numbers such that the sums in all the columns are different. After permuting the numbers in the table, let $S$ be the set of all possible sums of the rows and columns. Find the minimum possible value of $|S|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-0ee91ec92e3a447a052774a6a7f8277ef6f4f74d3b4a2877eff1edb6135817d2 | A $100 \times 200$ board has $k$ black cells. An operation consists of choosing a $2 \times 3$ or $3 \times 2$ sub-board having exactly 5 black cells and painting the remaining cell black. Find the least value of $k$ for which there exists an initial distribution of the black cells such that after some operations the b... | [
"5000"
] | [
"A $100 \\times 200$ board has $k$ black cells. An operation consists of choosing a $2 \\times 3$ or $3 \\times 2$ sub-board having exactly 5 black cells and painting the remaining cell black. Find the least value of $k$ for which there exists an initial distribution of the black cells such that after some operatio... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-c88f64b34b68c6346c0da6c34dada9590f93836923bc2668e2d0ef4cdcc30e9d | A $100$-gallon tank is initially full of pure water. Salt water with a concentration of $\frac{1}{2}$ lbs. of salt per gallon is added at a rate of $2$ gallons per minute, while the water in the tank is drained at the same rate, keeping the volume constant. After 10 minutes, the salt water is switched to fresh water. U... | [
"7.42 lbs of salt"
] | [
"A $100$-gallon tank is initially full of pure water. Salt water with a concentration of $\\frac{1}{2}$ lbs. of salt per gallon is added at a rate of $2$ gallons per minute, while the water in the tank is drained at the same rate, keeping the volume constant. After 10 minutes, the salt water is switched to fresh wa... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 24
} | 25 | 0 | 0 |
omr-ws-27fff2b2c7a5045d82c4a9b02def4befce2711ad8da80289264ea20e7071bc8f | A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3... | [
"2009"
] | [
"A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertice... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4077504f2beacbdf4b5d94a26824de2267c1e22d376c422537d4954ac45bb15 | A $100$-gon is drawn on the plane, not necessarily convex. $98$ diagonals that do not intersect at interior points divide it into triangles. What is the largest number of drawn segments (sides and diagonals of a $100$-gon) can intersect interior points at the same line? | [
"99"
] | [
"A $100$-gon is drawn on the plane, not necessarily convex. $98$ diagonals that do not intersect at interior points divide it into triangles. What is the largest number of drawn segments (sides and diagonals of a $100$-gon) can intersect interior points at the same line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-e61804e589564ea86b1aa36c5a419fa3a59221dac89ea1d292bd7db8b359ef88 | A $10\times10\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points. | [
"168"
] | [
"A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb628e21f6e5679f9c53893981f6f0d784480abd67717916d00802f51d37fe15 | A $12 \times 13$ chessboard is tiled with $1 \times 3$ mosaics such that there are exactly $k$ vertical mosaics in every column. Find the number of different possible values for $k$. | [
"5"
] | [
"A $12 \\times 13$ chessboard is tiled with $1 \\times 3$ mosaics such that there are exactly $k$ vertical mosaics in every column. Find the number of different possible values for $k$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2b5eee54195b47a227f623ce489cf4c0abc4b904afd72e5cb8e2dfc81c99c58 | A $14 \times 14$ square is to be filled using 6 squares of size $1 \times 1$, 5 squares of size $2 \times 2$, ..., and 1 square of size $6 \times 6$. Calculate the total number of distinct ways the $14 \times 14$ square can be filled under these conditions. | [
"1"
] | [
"A $14 \\times 14$ square is to be filled using 6 squares of size $1 \\times 1$, 5 squares of size $2 \\times 2$, ..., and 1 square of size $6 \\times 6$. Calculate the total number of distinct ways the $14 \\times 14$ square can be filled under these conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-75c30ae1e5c51fc465e0d14b6bbdc0255095f423b7156bde96a7ae7569bcbad1 | A $150\times 324\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes? | [
"768"
] | [
"A $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\\times 1\\times 1$ cubes?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f7e00d88c051cebe58cdcfaf2a45754829c3beb77e81d5da5b475271734cba1 | A $17 \times 17$ table is colored in black and white arbitrarily. In every move, we choose an $m \times n$ table which is included in the square and switch all of its colors (including the inside). What is the minimal number of moves such that, in the end of the process, we can promise that the whole square is colored ... | [
"289"
] | [
"A $17 \\times 17$ table is colored in black and white arbitrarily. In every move, we choose an $m \\times n$ table which is included in the square and switch all of its colors (including the inside). What is the minimal number of moves such that, in the end of the process, we can promise that the whole square is c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-af850b2994b61685f3a0439186a473c2867587b6872965670999971804a3e851 | A $180^\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$ What is the resulting complex number? | [
"6 + 3i"
] | [
