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-876f52c8d10b3c64ed3eba6f9bd8b61a035935bfb97c9d91308f7f863b8072f6 | A 160 ft rope is suspended at its two ends from the top of two 100 ft poles. If the lowest point to which the midpoint of the rope can be pulled from the ground is 72 ft, what is the distance between the poles? | [
"\\( 24\\sqrt{39} \\)"
] | [
"A 160 ft rope is suspended at its two ends from the top of two 100 ft poles. If the lowest point to which the midpoint of the rope can be pulled from the ground is 72 ft, what is the distance between the poles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-78eb6115aa4b74b20568c77a01334fc455d10018ca07b9957369c13053a4e3da | A 16x16 matrix is filled with 1 or -1. If the sum of the corresponding entries of any two different rows is at most 0, find the maximum possible number of entries of "+1" in the matrix. | [
"128"
] | [
"A 16x16 matrix is filled with 1 or -1. If the sum of the corresponding entries of any two different rows is at most 0, find the maximum possible number of entries of \"+1\" in the matrix."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d354cb9d0872ddb453677b80d82add709b3b9dcb8f8b17879a1628f980fedd40 | A 17 ft ladder is leaning against a wall. If the bottom of the ladder is pulled away from the wall at a constant rate of 5 ft/sec, how fast will the top of the ladder be moving down the wall when it is 8 ft above the ground? | [
"\\(\\frac{75}{8}\\) ft/sec"
] | [
"A 17 ft ladder is leaning against a wall. If the bottom of the ladder is pulled away from the wall at a constant rate of 5 ft/sec, how fast will the top of the ladder be moving down the wall when it is 8 ft above the ground?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-30c1ccc0651b0514a70f0b0bbfee21fb54ebf99dc36e18e795d488e6c1f5e3e2 | A 17-sided die has faces numbered 1 to 17, but is not fair. The number 17 comes up with probability 0.5, and each of the numbers from 1 to 16 comes up with a probability of $\frac{1}{32}$. Compute the probability that the sum of two rolls is either 20 or 12. | [
"\\(\\frac{7}{128}\\)"
] | [
"A 17-sided die has faces numbered 1 to 17, but is not fair. The number 17 comes up with probability 0.5, and each of the numbers from 1 to 16 comes up with a probability of $\\frac{1}{32}$. Compute the probability that the sum of two rolls is either 20 or 12."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1141e160f103848c17c74d0fe0af0c1f91f157ea8226af04d900d91d0f736b1d | A 1999-by-1999 chocolate bar has vertical and horizontal grooves which divide it into \( 1999^2 \) one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; and then... | [
"Brutus"
] | [
"A 1999-by-1999 chocolate bar has vertical and horizontal grooves which divide it into \\( 1999^2 \\) one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-614eebcb015f75b1562f7117f21a2399e15fb02823c282f80dfd2b4b941e8c84 | A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. [asy] size(7cm); defaultpen(linewidth(0.... | [
"\\dfrac{1001}{2003}"
] | [
"A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. [asy]\nsize(7cm);\ndefaultpen(li... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9777f57c3b41bdd1c3a39f362ce8030640adf74c03d6bd03b401bbcc826a08fb | A 2 by 2003 rectangle consists of unit squares. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. | [
"\\(\\frac{1001}{2003}\\)"
] | [
"A 2 by 2003 rectangle consists of unit squares. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-4239432e31374158ebe571edf6122ad11df7f77e22e4bfe291ce7792e697cdcf | A 2 x 3 rectangle and a 3 x 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? | [
"25"
] | [
"A 2 x 3 rectangle and a 3 x 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-52e8b3f32bc93ae9f412c2192f4b54b954b439276976b10ef696390ebdac9fc3 | A 2 x 4 rectangle is divided into 8 squares of side 1, resulting in a set \( M \) of 15 vertices. Find all points \( A \) in \( M \) for which it is possible to arrange the set \( \{A\} \) into pairs \((A_1, B_1), (A_2, B_2), \ldots, (A_7, B_7)\) such that the sum of the vectors \( A_1B_1 + A_2B_2 + \cdots + A_7B_7 = 0... | [
"\\((0,1), (2,1), (4,1)\\)"
] | [
