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-4cc0ec51f159fa87c663700861dbcf6c8ed8edbca44dbbb24830a403c87a92ac | A 500-foot tall pole is 1000 feet from a 400-foot tall pole. A taut wire runs from the top of each pole to the ground 1000 feet from the opposite pole. How high is the point where the wires cross? | [
"\\(\\frac{1000}{9}\\)"
] | [
"A 500-foot tall pole is 1000 feet from a 400-foot tall pole. A taut wire runs from the top of each pole to the ground 1000 feet from the opposite pole. How high is the point where the wires cross?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a9dbc33051807e9baee071b4e8ba385274c69a9497d97d8667f95f0b28c8d81 | A 51-digit number \( N \) consists of fifty \( 2 \)'s and one \( y \), where \( y \) is a digit from 0 to 9. If \( N \) is divisible by 17 and \( y \) is the 17th digit from the right, what is the value of \( y \)? | [
"1"
] | [
"A 51-digit number \\( N \\) consists of fifty \\( 2 \\)'s and one \\( y \\), where \\( y \\) is a digit from 0 to 9. If \\( N \\) is divisible by 17 and \\( y \\) is the 17th digit from the right, what is the value of \\( y \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94fcbd8fdde18535c491c23edfda49f0159ac54962c2f9c4c0e1440a8847f46b | A 55-gallon drum is designed to hold a minimum volume of 57.20 gallons. The drum is made with 18 gage steel disks (0.0428 inches thick) for the top and bottom, and a 20 gage steel sheet (0.0324 inches thick) for the cylindrical body. The pressing/sealing process requires approximately 1 inch from the cylinder and J inc... | [
"9.72"
] | [
"A 55-gallon drum is designed to hold a minimum volume of 57.20 gallons. The drum is made with 18 gage steel disks (0.0428 inches thick) for the top and bottom, and a 20 gage steel sheet (0.0324 inches thick) for the cylindrical body. The pressing/sealing process requires approximately 1 inch from the cylinder and ... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-d5bf0c25e45f860eb9e46f381b9e2192c23bab20225880f7f0f7c39a7ca5d3f6 | A 5in by 7in photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame. | [
"\\(-3 + \\frac{\\sqrt{71}}{2}\\)"
] | [
"A 5in by 7in photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-35dbe5579894b2018fafc57067a3a8d4428b318cfb1860e805ea9e400864e70f | A 5×5 square is divided into 25 unit squares. One of the numbers 1, 2, 3, 4, 5 is inserted into each of the unit squares in such a way that each row, each column, and each of the two diagonals contains each of the five numbers once and only once. The sum of the numbers in the four squares immediately below the diagonal... | [
"14"
] | [
"A 5×5 square is divided into 25 unit squares. One of the numbers 1, 2, 3, 4, 5 is inserted into each of the unit squares in such a way that each row, each column, and each of the two diagonals contains each of the five numbers once and only once. The sum of the numbers in the four squares immediately below the dia... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-4ffba6ba3ffec3d99099c3fda8333751137d7fb56f3215602e8b94b483b77d92 | A 5×5 square is divided into 25 unit squares. One of the numbers 1, 2, 3, 4, 5 is inserted into each of the unit squares in such a way that each row, each column, and each of the two diagonals contains each of the five numbers once and only once. The sum of the numbers in the four squares immediately below the diagonal... | [
"20"
] | [
"A 5×5 square is divided into 25 unit squares. One of the numbers 1, 2, 3, 4, 5 is inserted into each of the unit squares in such a way that each row, each column, and each of the two diagonals contains each of the five numbers once and only once. The sum of the numbers in the four squares immediately below the dia... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-ccad47839718a5a4a4293f9ef791de4ec97ada259bf2057cd9c532eeb5778a1e | A 6 ft tall person walks away from a 10 ft lamppost at a constant rate of 3 ft/sec. What is the rate that the tip of the shadow moves away from the post when the person is 10 ft away from the pole? | [
"5 ft/sec"
] | [
"A 6 ft tall person walks away from a 10 ft lamppost at a constant rate of 3 ft/sec. What is the rate that the tip of the shadow moves away from the post when the person is 10 ft away from the pole?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-025351804598c53e01c661b15db4310553880ce09e0af93e15e4b2a4ffb29836 | A 6*6 square is dissected into 9 rectangles by lines parallel to its sides such that all these rectangles have only integer sides. Let $S$ be the sum of the areas of all distinct rectangles (where congruent rectangles are counted only once). Find the maximum possible value of $S$. | [
"30"
] | [
"A 6*6 square is dissected into 9 rectangles by lines parallel to its sides such that all these rectangles have only integer sides. Let $S$ be the sum of the areas of all distinct rectangles (where congruent rectangles are counted only once). Find the maximum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-5d37a9c8ac10c64014fc2a41df3befefcd81f088933353e08557639934abc8fb | A 6-digit number $n$ is not a symmetric number. By changing exactly two digits, it becomes a symmetric number. How many such 6-digit numbers $n$ exist, given that the first digit of $n$ is not 0? | [
"10800"
] | [
"A 6-digit number $n$ is not a symmetric number. By changing exactly two digits, it becomes a symmetric number. How many such 6-digit numbers $n$ exist, given that the first digit of $n$ is not 0?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fae67afecd872852cd9986686b983bdc4e813b1ae301dc2b30b2060b194e91e4 | A 6-digit number \( x_1 x_2 \ldots x_6 \) where \( 0 \le x_i \le 9 \) is said to be lucky if the sum of the first three digits is equal to the sum of the last three digits. What is the probability that a randomly selected 6-digit number is lucky? | [
"0.055252"
] | [
"A 6-digit number \\( x_1 x_2 \\ldots x_6 \\) where \\( 0 \\le x_i \\le 9 \\) is said to be lucky if the sum of the first three digits is equal to the sum of the last three digits. What is the probability that a randomly selected 6-digit number is lucky?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 13
} | 26 | 0 | 0 |
omr-ws-ed9037c0491301c4e9243d95423265b94b23f121b52fab847a642423e7c4874f | A 6-digit number has its first digit as 9. If you move this digit to the last position, the resulting number is one fourth the size of the original number. What is the sum of the digits of the original number? | [
"27"
] | [
"A 6-digit number has its first digit as 9. If you move this digit to the last position, the resulting number is one fourth the size of the original number. What is the sum of the digits of the original number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-0703fb8a4d6f3b9f22e919f10553a26c434221f8487163c5e4f975a121d66f03 | A 6-digit number is in the form \( ABCDE6 \). Let this number be \( x \). If 4 times \( x \) is equal to \( 6abcde \), find all possible values of \( x \). | [
"153846"
] | [
"A 6-digit number is in the form \\( ABCDE6 \\). Let this number be \\( x \\). If 4 times \\( x \\) is equal to \\( 6abcde \\), find all possible values of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-2391da94dd1c07045158a9a3fcd47fa96392cd044797493f9a2015f169299d30 | A 6-digit palindrome \( n \) is chosen uniformly at random. What is the probability that \( \frac{n}{11} \) is also a palindrome? | [
"\\(\\frac{11}{30}\\)"
] | [
"A 6-digit palindrome \\( n \\) is chosen uniformly at random. What is the probability that \\( \\frac{n}{11} \\) is also a palindrome?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-17423ebfe6505de874bea4d1f4bcd74a53af19533af1f4a51a7e542e380edd96 | A 6-foot man standing in front of a 15-foot lamppost has a shadow of 8 feet. If he steps another 3 feet away from the lamppost, how much will his shadow lengthen? | [
"2"
] | [
