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-f3a944f4e7f2298584f4763b21420bea5e7311ee2e9d95e4332db4f8023c2ffd | A $40 \times 40$ white square is divided into $1 \times 1$ squares. Some of these $1 \times 1$ squares are colored red such that each $1 \times 1$ square shares a side with at most one red square (excluding itself). What is the largest possible number of red squares? | [
"420"
] | [
"A $40 \\times 40$ white square is divided into $1 \\times 1$ squares. Some of these $1 \\times 1$ squares are colored red such that each $1 \\times 1$ square shares a side with at most one red square (excluding itself). What is the largest possible number of red squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2aaef15b726d1cb3079157fca231043014f15a399d6569d9531628824d742431 | A $45^{\circ}$ set square $ABC$ has a right angle at $B$ and $AB=BC= a$. Initially $A$ is at the origin, $B$ is at the point with cartesian coordinates $(a, 0)$ and $C$ is at $(a, -a)$. The corner $C$ moves up the line $x=a$ until it reaches $(a, 0)$ and the set square moves so that the edge $AB$ always passes through ... | [
"a(\\sqrt{2} - 1)"
] | [
"A $45^{\\circ}$ set square $ABC$ has a right angle at $B$ and $AB=BC= a$. Initially $A$ is at the origin, $B$ is at the point with cartesian coordinates $(a, 0)$ and $C$ is at $(a, -a)$. The corner $C$ moves up the line $x=a$ until it reaches $(a, 0)$ and the set square moves so that the edge $AB$ always passes th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2b3243a40c5b2f0c206d52dd2d4e214b414022c376f7dbfe1ac2d67a35db798 | A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is "intriguing" if there is exactly $1$ red unit cube in every $1\times1\times4$ rectangular box composed of $4$ unit cubes. Determine the number of "intriguing" colorings. | [
"672"
] | [
"A $4\\times4\\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is \"intriguing\" if there is exactly $1$ red unit cube in every $1\\times1\\times4$ rectangular box composed of $4$ unit cubes. Determine the number of \"intriguing\" colorings."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2927f0b3fe380ec60360bfcaa97777f0a1eebfb69ba1ab87e62e21878bc560b0 | A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is $\mathfrak{Intriguing}$ if there is exactly $1$ red unit cube in every $1\times1\times4$ rectangular box composed of $4$ unit cubes. Determine the number of $\mathfrak{Intriguing}$ color... | [
"576"
] | [
"A $4\\times4\\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is $\\mathfrak{Intriguing}$ if there is exactly $1$ red unit cube in every $1\\times1\\times4$ rectangular box composed of $4$ unit cubes. Determine the number of $\\mathfrak{Intrigu... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7f3ed929d9fea11a618e2394a293d4e5230c94c83a39954e8250ce61d845b4b | A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are colored black and the rest are colored white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n... | [
"302"
] | [
"A $5 \\times 100$ table is divided into $500$ unit square cells, where $n$ of them are colored black and the rest are colored white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8d1ae97a2943c546146ba2bf9b1e162825a73c760cd2df9fabd2b7d7089cb64 | A $5 \times 5 \times 5$ cube is formed using $1 \times 1 \times 1$ cubes. A number of the smaller cubes are removed by punching out the 15 designated columns from front to back, top to bottom, and side to side. Find the number of smaller cubes that remain. | [
"68"
] | [
"A $5 \\times 5 \\times 5$ cube is formed using $1 \\times 1 \\times 1$ cubes. A number of the smaller cubes are removed by punching out the 15 designated columns from front to back, top to bottom, and side to side. Find the number of smaller cubes that remain."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7cec61a3f41c3dc70a290fb76dad778adfbab60b1ee2ec162bd5745d76930bb4 | A $5 \times 5 \times 5$ wooden cube is painted on all $6$ faces and then cut up into unit cubes. One unit is randomly selected and rolled. What is the probability that exactly one of the five visible faces is painted? | [
"\\(\\frac{57}{125}\\)"
] | [
"A $5 \\times 5 \\times 5$ wooden cube is painted on all $6$ faces and then cut up into unit cubes. One unit is randomly selected and rolled. What is the probability that exactly one of the five visible faces is painted?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-113f048e6995d5c9f60cb4a14d9f5bacae3493081fcbf35dfb0bf0d8fb05ae86 | A $5 \times 5$ matrix $A$ with entries of $0$ or $1$ has $\operatorname{tr}(A) = 0$ and $\operatorname{tr}(AA^T) = 10$. Let $S$ be the set of all even permutations $\sigma \in S_5$ such that $N_{\sigma} := (n_{ij} = \delta_{\sigma(i)j})_{1 \le i, j \le 5}$ satisfies $\operatorname{tr}(AN_{\sigma}A^TN_{\sigma}^T) = 4$. ... | [
"4"
] | [
"A $5 \\times 5$ matrix $A$ with entries of $0$ or $1$ has $\\operatorname{tr}(A) = 0$ and $\\operatorname{tr}(AA^T) = 10$. Let $S$ be the set of all even permutations $\\sigma \\in S_5$ such that $N_{\\sigma} := (n_{ij} = \\delta_{\\sigma(i)j})_{1 \\le i, j \\le 5}$ satisfies $\\operatorname{tr}(AN_{\\sigma}A^TN_{... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0aae24c87028559235e8774480ea029449cb615706790583a238149832c24b54 | A $5 \times 5$ square is divided into $25$ unit squares by lines drawn parallel to its sides. Some of these unit squares are colored blue and some are colored red. Determine the minimum number of unit squares that must be colored to ensure there exist 4 squares of the same color that are the corners of a rectangle with... | [
"25"
] | [
"A $5 \\times 5$ square is divided into $25$ unit squares by lines drawn parallel to its sides. Some of these unit squares are colored blue and some are colored red. Determine the minimum number of unit squares that must be colored to ensure there exist 4 squares of the same color that are the corners of a rectangl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-14720ba5211f29b8320c0bc41473dfabece278a029e84f1bd083c26c0fd51589 | A $5 \times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \times 2$ subtable. The sum of all numbers in a regular table is called the total sum of the table. With any four numbers, one constructs all possible r... | [
"60"
] | [
"A $5 \\times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \\times 2$ subtable. The sum of all numbers in a regular table is called the total sum of the table. With any four numbers, one constructs all pos... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56091f8db42985d9f6eb89c4a00d52fb6c2d051696cc41926fb629e6fa537fe4 | A $5 \times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction. | [
"\\frac{8}{5}"
] | [
"A $5 \\times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1c5d5a88bb09d2ef60597d513e05c7f0600396d7a89005daa4991748641cc68 | A $5,000,000$ gallon lake becomes polluted with sulfur with a concentration of $30$ mg per gallon. Water flows into the lake at a rate of $20,000$ gallons per day. Water flows out at a rate of $20,050$ gallons per day. If $y$ represents the amount of sulfur in the lake after $t$ days of contamination, then find an appr... | [