"A 2 x 4 rectangle is divided into 8 squares of side 1, resulting in a set \\( M \\) of 15 vertices. Find all points \\( A \\) in \\( M \\) for which it is possible to arrange the set \\( \\{A\\} \\) into pairs \\((A_1, B_1), (A_2, B_2), \\ldots, (A_7, B_7)\\) such that the sum of the vectors \\( A_1B_1 + A_2B_2 + ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 4
} | 23 | 0 | 0 |
omr-ws-fa3a564040e434a0120dc3c07865036391184512af85bb19053882e689f204a4 | A 2-d ant starts at the point $(0,0)$ of an $xy$-coordinate grid and moves 1 unit in a direction parallel to the coordinate axes every minute until it reaches a point where $|x|+|y|$ is prime. A 3-d ant starts at the point $(0,0,0)$ of an $xyz$ space and moves 1 unit in a direction parallel to the coordinate axes every... | [
"1"
] | [
"A 2-d ant starts at the point $(0,0)$ of an $xy$-coordinate grid and moves 1 unit in a direction parallel to the coordinate axes every minute until it reaches a point where $|x|+|y|$ is prime. A 3-d ant starts at the point $(0,0,0)$ of an $xyz$ space and moves 1 unit in a direction parallel to the coordinate axes ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eeb9568860cd4296a6cb5ccaee62120f45ce98a4769aca118c5270f7fed9d53e | A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number? | [
"9"
] | [
"A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0797815a7904d81eb4679b964a8a82304f3860ef61717c7a2e8954141f128b1f | A 20 liter vessel is filled with alcohol. Some of the alcohol is poured out into another vessel of an equal capacity, which is then made full by adding water. The mixture thus obtained is then poured into the first vessel to capacity. Then $6\frac{2}{3}$ liters is poured from the first vessel into the second. Both vess... | [
"10 liters"
] | [
"A 20 liter vessel is filled with alcohol. Some of the alcohol is poured out into another vessel of an equal capacity, which is then made full by adding water. The mixture thus obtained is then poured into the first vessel to capacity. Then $6\\frac{2}{3}$ liters is poured from the first vessel into the second. Bot... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-616daa50e7ab54d65bc9f767e979daa33ab24f21026b77449d90dcb31c3de6a7 | A 20-foot ladder is being pulled away from a wall at a rate of 1.5 feet per second. At the instant when the foot of the ladder is 12 feet away from the wall, at what rate (in radians per second) is the angle the ladder makes with the floor decreasing? | [
"-\\dfrac{3}{32}"
] | [
"A 20-foot ladder is being pulled away from a wall at a rate of 1.5 feet per second. At the instant when the foot of the ladder is 12 feet away from the wall, at what rate (in radians per second) is the angle the ladder makes with the floor decreasing?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-b68f3bd9f4ed298fc58dda352b91b9f800526752660557ee9df50fce8805175a | A 20-foot ladder rests vertically against a wall. A bug starts at the bottom of the ladder and climbs up at a rate of 1 foot per minute. At the same time, the foot of the ladder is being pulled along the ground at a rate of 1 foot per minute until the top reaches the ground. Let \( x \) be the distance of the bug from ... | [
"10\\sqrt{3}"
] | [
"A 20-foot ladder rests vertically against a wall. A bug starts at the bottom of the ladder and climbs up at a rate of 1 foot per minute. At the same time, the foot of the ladder is being pulled along the ground at a rate of 1 foot per minute until the top reaches the ground. Let \\( x \\) be the distance of the bu... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14b0e842d79763f9932abe660aeb7b16c5b00e21b86837df242312a55ed2d119 | A 20-foot ladder slides down a wall at a rate of 5 ft/sec. At what speed is the bottom of the ladder sliding out when the top of the ladder is 10 feet from the floor (in ft/sec)? | [
"\\(\\frac{5\\sqrt{3}}{3}\\)"
] | [
"A 20-foot ladder slides down a wall at a rate of 5 ft/sec. At what speed is the bottom of the ladder sliding out when the top of the ladder is 10 feet from the floor (in ft/sec)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-4bcf72de570e4b67c564bb727074ad82949eeeab01579dbda1dbba2d2d9169bb | A 20-meter string is cut into two pieces. The first piece is used to form a rectangle, and the second piece is used to form a triangle. What should be the length of the piece used to form the triangle in order to maximize the total area of the two geometric figures? | [