"A 6-foot man standing in front of a 15-foot lamppost has a shadow of 8 feet. If he steps another 3 feet away from the lamppost, how much will his shadow lengthen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a1372287e6fb855840601d9a13cbe30dd8b4861b62b665982a247165f546a8ba | A 6-letter word is formed using 16 letters of the English alphabet, with repetition allowed. Find the probability that exactly 2 letters are repeated. | [
"\\(\\frac{\\binom{16}{4}\\binom{4}{2}\\frac{6!}{2!2!}+\\binom{16}{3}\\binom{3}{2}\\binom 21\\frac{6!}{3!2!}+\\binom{16}{2}\\binom{2}{1}\\frac{6!}{4!2!}+\\binom{16}{2}\\frac{6!}{3!3!}}{16^6}\\)"
] | [
"A 6-letter word is formed using 16 letters of the English alphabet, with repetition allowed. Find the probability that exactly 2 letters are repeated."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-337b0e2551c3fb3d4db10228494e9eb46e1865882a2142fe69f500a0e662e5a1 | A 6-meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach? | [
"\\(\\frac{12\\sqrt{5}}{5}\\)"
] | [
"A 6-meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-68004ff7996cdabf9e142e839c0e68b5f05299fa652b459f3208e9d5d391fdb2 | A 6-sided die is rolled 600 times. What is the expected value of the longest streak of a number being rolled consecutively? | [
"7"
] | [
"A 6-sided die is rolled 600 times. What is the expected value of the longest streak of a number being rolled consecutively?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-9f60d9b2437cb03e88b25a54f4ecff4817d0a4d6a5008a4f53cc61ba921f7090 | A 7 meter long fallen tree is being cut up by the logging company, Logarithm Co. The time, \( t \), in minutes that it takes to saw through the tree trunk increases as the radius, \( r \), in centimeters, of the trunk increases according to the formula \( t = \frac{r}{2} + 1 \). The radius of the trunk increases as the... | [
"29.5"
] | [
"A 7 meter long fallen tree is being cut up by the logging company, Logarithm Co. The time, \\( t \\), in minutes that it takes to saw through the tree trunk increases as the radius, \\( r \\), in centimeters, of the trunk increases according to the formula \\( t = \\frac{r}{2} + 1 \\). The radius of the trunk incr... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a63a1b105a963b42817b806181602fb271cc859fdc634e49c4596e7f77b0403 | A 7-card hand is dealt from a standard deck of 52 cards. What is the probability that the hand will contain 3 clubs and 4 red cards? | [
"\\dfrac{715}{22372}"
] | [
"A 7-card hand is dealt from a standard deck of 52 cards. What is the probability that the hand will contain 3 clubs and 4 red cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ae5b1da8819cc9ea4adb809cc1300f157924dee565915239802b82ac66b712d | A 70 cm long loop of string is to be arranged into a shape consisting of two adjacent squares, with the side of the smaller square lying entirely within the side of the larger one. What is the minimum area of the resulting shape, in square centimetres? | [
"245"
] | [
"A 70 cm long loop of string is to be arranged into a shape consisting of two adjacent squares, with the side of the smaller square lying entirely within the side of the larger one. What is the minimum area of the resulting shape, in square centimetres?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-6f62a91d650e473ad143d85aaf65ab9aca74a4036d32eac991bb756476b35bb0 | A 70-foot pole stands vertically in a plane supported by three 490-foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of these anchor points? | [
"240"
] | [
"A 70-foot pole stands vertically in a plane supported by three 490-foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of these anchor points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f905b7015aaebbcc0eea3369b217dc927296f3820a01c8b6eb8ebf9e3c2edc1b | A 7x7 square is made up of 16 '1x3' tiles and 1 '1x1' tile. If the 1x1 tile does not lie at the center of the square, calculate the sum of all possible Manhattan distances from the 1x1 tile to the center of the square. | [
"60"
] | [
"A 7x7 square is made up of 16 '1x3' tiles and 1 '1x1' tile. If the 1x1 tile does not lie at the center of the square, calculate the sum of all possible Manhattan distances from the 1x1 tile to the center of the square."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe0bfd262f27c3ebadf8307312ef19aa10eb6e346f561131dee517958eda92e4 | A 7x7x7 cube is cut into smaller cubes of sizes 4x4x4, 3x3x3, 2x2x2, or 1x1x1. What is the minimum number of smaller cubes that must be cut out? | [
"20"
] | [
"A 7x7x7 cube is cut into smaller cubes of sizes 4x4x4, 3x3x3, 2x2x2, or 1x1x1. What is the minimum number of smaller cubes that must be cut out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-78558b2e2ca55eb866cc56b4c61dd12fc8c91fb0a451bb2ac2156406166a2061 | A 7×7 table is to be covered with rectangles of size 1×4, leaving one square uncovered. Where can this square be located? | [
"(i, j) \\text{ where } (i + j) \\equiv 0 \\pmod{4}"
] | [
"A 7×7 table is to be covered with rectangles of size 1×4, leaving one square uncovered. Where can this square be located?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-4126b240aca5c9ac2c720e1a2d112be6ed0b6198d77796aa2aac6095187f27c9 | A 95% confidence interval for the mean of a population is to be constructed and must be accurate within 0.3 units. A preliminary sample standard deviation is 1.4. What is the smallest sample size \( n \) that provides the desired accuracy? | [
"84"
] | [
"A 95% confidence interval for the mean of a population is to be constructed and must be accurate within 0.3 units. A preliminary sample standard deviation is 1.4. What is the smallest sample size \\( n \\) that provides the desired accuracy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2004ad5c2b87b8b8e20f49a5006ff927b4ab2a86340e7704407392d2e07dde1c | A 98 x 98 chessboard has squares colored alternately black and white. A move consists of selecting a rectangular subset of the squares (with boundaries parallel to the sides of the board) and changing their color. What is the smallest number of moves required to make all the squares black? | [
"98"
] | [
"A 98 x 98 chessboard has squares colored alternately black and white. A move consists of selecting a rectangular subset of the squares (with boundaries parallel to the sides of the board) and changing their color. What is the smallest number of moves required to make all the squares black?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-5a0bb2ed66d2b6d1230f8add08ddac4a4cb5798e5c6fe77bdef4f63b0b9e2553 | A 999-digit number \( N \) starts with 9. Every block of 2 consecutive digits forms an integer that is divisible by 17 or 23. There are two possibilities for the number formed by the last 3 digits of \( N \). What is the sum of these two possibilities? | [
"751"
] | [
"A 999-digit number \\( N \\) starts with 9. Every block of 2 consecutive digits forms an integer that is divisible by 17 or 23. There are two possibilities for the number formed by the last 3 digits of \\( N \\). What is the sum of these two possibilities?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 7
} | 24 | 0 | 0 |
omr-ws-e86d12fd9e9762f4cad18416e8d1559964c8c69fc8c458c540e7a8ba09e23882 | A 99×99 table is covered with 1×3 dominoes. Each domino is labeled with the number of the row if it is horizontal or the number of the column if it is vertical. Calculate the remainder when the sum of these numbers is divided by 3. | [
"0"
] | [
"A 99×99 table is covered with 1×3 dominoes. Each domino is labeled with the number of the row if it is horizontal or the number of the column if it is vertical. Calculate the remainder when the sum of these numbers is divided by 3."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c0502e108692c650f03456ae59b4410a8fd3524cbd700ed0b0c52e317e2dcf7 | A 9x9x9 cube is composed of twenty-seven 3x3x3 cubes. The big cube is "tunneled" as follows: First, the six 3x3x3 cubes which make up the center of each face as well as the center 3x3x3 cube are removed. Second, each of the twenty remaining 3x3x3 cubes is diminished in the same way, with the center facial unit cubes as... | [