"\\frac{dy}{dt} = -\\frac{401}{100000 - t} y \\quad \\text{with} \\quad y(0) = 150000000"
] | [
"A $5,000,000$ gallon lake becomes polluted with sulfur with a concentration of $30$ mg per gallon. Water flows into the lake at a rate of $20,000$ gallons per day. Water flows out at a rate of $20,050$ gallons per day. If $y$ represents the amount of sulfur in the lake after $t$ days of contamination, then find an... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-80814ba80251f0e589de92047de60e8564fc262a8b0bd01674a849128e7e98ec | A $50 \times 50$ square board is tiled using tetrominoes of three types: straight, square, and L-shaped. Find the greatest possible number of L-shaped tetrominoes in the tiling. | [
"624"
] | [
"A $50 \\times 50$ square board is tiled using tetrominoes of three types: straight, square, and L-shaped. Find the greatest possible number of L-shaped tetrominoes in the tiling."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-d9e92dd4203b536716c117b62b8b0140d689ff9c589254bff707426b84de1094 | A $50 \times 50$ square board is tiled using tetrominoes of three types: straight, square, and L-shaped. Find the smallest possible number of L-shaped tetrominoes in the tiling. | [
"0"
] | [
"A $50 \\times 50$ square board is tiled using tetrominoes of three types: straight, square, and L-shaped. Find the smallest possible number of L-shaped tetrominoes in the tiling."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-fabac6af7933ccd31a4ea1f66f2067988ec96d86cbfd956d7a5e89fd903d37a1 | A $50$ litre tank is initially filled with $10$ litres of brine solution containing $20$ kg of salt. Starting from time $t=0$, distilled water is poured into the tank at a constant rate of $4$ litres per minute. At the same time, the mixture leaves the tank at a constant rate of $\sqrt{k}$ litres per minute, where $k>0... | [
"\\dfrac{5}{2}"
] | [
"A $50$ litre tank is initially filled with $10$ litres of brine solution containing $20$ kg of salt. Starting from time $t=0$, distilled water is poured into the tank at a constant rate of $4$ litres per minute. At the same time, the mixture leaves the tank at a constant rate of $\\sqrt{k}$ litres per minute, wher... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-159f89492876408896a3877b63284e98970a1941251e149ae617ce5700586448 | A $6 \times 6$ board is filled with positive integers. Each move consists of selecting a square larger than $1 \times 1$, consisting of entire cells, and increasing all numbers inside the selected square by $1$. Determine the minimum number of moves required to reach a situation where all numbers on the board are divis... | [
"2"
] | [
"A $6 \\times 6$ board is filled with positive integers. Each move consists of selecting a square larger than $1 \\times 1$, consisting of entire cells, and increasing all numbers inside the selected square by $1$. Determine the minimum number of moves required to reach a situation where all numbers on the board ar... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0730382ec2623dc2dfeceb9c704e9d10f7f45e374b24bdbaaa6bb141e700f406 | A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal pos... | [
"20"
] | [
"A $6 \\times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \\times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maxi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-f399e052906018161ef684fa0ce579082c3999b6e30a4c63aee110bf2018e5e4 | A $6 \times 6$ grid is made up of $36$ unit squares. Each $1 \times 1$ square is colored red or blue so that each $2 \times 2$ square has exactly $2$ reds and $2$ blues. How many ways can the grid be colored if rotations are considered different? | [
"126"
] | [
"A $6 \\times 6$ grid is made up of $36$ unit squares. Each $1 \\times 1$ square is colored red or blue so that each $2 \\times 2$ square has exactly $2$ reds and $2$ blues. How many ways can the grid be colored if rotations are considered different?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-26bbba58b0034fa3f40106a912f2f425a3cf58303885f1cd04ff1586c4fc0e23 | A $6 \times 6$ square contains thirty-six individual $1 \times 1$ squares. Twelve squares are to be colored red, twelve are to be colored blue, and twelve are to be colored green uniformly at random. Define a [i]snake[/i] to be a connection of twelve adjacent squares (squares sharing at least one edge). What is the pro... | [
"0"
] | [
"A $6 \\times 6$ square contains thirty-six individual $1 \\times 1$ squares. Twelve squares are to be colored red, twelve are to be colored blue, and twelve are to be colored green uniformly at random. Define a [i]snake[/i] to be a connection of twelve adjacent squares (squares sharing at least one edge). What is ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-891e31b8a1b128a58fea0ebdc623cd8c528befe36e07e7fc74ca3a3419815db1 | A $6 \times 6$ square grid is tiled with 18 dominoes. Let $L$ be the number of distinct lines that can be drawn through the grid such that each line intersects no dominoes. Calculate the maximum possible value of $L$. | [
"7"
] | [
"A $6 \\times 6$ square grid is tiled with 18 dominoes. Let $L$ be the number of distinct lines that can be drawn through the grid such that each line intersects no dominoes. Calculate the maximum possible value of $L$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-3e1b5685b132856811b985a43de9b40ec1d31b784ae96152f6f050f963aec192 | A $6 \times 6$ square is dissected into 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Let $A$ be the total number of pairs of congruent rectangles that can be formed. Find the minimum possible value of $A$. | [
"3"
] | [
"A $6 \\times 6$ square is dissected into 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Let $A$ be the total number of pairs of congruent rectangles that can be formed. Find the minimum possible value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-97d02546608eeeac91d05512cde4e6e8af8d6301f23d36893c9530b2ec11f48e | A $6 \times 6$ square is dissected into 9 rectangular regions by lines parallel to its sides such that all these rectangles have only integral side lengths. If $A$ is the sum of the areas of all distinct rectangles (up to congruence), find the maximum possible value of $A$. | [
"30"
] | [
"A $6 \\times 6$ square is dissected into 9 rectangular regions by lines parallel to its sides such that all these rectangles have only integral side lengths. If $A$ is the sum of the areas of all distinct rectangles (up to congruence), find the maximum possible value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-6fc37b6a929c0be377ff088caf27c4a4d7dfa1a92aefc391e6756dc0b9627ae8 | A $6 \times 6$ square is partitioned into $36$ $1 \times 1$ squares. In each of these unit squares, the number of squares (along the grid lines) containing that unit square is written. For example, the number $6$ would be written in a corner square, as it is contained in exactly $6$ squares. What is the sum of all $36$... | [
"560"
] | [