"A $180^\\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$ What is the resulting complex number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09abcee819acdf4694c8d8958d7327ac31ce97c08b89a7b422f2105eabd00fea | A $1\times 1\times 1$ cube is placed on an $8\times 8$ chessboard so that its bottom face coincides with a square of the chessboard. The cube rolls over a bottom edge so that the adjacent face now lands on the chessboard. In this way, the cube rolls around the chessboard, landing on each square at least once. If it is ... | [
"0"
] | [
"A $1\\times 1\\times 1$ cube is placed on an $8\\times 8$ chessboard so that its bottom face coincides with a square of the chessboard. The cube rolls over a bottom edge so that the adjacent face now lands on the chessboard. In this way, the cube rolls around the chessboard, landing on each square at least once. I... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0159f9a488ec78ee16f125865391d537cf2de06596f6a325624613b7adc03255 | A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? | [
"25"
] | [
"A $2 \\times 3$ rectangle and a $3 \\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-ac5e76f27e1dc0de20e8f35a112bfb94d973e0a0b3201cb28e427e14557f9c47 | A $2$ by $1$ strip of paper is folded so that a corner meets a point a distance $x$ along on the opposite edge. Let $x$ be the value such that the area of the resulting shape is maximized. Suppose $x^4$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"10"
] | [
"A $2$ by $1$ strip of paper is folded so that a corner meets a point a distance $x$ along on the opposite edge. Let $x$ be the value such that the area of the resulting shape is maximized. Suppose $x^4$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-4f00d7878fb1d32b631c247239b8d8e12ae52b46f5b5d8e285d56f15d83b7d75 | A $20 \times 20 \times 20$ block is cut into 8000 unit cubes, each assigned a number. In each column of 20 cubes parallel to any edge, the sum of the numbers is 1. One of the unit cubes has the number 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Find the sum of all... | [
"333"
] | [
"A $20 \\times 20 \\times 20$ block is cut into 8000 unit cubes, each assigned a number. In each column of 20 cubes parallel to any edge, the sum of the numbers is 1. One of the unit cubes has the number 10. Three $1 \\times 20 \\times 20$ slices parallel to the faces of the block contain this unit cube. Find the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 43
} | 43 | 0 | 0 |
omr-ws-a0e51aa9b6b7fd3d20f1ab267b97d87e6d67679f4c166ec14bc2f80a7b5d392f | A $20 \times 20$ square pie is divided into unit squares. There are 200 grapes and 200 strawberries distributed in these squares such that each square contains exactly one berry. Calculate the number of distinct ways to divide this pie into two pieces by the sides of the squares such that each piece contains an equal n... | [
"2"
] | [
"A $20 \\times 20$ square pie is divided into unit squares. There are 200 grapes and 200 strawberries distributed in these squares such that each square contains exactly one berry. Calculate the number of distinct ways to divide this pie into two pieces by the sides of the squares such that each piece contains an e... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f2f399f7956cd2a963ec336a788c4c32702b8f9a59387d417095e1b4f3b1366 | A $20 \times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1 \times 4$ or $4 \times 1$ rectangles to determine if the treasure is in any of these rectangles. The answers are given only after all rectangles are set. What is the least number of questions nee... | [
"160"
] | [
"A $20 \\times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1 \\times 4$ or $4 \\times 1$ rectangles to determine if the treasure is in any of these rectangles. The answers are given only after all rectangles are set. What is the least number of quest... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b57d6c41a22394f17b654a211a8fad5c027fe9528bb3c366365a4721aa6c0c4 | A $2002 \times 1000$ rectangle (with the side of length $2002$ placed horizontally) is cut into $1 \times 2$ dominos and Z tetraminos. Given that the number of dominos is not greater than 600, find the minimum number of Z tetraminos that must be placed horizontally. | [
"334"
] | [
"A $2002 \\times 1000$ rectangle (with the side of length $2002$ placed horizontally) is cut into $1 \\times 2$ dominos and Z tetraminos. Given that the number of dominos is not greater than 600, find the minimum number of Z tetraminos that must be placed horizontally."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e324e38eb557435bbc9907aaf39a24cc92796e173619d4ada8e6a387773398af | A $2005 \times 2005$ square matrix is filled with 1s and 0s. For each 1 in the matrix, the total number of 1s in the same row or the same column is at most 40% of the row or column length. What is the maximum number of 1s that can be in the matrix? | [
"\\(1403 \\cdot 1404\\)"
] | [
"A $2005 \\times 2005$ square matrix is filled with 1s and 0s. For each 1 in the matrix, the total number of 1s in the same row or the same column is at most 40% of the row or column length. What is the maximum number of 1s that can be in the matrix?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-bd6d35402fb1d8f10c91531c26e4538009a3745ac8e062e89a6102520f493d8e | A $2007 \times 2007$ board is filled with pluses. In the first move, we change all signs of the rows whose number is divisible by $1$. In the second move, we change all signs of the columns whose number is divisible by $2$. In the third move, we change all signs of the rows whose number is divisible by $3$, and so on. ... | [
"The number of pluses is bigger."