"864"
] | [
"A 9x9x9 cube is composed of twenty-seven 3x3x3 cubes. The big cube is \"tunneled\" as follows: First, the six 3x3x3 cubes which make up the center of each face as well as the center 3x3x3 cube are removed. Second, each of the twenty remaining 3x3x3 cubes is diminished in the same way, with the center facial unit c... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6de7edbd20e5d0630ebae5f007e77bc2c25670356bb395cb22eec135535ba27a | A Bidiophantine triangle is a triangle with integer side lengths $a$, $b$, and $c$ where each side length is greater than 1 and the product of any two sides minus the square of the remaining side is a perfect square. Find the smallest possible perimeter of such a Bidiophantine triangle. | [
"6"
] | [
"A Bidiophantine triangle is a triangle with integer side lengths $a$, $b$, and $c$ where each side length is greater than 1 and the product of any two sides minus the square of the remaining side is a perfect square. Find the smallest possible perimeter of such a Bidiophantine triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0291b59eacf1eefad3ae2dc296472b38f5fc1daf8eb2bc55678634752b1668c | A Bidiophantine triangle is a triangle with integer side lengths \(a\), \(b\), and \(c\) such that the area of the triangle is also an integer. Let \(T\) be the set of all Bidiophantine triangles where each side length is greater than 2. For each triangle in \(T\), calculate the sum of the reciprocals of the side lengt... | [
"0"
] | [
"A Bidiophantine triangle is a triangle with integer side lengths \\(a\\), \\(b\\), and \\(c\\) such that the area of the triangle is also an integer. Let \\(T\\) be the set of all Bidiophantine triangles where each side length is greater than 2. For each triangle in \\(T\\), calculate the sum of the reciprocals of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-8317b5af230a5f74952685cf412f6444933be9aa80069e4456c78c48dead1520 | A Boy Scouts camp has scarfs of \( m \) colors with \( n \) scarves of each color, and gives each of its \( mn \) scouts a scarf. The scouts are initially seated in a circle such that scouts in the same troop (same scarf color) are seated next to each other. The camp organizer selects representatives to perform a show ... | [
"m(2^n - n - 1)"
] | [
"A Boy Scouts camp has scarfs of \\( m \\) colors with \\( n \\) scarves of each color, and gives each of its \\( mn \\) scouts a scarf. The scouts are initially seated in a circle such that scouts in the same troop (same scarf color) are seated next to each other. The camp organizer selects representatives to perf... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-04014e33e4512da5e25f56a3a286765a314c8df6b919a50cf39ead431dd8c2ad | A Chinese restaurant has many boxes of food. The manager notices that: - He can divide the boxes into groups of \( M \) where \( M \) is 19, 20, or 21. - There are exactly 3 integers \( x \) less than 16 such that grouping the boxes into groups of \( x \) leaves 3 boxes left over. Find the smallest possible number of b... | [
"39900"
] | [
"A Chinese restaurant has many boxes of food. The manager notices that:\n- He can divide the boxes into groups of \\( M \\) where \\( M \\) is 19, 20, or 21.\n- There are exactly 3 integers \\( x \\) less than 16 such that grouping the boxes into groups of \\( x \\) leaves 3 boxes left over.\nFind the smallest poss... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-db5b1e303bfe63c71045b28070f3264b3f92a9f1ed602c6b406538be4df10d89 | A Coast Guard cutter detects an unidentified ship at a distance of $25$ km in the direction $20^{\circ}$ east of north. The ship is traveling at $30$ km/h on a course at $35^{\circ}$ east of north. The Coast Guard wishes to send a speedboat to intercept the vessel and investigate it. If the speedboat travels $60$ km/h,... | [
"\\(20^{\\circ} + \\sin^{-1}\\left(\\frac{\\sin 15^{\\circ}}{2}\\right)\\) east of north."
] | [
"A Coast Guard cutter detects an unidentified ship at a distance of $25$ km in the direction $20^{\\circ}$ east of north. The ship is traveling at $30$ km/h on a course at $35^{\\circ}$ east of north. The Coast Guard wishes to send a speedboat to intercept the vessel and investigate it. If the speedboat travels $60... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e2c32f819bf509133e0a121c0214e549bdc02bcef5cde68bd71f22e7d0269f9 | A Colorado beetle runs along the propeller blade of a helicopter from its axis at a constant speed $u$. The propeller rotates at a constant angular velocity $w$. Determine the length of the beetle's trajectory during one complete revolution of the propeller, given that the beetle starts at the propeller axis. | [
"\\dfrac{u}{w} \\left( \\pi \\sqrt{1 + 4\\pi^2} + \\dfrac{1}{2} \\ln\\left(2\\pi + \\sqrt{4\\pi^2 + 1}\\right) \\right)"
] | [
"A Colorado beetle runs along the propeller blade of a helicopter from its axis at a constant speed $u$. The propeller rotates at a constant angular velocity $w$. Determine the length of the beetle's trajectory during one complete revolution of the propeller, given that the beetle starts at the propeller axis."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-6bf3460eb891974a984d082f1309f81a74f4d6f46198db8a81019c1004517c68 | A DJ has 7 songs named 1, 2, 3, 4, 5, 6, and 7. He decides that no two even-numbered songs can be played one after the other. In how many different orders can the DJ play the 7 songs? | [
"1440"
] | [
"A DJ has 7 songs named 1, 2, 3, 4, 5, 6, and 7. He decides that no two even-numbered songs can be played one after the other. In how many different orders can the DJ play the 7 songs?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba92861be48a5d3937ecc9859fb7602c8a477911005d6d1d2b57325201b9db2e | A Dyck path of length $2n$ ($n\in\mathbb{Z}^{+}$) is a sequence of $n$ U's and $n$ D's such that the number of D's never exceeds the number of U's at any point in the sequence (when read from left to right). A Dyck path $Q$ contains a Dyck path $S$ if $S$ is a subsequence of $Q$. A Dyck path is good if it avoids the su... | [
"000"
] | [
"A Dyck path of length $2n$ ($n\\in\\mathbb{Z}^{+}$) is a sequence of $n$ U's and $n$ D's such that the number of D's never exceeds the number of U's at any point in the sequence (when read from left to right). A Dyck path $Q$ contains a Dyck path $S$ if $S$ is a subsequence of $Q$. A Dyck path is good if it avoids... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-ecc8d79b0b80d1cb6493c595d835ccfdfb77ad6b7e2d989962f604516ac0d23b | A Farmer has fencing of length $80$ m which he uses to make a rectangular pen for his sheep. Since the pen is against a long wall, he only fences $3$ sides. Let $x$ be the length of the rectangular pen. Find the value of $x$ that maximizes the area $A$. | [
"20"
] | [
"A Farmer has fencing of length $80$ m which he uses to make a rectangular pen for his sheep. Since the pen is against a long wall, he only fences $3$ sides. Let $x$ be the length of the rectangular pen. Find the value of $x$ that maximizes the area $A$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c80cea2126366c7f9df21e606999ce3c41d7557e659e906c7599bacc5ee5b89 | A Farmer has fencing of length $80$ m which he uses to make a rectangular pen for his sheep. Since the pen is against a long wall, he only fences $3$ sides. Let $x$ be the length of the rectangular pen. Write down the area $A$ in terms of $x$. | [
"\\( -2x^2 + 80x \\)"
] | [