"A $6 \\times 6$ square is partitioned into $36$ $1 \\times 1$ squares. In each of these unit squares, the number of squares (along the grid lines) containing that unit square is written. For example, the number $6$ would be written in a corner square, as it is contained in exactly $6$ squares. What is the sum of a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-bd5842b0ea9a26c26f3525b98426d5cd6f2600b8c039ca313efc6dfdc54d5b64 | A $6$-sided die is rolled $3$ times. If the probability that the product of all three numbers is a square number is $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, what is $a + b$? | [
"127"
] | [
"A $6$-sided die is rolled $3$ times. If the probability that the product of all three numbers is a square number is $\\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, what is $a + b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-c020a2ad4bcca9f4f9edd84f7b07dfdd5e34ce5011af10afa18c9e7ddb89cf4c | A $600 \times 600$ square is divided into figures of 4 cells each, with the first two types of figures having shaded cells. In these shaded cells, the number $2^k$ is written, where $k$ is the column number of the cell. Calculate the remainder when the sum of all the numbers written is divided by $9$. | [
"0"
] | [
"A $600 \\times 600$ square is divided into figures of 4 cells each, with the first two types of figures having shaded cells. In these shaded cells, the number $2^k$ is written, where $k$ is the column number of the cell. Calculate the remainder when the sum of all the numbers written is divided by $9$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86b293fcc68041d6c8c3fd2c0859d9f00fb75a4a8f9f336e788547482e00a45a | A $600 \times 600$ square is divided into figures of four types. In the figures of the two types shown on the left, each black cell is labeled with the number $2^k$, where $k$ is the column number (columns are numbered from left to right from $1$ to $600$). Calculate the remainder when the sum of all recorded numbers i... | [
"0"
] | [
"A $600 \\times 600$ square is divided into figures of four types. In the figures of the two types shown on the left, each black cell is labeled with the number $2^k$, where $k$ is the column number (columns are numbered from left to right from $1$ to $600$). Calculate the remainder when the sum of all recorded num... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc3e578c6a8e27d46e993bc52cf94f0a031c733e74755c7a69a3c82051e9ab6e | A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at l... | [
"106"
] | [
"A $7 \\times 1$ board is completely covered by $m \\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \\times 1$ board in which all three colors are u... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 19
} | 58 | 0 | 0 |
omr-ws-330bd57b8bcc96f15f322152708df6059a61d473540ba7bab65052c3b693e48a | A $7 \times 7 \times 7$ box $B$ is packed with 114 $1 \times 1 \times 3$ bricks, leaving a single unit-cell hole $H$ that is not on the outside surface but buried somewhere in the $5 \times 5 \times 5$ inner core. Calculate the coordinates of $H$ if the bottom-left-front corner of the box is at $(0,0,0)$. Express your ... | [
"9"
] | [
"A $7 \\times 7 \\times 7$ box $B$ is packed with 114 $1 \\times 1 \\times 3$ bricks, leaving a single unit-cell hole $H$ that is not on the outside surface but buried somewhere in the $5 \\times 5 \\times 5$ inner core. Calculate the coordinates of $H$ if the bottom-left-front corner of the box is at $(0,0,0)$. Ex... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-a236977ff9ba15edb9789fe1f040319b553bd1b27ec7063a518d928096c40eb0 | A $7 \times 7 \times 7$ box is partitioned into some $1 \times 1 \times 3$ parallelepipeds and a unit cube. If the unit cube is not on the surface of the $7 \times 7 \times 7$ box, calculate the coordinates of the unit cube assuming the box is aligned with the coordinate axes and one corner of the box is at the origin. | [
"(3,3,3)"
] | [
"A $7 \\times 7 \\times 7$ box is partitioned into some $1 \\times 1 \\times 3$ parallelepipeds and a unit cube. If the unit cube is not on the surface of the $7 \\times 7 \\times 7$ box, calculate the coordinates of the unit cube assuming the box is aligned with the coordinate axes and one corner of the box is at ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-137a6289456dcbfd15af2c40408200bc61e1ea13f4c2a16b7da515a17cfeace5 | A $7 \times 7 \times 7$ cube is completely broken down into smaller cubes with side lengths of 4, 3, 2, or 1. What is the minimum number of smaller cubes that must be cut out? | [
"247"
] | [
"A $7 \\times 7 \\times 7$ cube is completely broken down into smaller cubes with side lengths of 4, 3, 2, or 1. What is the minimum number of smaller cubes that must be cut out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a5d26d0244470a5a52a4d77958df2827968e31fd8963c5a7981e7219c6b07920 | A $7 \times 7$ square is covered by sixteen $3 \times 1$ tiles and one $1 \times 1$ tile. What are the possible positions of the $1 \times 1$ tile? | [
"\\text{All squares where the sum of the row and column numbers is a multiple of }3"
] | [
"A $7 \\times 7$ square is covered by sixteen $3 \\times 1$ tiles and one $1 \\times 1$ tile. What are the possible positions of the $1 \\times 1$ tile?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-66f39cd8737937f0470339d019e2ec34cf83041ab776fade8da62a3386eef168 | A $7 \times 7$ table is overlapped by $16$ rectangles of size $1 \times 3$ such that one cell remains uncovered. Identify the position of this uncovered cell. | [
"(1,1), (1,4), (1,7), (4,1), (4,4), (4,7), (7,1), (7,4), (7,7)"
] | [
"A $7 \\times 7$ table is overlapped by $16$ rectangles of size $1 \\times 3$ such that one cell remains uncovered. Identify the position of this uncovered cell."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34220a2d58be00f275bf1815a77b1ba78d8581be41a7f663bbf13c3ad9ff3b7e | A $7 \times 7$ table is tiled by sixteen $1 \times 3$ rectangles such that only one square remains uncovered. What are the possible positions of this uncovered square? | [
"$(p, q)$ where $p \\equiv q \\equiv 0 \\pmod{3}$",
"The possible positions of the uncovered square are those where both the row and column indices are congruent to $1 \\bmod 3$."
] | [
"A $7 \\times 7$ table is tiled by sixteen $1 \\times 3$ rectangles such that only one square remains uncovered. What are the possible positions of this uncovered square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 28
} | 29 | 0 | 0 |
omr-ws-8a15641281fde7412cc4920dbe7e2bb951066979d6c7702f83f1c6dde6517125 | A $7 \times 7$ table is tiled with $16$ rectangles of size $1 \times 3$ such that only one square remains uncovered. What is the position of this uncovered square? | [
"(4,4)"
] | [
"A $7 \\times 7$ table is tiled with $16$ rectangles of size $1 \\times 3$ such that only one square remains uncovered. What is the position of this uncovered square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7978fdf4d5af72a69c6bcef4f91447684eb4bb3d81c5e620ce96fd050764f8d | A $77 \times 77 \times 77$ cube is filled using horizontal $3 \times 1 \times 1$ bricks, vertical $1 \times 1 \times 4$ bricks, and unit $1 \times 1 \times 1$ bricks. Determine the minimum number of unit bricks required to fill the cube. | [
"1."