] | [
"A $2007 \\times 2007$ board is filled with pluses. In the first move, we change all signs of the rows whose number is divisible by $1$. In the second move, we change all signs of the columns whose number is divisible by $2$. In the third move, we change all signs of the rows whose number is divisible by $3$, and s... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7a61597d9b4ab18ed227dacedd6ace34fb215a5543a91d8bb1087ce2ebe7dd78 | A $2007 \times 2007$ chessboard is completely tiled with $1 \times 1$ and $2 \times 2$ tiles. Let $S$ be the sum of the number of tiles intersected by each row of the chessboard. Find the remainder when $S$ is divided by $2$. | [
"1"
] | [
"A $2007 \\times 2007$ chessboard is completely tiled with $1 \\times 1$ and $2 \\times 2$ tiles. Let $S$ be the sum of the number of tiles intersected by each row of the chessboard. Find the remainder when $S$ is divided by $2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-55a9b295d30f829c2c9b2328bbf4639a9c3ec12ad037ef7addc6e6c152bbd952 | A $2007 \times 2007$ chessboard is filled with odd integers. For every $1 \leq i \leq 2007$, let $Z_i$ be the sum of the $i$-th row, and for every $1 \leq j \leq 2007$, let $S_j$ be the sum of the $j$-th column. Let $A$ be the product of all $Z_i$ and $B$ be the product of all $S_j$. Calculate the parity (even or odd) ... | [
"even"
] | [
"A $2007 \\times 2007$ chessboard is filled with odd integers. For every $1 \\leq i \\leq 2007$, let $Z_i$ be the sum of the $i$-th row, and for every $1 \\leq j \\leq 2007$, let $S_j$ be the sum of the $j$-th column. Let $A$ be the product of all $Z_i$ and $B$ be the product of all $S_j$. Calculate the parity (eve... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-101b987e7fa802c9bc2a5983ab1e37c2296d95d74012c604d7674522d3f91d16 | A $2010 \times 2010$ board is divided into corner-shaped figures of three cells. Mark one cell in each figure such that each row and each column will have the same number of marked cells. If $k$ is the number of marked cells in each row and each column, find the value of $k$. | [
"670"
] | [
"A $2010 \\times 2010$ board is divided into corner-shaped figures of three cells. Mark one cell in each figure such that each row and each column will have the same number of marked cells. If $k$ is the number of marked cells in each row and each column, find the value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf79f5b1b9616695d0d61c2224d445d4749a20fb227d2dbd321e0c43ec947bc5 | A $2010 \times 2010$ rectangular table is tiled completely with $4040100$ $1 \times 1$ tiles. A circular coin with diameter $2$ is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | [
"2 - \\sqrt{3}"
] | [
"A $2010 \\times 2010$ rectangular table is tiled completely with $4040100$ $1 \\times 1$ tiles. A circular coin with diameter $2$ is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-ac9a8df94e0ea2e3ac0fc2b4aab3e7fc970fd253ab149b360fde35f7d990d8b0 | A $2010 \times 2010$ table is divided into $L$-triminoes. Paint exactly one cell of each $L$-trimino such that each row and each column has the same number of painted cells. Calculate the number of painted cells in each row and each column. | [
"670"
] | [
"A $2010 \\times 2010$ table is divided into $L$-triminoes. Paint exactly one cell of each $L$-trimino such that each row and each column has the same number of painted cells. Calculate the number of painted cells in each row and each column."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbe1d610bcce5085a5e5777223c38a4946f533aba52a87d2bc0d91065eaa8aa9 | A $2014 \times 2014$ chessboard is given. A move consists of choosing a column or a row and coloring $1022$ consecutive cells. If no cell can be colored more than once, what is the maximum number of moves one can perform? | [
"3968"
] | [
"A $2014 \\times 2014$ chessboard is given. A move consists of choosing a column or a row and coloring $1022$ consecutive cells. If no cell can be colored more than once, what is the maximum number of moves one can perform?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-c3ac49adb7c42b6afa29fc9bf328b5a79fc207e4ad8f9ca638a1ff7feef8db3a | A $2015 \times 2015$ chessboard is given, with cells painted white and black alternately such that the corner cells are black. There are $n$ L-trominoes placed on the board, no two of which overlap and which cover all of the black cells. Find the smallest possible value of $n$. | [
"\\(\\frac{2015^2}{6}\\)"
] | [
"A $2015 \\times 2015$ chessboard is given, with cells painted white and black alternately such that the corner cells are black. There are $n$ L-trominoes placed on the board, no two of which overlap and which cover all of the black cells. Find the smallest possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-2be7cdb45a084bf28df121900909c126bd75523f7455d70db11f8ced59d62163 | A $2019 \times 2019$ board is alternately colored black and white, with the four corners colored black. A tromino is an L-shaped tile formed by three connected unit squares. What is the smallest number of trominos needed to cover all the black squares of the board, where trominos are entirely on the board and do not ov... | [