"A Farmer has fencing of length $80$ m which he uses to make a rectangular pen for his sheep. Since the pen is against a long wall, he only fences $3$ sides. Let $x$ be the length of the rectangular pen. Write down the area $A$ in terms of $x$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f897152319395ae6bbc27fe95729f0bb16ab3d6c2d5eaa1d026701f708357259 | A Fermi Question is a question where you need to make educated guesses in order to estimate the order of magnitude of a certain quantity. Estimate the volume of the Sun compared to that of a Hindenburg class airship. You need to give your answer as the closest smaller power of 10, i.e., if your answer is 999 then you s... | [
"\\(10^{12}\\)"
] | [
"A Fermi Question is a question where you need to make educated guesses in order to estimate the order of magnitude of a certain quantity. Estimate the volume of the Sun compared to that of a Hindenburg class airship. You need to give your answer as the closest smaller power of 10, i.e., if your answer is 999 then ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-5fd1393ad42e525cd0417a641932ac2a19a14d10e0f780c2467f1597d9e5e645 | A Fermi question is a question where you need to make educated guesses in order to estimate the order of magnitude of a certain quantity. Estimate the order of magnitude of how many times the mass of Earth is bigger than that of an adult Asian elephant. Give your answer as the multiplicatively closest power of $10$, i.... | [
"10^{21}"
] | [
"A Fermi question is a question where you need to make educated guesses in order to estimate the order of magnitude of a certain quantity. Estimate the order of magnitude of how many times the mass of Earth is bigger than that of an adult Asian elephant. Give your answer as the multiplicatively closest power of $10... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-463ade9616c58c2de78646d7b67ef5b76fcd29abc8034be7ab06cb91f7cd14f3 | A Ferris wheel has a radius of 20 feet and completes one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point 10 vertical feet above the bottom? | [
"10"
] | [
"A Ferris wheel has a radius of 20 feet and completes one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point 10 vertical feet above the bottom?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4ed4d31cf6979f605cd824106ac2e333dcaac29340681ca6dcae3ad2ffaa955 | A Ferris wheel has a radius of 20 meters and completes one revolution every 24 minutes. The Ferris wheel is supported by two pillars, each intersecting at the center of the Ferris wheel and slanted at an elevation of 54.14°. The distance between the bases of the pillars is 60 meters. Create a sine function that gives t... | [
"h(t) = 20 \\sin\\left(\\frac{\\pi}{12}(t - 6)\\right) + 41.5"
] | [
"A Ferris wheel has a radius of 20 meters and completes one revolution every 24 minutes. The Ferris wheel is supported by two pillars, each intersecting at the center of the Ferris wheel and slanted at an elevation of 54.14°. The distance between the bases of the pillars is 60 meters. Create a sine function that gi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f69562ca1973093069524bbabd6520fe645ff84d518a79c1b3feb93f3df117e1 | A Ferris wheel with a diameter of 24 m rotates at a rate of 3 minutes per revolution. Riders board the Ferris wheel 2 m above the ground. A couple boards the Ferris wheel and the ride starts. a) How far above the ground is the couple 30 seconds after the ride starts? | [
"8"
] | [
"A Ferris wheel with a diameter of 24 m rotates at a rate of 3 minutes per revolution. Riders board the Ferris wheel 2 m above the ground. A couple boards the Ferris wheel and the ride starts.\n\na) How far above the ground is the couple 30 seconds after the ride starts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-71c4838ec894f93f91eff767db19b07799d3b9d479b9280a2f32a7bce7b228c3 | A Ferris wheel with a diameter of 24 m rotates at a rate of 3 minutes per revolution. Riders board the Ferris wheel 2 m above the ground. A couple boards the Ferris wheel and the ride starts. b) How many seconds will it be (after the ride starts) before the couple is 17 m above the ground for the second time? | [
"128"
] | [
"A Ferris wheel with a diameter of 24 m rotates at a rate of 3 minutes per revolution. Riders board the Ferris wheel 2 m above the ground. A couple boards the Ferris wheel and the ride starts.\n\nb) How many seconds will it be (after the ride starts) before the couple is 17 m above the ground for the second time?"
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c21becc05b8d6f5a391ca5f702d990db3dfa617ee182e06733741cc993a6148e | A Ferris wheel with a diameter of 40 meters has its center 21 meters above the ground. The wheel rotates once every 30 seconds. Starting from the lowest position, draw a graph showing a rider's height above the ground during a 4-minute ride and write an equation for the graph. | [
"\\( y = -20 \\cos \\left( \\frac{\\pi}{15} t \\right) + 21 \\)"
] | [
"A Ferris wheel with a diameter of 40 meters has its center 21 meters above the ground. The wheel rotates once every 30 seconds. Starting from the lowest position, draw a graph showing a rider's height above the ground during a 4-minute ride and write an equation for the graph."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-bc3e2254cc209b34217854a6630d33e92da85dc7b97898b8d4f51bb39c133d00 | A Ferris wheel with a radius of 12 meters rotates clockwise at a rate of 10 degrees per second. Initially, Richard is at the very top of the wheel, which is 27 meters above the ground. How high above the ground will Richard be after the wheel has rotated for exactly one minute? | [
"9 meters"
] | [
"A Ferris wheel with a radius of 12 meters rotates clockwise at a rate of 10 degrees per second. Initially, Richard is at the very top of the wheel, which is 27 meters above the ground. How high above the ground will Richard be after the wheel has rotated for exactly one minute?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba95fd7841fe8a83af0ec6fb14d26f3794498bc56aa8ad7bc5a2d03c5dc71eb7 | A Ferris wheel with a radius of 25 meters makes one rotation every 36 seconds. At the bottom of the ride, the passenger is 1 meter above the ground. a) Determine the height \( h \) as a function of time if \( h = 51 \) meters at \( t = 0 \). b) Find the height \( h \) after 45 seconds. | [
"26"
] | [
"A Ferris wheel with a radius of 25 meters makes one rotation every 36 seconds. At the bottom of the ride, the passenger is 1 meter above the ground.\na) Determine the height \\( h \\) as a function of time if \\( h = 51 \\) meters at \\( t = 0 \\).\nb) Find the height \\( h \\) after 45 seconds."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d347b08cf7e511082e0ba0bf80b201616a46912a55114fc4514f4a82f887cd0 | A Fibonacci Spiral is constructed by adding quarter-circles of radius \( n \), where each \( n \) is a term of the Fibonacci series. What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area 714, whose side lengths are terms in the Fibonacci series? | [
"27\\pi"
] | [
"A Fibonacci Spiral is constructed by adding quarter-circles of radius \\( n \\), where each \\( n \\) is a term of the Fibonacci series. What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area 714, whose side lengths are terms in the Fibonacci series?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-6a63c7b7bf439de8f22e23a98b0e463bce9d772b72c2570314d486fb41ec6678 | A Fibonacci sequence is defined by $F_0 = F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n = 2, 3, 4, \ldots$. Find the value of $S$ such that $S = \sum_{n=0}^{2019} \left\lfloor \frac{F_{n+3}}{F_n} \right\rfloor$. | [
"8080"
] | [
"A Fibonacci sequence is defined by $F_0 = F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n = 2, 3, 4, \\ldots$. Find the value of $S$ such that $S = \\sum_{n=0}^{2019} \\left\\lfloor \\frac{F_{n+3}}{F_n} \\right\\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8d49a2e6a87fc11c3d8580e40e6f56bebb1c7efb05ed8f3dab54ae328f6fea0b | A French Club collected the same amount from each student going on a trip to Le Cercle Moliere in Winnipeg. When six students could not go, each of the remaining students was charged an extra $3. If the total cost was $540, how many students went on the trip? | [