"\\(\\frac{240\\sqrt{3} - 540}{11}\\)"
] | [
"A 20-meter string is cut into two pieces. The first piece is used to form a rectangle, and the second piece is used to form a triangle. What should be the length of the piece used to form the triangle in order to maximize the total area of the two geometric figures?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-5ccf5987a93552b4251905d06d213e77194f52679edef52e38f31da683be5b2a | A 20-mile stretch of a 3-lane highway is filled with cars, each 60 yards apart and traveling at 60 miles per hour. The front of the group reaches point A, where the road narrows to 2 lanes. Assuming perfect driving conditions and that drivers can adjust instantly, how long does it take for the entire group of cars to p... | [
"20 minutes"
] | [
"A 20-mile stretch of a 3-lane highway is filled with cars, each 60 yards apart and traveling at 60 miles per hour. The front of the group reaches point A, where the road narrows to 2 lanes. Assuming perfect driving conditions and that drivers can adjust instantly, how long does it take for the entire group of cars... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-a943a3e1606b1d9322100768875138c1c7965976880c7517de30452297049525 | A 20-sided die is rolled twice, showing numbers \(a\) and \(b\). Find the probability that the quadratic equation \(x^2 - ax + b = 0\) has integer roots. | [
"\\(\\frac{17}{200}\\)"
] | [
"A 20-sided die is rolled twice, showing numbers \\(a\\) and \\(b\\). Find the probability that the quadratic equation \\(x^2 - ax + b = 0\\) has integer roots."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2c574777f0e0880ccdc20a87b22b4f330233c9aaf6f0bd6416f3f509f025489c | A 200-digit decimal number is selected at random (possibly starting with 0). Let \( Y \) be the number of pairs of consecutive digits whose positive difference is less than or equal to 1. Find the expected value of \( Y \). | [
"\\(\\frac{1393}{25}\\)"
] | [
"A 200-digit decimal number is selected at random (possibly starting with 0). Let \\( Y \\) be the number of pairs of consecutive digits whose positive difference is less than or equal to 1. Find the expected value of \\( Y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-731f6005a5c8ba79574b985ef552d5bf71bdab237b1850864fd1831f4c7b60ab | A 2010 by 2010 rectangular table is tiled completely with 4040100 1x1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | [
"1 + \\dfrac{\\pi}{3} - \\sqrt{3}"
] | [
"A 2010 by 2010 rectangular table is tiled completely with 4040100 1x1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-b57a376e2f591d317e339617eecab704237913bdb329f06f479765abe4a5927f | A 2010 x 2010 rectangular grid is divided into squares of side length 1. How many of these squares are crossed by the diagonal of the rectangle? (Squares that only touch the diagonal at a single point are not counted.) | [
"3000"
] | [
"A 2010 x 2010 rectangular grid is divided into squares of side length 1. How many of these squares are crossed by the diagonal of the rectangle? (Squares that only touch the diagonal at a single point are not counted.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40a08feca2ccfe6d8917f9efb09cbed9f5eaf5ea19030da8a6fbd28f0b125e1f | A 2010 x 2010 rectangular table is tiled completely with 4040100 1 x 1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | [
"2 - \\sqrt{3}"
] | [
"A 2010 x 2010 rectangular table is tiled completely with 4040100 1 x 1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-85d14a6135f9b26899add4186ca9d17603883d948713c90e7c39636c52211dd9 | A 2010 × 2010 rectangular table is tiled completely with 4040100 1 × 1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A 2010 × 2010 rectangular table is tiled completely with 4040100 1 × 1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a4c754c9d400f6ee318e71eef188d5264c1f308efcfdc0184d8b95aaabd4d84 | A 2010*2010 rectangular table is tiled completely with 1*1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | [
"2 - \\sqrt{3}",
"\\( 2 - \\sqrt{3} \\)"
] | [
"A 2010*2010 rectangular table is tiled completely with 1*1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 12,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-17fbc3c749deca46eec5cbd066fe7d70c17509e815131e3572c93aabf91c69d2 | A 2010*2010 rectangular table is tiled completely with 4040100 1*1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | [
"2 - \\sqrt{3}",
"3 - 2\\sqrt{2}"
] | [
"A 2010*2010 rectangular table is tiled completely with 4040100 1*1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-2eb5208ec3e20b02e9ec1209d18a0184c8331f96b06851b7f7f8ef5ac79c716e | A 2010-sided die is biased such that the probability of rolling a number divisible by 3 is 0.5. If each number divisible by 3 has an equal chance of being rolled, find the probability of rolling a number whose digits sum up to 9. | [
"\\(\\frac{101}{1340}\\)"
] | [
"A 2010-sided die is biased such that the probability of rolling a number divisible by 3 is 0.5. If each number divisible by 3 has an equal chance of being rolled, find the probability of rolling a number whose digits sum up to 9."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-b87d69f9dc41717ee9add74e66103eb3f0fe9c117eafe1cbb1109dd98a003e09 | A 2011-sided regular polygon has both its inscribed and circumscribed circles drawn. If the length of each side of the polygon is 2, what is the area of the region between the two circles? | [
"\\pi"
] | [
"A 2011-sided regular polygon has both its inscribed and circumscribed circles drawn. If the length of each side of the polygon is 2, what is the area of the region between the two circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d2886ad9fdac1632aa12fb08512dd288b1ee47b5374b4a23002cc4b80176ace | A 2019 × 2019 square grid is made up of 2019^2 unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of k^2 cells for 1 ≤ k ≤ 2019, the number of black cells differs from the number of white cells by at most one. How many different balanced colorin... | [
"2"
] | [
"A 2019 × 2019 square grid is made up of 2019^2 unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of k^2 cells for 1 ≤ k ≤ 2019, the number of black cells differs from the number of white cells by at most one. How many different balanced co... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e5dc29be4fd1ac3973f2fb296ed72c7b3f2e350c435cd062d8bf8b2796e66e1 | A 236 mile trip is made by boat and by train. The boat averages 32 mph and the train averages 52 mph. If the total trip took 5.5 hours, how many miles was the train trip? | [
"156 miles"
] | [
"A 236 mile trip is made by boat and by train. The boat averages 32 mph and the train averages 52 mph. If the total trip took 5.5 hours, how many miles was the train trip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-85ed4ad6e55eb2725de84d47ab7ede0eb4af118ad425cd4bb52d054a6ce7dc4c | A 23×23 table is to be covered with 1×1, 2×2, and 3×3 tiles. How many 1×1 tiles can be used? | [
"516"
] | [
"A 23×23 table is to be covered with 1×1, 2×2, and 3×3 tiles. How many 1×1 tiles can be used?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-fb558b0d978df6d1648029e90f449ec39f15c3c953e910b3a8ec514da32bcf0d | A 24 x 52 field is fenced. An additional 1994 units of fencing is available. It is desired to divide the entire field into identical square (fenced) plots. What is the largest number of such square plots that can be obtained? | [
"702"
] | [
"A 24 x 52 field is fenced. An additional 1994 units of fencing is available. It is desired to divide the entire field into identical square (fenced) plots. What is the largest number of such square plots that can be obtained?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 8,
"tir": 3
} | 34 | 0 | 0 |
omr-ws-72fe128c00262c70706fb63ff5e5e6c1059124fd0a2765748fb5ec4e42f7d9bc | A 24-hour clock that runs fast gains 90 seconds every hour. If it is set correctly at 9:00 a.m., in how many days will it next show the correct time? | [
"40"
] | [