"1019091"
] | [
"A $2019 \\times 2019$ board is alternately colored black and white, with the four corners colored black. A tromino is an L-shaped tile formed by three connected unit squares. What is the smallest number of trominos needed to cover all the black squares of the board, where trominos are entirely on the board and do ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f4a39b3e6127b5e0fb974eaee9452529bc9db46824887a93750affe299d96d81 | A $2019 \times 2019$ grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \leq k \leq 2019$, the number of black cells differs from the number of white cells by at most one. How many different bala... | [
"\\(2^{1012} - 6\\)"
] | [
"A $2019 \\times 2019$ grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \\leq k \\leq 2019$, the number of black cells differs from the number of white cells by at most one. How many differ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e8e2c15d154a1d6e8c3832bb6906100bdd0213b64021778e98a7f209e785c11 | A $2019 \times 2019$ square grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \leq k \leq 2019$, the number of black cells differs from the number of white cells by at most one. How many differe... | [
"2"
] | [
"A $2019 \\times 2019$ square grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \\leq k \\leq 2019$, the number of black cells differs from the number of white cells by at most one. How many... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe5b04630d6c8f5c6ca61729201c6c39207fe8092cdc78992a7082124695e21c | A $2019 \times 2019$ square grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \leq k \leq 2019$, the number of black cells differs from the number of white cells by at most one. How many differe... | [
"2"
] | [
"A $2019 \\times 2019$ square grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \\leq k \\leq 2019$, the number of black cells differs from the number of white cells by at most one. How many... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f8c2f7d6be40ad192259d4b346b6016cd5697fe5b78158247c9e0c02311c2781 | A $2020k \times 2020k$ square is perfectly tiled with rectangles of dimensions $1 \times k$ (rotation of tiles is allowed). Let $N(k)$ be the number of $k \times k$ sub-squares that are perfectly covered by $k$ tiles. Find the minimum possible value of $N(k)$. | [
"2020^2"
] | [
"A $2020k \\times 2020k$ square is perfectly tiled with rectangles of dimensions $1 \\times k$ (rotation of tiles is allowed). Let $N(k)$ be the number of $k \\times k$ sub-squares that are perfectly covered by $k$ tiles. Find the minimum possible value of $N(k)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ce8b6099d9ed470e375992d4e64e65079be7f7aab62c6eb7b4e65bd3b369761b | A $24 \times 60$ rectangle is divided into unit squares by lines parallel to its sides. Draw another straight line so that the rectangle is divided into the largest possible number of parts. | [
"1512"
] | [
"A $24 \\times 60$ rectangle is divided into unit squares by lines parallel to its sides. Draw another straight line so that the rectangle is divided into the largest possible number of parts."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-48a76bca1b602a5a824f9c98d002c3cd23b6c92b0b3ff9aeaad43fd559692513 | A $29 \times 29$ square is covered with 99 non-overlapping $2 \times 2$ tiles. What is the maximum number of additional $2 \times 2$ tiles that can be placed on the square without overlapping any existing tiles? | [
"111"
] | [
"A $29 \\times 29$ square is covered with 99 non-overlapping $2 \\times 2$ tiles. What is the maximum number of additional $2 \\times 2$ tiles that can be placed on the square without overlapping any existing tiles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-5311da79297a523ab0770dc2a4506b12645e7fb6f49163c1c18077224245af7e | A $2N \times 2N$ board is covered with non-overlapping dominoes. A "lazy" rook visits each cell exactly once, moving to adjacent cells. A move is called "good" if it transitions from one cell to another cell within the same domino. Determine the largest and smallest possible number of "good" moves. | [
"2N"
] | [
"A $2N \\times 2N$ board is covered with non-overlapping dominoes. A \"lazy\" rook visits each cell exactly once, moving to adjacent cells. A move is called \"good\" if it transitions from one cell to another cell within the same domino. Determine the largest and smallest possible number of \"good\" moves."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-234adec9036be0af0ac68dc84e89af7091accb2f1ca3431b26bd74d9e6a3b25e | A $2\times 3$ rectangle and a $3\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? | [
"25"
] | [