"30"
] | [
"A French Club collected the same amount from each student going on a trip to Le Cercle Moliere in Winnipeg. When six students could not go, each of the remaining students was charged an extra $3. If the total cost was $540, how many students went on the trip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-faffb1f8177325fc4335c6c3d240848a07d190c47020e33ce6e4770f94b3db5e | A GMO orange is a perfect sphere with radius 12 cm. If the fruit is cut vertically into 6 equivolume sections, each cut passing through the center of the orange, then what is the union of the total surface area of the resulting pieces? | [
"864\\pi"
] | [
"A GMO orange is a perfect sphere with radius 12 cm. If the fruit is cut vertically into 6 equivolume sections, each cut passing through the center of the orange, then what is the union of the total surface area of the resulting pieces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-cb40ff96398c91fdcf4b5a4cbef1eb1fa3f091f13b0bc9cc2bac74532e6081df | A Gaussian integer \(a+bi\) (where \(a, b > 0\)) divides both \(8-i\) and \(11+3i\). Additionally, any other Gaussian integer that divides both \(8-i\) and \(11+3i\) also divides \(a+bi\). What is \(a+b\)? | [
"3"
] | [
"A Gaussian integer \\(a+bi\\) (where \\(a, b > 0\\)) divides both \\(8-i\\) and \\(11+3i\\). Additionally, any other Gaussian integer that divides both \\(8-i\\) and \\(11+3i\\) also divides \\(a+bi\\). What is \\(a+b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6cf1bb6bf3882d465e25251f179430f66e6c18177eac79c5ef3d6cd768b6a28d | A Habitat for Humanity crew can build a house in 3 days less time than it takes a local construction company. When both crews work together, a house is built in 2 days. How long does it take each crew working alone? | [
"6 days (local construction company), 3 days (Habitat for Humanity crew)"
] | [
"A Habitat for Humanity crew can build a house in 3 days less time than it takes a local construction company. When both crews work together, a house is built in 2 days. How long does it take each crew working alone?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-d5c387e50434679cdaaaa414f4a0c5d4b85d3f5bc04aff539bcd6ca08839de0c | A Heron triangle is a triangle with integer sides and integer area. Let $T_n$ denote the $n$-th primitive Heron triangle with two sides having a common factor of 5, ordered by increasing perimeter. Find the smallest perimeter of $T_5$. | [
"90"
] | [
"A Heron triangle is a triangle with integer sides and integer area. Let $T_n$ denote the $n$-th primitive Heron triangle with two sides having a common factor of 5, ordered by increasing perimeter. Find the smallest perimeter of $T_5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4bf55a135ad029a430a2d7b0117789ef5b3afd1cc20a6752d9ce365fef6a937 | A Hotel corridor has 10 rooms on each side. A math olympiad team leader wants to reserve 7 rooms such that, on each side of the corridor, no two rooms are adjacent. In how many ways can the reservation be made? | [
"4352"
] | [
"A Hotel corridor has 10 rooms on each side. A math olympiad team leader wants to reserve 7 rooms such that, on each side of the corridor, no two rooms are adjacent. In how many ways can the reservation be made?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 14
} | 62 | 0 | 0 |
omr-ws-fb993b50c94b0b84867a23ba3726d92c7f627dca2f69e748d3ada629bb76d2c3 | A King is placed in the middle of a 15x15 chess board. He can move in any direction (forward, backward, left, right, or diagonally) but only one square at a time. He starts moving from his initial position and returns to the middle point in 6 steps. How many different ways can he return to the starting point? | [
"5120"
] | [
"A King is placed in the middle of a 15x15 chess board. He can move in any direction (forward, backward, left, right, or diagonally) but only one square at a time. He starts moving from his initial position and returns to the middle point in 6 steps. How many different ways can he return to the starting point?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3d25fa3e9720d722adff76e15204bf1e4648729382bf7e1a38a0e0cdd1924fd | A Korean restaurant offers one kind of soup each day, which can be fish soup, beef soup, or ginseng chicken soup. The restaurant will not offer ginseng chicken soup three days in a row. Determine the number of different seven-day menus. | [
"1892"
] | [
"A Korean restaurant offers one kind of soup each day, which can be fish soup, beef soup, or ginseng chicken soup. The restaurant will not offer ginseng chicken soup three days in a row. Determine the number of different seven-day menus."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-cfec12a73871e80eb23e922d1590fc094b8ee8cff8dc7165da5538e62ea3aa17 | A Mathematical Challenge consists of five problems, each worth a different whole number of marks. Carl solved all five problems correctly. He scored 10 marks for the two problems with the lowest marks and 18 marks for the two problems with the highest number of marks. How many marks did he score for all five problems? | [
"35"
] | [
"A Mathematical Challenge consists of five problems, each worth a different whole number of marks. Carl solved all five problems correctly. He scored 10 marks for the two problems with the lowest marks and 18 marks for the two problems with the highest number of marks. How many marks did he score for all five probl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-dda965bfb0c25be9bd38b123f0c4f8fe8225deec58cfd6296e4683580fbba78c | A Mathematics test consists of 3 problems, each worth an integral number of marks between 1 and 10 inclusive. Each student scores more than 15 marks, and for any two students, they obtain different numbers of marks for at least one problem. Find the maximum number of students. | [
"575"
] | [
"A Mathematics test consists of 3 problems, each worth an integral number of marks between 1 and 10 inclusive. Each student scores more than 15 marks, and for any two students, they obtain different numbers of marks for at least one problem. Find the maximum number of students."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-a178d028bab21537596c9693fb0f0f9330487a2c9d4984267bf2650e9b3c69e7 | A Mega Kangaskhan card does 100 base damage and increases its damage by flipping a coin until tails appears. For each heads, the damage increases by 30. What is the expected value of the damage? | [
"130"
] | [
"A Mega Kangaskhan card does 100 base damage and increases its damage by flipping a coin until tails appears. For each heads, the damage increases by 30. What is the expected value of the damage?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-31ad65967375102b9a49d90dfb98d8111ea0893175e927efe61f93b8cc16975a | A Merida path of order $2n$ is a lattice path in the first quadrant of the $xy$-plane joining $(0,0)$ to $(2n,0)$ using steps $U(1,1)$, $D(1,-1)$, and $L(2,0)$. An 'ascent' in a Merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of Merida paths of order $2n$ with exactl... | [
"$S(n, 1) = 2^{n + 1} - n - 2$ and $S(n, n - 1) = n^2$"
] | [
"A Merida path of order $2n$ is a lattice path in the first quadrant of the $xy$-plane joining $(0,0)$ to $(2n,0)$ using steps $U(1,1)$, $D(1,-1)$, and $L(2,0)$. An 'ascent' in a Merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of Merida paths of order $2n$ with e... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c70306483b880ec722a3f25189d953ee45668e4d815c5a36d0edf8570a056fb0 | A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. For example, since $2^3 - 1 = 7$, and 3 is a prime number, 7 is a Mersenne prime. What is the largest Mersenne prime less than 200? | [
"127"
] | [