] | [
"A $77 \\times 77 \\times 77$ cube is filled using horizontal $3 \\times 1 \\times 1$ bricks, vertical $1 \\times 1 \\times 4$ bricks, and unit $1 \\times 1 \\times 1$ bricks. Determine the minimum number of unit bricks required to fill the cube."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-47a07769aa0562c3cd4891e34ca4e0896bae43085e234d52051663f8cbb13129 | A $7\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\times 1$ board in which all three colors are used at leas... | [
"106"
] | [
"A $7\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\\times 1$ board in which all three colors are used... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-23f9182ad4eeff53ed33e567b95568bca7db751fe1be8c44fae0152867a5a4b5 | A $8 \times 7$ piece of chocolate is made of 56 congruent square cells. Kevin selects a random cell and eats it, along with all cells to the bottom and/or right of it. What is the expected number of cells of chocolate Kevin eats? | [
"32"
] | [
"A $8 \\times 7$ piece of chocolate is made of 56 congruent square cells. Kevin selects a random cell and eats it, along with all cells to the bottom and/or right of it. What is the expected number of cells of chocolate Kevin eats?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b74b5ac963434a557fd773d093acfb9019a58f88c1cf3d6716abbb82ee60e00 | A $9 \times 1$ rectangle is divided into unit squares. A broken line from the lower left to the upper right corner goes through all 20 vertices of the unit squares and consists of 19 line segments. How many such lines are there? | [
"\\(\\binom{19}{9}\\)"
] | [
"A $9 \\times 1$ rectangle is divided into unit squares. A broken line from the lower left to the upper right corner goes through all 20 vertices of the unit squares and consists of 19 line segments. How many such lines are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fe62bd5ab6739a2cb193ea63f0bb1d35f11d472b350c50c14d756de6dc4c715 | A $9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are colored red. Label these 96 red centers $C_1, C_2, \ldots, C_{96}$ such that the distances $C_1C_2, \ldots, C_{95}C_{96}, C_{9... | [
"(6, 4.5)"
] | [
"A $9 \\times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are colored red. Label these 96 red centers $C_1, C_2, \\ldots, C_{96}$ such that the distances $C_1C_2, \\ldots, C_{95}C_{9... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-5c0bb3a197685a8a80d28224311c378d03abdf001a4b8d48a312b8e540ec50a2 | A $9 \times 7$ rectangle is tiled using L-shaped tiles (each composed of three unit squares) and square tiles (each composed of four unit squares). Let $n \geq 0$ be the number of $2 \times 2$ square tiles used in such a tiling. Find all possible values of $n$. | [
"\\( n \\in \\{0, 3\\} \\)"
] | [
"A $9 \\times 7$ rectangle is tiled using L-shaped tiles (each composed of three unit squares) and square tiles (each composed of four unit squares). Let $n \\geq 0$ be the number of $2 \\times 2$ square tiles used in such a tiling. Find all possible values of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee4e6e5020ae280adcce5511d2ba9c38eabc2889074b0639bbdcf1b71feda165 | A $9 \times 9 \times 9$ cube is composed of twenty-seven $3 \times 3 \times 3$ cubes. The big cube is 'tunneled' as follows: First, the six $3 \times 3 \times 3$ cubes which make up the center of each face as well as the center $3 \times 3 \times 3$ cube are removed. Second, each of the twenty remaining $3 \times 3 \ti... | [
"1056"
] | [
"A $9 \\times 9 \\times 9$ cube is composed of twenty-seven $3 \\times 3 \\times 3$ cubes. The big cube is 'tunneled' as follows: First, the six $3 \\times 3 \\times 3$ cubes which make up the center of each face as well as the center $3 \\times 3 \\times 3$ cube are removed. Second, each of the twenty remaining $3... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-cf4bb09a1eca1e9b0230ca56bb95447347169f8669e14e387fff68129156ac89 | A $9 \times 9$ chessboard has one corner square removed, leaving it to be covered entirely by $1 \times 2$ dominoes. Calculate the number of distinct configurations of dominoes where the uncovered square can be transferred to any other corner by shifting the dominoes about on the board. | [
"2"
] | [
"A $9 \\times 9$ chessboard has one corner square removed, leaving it to be covered entirely by $1 \\times 2$ dominoes. Calculate the number of distinct configurations of dominoes where the uncovered square can be transferred to any other corner by shifting the dominoes about on the board."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-21f6acfc866b0fc029a9dc7bbabcd22db5f644dc42f5bb0c7b7cf6fe1a1400bb | A $9 \times 9$ rectangle is cut into fourteen rectangles with integral dimensions and whose sides are parallel to the sides of the initial rectangle. Let $N$ be the minimum number of pairs of congruent rectangles among these fourteen rectangles. Find $N$. | [
"3"
] | [
"A $9 \\times 9$ rectangle is cut into fourteen rectangles with integral dimensions and whose sides are parallel to the sides of the initial rectangle. Let $N$ be the minimum number of pairs of congruent rectangles among these fourteen rectangles. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-49c29f20f630ce6e888feeff79b691ad9d273c0e44ebc15a98f455f1d94604ae | A $9 \times 9$ table is tiled by $26$ $(1 \times 3)$ tiles such that only $3$ squares remain uncovered. Let $N$ be the number of possible configurations of these $3$ squares, without regard to rotations. Find $N \mod 1000$. | [
"683"
] | [
"A $9 \\times 9$ table is tiled by $26$ $(1 \\times 3)$ tiles such that only $3$ squares remain uncovered. Let $N$ be the number of possible configurations of these $3$ squares, without regard to rotations. Find $N \\mod 1000$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-5058ec258e443d8d7b0792127a99b085a5083141c91d85d63b38bac163c1ebed | A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices. How many diagonals does a convex polygon with 20 sides have? | [
"170"
] | [
"A $\\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices. How many diagonals does a convex polygon with 20 sides have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6896a074a4db74c3edd8e0fea98b0c706e332fc4a60d507c9b5e9c2c75f645dd | A $\emph{hailstone}$ number $n = d_1d_2 \cdots d_k$, where $d_i$ denotes the $i$th digit in the base-$10$ representation of $n$ for $i = 1,2, \ldots,k$, is a positive integer with distinct nonzero digits such that $d_m < d_{m-1}$ if $m$ is even and $d_m > d_{m-1}$ if $m$ is odd for $m = 1,2,\ldots,k$ (and $d_0 = 0$). L... | [
"798"
] | [
"A $\\emph{hailstone}$ number $n = d_1d_2 \\cdots d_k$, where $d_i$ denotes the $i$th digit in the base-$10$ representation of $n$ for $i = 1,2, \\ldots,k$, is a positive integer with distinct nonzero digits such that $d_m < d_{m-1}$ if $m$ is even and $d_m > d_{m-1}$ if $m$ is odd for $m = 1,2,\\ldots,k$ (and $d_0... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 22
} | 41 | 0 | 0 |
omr-ws-50abaa1d1a2db6412097041b930cf764271e6547c038b090ed0489075235e036 | A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows: \begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108}... | [
"1"
] | [
"A $\\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\\frac 13$ -array is constructed as follows:\n\\begin{align*} 1 \\qquad \\frac 13\\,\\ \\qquad \\frac 19\\,\\ \\qquad \\frac 1{27} \\qquad &\\cdots\\\\ \\frac 16 \\qquad \\frac 1{18}\\,\\ \\qquad \\frac{1}{54} \\qquad &\\cdots\\... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81fa18ac528a0cedb9365e3ba08ac9e0d580af42097344b991d4e3959471917e | A $\mathrm{PCl}_{5}$ molecule consists of a $P$ atom at the centre of an equilateral triangle (side length $\sqrt{3}$ units), with $C l$ atoms at the 3 vertices. Two more $C l$ atoms lie on a line containing $P$ and perpendicular to the plane of the equilateral triangle. Consider $S=$ the set of $P-C l$ distances, and ... | [
"14"
] | [
"A $\\mathrm{PCl}_{5}$ molecule consists of a $P$ atom at the centre of an equilateral triangle (side length $\\sqrt{3}$ units), with $C l$ atoms at the 3 vertices. Two more $C l$ atoms lie on a line containing $P$ and perpendicular to the plane of the equilateral triangle. Consider $S=$ the set of $P-C l$ distance... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c11d1245e3d5bbd40de66020e62d82ee14f91f78d37d0dd45f5a66cf10df8a1e | A $\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? | [
"3"
] | [
"A $\\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a44d0767f1f36a0f64e1df71392531ffa0d47f64ee7755aba173d3edda349c36 | A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Theref... | [
"57"
] | [
"A $\\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c5ed86358eba347d35418b19f4352c8562604a276ddf05f4bc9a97db345abef | A $\textit{palindrome}$ is a number which reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 40,305 to create a palindrome? | [
"99"
] | [
"A $\\textit{palindrome}$ is a number which reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 40,305 to create a palindrome?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dae18664bc1f2ce0f5da34c3ba68029cdfc448af7df2cf6001e9c8cac1af4074 | A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$. How many $4$-digit palindromes are there? | [
"90"
] | [
"A $\\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.\n\nHow many $4$-digit palindromes are there?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-020436b0d4fefd75d3e899e3e1f80e89cd33071bb3372fc33fc27e5994dca26c | A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)? | [
"\\(3^{n + 1} - 1\\)."