"A $2\\times 3$ rectangle and a $3\\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c9d3c90f04481f6ed8cb525d0e60e19e2e3601b72d128dbe1fbc93f47af53c8 | A $2n \times 2n$ board is divided into $4n^2$ small squares in the manner of a chessboard. Each small square is painted in one of four colors so that every $2 \times 2$ block of four small squares involves all four colors. Let $C(n)$ denote the number of distinct colorings of the four corner squares. Find $C(n)$. | [
"24"
] | [
"A $2n \\times 2n$ board is divided into $4n^2$ small squares in the manner of a chessboard. Each small square is painted in one of four colors so that every $2 \\times 2$ block of four small squares involves all four colors. Let $C(n)$ denote the number of distinct colorings of the four corner squares. Find $C(n)$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a853eacee241d4e804b97ef272eb222c7168726fbb4788bab9745b378382d5a | A $2n \times 2n$ board is divided into $4n^2$ small squares. Each small square is painted with one of four colors such that every $2 \times 2$ block of 4 small squares contains all four colors. If the top-left corner square is painted red, the top-right corner square is painted blue, the bottom-left corner square is pa... | [
"1"
] | [
"A $2n \\times 2n$ board is divided into $4n^2$ small squares. Each small square is painted with one of four colors such that every $2 \\times 2$ block of 4 small squares contains all four colors. If the top-left corner square is painted red, the top-right corner square is painted blue, the bottom-left corner squar... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bad89e28f4f765746bfc034ca304c517f531a32363d0e4cfb1500e3644a4f429 | A $2n \times 2n$ board is divided into $4n^2$ small squares. Each small square is painted with one of four colors such that every $2 \times 2$ block of four small squares contains all four colors. Let $C(n)$ denote the number of distinct colorings of the four corner squares. Find $C(5)$. | [
"24"
] | [
"A $2n \\times 2n$ board is divided into $4n^2$ small squares. Each small square is painted with one of four colors such that every $2 \\times 2$ block of four small squares contains all four colors. Let $C(n)$ denote the number of distinct colorings of the four corner squares. Find $C(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-90b6f5f2c00622553ad9550459390479fb5bc33e8f86b3410a6bd8b8163d8b55 | A $2n \times 2n$ chessboard is filled with $4n^2$ real numbers whose sum is zero. The absolute value of each number does not exceed 1. Calculate the maximum possible value of $n$ such that the absolute value of the sum of all the numbers in at least one row or one column is guaranteed not to exceed $n$. | [
"2"
] | [
"A $2n \\times 2n$ chessboard is filled with $4n^2$ real numbers whose sum is zero. The absolute value of each number does not exceed 1. Calculate the maximum possible value of $n$ such that the absolute value of the sum of all the numbers in at least one row or one column is guaranteed not to exceed $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c61ae79a3c6b2aa6ee6cb52cd07df9683906564361dd2c4711ab8ba0b3bbcc85 | A $3 \times 11$ cm rectangular box has one vertex at the origin, and the other vertices are above the $x$-axis. Its sides lie on the lines $y = x$ and $y = -x$. What is the $y$-coordinate of the highest point on the box, in centimeters? | [
"\\( 7\\sqrt{2} \\)"
] | [
"A $3 \\times 11$ cm rectangular box has one vertex at the origin, and the other vertices are above the $x$-axis. Its sides lie on the lines $y = x$ and $y = -x$. What is the $y$-coordinate of the highest point on the box, in centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2007ed2a587f07b37775d6d8bf95cc1e3c0c35413afce534a8b66ed9c73b6d12 | A $3 \times 3 \times 3$ cube is composed of 27 identical small cubes. A plane is perpendicular to a diagonal of the large cube and passes through its center. How many small cubes does that plane intersect? | [
"19"
] | [
"A $3 \\times 3 \\times 3$ cube is composed of 27 identical small cubes. A plane is perpendicular to a diagonal of the large cube and passes through its center. How many small cubes does that plane intersect?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-43fb34edc69747d088e897a7bdbf13f5f5ed88f58d969a5665a18d57d4bba6d4 | A $3 \times 3 \times 3$ cube is composed of 27 identical small cubes. A plane is perpendicular to a diagonal of the large cube and passes through its center. How many small cubes does the plane intersect? | [
"19"
] | [
"A $3 \\times 3 \\times 3$ cube is composed of 27 identical small cubes. A plane is perpendicular to a diagonal of the large cube and passes through its center. How many small cubes does the plane intersect?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 12
} | 29 | 0 | 0 |
omr-ws-35158f68d9f9d9d300384b07838c82c62f263cc5863f0fa806b28826a584c8e4 | A $3 \times 3 \times 3$ cube is made of 27 unit cube pieces, each containing a lamp that can be on or off. Pressing a piece (except the center piece) toggles the state of that piece and the pieces that share a face with it. Initially, all lamps are off. Determine if the following states are achievable: (1) All lamps ar... | [
"(1) No, (2) Yes, (3) No."