"A 24-hour clock that runs fast gains 90 seconds every hour. If it is set correctly at 9:00 a.m., in how many days will it next show the correct time?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-270d14c606a29517758d7ac56596032f6dc1cd7f21a203fbb9f6908e8e966a44 | A 24-letter string is made of As and Bs. Each index is an A with probability $p$ and a B with probability $1-p$, with each index being independently selected. What is the probability that there is at least one block of 3 As in a row? | [
"1 - T(24)"
] | [
"A 24-letter string is made of As and Bs. Each index is an A with probability $p$ and a B with probability $1-p$, with each index being independently selected. What is the probability that there is at least one block of 3 As in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-78f686a6416975b217484272f2cdddee0712136c52b85971c562f06612b3938f | A 28-digit number is partially visible as 5-383-8-2-936-5-8-203-9-3-76, with 10 digits missing. These blanks are filled using the digits $0$ to $9$ exactly once. What is the probability that the number formed is divisible by $396$? | [
"1"
] | [
"A 28-digit number is partially visible as 5-383-8-2-936-5-8-203-9-3-76, with 10 digits missing. These blanks are filled using the digits $0$ to $9$ exactly once. What is the probability that the number formed is divisible by $396$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd9210fec0ecd317ea18da9f95c08024339e314a1d18982815315429f5ed5d79 | A 2D view of a conveyor belt consists of two circles with radii 7 and 21, wrapped by a belt of length \(78\pi\). Find the area enclosed by the belt, divided by \(\pi\). | [
"392"
] | [
"A 2D view of a conveyor belt consists of two circles with radii 7 and 21, wrapped by a belt of length \\(78\\pi\\). Find the area enclosed by the belt, divided by \\(\\pi\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-a432c2c7b39df31640863d0127bc8b8f9929400e0d39131fae6f563fb8c4c959 | A 2x1 rectangle lies in the 2D plane. You are viewing the plane from an angle that makes the rectangle appear as a line of length 1. What is the minimum number of radians you must rotate the rectangle around a line perpendicular to the plane intersecting the center point of the rectangle such that the length of the lin... | [
"\\( \\cos^{-1}\\left(\\frac{1}{\\sqrt{5}}\\right) \\)"
] | [
"A 2x1 rectangle lies in the 2D plane. You are viewing the plane from an angle that makes the rectangle appear as a line of length 1. What is the minimum number of radians you must rotate the rectangle around a line perpendicular to the plane intersecting the center point of the rectangle such that the length of th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-5c250b20bacb065befd73f63c803dae7ac357bf28ba51c58bb12b85b2603222d | A 2x1x1 rectangular prism lies in the 3D space. You are viewing the space from an angle that makes the prism appear as a square of area 1. What is the minimum number of radians you must rotate the prism around any line such that the area of the 2D shape you see is maximized? | [
"\\dfrac{\\pi}{3}"
] | [
"A 2x1x1 rectangular prism lies in the 3D space. You are viewing the space from an angle that makes the prism appear as a square of area 1. What is the minimum number of radians you must rotate the prism around any line such that the area of the 2D shape you see is maximized?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-f1e5bf5899ba9a0a9d7df19c931e3fc58aeff797a9c827f7042ee29a0ad4c410 | A 2x1x1x... hyperrectangle lies in the N-dimensional space. You are viewing the space from an angle that makes the hyperrectangle appear as an (N-1)-dimensional object of minimum possible area. What is the minimum number of radians you must rotate this hyperrectangle around any line such that the area of the (N-1)-dime... | [
"\\dfrac{\\pi}{2}"
] | [
"A 2x1x1x... hyperrectangle lies in the N-dimensional space. You are viewing the space from an angle that makes the hyperrectangle appear as an (N-1)-dimensional object of minimum possible area. What is the minimum number of radians you must rotate this hyperrectangle around any line such that the area of the (N-1)... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-321b43d3d1d5976de25b6ce1b56f11014129e687f43ba064b2d6259eb4e5b307 | A 3 by 4 by 12 rectangular prism is on the ground. What is the sum of the third smallest possible distance between two vertices and the longest distance between two vertices? | [