] | [
"A $k \\times k$ array contains each of the numbers $1, 2, \\dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \\in \\mathbb{N}^+$)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0d9af28b99781337df2180929d1e4f7d38ed420b24f3a90c9078a1e333835c8 | A $k$-cross is defined as a geometric shape consisting of a central cube and $k-1$ cubes attached to its faces. For $k \geq 5$, determine the maximum number of $k$-crosses that can be placed in a $10 \times 10 \times 10$ cubic grid without overlapping or extending beyond the grid boundaries. | [
"125"
] | [
"A $k$-cross is defined as a geometric shape consisting of a central cube and $k-1$ cubes attached to its faces. For $k \\geq 5$, determine the maximum number of $k$-crosses that can be placed in a $10 \\times 10 \\times 10$ cubic grid without overlapping or extending beyond the grid boundaries."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9680d53faf64001e002c3c513ea84ce4cec11e6a95b05a302fdaec029f8e695e | A $k$-term arithmetic progression or $k$-AP is a sequence $a, a + d, a + 2d, \ldots, a + (k - 1)d$ of length $k$. Fix a $k$-AP in $\{1, \ldots, n\}$ and find an upper bound on the number of other $k$-APs in $\{1, \ldots, n\}$ that share at least one term with the fixed $k$-AP. You may assume that $k$ is even or $k$ is ... | [
"\\frac{2k(n - 1)}{k - 1} - 1"
] | [
"A $k$-term arithmetic progression or $k$-AP is a sequence $a, a + d, a + 2d, \\ldots, a + (k - 1)d$ of length $k$. Fix a $k$-AP in $\\{1, \\ldots, n\\}$ and find an upper bound on the number of other $k$-APs in $\\{1, \\ldots, n\\}$ that share at least one term with the fixed $k$-AP. You may assume that $k$ is eve... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-a7c039ea807fb1565dc7959181df706c550380b33fc51da07bd83ab7017f3391 | A $m \times n \times l$-cuboid ($mn > 1$, $nl > 1$, $lm > 1$) is divided into $1 \times 1 \times 1$-cubes. We place some $1 \times 1 \times 2$-cuboids in the cuboid such that each $1 \times 1 \times 2$-cuboid covers two $1 \times 1 \times 1$-cubes, and no more $1 \times 1 \times 2$-cuboids can be added. Let $x$ be the ... | [
"15"
] | [
"A $m \\times n \\times l$-cuboid ($mn > 1$, $nl > 1$, $lm > 1$) is divided into $1 \\times 1 \\times 1$-cubes. We place some $1 \\times 1 \\times 2$-cuboids in the cuboid such that each $1 \\times 1 \\times 2$-cuboid covers two $1 \\times 1 \\times 1$-cubes, and no more $1 \\times 1 \\times 2$-cuboids can be added... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f71d5ee1fc4304bb9f5c6cd5e239a1a4b685247f8bfae18c07237770333cbe4a | A $n \times n$ grid of unit squares is tiled with $3 \times 1$ rectangles such that all the tiles lie strictly inside the grid, no tile overlaps, and one of the two main diagonals of the square grid is not covered by any tile. For each $n$, let $g(n)$ denote the minimum number of empty cells in the square grid of a til... | [
"116"
] | [
"A $n \\times n$ grid of unit squares is tiled with $3 \\times 1$ rectangles such that all the tiles lie strictly inside the grid, no tile overlaps, and one of the two main diagonals of the square grid is not covered by any tile. For each $n$, let $g(n)$ denote the minimum number of empty cells in the square grid o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-074b2368490c6305c19bcee547edd66acc6960d9288a52c1169d51433a9a8d75 | A $n \times n$ square has been divided into some rectangles such that the sides of each rectangle are powers of 2. Determine the minimum number of such rectangles required to cover the entire square, and express your answer as a function of $n$. Find the value of this function when $n = 2^{10}$. | [
"1"
] | [
"A $n \\times n$ square has been divided into some rectangles such that the sides of each rectangle are powers of 2. Determine the minimum number of such rectangles required to cover the entire square, and express your answer as a function of $n$. Find the value of this function when $n = 2^{10}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b388a5b703a0825b3f8475bb2499edcfdcdd875bc1ae0c255146a040142e5ba4 | A $n \times n$ table is filled such that the first column contains $1$'s, the second column contains $2$'s, and so on. The numbers on the main diagonal (from top-left to bottom-right) are then erased. Calculate the difference between the total sums of the numbers on the upper triangle (above the main diagonal) and the ... | [
"165"
] | [
"A $n \\times n$ table is filled such that the first column contains $1$'s, the second column contains $2$'s, and so on. The numbers on the main diagonal (from top-left to bottom-right) are then erased. Calculate the difference between the total sums of the numbers on the upper triangle (above the main diagonal) an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be2c45b22d461fb8020f2a76e4b21496083b7ae2b4e65ee5971589fc1ba42154 | A $p \times p$ matrix is called "good" if its elements are the numbers $1, 2, \ldots, p^2$, each used exactly once, and if it is possible to add integers to its rows and columns to obtain a null matrix. Find the number of good matrices. | [
"(p!)^2"
] | [
"A $p \\times p$ matrix is called \"good\" if its elements are the numbers $1, 2, \\ldots, p^2$, each used exactly once, and if it is possible to add integers to its rows and columns to obtain a null matrix. Find the number of good matrices."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-be2db09af1fbdbca4db99d63cb2a3460ec955d045160076fa91c3b96f99e59d5 | A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it travelled? | [
"65"
] | [
"A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it travelled?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e897b402d105869629ee4afb3ae96f1a37d9feb20e144b57abe2287d55c62d0e | A 'triline' is a line with the property that three times its slope is equal to the sum of its $x$-intercept and its $y$-intercept. For how many integers $q$ with $1 \leq q \leq 10,000$ is there at least one integer $p$ so that there is exactly one triline through $(p,q)$? | [
"57"
] | [
"A 'triline' is a line with the property that three times its slope is equal to the sum of its $x$-intercept and its $y$-intercept. For how many integers $q$ with $1 \\leq q \\leq 10,000$ is there at least one integer $p$ so that there is exactly one triline through $(p,q)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 7
} | 52 | 0 | 0 |
omr-ws-a35057fe888f81e6cd6cb552616418f3aa59f4998124e6e399730274fcc8e19a | A (base ten) number \(N\) is formed by a strictly increasing sequence of digits. Compute the sum of the digits of \(9N\). | [
"9"
] | [
"A (base ten) number \\(N\\) is formed by a strictly increasing sequence of digits. Compute the sum of the digits of \\(9N\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67069f7854064a3f86d7b00d6831b10d6f28c4c0c06e73e03d67e36ca13daa4c | A (non-degenerate) triangle with integral sides has a perimeter of 8. Compute all possible values of its area \( K \). | [
"\\(\\sqrt{6}\\)"
] | [