] | [
"A $3 \\times 3 \\times 3$ cube is made of 27 unit cube pieces, each containing a lamp that can be on or off. Pressing a piece (except the center piece) toggles the state of that piece and the pieces that share a face with it. Initially, all lamps are off. Determine if the following states are achievable:\n(1) All ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 12
} | 14 | 0 | 0 |
omr-ws-692e9c211df184dc642c7633c3b775dbab3cbaebd7810b5d7e10942790e0bb18 | A $3 \times 3 \times 3$ cube is made up of 27 unit cubes. How many of these unit cubes cross a plane that is perpendicular to the diagonal connecting opposite vertices of the cube and passes through its center? | [
"19"
] | [
"A $3 \\times 3 \\times 3$ cube is made up of 27 unit cubes. How many of these unit cubes cross a plane that is perpendicular to the diagonal connecting opposite vertices of the cube and passes through its center?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 15
} | 33 | 0 | 0 |
omr-ws-0aed2a1b73a79bf67cca02467ad009d65d9bfd631bae6b34db06b0cafa88b2ce | A $3 \times 3 \times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After dismantling the cuboid into its constituent unit cubes and randomly reconstructing it, determine the probability that all of its apparent sides are still red-coloured. (The cubo... | [
"\\(\\frac{3^8\\cdot 8!2^{16}16!4^{10}10!24^{2}2!}{24^{36}\\cdot 36!}\\)"
] | [
"A $3 \\times 3 \\times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After dismantling the cuboid into its constituent unit cubes and randomly reconstructing it, determine the probability that all of its apparent sides are still red-coloured. (T... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 3
} | 6 | 0 | 0 |
omr-ws-ff164cdb3984a40e0907eabf07b14cab48532f48cae62e74ea6b58c6b7e37818 | A $3 \times 3 \times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After dismantling the cuboid into its constituent unit cubes, how many ways are there to position eight of its corner cubes so that the apparent sides of these corner cubes are still ... | [
"\\(3^8 \\cdot 8!\\)"
] | [
"A $3 \\times 3 \\times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After dismantling the cuboid into its constituent unit cubes, how many ways are there to position eight of its corner cubes so that the apparent sides of these corner cubes are... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 10
} | 20 | 0 | 0 |
omr-ws-6c53f766950ddf18cbef0d4e01159f6673512aaa43f5c46522a604d0d1c40813 | A $3 \times 3$ grid is filled with integers (positive or negative) such that the product of the integers in any row or column is equal to $20$. How many ways can this be done? | [
"46080"
] | [
"A $3 \\times 3$ grid is filled with integers (positive or negative) such that the product of the integers in any row or column is equal to $20$. How many ways can this be done?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ae865262b5e56aa25a471c5a22874f6381b88ef29cadde3d59490313f5c6fa7 | A $3 \times 3$ matrix is given where each cell contains either the symbol $a$ or the symbol $b$. We are allowed to choose a cell and flip the symbols in all cells that share a common side with the selected cell (i.e., replace $a$ with $b$ and $b$ with $a$). How many different matrices can be obtained by applying this o... | [
"64"
] | [
"A $3 \\times 3$ matrix is given where each cell contains either the symbol $a$ or the symbol $b$. We are allowed to choose a cell and flip the symbols in all cells that share a common side with the selected cell (i.e., replace $a$ with $b$ and $b$ with $a$). How many different matrices can be obtained by applying ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 7,
"tir": 27
} | 51 | 0 | 0 |
omr-ws-ffc66d5d41e81da8c8843a7769bffc08523b371f4edab3ba6ebfa7fa4b1d886e | A $3 \times 3$ square is divided into nine $1 \times 1$ unit squares. Different integers from 1 to 9 are written into these nine unit squares. Consider the pairs of numbers in the squares sharing a common edge. What is the largest number of pairs where one number is a factor of the other number? | [
"9"
] | [
"A $3 \\times 3$ square is divided into nine $1 \\times 1$ unit squares. Different integers from 1 to 9 are written into these nine unit squares. Consider the pairs of numbers in the squares sharing a common edge. What is the largest number of pairs where one number is a factor of the other number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-a7fed7aa950eb52028f6a335e6ce9cb65d930599dda767ceb7fa166effe5ebeb | A $3 \times 3$ square is divided into three $1 \times 1$ squares, labelled $A, B, C$ (from left to right) above a $2 \times 2$ square labelled $F$ to the left of two $1 \times 1$ squares above each other, $D$ above $E$. Two squares are considered to be adjacent if they have more than one point in common. The numbers $1... | [
"96"
] | [
"A $3 \\times 3$ square is divided into three $1 \\times 1$ squares, labelled $A, B, C$ (from left to right) above a $2 \\times 2$ square labelled $F$ to the left of two $1 \\times 1$ squares above each other, $D$ above $E$. Two squares are considered to be adjacent if they have more than one point in common. The n... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 22
} | 39 | 0 | 0 |
omr-ws-982b88a731b8add3ceb7264248967fc90315cba2d2df36fd897a38b531576cda | A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black squa... | [
"\\(\\frac{49}{512}\\)"
] | [
"A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^\\circ$ clockwise about its center, and every white square in a position formerly occupied by a bla... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 32
} | 71 | 0 | 0 |
omr-ws-abf43de9f4e10e4c92be5116b061f32969d768aaf816cf564e1a2f3a22898a09 | A $3$ by $6$ grid is filled with the numbers in the list $(1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9)$ according to the following rules: - Both the first three columns and the last three columns contain the integers $1$ through $9$. - No number appears more than once in a given row. Find the number of ways t... | [