"18"
] | [
"A 3 by 4 by 12 rectangular prism is on the ground. What is the sum of the third smallest possible distance between two vertices and the longest distance between two vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-3de8b580f0bdeb51e8b189632e75edac204f66a6a48070fbf68f9942663d7ce3 | A 3 x 3 cube is assembled from 27 white unit cubes. The large cube is then painted black on the outside and then disassembled. If it is reassembled at random, what is the probability that the large cube is still completely black on the outside? | [
"} \\frac{8! \\times 12! \\times 6!}{27!} \\]"
] | [
"A 3 x 3 cube is assembled from 27 white unit cubes. The large cube is then painted black on the outside and then disassembled. If it is reassembled at random, what is the probability that the large cube is still completely black on the outside?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 19
} | 25 | 0 | 0 |
omr-ws-33db63ce4f47a31776c52b5ba31ca520bee000da6e9dfe0f0fbd185d2b37b5a4 | A 3 x 3 x 3 cube is formed by gluing together 27 standard cubical dice. On a standard die, the sum of the numbers on any pair of opposite faces is 7. What is the smallest possible sum of all the numbers showing on the surface of the cube? | [
"90"
] | [
"A 3 x 3 x 3 cube is formed by gluing together 27 standard cubical dice. On a standard die, the sum of the numbers on any pair of opposite faces is 7. What is the smallest possible sum of all the numbers showing on the surface of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ad587279e8e3fb5f866e18745f9437f0e8f4557271eb745d705416a64703f2d6 | A 3*3*3 cube is cut into 27 smaller cubes. Each face of the smaller cubes is to be painted with one of three colors (red, blue, green). In how many ways can these smaller cubes be painted such that they can be reassembled to form a larger 3*3*3 cube with all faces having the same color? | [
"3 \\times 2^{54}",
"\\( 3^{27} \\)"
] | [
"A 3*3*3 cube is cut into 27 smaller cubes. Each face of the smaller cubes is to be painted with one of three colors (red, blue, green). In how many ways can these smaller cubes be painted such that they can be reassembled to form a larger 3*3*3 cube with all faces having the same color?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-af7a74c4a67148fcc354f54419130225b238ae1c958b9b492c4e42e303875b50 | A 3-by-4 rectangle is rotated about its diagonal to form a 3D solid. What is the volume of the figure, in cubic units? Express your answer as a decimal to the nearest hundredth. | [
"41.91"
] | [
"A 3-by-4 rectangle is rotated about its diagonal to form a 3D solid. What is the volume of the figure, in cubic units? Express your answer as a decimal to the nearest hundredth."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fa5b2d4e785e34ce882e94307ba9d3344647d1e45bbf7e3352f698f83a884cd | A 3-digit number \( M \) has three distinct non-zero digits. If the digits of \( M \) are reversed, we obtain another 3-digit number \( N \). If \( M - N = 495 \), how many possible values are there for \( M \)? | [
"50"
] | [
"A 3-digit number \\( M \\) has three distinct non-zero digits. If the digits of \\( M \\) are reversed, we obtain another 3-digit number \\( N \\). If \\( M - N = 495 \\), how many possible values are there for \\( M \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98b7053e82fd0146f0157ebd37d80a192479d18f6908e9c0baf4be19023e39aa | A 3-digit number \(\alpha\beta\gamma\) is given, where \(\alpha, \beta, \gamma \in \{1, 2, 3, \ldots, 9\}\). The student writes the numbers \(\alpha\gamma\beta\), \(\beta\alpha\gamma\), \(\beta\gamma\alpha\), \(\gamma\alpha\beta\), and \(\gamma\beta\alpha\) and adds them to get a sum \(\Sigma = 3194\). Find the initial... | [
"358"
] | [