"A (non-degenerate) triangle with integral sides has a perimeter of 8. Compute all possible values of its area \\( K \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2842dff31e6640e5509b68becb4997c70470bf3ec7fdbbc2fd76c4f2c812acda | A 0 or 1 is to be placed in each of the nine 1 x 1 squares in a 3 x 3 grid so that each row contains at least one 0 and at least one 1, and each column contains at least one 0 and at least one 1. How many ways can this be done? | [
"174"
] | [
"A 0 or 1 is to be placed in each of the nine 1 x 1 squares in a 3 x 3 grid so that each row contains at least one 0 and at least one 1, and each column contains at least one 0 and at least one 1. How many ways can this be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-2636fcdaa1e96d850881383baef17c1415d3e6822db4b0f3c9bdb6b1aa0c9b48 | A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence and reveal it (its place and its value) to Bob. Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the memb... | [
"\\( k+1 \\)"
] | [
"A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence and reveal it (its place and its value) to Bob. Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-1ec59c9f4319fe068cea62cf8fe8f2145a84025a68df6d7d3266d8e74e386e53 | A 1 km racetrack is to be built with two straight sides and semi-circles at the ends. Find the dimensions of the track that encloses the maximum area. Given: - Let \( y \) represent the length of one straight side. - Let \( x \) represent the radius of the semi-circles (so \( 2x \) is the width of the rectangle). Const... | [
"\\( x = \\frac{500}{\\pi}, y = 0 \\)"
] | [
"A 1 km racetrack is to be built with two straight sides and semi-circles at the ends. Find the dimensions of the track that encloses the maximum area.\n\nGiven:\n- Let \\( y \\) represent the length of one straight side.\n- Let \\( x \\) represent the radius of the semi-circles (so \\( 2x \\) is the width of the r... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 28
} | 39 | 0 | 0 |
omr-ws-533482a297eb052e5d931bbd4781ff11d2297377a2f3cac7d35a4f9eb2330f1c | A 1 m × 1 m square can be covered by three circular discs of equal size. If the diameter of each disc is 1008 mm, find the minimum radius, in millimeters, of these discs that allows the square to be completely covered. | [
"504"
] | [
"A 1 m × 1 m square can be covered by three circular discs of equal size. If the diameter of each disc is 1008 mm, find the minimum radius, in millimeters, of these discs that allows the square to be completely covered."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc6e1df7d149dae9086dec4ef828cbe7f2dab6515c7889981f0c87613dfc81d3 | A 1 meter long stick has one end touching the ground at point $P$. The sun is 30 degrees above the ground in the south direction of $P$. There is an east-west wall 2 meters north of $P$. What is the probability that part of the stick's shadow falls on the wall? | [
"\\(\\frac{2}{3}\\)"
] | [
"A 1 meter long stick has one end touching the ground at point $P$. The sun is 30 degrees above the ground in the south direction of $P$. There is an east-west wall 2 meters north of $P$. What is the probability that part of the stick's shadow falls on the wall?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cba67a71412611df0ef9d03d0c5c9338b1cb0ed83be23adf504945d4139398a | A 1$\%$ late charge was added to Jenna's bill on the $30^{\text{th}}$ day past its due date. The resulting total was then increased by 1$\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\$400$. Exactly how much is the bill now? | [
"408.04"
] | [
"A 1$\\%$ late charge was added to Jenna's bill on the $30^{\\text{th}}$ day past its due date. The resulting total was then increased by 1$\\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\\$400$. Exactly how much is the bill now?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b15a425664783864b0072d270cdd959ffaf04a836b284b54b794e58f812b4512 | A 1-inch toothpick is randomly dropped onto a plane at a random position and orientation. An infinite number of parallel lines are drawn on the plane, each separated by 1 inch. What is the probability that the toothpick intersects a line? | [
"\\(\\frac{2}{\\pi}\\)"
] | [
"A 1-inch toothpick is randomly dropped onto a plane at a random position and orientation. An infinite number of parallel lines are drawn on the plane, each separated by 1 inch. What is the probability that the toothpick intersects a line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-08c85d78ba9dce7f06d306c0601f024f59b26e9c28cf9d2254c2f87636a2ae2e | A 10 by 10 by 10 cube is composed of white and black unit cubes that are alternating colors. What is the positive difference between the number of white cubes and black cubes? | [
"0"
] | [
"A 10 by 10 by 10 cube is composed of white and black unit cubes that are alternating colors. What is the positive difference between the number of white cubes and black cubes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e3ae0665fed040a3c85d788497a9641427d008a4db106cfcb6aeb69d73955b34 | A 10 digit positive integer is called a cute number if its digits are from the set $\{1,2,3\}$ and every two consecutive digits differ by 1. a) Calculate the number of ways to arrange the digits of a cute number such that exactly 5 digits are equal to 2. b) Find the total number of cute numbers. c) Calculate the remain... | [
"0"
] | [
"A 10 digit positive integer is called a cute number if its digits are from the set $\\{1,2,3\\}$ and every two consecutive digits differ by 1.\n\na) Calculate the number of ways to arrange the digits of a cute number such that exactly 5 digits are equal to 2.\n\nb) Find the total number of cute numbers.\n\nc) Calc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-0ca06747c37f8a6133bd778cdaccc9d6b17da9aa1474b85b4f5b6d6264197e9a | A 10 inch by 10 inch square is divided into 100 unit squares. The vertices of the unit squares are colored black and white. How many colorings exist in which each 1 inch by 1 inch square has 2 white vertices and 2 black vertices? Rotation and reflections of a configuration are considered the same. | [
"2"
] | [
"A 10 inch by 10 inch square is divided into 100 unit squares. The vertices of the unit squares are colored black and white. How many colorings exist in which each 1 inch by 1 inch square has 2 white vertices and 2 black vertices? Rotation and reflections of a configuration are considered the same."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-601f20c84c3617bd407ac128c6f7ffcf544076f75b04eb6a4ac12823cc0530ad | A 10 x 10 x 10 cube is formed by small unit cubes. A grasshopper starts at the center of one of the corner cubes. At any time, it can jump to the center of any adjacent cube that increases its distance from the starting corner. How many ways can the grasshopper reach the unit cube in the opposite corner? | [
"$\\frac{27!}{(9!)^3}$"
] | [