"A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. For example, since $2^3 - 1 = 7$, and 3 is a prime number, 7 is a Mersenne prime. What is the largest Mersenne prime less than 200?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-19ba9bd354360c7ded80f4a60ea84f3ff84d4852022f9637a66a8fa92f27a39c | A Münchhausen number is a number that is equal to the sum of its digits each raised to the power of itself. For example, $3435 = 3^3 + 4^4 + 3^3 + 5^5$. Find the sum of all Münchhausen numbers. | [
"3436"
] | [
"A Münchhausen number is a number that is equal to the sum of its digits each raised to the power of itself. For example, $3435 = 3^3 + 4^4 + 3^3 + 5^5$. Find the sum of all Münchhausen numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-ac00ea6bfb881d0d11b09163b74bba3c51f51c096eb6daa949760c48ca5029d9 | A Nishop moves like a knight on odd-numbered turns and like a bishop on even-numbered turns. Starting from the bottom-left square of a \( 3 \times 3 \) chessboard, how many ways can it travel to touch each square exactly once? | [
"4"
] | [
"A Nishop moves like a knight on odd-numbered turns and like a bishop on even-numbered turns. Starting from the bottom-left square of a \\( 3 \\times 3 \\) chessboard, how many ways can it travel to touch each square exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-59a35fc1b31aa838a09178a7d6bffaa164196e5ba8dc886e49823e3d55d7f008 | A Norman window consists of a half-circle on top of a rectangle. The vertical side of the rectangular part is r feet, the radius of the circular part is also r feet, the perimeter of the window is p feet, and the area of the window is p square feet. Determine the radius and perimeter of the window. | [
"\\( r = 2 \\), \\( P = 2\\pi + 8 \\)"
] | [
"A Norman window consists of a half-circle on top of a rectangle. The vertical side of the rectangular part is r feet, the radius of the circular part is also r feet, the perimeter of the window is p feet, and the area of the window is p square feet. Determine the radius and perimeter of the window."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-81946bc6fad7826a546547ed64c69a4a40c82a5603694ed53ec65bb3807282b9 | A Norman window has the shape of a rectangle surmounted by a semicircle whose diameter is the same as the length of the bottom side of the rectangle. If the perimeter of the Norman window is 20 feet, what should the length of the bottom side be to maximize the area of the window? | [
"\\( \\frac{20}{4 + \\pi} \\)"
] | [
"A Norman window has the shape of a rectangle surmounted by a semicircle whose diameter is the same as the length of the bottom side of the rectangle. If the perimeter of the Norman window is 20 feet, what should the length of the bottom side be to maximize the area of the window?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-29d254eccdd58d4ea9d69abc1232d552ec16033c7dc077efbb28f80b494d17cb | A PUMaC grader is grading the submissions of forty students \( s_1, s_2, \cdots, s_{40} \) for the individual finals round, which has three problems. After grading a problem of student \( s_i \), the grader either grades another problem of the same student or grades the same problem of the student \( s_{i-1} \) or \( s... | [
"88"
] | [
"A PUMaC grader is grading the submissions of forty students \\( s_1, s_2, \\cdots, s_{40} \\) for the individual finals round, which has three problems. After grading a problem of student \\( s_i \\), the grader either grades another problem of the same student or grades the same problem of the student \\( s_{i-1}... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-ee0bcc387f765d9b8d1f2abc23ad6d64256b711fbc2f4d26a4c5a00145eb9e82 | A PUMaC grader is grading the submissions of forty students \( s_1, s_2, \ldots, s_{40} \) for the individual finals round, which has three problems. After grading a problem of student \( s_i \), the grader either grades another problem of the same student or grades the same problem of the student \( s_{i-1} \) or \( s... | [
"88"
] | [
"A PUMaC grader is grading the submissions of forty students \\( s_1, s_2, \\ldots, s_{40} \\) for the individual finals round, which has three problems. After grading a problem of student \\( s_i \\), the grader either grades another problem of the same student or grades the same problem of the student \\( s_{i-1}... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-d58873bcc40e4ee8d10e90b55c8829f749d33a1fd8301e572e94bfc3413a73ec | A Pokemon Go player starts at $(0,0)$ and takes steps of length $1$ unit in the north, south, east, or west direction such that each move after the first is perpendicular to the move before it. The player eventually returns to $(0, 0)$ without visiting any other point twice. Let $n$ be the number of steps taken when th... | [
"504"
] | [
"A Pokemon Go player starts at $(0,0)$ and takes steps of length $1$ unit in the north, south, east, or west direction such that each move after the first is perpendicular to the move before it. The player eventually returns to $(0, 0)$ without visiting any other point twice. Let $n$ be the number of steps taken wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7257b233fdb4581a1e7f0da987506755e0896098742cafc15cdfbca939ebf4b0 | A Rational Unit Jumping Frog starts at $(0, 0)$ and can jump exactly 1 unit to any point with rational coordinates. Let $P_n$ be the set of all points the frog can reach after $n$ jumps. Determine the smallest positive integer $n$ such that the point $\left(0, \frac{1}{2}\right)$ is not in $P_n$. | [
"1"
] | [
"A Rational Unit Jumping Frog starts at $(0, 0)$ and can jump exactly 1 unit to any point with rational coordinates. Let $P_n$ be the set of all points the frog can reach after $n$ jumps. Determine the smallest positive integer $n$ such that the point $\\left(0, \\frac{1}{2}\\right)$ is not in $P_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-6c8eebb2526c5e8508bb57e71e645563c4ce310987fe602974904ae5cb5c3f28 | A Rational Unit Jumping Frog starts at $(0, 0)$. Each jump is exactly 1 unit to a point with rational coordinates. Let $P_n$ be the set of all points that can be reached by the frog in exactly $n$ jumps. Find the smallest positive integer $n$ such that $\left(\frac{1}{5}, \frac{1}{17}\right) \in P_n$. | [
"no_valid_answer_found"
] | [
"A Rational Unit Jumping Frog starts at $(0, 0)$. Each jump is exactly 1 unit to a point with rational coordinates. Let $P_n$ be the set of all points that can be reached by the frog in exactly $n$ jumps. Find the smallest positive integer $n$ such that $\\left(\\frac{1}{5}, \\frac{1}{17}\\right) \\in P_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-922e6de7adc76a993eab957dc712d2a488abcd3a1772b5043bd95b38e5a54055 | A Retired Linguist (R.L.) writes a word consisting of $n$ different letters in the first move. In each subsequent move, R.L. determines the maximum $i$ such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Let $f(n)$ be the maximum number o... | [
"120"
] | [
"A Retired Linguist (R.L.) writes a word consisting of $n$ different letters in the first move. In each subsequent move, R.L. determines the maximum $i$ such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Let $f(n)$ be the maximum num... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-b55ca35484ccc40519a8a26a6733add38831c1f8bace19da960e10acddd5847a | A Rubik's cube has one visible face (the "good face"). Two moves are repeated until the cube returns to its initial form: the first move sends the third column of the good face upwards, and the second move sends the third row of the good face to the right. How many times must these two moves be repeated until the cube ... | [
"12"
] | [
"A Rubik's cube has one visible face (the \"good face\"). Two moves are repeated until the cube returns to its initial form: the first move sends the third column of the good face upwards, and the second move sends the third row of the good face to the right. How many times must these two moves be repeated until th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-70a8b8ffaedc6d41ecf14eae88e7e40587f34c3072ab0f3c6bd3c6f3f1de27ce | A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.) | [