"A 3-digit number \\(\\alpha\\beta\\gamma\\) is given, where \\(\\alpha, \\beta, \\gamma \\in \\{1, 2, 3, \\ldots, 9\\}\\). The student writes the numbers \\(\\alpha\\gamma\\beta\\), \\(\\beta\\alpha\\gamma\\), \\(\\beta\\gamma\\alpha\\), \\(\\gamma\\alpha\\beta\\), and \\(\\gamma\\beta\\alpha\\) and adds them to g... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-dd21ed3717229e91dc83aaa4351fdddbe9e26ac8528181136370a92655a3afa7 | A 3-digit number \(abc\) in base 6 is equal to \(cba\) in base 9. Find the digits \(a\), \(b\), and \(c\). | [
"\\(a = 5\\), \\(b = 5\\), \\(c = 2\\)"
] | [
"A 3-digit number \\(abc\\) in base 6 is equal to \\(cba\\) in base 9. Find the digits \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 31
} | 37 | 0 | 0 |
omr-ws-861f6fc8bf65d8085c444ebd68d31a9a5f59e6ef9bd8a8cfb809d9c6ffd11327 | A 3-digit number is subtracted from a 4-digit number to produce a 3-digit number. If all the digits are different, what is the smallest possible result? | [
"246"
] | [
"A 3-digit number is subtracted from a 4-digit number to produce a 3-digit number. If all the digits are different, what is the smallest possible result?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-201c5638a205bde7c486d86d5f35069636bb7f74ae49f77276f795c2c5375a07 | A 3-digit number is subtracted from a 4-digit number, and the result is a 3-digit number. All 10 digits used are different. What is the smallest possible result? | [
"246"
] | [
"A 3-digit number is subtracted from a 4-digit number, and the result is a 3-digit number. All 10 digits used are different. What is the smallest possible result?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 21
} | 32 | 0 | 0 |
omr-ws-af28802a83587b6b76888bdb7d19e5d9752afb26f3dafef58c725a9ff9ce2b22 | A 3-digit number is subtracted from a 4-digit number, resulting in a 3-digit number. All 10 digits involved are different. What is the smallest possible value of the result? | [
"289"
] | [
"A 3-digit number is subtracted from a 4-digit number, resulting in a 3-digit number. All 10 digits involved are different. What is the smallest possible value of the result?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-98a5bddbefd0c8b5d7dae0cf7dd2465d8d88a0c298595b52d7a3be458133c57d | A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it. The cross is inscribed in a circle with radius 1. What is its volume? | [
"\\(\\frac{56}{11\\sqrt{11}}\\)"
] | [
"A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it. The cross is inscribed in a circle with radius 1. What is its volume?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 8,
"tir": 28
} | 77 | 0 | 0 |
omr-ws-b9b776253e5f06a6647e5e2e45bada8eaf47da7d261df3798548e9c8cdf3d1ec | A 3-dimensional cube contains "cubes" of smaller dimensions, i.e., faces (2-cubes), edges (1-cubes), and vertices (0-cubes). How many 3-cubes are in a 5-cube? | [
"40"
] | [
"A 3-dimensional cube contains \"cubes\" of smaller dimensions, i.e., faces (2-cubes), edges (1-cubes), and vertices (0-cubes). How many 3-cubes are in a 5-cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c03cf82249017e3e2af98a1381a72f3b3e9c87b6516408e99eb29dee0ecc5016 | A 30-60-90 right triangle with hypotenuse 1 has an equilateral triangle constructed outwards from each side. Find the side length of the equilateral triangle formed by the centroids of the three constructed equilateral triangles. | [
"\\(\\frac{\\sqrt{21}}{6}\\)"
] | [
"A 30-60-90 right triangle with hypotenuse 1 has an equilateral triangle constructed outwards from each side. Find the side length of the equilateral triangle formed by the centroids of the three constructed equilateral triangles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 38
} | 68 | 0 | 0 |
omr-ws-aacf86798f6b5c3e705bca3d564c1e4fd91f5346a390c9bcb1e158e351e8d591 | A 31×31 table is to be covered with 3×3 and 5×5 tiles. At least how many squares will remain uncovered? | [
"7"
] | [
"A 31×31 table is to be covered with 3×3 and 5×5 tiles. At least how many squares will remain uncovered?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-c7769cbf654a8257256013c682dbe0c06a4a867e20137710f38a627420448a3b | A 32-foot ladder is leaned against a vertical wall, forming a 30-degree angle with the wall. To what height does the ladder reach? | [
"\\( 16\\sqrt{3} \\)"
] | [