"A 10 x 10 x 10 cube is formed by small unit cubes. A grasshopper starts at the center of one of the corner cubes. At any time, it can jump to the center of any adjacent cube that increases its distance from the starting corner. How many ways can the grasshopper reach the unit cube in the opposite corner?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4cc5675909b4a07a305ec4f7b3ab6bceee6b7d78c09a72000970e9f8feaf811d | A 10 × 10 square table is divided into 1 × 1 squares. Each square contains one of the numbers {1, 2, 3, ..., 9, 10}. Numbers in any two adjacent or diagonally adjacent squares are coprime. Let $N$ be the minimum number of times any number must appear in the table to satisfy these conditions. Calculate $N$. | [
"25"
] | [
"A 10 × 10 square table is divided into 1 × 1 squares. Each square contains one of the numbers {1, 2, 3, ..., 9, 10}. Numbers in any two adjacent or diagonally adjacent squares are coprime. Let $N$ be the minimum number of times any number must appear in the table to satisfy these conditions. Calculate $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-f9680d1b5ee4ad543d176f0df2ea329299fab84f2db601118e902fbddbf4e3de | A 10*10*10 cube is formed of small unit cubes. A grasshopper sits in the center 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 the starting point and the current position. Ho... | [
"\\dfrac{27!}{(9!)^3}"
] | [
"A 10*10*10 cube is formed of small unit cubes. A grasshopper sits in the center 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 the starting point and the current positio... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5fe0673ffbdf41c94881319b457e832d2f87d3954f84f285cae7287224813a49 | A 10-digit number has distinct digits. Using all of its digits, two new numbers are created such that their sum is 99999 and their product is the same 10-digit number. Find this 10-digit number. | [
"1234758690"
] | [
"A 10-digit number has distinct digits. Using all of its digits, two new numbers are created such that their sum is 99999 and their product is the same 10-digit number. Find this 10-digit number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-355a5d9131641531263bfaa3b4572a20d46d13587fe08fc1a3287483e4d80687 | A 10-foot ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 5 feet from the wall? | [
"\\( \\frac{2\\sqrt{3}}{3} \\)"
] | [
"A 10-foot ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 5 feet from the wall?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49edec2aaf61d112b67b67fa9e5afb4b2f9c1b6ce86b0c243b748162da5947ad | A 10-foot ladder on level ground is leaning up against a vertical wall. The base of the ladder begins to slide away from the wall at a rate of 3 ft/sec. How fast is the angle of elevation between the ladder and the ground changing when the ladder base is four feet from the wall? Leave your answer in radical form. | [
"-\\dfrac{\\sqrt{21}}{14}"
] | [
"A 10-foot ladder on level ground is leaning up against a vertical wall. The base of the ladder begins to slide away from the wall at a rate of 3 ft/sec. How fast is the angle of elevation between the ladder and the ground changing when the ladder base is four feet from the wall? Leave your answer in radical form."... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f0cfdd02da8fc63059128b9dd5e3cb4e0e7901c94e6fa330fbb310957d4e7e3f | A 10-man math team must pick categories for its members according to the following rules: - There are exactly 5 distinct categories. - Each individual must participate in exactly 3 of the categories. - Each category must have exactly 6 people from the team participating in it. In how many ways can the team arrange itse... | [
"3628800"
] | [
"A 10-man math team must pick categories for its members according to the following rules:\n- There are exactly 5 distinct categories.\n- Each individual must participate in exactly 3 of the categories.\n- Each category must have exactly 6 people from the team participating in it.\n\nIn how many ways can the team a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-97398e2e3e711855cf3bec673d25cad543f8438940956e983030c961cafdca8c | A 10-meter string is cut into two pieces. The first piece is used to form a square, and the second piece is used to form a circle. What should be the length of the piece used to form the circle in order to maximize the total area of the two geometric figures? | [
"10"
] | [
"A 10-meter string is cut into two pieces. The first piece is used to form a square, and the second piece is used to form a circle. What should be the length of the piece used to form the circle in order to maximize the total area of the two geometric figures?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-6542f4e850b1b9c2c3e3a3fcce8ac99c5c7603c68a30817599e8375989d10ddf | A 100 by 100 by 100 cube PQRSTUVW is made of 1 by 1 by 1 non-overlapping cubes. Z is a point on PS such that PZ = 33. Through how many of these 1 by 1 by 1 cubes does VZ pass? | [
"166"
] | [
"A 100 by 100 by 100 cube PQRSTUVW is made of 1 by 1 by 1 non-overlapping cubes. Z is a point on PS such that PZ = 33. Through how many of these 1 by 1 by 1 cubes does VZ pass?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f54348c8ccc7fd0741d249cc26b9f6619266ada182e5cc87a2a38a63748ba82c | A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the wa... | [
"052"
] | [
"A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start o... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-570c786d17a71a9b4f47f6d7e53e59fa1ca19d6296fe4dfde8d439d661400252 | A 100-foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after Bob, Cy reaches the start of the wal... | [
"48 feet"
] | [
"A 100-foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after Bob, Cy reaches the start of th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de2e0c18be3d0e1df7102a6eedad37c22905a4d08346c379a1c99481b060c44b | A 100-foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the wa... | [
"52"
] | [
"A 100-foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of t... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 7
} | 55 | 0 | 0 |
omr-ws-877dfc037f25e9d42d990c71e4f7c50f56d1e390bc886d5aba5433b5a1b5acb5 | A 100-lb weight is being lifted to a height of 10 ft off the ground by a rope that weighs 3 lbs/ft. How much work does it take to lift the weight, taking into account the weight of the rope? | [
"1450 ft*lbs"
] | [
"A 100-lb weight is being lifted to a height of 10 ft off the ground by a rope that weighs 3 lbs/ft. How much work does it take to lift the weight, taking into account the weight of the rope?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee2c0333122f5def7dbcbcd509375564a0d61c6f2a0416aa33c25dbeee7b5f8b | A 1001-digit number begins with 6. The number formed by any two adjacent digits is divisible by 17 or 23. Write down the last six digits. | [
"692346"
] | [
"A 1001-digit number begins with 6. The number formed by any two adjacent digits is divisible by 17 or 23. Write down the last six digits."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 16
} | 63 | 0 | 0 |