"362,\\!880"
] | [
"A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9f26b09cab32c6876fd962ba0da70e4f83ab96845d87db867c0511403cc38d41 | A Senate committee has 5 Democrats and 5 Republicans. In how many ways can they sit around a circular table if each member sits next to two members of the other party? (Two seatings are the same if one is a rotation of the other.) | [
"2,\\!880"
] | [
"A Senate committee has 5 Democrats and 5 Republicans. In how many ways can they sit around a circular table if each member sits next to two members of the other party? (Two seatings are the same if one is a rotation of the other.)"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-cec4b803b60d28e3eb83aa75c6f01aca07157ec02293214fc88a2e7adaa8ad25 | A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats? | [
"840"
] | [
"A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-321d5451b49498f06c01edcc014a4a77901fea2a72f278061d766748f939911d | A Sierpinski's triangle is formed by taking a triangle with side length 1 and drawing an upside-down triangle inside each upright triangle that appears. A circle is inscribed in every upside-down triangle. What is the total area of all the individual circles? | [
"\\(\\frac{\\pi}{12}\\)"
] | [
"A Sierpinski's triangle is formed by taking a triangle with side length 1 and drawing an upside-down triangle inside each upright triangle that appears. A circle is inscribed in every upside-down triangle. What is the total area of all the individual circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-1ee205c95b4115ff9b73d11f57303751b9ed166ec6a358bea8caf6a6dacda879 | A Soldier and a 3-headed dragon are fighting. Every turn, the Soldier tries to swing his sword at the dragon. The outcomes are as follows: - The Soldier misses, and the dragon grows a head. - The Soldier cuts one head, which immediately regrows. - The Soldier cuts off two heads, neither of which regrow. The soldier is ... | [
"\\(\\frac{3}{5}\\)"
] | [
"A Soldier and a 3-headed dragon are fighting. Every turn, the Soldier tries to swing his sword at the dragon. The outcomes are as follows:\n- The Soldier misses, and the dragon grows a head.\n- The Soldier cuts one head, which immediately regrows.\n- The Soldier cuts off two heads, neither of which regrow.\n\nThe ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 15
} | 63 | 0 | 0 |
omr-ws-924c73409a851087f83277e967451ba763d86ad2b036f5036ff59d8324dd8355 | A Sudoku matrix is defined as a $9 \times 9$ array with entries from $(1, 2, \cdots, 9)$ and with the constraint that each row, each column, and each of the nine $3 \times 3$ boxes that tile the array contains each digit from $1$ to $9$ exactly once. Given the following partial Sudoku matrix, what is the probability th... | [
"\\(\\frac{2}{21}\\)"
] | [
"A Sudoku matrix is defined as a $9 \\times 9$ array with entries from $(1, 2, \\cdots, 9)$ and with the constraint that each row, each column, and each of the nine $3 \\times 3$ boxes that tile the array contains each digit from $1$ to $9$ exactly once. Given the following partial Sudoku matrix, what is the probab... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a53f11992993d760f54489d9f4787ee42ddc9a4e2a1b64691cb5c9da1868f513 | A Sudoku matrix is defined as a $9 \times 9$ array with entries from $\{1, 2, \ldots, 9\}$ such that each row, each column, and each of the nine $3 \times 3$ boxes contains each digit from $1$ to $9$ exactly once. Given a Sudoku matrix where two squares are known (with the digit $1$ in the top-left corner and the digit... | [
"\\(\\frac{2}{21}\\)"
] | [
"A Sudoku matrix is defined as a $9 \\times 9$ array with entries from $\\{1, 2, \\ldots, 9\\}$ such that each row, each column, and each of the nine $3 \\times 3$ boxes contains each digit from $1$ to $9$ exactly once. Given a Sudoku matrix where two squares are known (with the digit $1$ in the top-left corner and... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-3d761c416fd7fd6c2b0c40badaf0f8c1459e3360c9eaaecb5d86e8b062924921 | A Sultan had many children. His oldest son was a twin. Moreover, all his children, except for 41 of them, were twins. Also, all his children except for 41 of them were triplets. And finally, all his children, except for 41 of them, were quadruplets. How many children did the Sultan have? (Quadruplets are not counted as... | [
"53"
] | [
"A Sultan had many children. His oldest son was a twin. Moreover, all his children, except for 41 of them, were twins. Also, all his children except for 41 of them were triplets. And finally, all his children, except for 41 of them, were quadruplets. How many children did the Sultan have? (Quadruplets are not count... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-25256a635f971adbe5f80b2c34e1f7f39b1c1df50da4535c2d1ce84258769374 | A T-shaped figure composed of five unit squares is inscribed in a circle. Find the radius \( r \) of the circle. | [
"\\(\\sqrt{\\frac{65}{18}}\\)"
] | [
"A T-shaped figure composed of five unit squares is inscribed in a circle. Find the radius \\( r \\) of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 8
} | 18 | 0 | 0 |
omr-ws-bce4adc3b5bfef1a97a202de339eb018a2f26cb41ee2f8eb28fb2ba189ff3fa8 | A TV cable company has 4800 subscribers who each pay \$28 per month. The company can gain 120 more subscribers for each \$0.50 decrease in the monthly fee. What monthly fee will yield the maximum revenue, and what will this maximum revenue be? | [
"138240"
] | [
"A TV cable company has 4800 subscribers who each pay \\$28 per month. The company can gain 120 more subscribers for each \\$0.50 decrease in the monthly fee. What monthly fee will yield the maximum revenue, and what will this maximum revenue be?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e893456c4ae3c7e40d796a4eb79cd4c5b36ad761febfe974e63628613ab8aac3 | A Tibetan monk leaves the monastery at 7:00 A.M. and takes his usual path to the top of the mountain, arriving at 7:00 P.M. The following morning, he starts at 7:00 A.M. at the top and takes the same path back, arriving at the monastery at 7:00 P.M. Let $f(t)$ represent the monk's position on the path at time $t$ on th... | [
"0"
] | [
"A Tibetan monk leaves the monastery at 7:00 A.M. and takes his usual path to the top of the mountain, arriving at 7:00 P.M. The following morning, he starts at 7:00 A.M. at the top and takes the same path back, arriving at the monastery at 7:00 P.M. Let $f(t)$ represent the monk's position on the path at time $t$ ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9345f2711fb0d7deb08a2c70621350106d340307b3fc94ae310d774b21bf866d | A Tribonacci sequence is defined as follows: $T_0 = 0$, $T_1 = 0$, $T_2 = 1$, and for $n \ge 3$, $T_n = T_{n - 1} + T_{n - 2} + T_{n - 3}$. There are $m$ jars, each containing $T_3$, $T_4$, $\dots$, $T_{m+2}$ cookies. In one move, Ilhan can select several jars, and from every selected jar, he takes the same amount of c... | [
"m"
] | [
"A Tribonacci sequence is defined as follows: $T_0 = 0$, $T_1 = 0$, $T_2 = 1$, and for $n \\ge 3$, $T_n = T_{n - 1} + T_{n - 2} + T_{n - 3}$. There are $m$ jars, each containing $T_3$, $T_4$, $\\dots$, $T_{m+2}$ cookies. In one move, Ilhan can select several jars, and from every selected jar, he takes the same amou... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a5865e2200eb2f41f6c6fc13c29db2f7e169d54300400c2fcf6d959bcb2c6ca2 | A V-gauge with an angle AVB of 54° is used to measure the diameter of pipes. The distance VP is used to predict the diameter d. a. If the diameter of a pipe is 2 cm, what is the distance VP? b. If the distance VP is 3.93 cm, what is the diameter of the pipe? c. Find the formula for d in terms of VP. d. Find a formula f... | [