"4389396480"
] | [
"A $3$ by $6$ grid is filled with the numbers in the list $(1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9)$ according to the following rules:\n- Both the first three columns and the last three columns contain the integers $1$ through $9$.\n- No number appears more than once in a given row.\nFind the number o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-769fb1396ed3b9de30adb4b68d8146580f91dff9b4563c34f4ee53ee0037c545 | A $30 \times 30$ square is partitioned into unit squares, each colored either black or white. Two unit squares are adjacent if they share a side. A white unit square changes to black if all its adjacent squares are white, and a black unit square changes to white if all its adjacent squares are black. Assuming the array... | [
"697"
] | [
"A $30 \\times 30$ square is partitioned into unit squares, each colored either black or white. Two unit squares are adjacent if they share a side. A white unit square changes to black if all its adjacent squares are white, and a black unit square changes to white if all its adjacent squares are black. Assuming the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92abb46f26f9b86adecbd450b63d7ddc6664ddd7010d8bdcd690d14586499deb | A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. For $1 \leq k \leq 29$, choose a point $B_k$ on the arc $A_kA_{k+1}$. Let $P$ be the polygon formed by the vertices $B_1, B_2, \ldots, B_{29}$. Find the maximum possible area of $P$. | [
"58 \\sin \\dfrac{2\\pi}{29}"
] | [
"A $30$-gon $A_1A_2\\cdots A_{30}$ is inscribed in a circle of radius $2$. For $1 \\leq k \\leq 29$, choose a point $B_k$ on the arc $A_kA_{k+1}$. Let $P$ be the polygon formed by the vertices $B_1, B_2, \\ldots, B_{29}$. Find the maximum possible area of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02788a7f6438004ab0b9fe3da6f3b0224ddef381654ec3c332cb7b6e75a1378c | A $3000 \times 3000$ square is tiled by dominoes ($1 \times 2$ rectangles) in an arbitrary way. Find the number of ways to color the dominoes in three colors such that the number of dominoes of each color is the same, and each domino has at most two neighbors of the same color. (Two dominoes are said to be neighbors if... | [
"1500000"
] | [
"A $3000 \\times 3000$ square is tiled by dominoes ($1 \\times 2$ rectangles) in an arbitrary way. Find the number of ways to color the dominoes in three colors such that the number of dominoes of each color is the same, and each domino has at most two neighbors of the same color. (Two dominoes are said to be neigh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-c58c1e28f3c36d4b4c5ac5075119949ffbb58e8c4c08b1556c0d419be24d8649 | A $317$th degree monic polynomial $P(x)$ has roots $1, 3, 6, 10, \ldots, \binom{318}{2}$. If $P(\binom{319}{2})$ can be represented as $\frac{a!b!}{c! \cdot 2^d}$ with $a, b, c, d \in \mathbb{N^*}$, compute the minimum possible value of $a+b+c+d$. | [
"1589"
] | [
"A $317$th degree monic polynomial $P(x)$ has roots $1, 3, 6, 10, \\ldots, \\binom{318}{2}$. If $P(\\binom{319}{2})$ can be represented as $\\frac{a!b!}{c! \\cdot 2^d}$ with $a, b, c, d \\in \\mathbb{N^*}$, compute the minimum possible value of $a+b+c+d$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 13
} | 25 | 0 | 0 |
omr-ws-fff7136a56fb96575dfaac4da0837a0e95bd6b9917ae72175351f033e69a5de3 | A $32$-page magazine has one page ripped off and has $2$ lined pages. The product of the numbers of the hatched and missing pages is $6144$. Determine the missing pages. | [
"24"
] | [
"A $32$-page magazine has one page ripped off and has $2$ lined pages. The product of the numbers of the hatched and missing pages is $6144$. Determine the missing pages."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-e18a28f45d9e702a4cce88fd2d72fe5e2e2730916d409f364d384e5fe981b750 | A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$. | [
"99"
] | [
"A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d786f006155b6fefaf54d414b7b24746ecdbedd884794f3845d9d346a53f9269 | A $4 \times 13$ board is given, and 52 usual playing cards are also given (4 suits, and 13 numbers on each suit). We have to place these cards on the board such that each square contains one card. The placement must ensure that if two squares share a side, their cards have either the same suit or the same number. Find ... | [
"4! \\times 13!"
] | [
"A $4 \\times 13$ board is given, and 52 usual playing cards are also given (4 suits, and 13 numbers on each suit). We have to place these cards on the board such that each square contains one card. The placement must ensure that if two squares share a side, their cards have either the same suit or the same number.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-fc01adb246408acdc0bd52711bf3f8ea163f47be0534be07005964fe5c5327a6 | A $4 \times 4 \times 4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be colored red. A coloring is called interesting if there is exactly $1$ red unit cube in every $1 \times 1 \times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colorings. | [
"576"
] | [
"A $4 \\times 4 \\times 4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be colored red. A coloring is called interesting if there is exactly $1$ red unit cube in every $1 \\times 1 \\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colorings."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5932b643d1e0c1a84d5ed1c36355db83c6848269a2699281d2c65a84f03323d | A $4 \times 4 \times 4$ inch cube cake has icing on the top and four side faces, but not on the bottom. The cake is cut into $64$ smaller $1 \times 1 \times 1$ inch cubes. How many of the small pieces will have icing on exactly two sides? | [
"20"
] | [