omr-ws-2d80f4e65c5bde0afca25771e7fe5f61f71ba75bee6793bfa5d29668f0b464c4 | A 100m long railroad rail lies flat along the level ground, fastened at both ends. Heat causes the rail to expand by 1% and rise into a circular arc. To the nearest meter, how far above the ground is the midpoint of the rail? | [
"6"
] | [
"A 100m long railroad rail lies flat along the level ground, fastened at both ends. Heat causes the rail to expand by 1% and rise into a circular arc. To the nearest meter, how far above the ground is the midpoint of the rail?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-a653b4d1bbb9a210e0f893373ff7b5e96ceb13810554a87b28fb3eb3427fa725 | A 100mm cube has a thin and rigid wire inserted 20mm from each of two adjacent edges. What is the largest possible radius, to the nearest mm, of a sphere that can fit inside the cube without being cut by the wire and that just touches the wire? | [
"47"
] | [
"A 100mm cube has a thin and rigid wire inserted 20mm from each of two adjacent edges. What is the largest possible radius, to the nearest mm, of a sphere that can fit inside the cube without being cut by the wire and that just touches the wire?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-8264334f40222e661fcb66e34086d457c72aa6ff60a9ea8cd3ab5efe1de082cf | A 10lb weight is hung on the lower end of a spring with a spring constant of 20 lb/ft. The weight comes to rest at equilibrium, and starting at time \( t = 0 \), an external force given by \( f(t) = 10 \cos{8t} \) is applied. The medium of resistance (in lbs) is numerically equal to 5 times the instantaneous velocity. ... | [
"\\( x(t) = \\frac{1}{4}\\sin{8t} - 2te^{-8t} \\)"
] | [
"A 10lb weight is hung on the lower end of a spring with a spring constant of 20 lb/ft. The weight comes to rest at equilibrium, and starting at time \\( t = 0 \\), an external force given by \\( f(t) = 10 \\cos{8t} \\) is applied. The medium of resistance (in lbs) is numerically equal to 5 times the instantaneous ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-6560fb9de51f9f45ba8736b0f92817fefb28e924a4ac215697e5a0acbb75d1ab | A 12 cm long wire is bent to form a triangle with one angle of $60^\circ$. Find the sides of the triangle if its area is maximized. | [
"4"
] | [
"A 12 cm long wire is bent to form a triangle with one angle of $60^\\circ$. Find the sides of the triangle if its area is maximized."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d496e5e5fac864e58c32c11d0dea947c80b6df4d3074945e8b550921e2a7984d | A 12 in. by 16 in. painting is surrounded by a frame of uniform width. The area of the frame equals the area of the painting. Find the width of the frame. | [
"\\(\\sqrt{97} - 7\\)"
] | [
"A 12 in. by 16 in. painting is surrounded by a frame of uniform width. The area of the frame equals the area of the painting. Find the width of the frame."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e95fc3af703c57a1ce3766bb57f8a5f209126198a2e8203bf04a1b3252041092 | A 12 x 12 square is divided into two pieces by joining two adjacent side midpoints. Copies of the triangular piece are placed on alternate edges of a regular hexagon, and copies of the other piece are placed on the other edges. The resulting figure is then folded to give a polyhedron with 7 faces. What is the volume of... | [
"864"
] | [
"A 12 x 12 square is divided into two pieces by joining two adjacent side midpoints. Copies of the triangular piece are placed on alternate edges of a regular hexagon, and copies of the other piece are placed on the other edges. The resulting figure is then folded to give a polyhedron with 7 faces. What is the volu... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d81cd0d88c740f6867b862c8cd2bc9be2b87a6564e9e6d41b5c84b8180a248a | A 12 x 12 x 12 cube is constructed by 1 x 1 x 1 unit cubes. A cricket is in the center \( K \) of one of the corner cubes. At noon, it jumps to the center of the other cubes which have a common face with the cube it sits, as long as its jump increases the distance from \( K \) to its current position (thus, a cricket c... | [
"\\dfrac{33!}{(11!)^3}"
] | [
"A 12 x 12 x 12 cube is constructed by 1 x 1 x 1 unit cubes. A cricket is in the center \\( K \\) of one of the corner cubes. At noon, it jumps to the center of the other cubes which have a common face with the cube it sits, as long as its jump increases the distance from \\( K \\) to its current position (thus, a ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-a2fc87dd2f36d31de38667c402a0807451be1da4d74cd190dcda90148f4f3009 | A 12-liter vessel was filled with acid. Some of the acid was poured from this vessel into another of the same capacity, and the second vessel filled with water. After this the first vessel was topped up with the solution from the second vessel. Then 4 liters of the solution was poured from the first vessel into the sec... | [
"6 liters"
] | [
"A 12-liter vessel was filled with acid. Some of the acid was poured from this vessel into another of the same capacity, and the second vessel filled with water. After this the first vessel was topped up with the solution from the second vessel. Then 4 liters of the solution was poured from the first vessel into th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-763426eeac0e6baad64d6311e5461f696f1dd29d921929b00f946ed4315e6738 | A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? | [
"4"
] | [
"A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f0aba10a920ed36c52dcc6f44394b54ba143da0a7a58b3af0316404d7197926 | A 12m ladder leans across a 5m fence and touches a high wall located 3m behind the fence. Find the distance \( X \) from the ladder base to the fence bottom. | [
"2.7"
] | [
"A 12m ladder leans across a 5m fence and touches a high wall located 3m behind the fence. Find the distance \\( X \\) from the ladder base to the fence bottom."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-c02a18c7708e55802e9c9f6d1e0942a164b5d83f4fba46a1f037afe2edc4bfb4 | A 15 × 15 square is tiled with unit squares. Each vertex is colored either red or blue. There are 133 red points, with two of those red points being corners of the original square and another 32 red points on the sides. The sides of the unit squares are colored according to the following rule: If both endpoints are red... | [
"134"
] | [
"A 15 × 15 square is tiled with unit squares. Each vertex is colored either red or blue. There are 133 red points, with two of those red points being corners of the original square and another 32 red points on the sides. The sides of the unit squares are colored according to the following rule: If both endpoints ar... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-871318c35632a3703cae58c0f694fc36541e09db00c7f4506d10f0b6a09f42ae | A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step? | [
"1698"
] | [
"A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \\le x \\le 2$, $-2 \\le y \\le 2$ at each step?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 21
} | 37 | 0 | 0 |
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