"a. VP = 1.96 cm, b. d = 4.00 cm, c. \\( d = 1.02 \\cdot VP \\), d. \\( VP = \\frac{d}{2 \\tan 27^\\circ} = 0.981d \\)"
] | [
"A V-gauge with an angle AVB of 54° is used to measure the diameter of pipes. The distance VP is used to predict the diameter d.\n\na. If the diameter of a pipe is 2 cm, what is the distance VP?\nb. If the distance VP is 3.93 cm, what is the diameter of the pipe?\nc. Find the formula for d in terms of VP.\nd. Find ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 13
} | 15 | 0 | 0 |
omr-ws-1198b74ba06378f0d29663d481011097a7016722f3b2cb9109648d1ebc6ef4cc | A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$. | [
"100"
] | [
"A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\\pi$. Compute $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-6c0fbcb5ebb093a15b24460ac454f4a63a854be0d46a1e947272f558961e8a12 | A [i]golden permutation[/i] of $\{1,2,3,...,n\}$ is a permutation $\{a_1, a_2, ..., a_n\}$ for which $\gcd(a_k, a_{k+1}) = 1$ for all $1 \le k \le n-1$. If a random permutation is chosen, what is the probability that it is golden, as $n$ approaches infinity? | [
"0"
] | [
"A [i]golden permutation[/i] of $\\{1,2,3,...,n\\}$ is a permutation $\\{a_1, a_2, ..., a_n\\}$ for which $\\gcd(a_k, a_{k+1}) = 1$ for all $1 \\le k \\le n-1$. If a random permutation is chosen, what is the probability that it is golden, as $n$ approaches infinity?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-53c1703937a1556c7a2bb4c940021236ddca2743a23a49d07fac2b581233785f | A [i]resplitment[/i] of a parent sequence of numbers $(a_1,a_2,\ldots ,a_n)$ generates a pair of junior sequences $(b_1,b_2,\ldots ,b_n)$ and $(c_1,c_2, \ldots ,c_n)$ such that there exists an integer $0< j <n$ that satisfies $b_i \cdot c_i = a_{i+j}$ for all integers $1 \leq i \leq n.$ If $i+j > n,$ define $a_{i+j}=a_... | [
"145"
] | [
"A [i]resplitment[/i] of a parent sequence of numbers $(a_1,a_2,\\ldots ,a_n)$ generates a pair of junior sequences $(b_1,b_2,\\ldots ,b_n)$ and $(c_1,c_2, \\ldots ,c_n)$ such that there exists an integer $0< j <n$ that satisfies $b_i \\cdot c_i = a_{i+j}$ for all integers $1 \\leq i \\leq n.$ If $i+j > n,$ define ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8088271f5b04a3addca5a53bc547b71df07181534542f8b73ac92923a1d83f5d | A [i]squiggle[/i] is composed of six equilateral triangles with side length $1$. Determine all possible integers $n$ such that an equilateral triangle with side length $n$ can be fully covered with [i]squiggle[/i]s (rotations and reflections of [i]squiggle[/i]s are allowed, overlappings are not). | [
"\\( n \\equiv 0 \\pmod{12} \\)"
] | [
"A [i]squiggle[/i] is composed of six equilateral triangles with side length $1$. Determine all possible integers $n$ such that an equilateral triangle with side length $n$ can be fully covered with [i]squiggle[/i]s (rotations and reflections of [i]squiggle[/i]s are allowed, overlappings are not)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a2dd67dbda62527836c70db04b23c9814edbd84344874f211d75600c5bdfb9b | A [i]trio[/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio[/i] contained in $\{1, 2, \ldots, 2002\}$ has the largest possible sum? Find all [i]trios[/i] with the maximum sum. | [
"4004"
] | [
"A [i]trio[/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio[/i] contained in $\\{1, 2, \\ldots, 2002\\}$ has the largest possible sum? Find all [i]trios[/i] with the maximum sum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 8,
"tir": 8
} | 36 | 0 | 0 |
omr-ws-92c77a617784d3d1a3adee88f9bdf0413c87fa5d1d4acee9f5a638aedb33db40 | A \( 150 \times 324 \times 375 \) block is made up of unit cubes. Find the number of unit cubes whose interior is cut by a long diagonal of the block. | [
"768"
] | [
"A \\( 150 \\times 324 \\times 375 \\) block is made up of unit cubes. Find the number of unit cubes whose interior is cut by a long diagonal of the block."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a588a6dd0a3395c38c11e6f4b8df201e4e0a7e888b23a10efa04b9ee29d60021 | A \( 2 \times 2 \) ship is located on an \( 8 \times 8 \) field. The ship can be found by pointing to certain cells, to which the answer is "hit" or "miss," depending on whether the ship contains this cell or not. What is the minimum number of shots required to guarantee finding the ship? | [
"16"
] | [
"A \\( 2 \\times 2 \\) ship is located on an \\( 8 \\times 8 \\) field. The ship can be found by pointing to certain cells, to which the answer is \"hit\" or \"miss,\" depending on whether the ship contains this cell or not. What is the minimum number of shots required to guarantee finding the ship?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eaf53e263d2e0ac715f3f2949e6354ad7b7e1fa901e71d3961144b0dcecaaf91 | A \( 3 \times 3 \) grid is to be filled from integers \( 1 \) to \( 9 \) so that there are no arithmetic sequences in any column. In how many ways is this possible? | [
"46080"
] | [
"A \\( 3 \\times 3 \\) grid is to be filled from integers \\( 1 \\) to \\( 9 \\) so that there are no arithmetic sequences in any column. In how many ways is this possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-9f69b10aae7540eb16015d095adbd1c90c585a02dfce48f3854fce889c034179 | A \( 9 \times 12 \) rectangle is partitioned into unit squares. The centers of all the squares, except for the four corner squares and the eight squares orthogonally adjacent to them, are colored in red. Determine if it is possible to draw a closed broken line such that (a) a point is a vertex of the broken line if and... | [
"\\text{Yes}"
] | [
"A \\( 9 \\times 12 \\) rectangle is partitioned into unit squares. The centers of all the squares, except for the four corner squares and the eight squares orthogonally adjacent to them, are colored in red. Determine if it is possible to draw a closed broken line such that (a) a point is a vertex of the broken lin... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-c85899c08c9be051ab8d709c696b4b726d2d067d310062de3bf6ec403d0e88ba | A \( n \times n \) square grid is composed of \( n^2 \) unit squares, for a positive integer \( n \). For each unit square in the grid, all of its sides are drawn, and some diagonals of some unit squares are also drawn, so that no unit square has both diagonals drawn and no two unit squares that share a side have diago... | [
"\\( n \\in \\{1, 2, 3\\} \\)"
] | [
"A \\( n \\times n \\) square grid is composed of \\( n^2 \\) unit squares, for a positive integer \\( n \\). For each unit square in the grid, all of its sides are drawn, and some diagonals of some unit squares are also drawn, so that no unit square has both diagonals drawn and no two unit squares that share a sid... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-003b2bf27e7a6bcbfa3766c66d4f2a85fec6abb3df3f601879b1914b950c0afd | A \(2 \times 2\) matrix is filled with randomly generated, uniformly distributed integers from 1 to 2008 inclusive. What is the probability that the discriminant is even? | [
"\\dfrac{1}{2}"
] | [
"A \\(2 \\times 2\\) matrix is filled with randomly generated, uniformly distributed integers from 1 to 2008 inclusive. What is the probability that the discriminant is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a3c142f054e4ead5f07a9e603845addbd2ad07f41f7fb726095a471685c9ac71 | A \(24 \times 60\) rectangle is divided by lines parallel to its sides into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? | [
"1512"
] | [
"A \\(24 \\times 60\\) rectangle is divided by lines parallel to its sides into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-2a352f683456e440b82dbc18fd83aafe36b455482381d48fb2073a4dfd4feda8 | 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 ... | [
"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 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 19
} | 41 | 0 | 0 |
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