"A $4 \\times 4 \\times 4$ inch cube cake has icing on the top and four side faces, but not on the bottom. The cake is cut into $64$ smaller $1 \\times 1 \\times 1$ inch cubes. How many of the small pieces will have icing on exactly two sides?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 60,
"genselect": 8,
"tir": 12
} | 80 | 0 | 0 |
omr-ws-c7b7470011a114f781a3f9045a3c34a628e5d19f46aa677caf2fb3ca57ea3067 | A $4 \times 4 \times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? | [
"\\(2 + 2\\sqrt{7}\\)"
] | [
"A $4 \\times 4 \\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 31
} | 79 | 0 | 0 |
omr-ws-420a3637dc8945348ffb7b5c20edeea57a8f7aeac662a7dfb305297804566cc9 | A $4 \times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains exactly 3 squares and paint each of these squares as follows: a) If the square is white, it is painted black. b) If the square is black, it is painted white. Calculate the minimum number of squares that w... | [
"1"
] | [
"A $4 \\times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains exactly 3 squares and paint each of these squares as follows:\na) If the square is white, it is painted black.\nb) If the square is black, it is painted white.\nCalculate the minimum number of squar... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-274dea95893ee7a0d9df9ea23ea845f696799ac0dfadf6b51ee8e438dc940d8d | A $4 \times 4$ grid has its 16 cells colored arbitrarily in three colors. A swap is an exchange between the colors of two cells. Determine the maximum number of swaps required to produce a line of symmetry, regardless of the grid's initial coloring. If it is not always possible to achieve a line of symmetry within a ce... | [
"3"
] | [
"A $4 \\times 4$ grid has its 16 cells colored arbitrarily in three colors. A swap is an exchange between the colors of two cells. Determine the maximum number of swaps required to produce a line of symmetry, regardless of the grid's initial coloring. If it is not always possible to achieve a line of symmetry withi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-14456b1553028cd647ae0cdb67b280f6c43af1e2e6dee99b651b0da8508fffd0 | A $4 \times 4$ grid is divided into four $2 \times 2$ squares. Each square contains the numbers 1, 2, 3, or 4. An arrangement is called "sudoku-like" if there is exactly one of 1, 2, 3, and 4 in each row, column, and $2 \times 2$ box. How many sudoku-like arrangements are there? | [
"288"
] | [
"A $4 \\times 4$ grid is divided into four $2 \\times 2$ squares. Each square contains the numbers 1, 2, 3, or 4. An arrangement is called \"sudoku-like\" if there is exactly one of 1, 2, 3, and 4 in each row, column, and $2 \\times 2$ box. How many sudoku-like arrangements are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb90b8bac16bb00b8f76c27565823c2c4a1bb7fc023737a2d1f1987c0c65bc73 | A $4 \times 4$ square is cut with a straight line into two congruent quadrilaterals. If each quadrilateral has a perimeter of $13$, what is the length of the shortest side of the quadrilaterals? | [
"\\(\\frac{1}{2}\\)"
] | [
"A $4 \\times 4$ square is cut with a straight line into two congruent quadrilaterals. If each quadrilateral has a perimeter of $13$, what is the length of the shortest side of the quadrilaterals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-c23f824e99b4a3333205b8266c130510510d84d856ede52e00299a3469d6cf41 | A $4 \times 4$ square is divided into $1 \times 1$ squares. You have 3 balls to place into 3 of these 16 squares. A good position is defined as one where no two balls lie on the same row or column. How many good positions can be made? | [
"96"
] | [
"A $4 \\times 4$ square is divided into $1 \\times 1$ squares. You have 3 balls to place into 3 of these 16 squares. A good position is defined as one where no two balls lie on the same row or column. How many good positions can be made?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3636ae4f72a1088fc2e97cd64b7af83d3b2c39f4fca5e6b7c673c2910fd5de15 | A $4 \times 4$ table is divided into 16 white unit square cells. Two cells are called neighbors if they share a common side. A move consists of choosing a cell and toggling the colors of its neighbors from white to black or from black to white. After exactly $n$ moves, all 16 cells are black. Find all possible values o... | [
"\\( 4 + 2k \\) where \\( k \\) is a non-negative integer.",
"\\( n \\geq 4 \\) and \\( n \\) is even."
] | [
"A $4 \\times 4$ table is divided into 16 white unit square cells. Two cells are called neighbors if they share a common side. A move consists of choosing a cell and toggling the colors of its neighbors from white to black or from black to white. After exactly $n$ moves, all 16 cells are black. Find all possible va... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 6
} | 7 | 0 | 0 |
omr-ws-234de59d52b9537df7b2768fd04275be7e9804e971319e20c069f943ec5e3fa5 | A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$. | [
"2765"
] | [
"A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aea8509e8292cae6fc0567d0bdfe879b0238157ceb477bd88aadb54228e9084d | A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by ... | [
"\\dfrac{4}{3}"
] | [
"A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane give... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-308885baf0220ad2e28bd49b5749d0fd216e728555d042a1bfdefbb01a49734f | A $4$-dimensional hypercube with edge length $1$ has vertices at coordinates $(a, b, c, d)$ where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by this i... | [
"\\(\\frac{4}{3}\\)"
] | [
"A $4$-dimensional hypercube with edge length $1$ has vertices at coordinates $(a, b, c, d)$ where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 16
} | 28 | 0 | 0 |
omr-ws-1a74d85d533ce1f28340fe7df520733f4b1acf2d08a83ae3920fbdcdaf15c4e0 | A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece? | [
"6"
] | [
"A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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