"A 32-foot ladder is leaned against a vertical wall, forming a 30-degree angle with the wall. To what height does the ladder reach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-a8837ed22b7dfcf8aa65cfe29ec0ad498ae9e872eb77528f6638d17387b27a9f | A 33×33 table is colored with three colors, each color covering 363 grids. A side of a grid is called "good" if the two neighboring grids have different colors. Find the least number of good sides. | [
"66"
] | [
"A 33×33 table is colored with three colors, each color covering 363 grids. A side of a grid is called \"good\" if the two neighboring grids have different colors. Find the least number of good sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42d34890a66aeb53b2242269c886f90cf03b7df5c6dc26294bec7db70c0a98de | A 360-foot long passenger train completely passes a 1400-foot long freight train traveling in the same direction in 60 seconds. When moving in opposite directions, the trains pass each other in 12 seconds. What is the speed of the passenger train in feet/second? | [
"88"
] | [
"A 360-foot long passenger train completely passes a 1400-foot long freight train traveling in the same direction in 60 seconds. When moving in opposite directions, the trains pass each other in 12 seconds. What is the speed of the passenger train in feet/second?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-7a687c2290335fedc28f595be625567a5e8b5698dc16a99cd4cb3e679c58bd5d | A 3x3 grid containing the numbers 1-9 is drawn. Define a "nice sequence" to be the following: start with any number. Move to another number that is next to, or diagonal to the current number. Repeat this process with the restriction that a number cannot be visited twice. Once every number has been visited, the sequence... | [
"1296"
] | [
"A 3x3 grid containing the numbers 1-9 is drawn. Define a \"nice sequence\" to be the following: start with any number. Move to another number that is next to, or diagonal to the current number. Repeat this process with the restriction that a number cannot be visited twice. Once every number has been visited, the s... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-00d8d3326119577c9f9807a026a23250ea1c3370794aada0bd771125bd49fc3a | A 3x3 grid is filled with the numbers 1 through 9. For each column, the second largest number is colored. How many ways can the numbers be arranged such that 5 is colored in each column? | [
"32768"
] | [
"A 3x3 grid is filled with the numbers 1 through 9. For each column, the second largest number is colored. How many ways can the numbers be arranged such that 5 is colored in each column?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fb1a6a43c570e3cf82f8a5bb683dda7855b019172b5e3f22d4f14ae0e54acc9 | A 3x3 square is divided into 9 1x1 squares. Different integers from 1 to 9 are placed in these 9 squares. For each two squares sharing a common edge, the sum of the integers in them is calculated. What is the minimum possible number of different sums? | [
"4"
] | [
"A 3x3 square is divided into 9 1x1 squares. Different integers from 1 to 9 are placed in these 9 squares. For each two squares sharing a common edge, the sum of the integers in them is calculated. What is the minimum possible number of different sums?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-5916d8c44d7508c6168703e48117cf337179265a36f14b4c75c5dd7e66b03962 | A 3x3 square is divided into 9 1x1 squares. Five of the nine 1x1 squares must be colored so that there is at least one colored square on each row and each column. In how many ways can the coloring be done? | [
"90"
] | [
"A 3x3 square is divided into 9 1x1 squares. Five of the nine 1x1 squares must be colored so that there is at least one colored square on each row and each column. In how many ways can the coloring be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-80063dbbebae30385926da480883752062f01e863b3bf899f86e61dc22cd0134 | A 3x3x3 cube is composed of 27 small 1x1x1 cubes with passageways allowing movement between adjacent cubes. A mouse starts at a corner and a piece of cheese is placed in the center cube. Each minute, the mouse moves randomly to an adjacent cube. Find the expected number of minutes before the mouse reaches the cheese. | [
"21"
] | [
"A 3x3x3 cube is composed of 27 small 1x1x1 cubes with passageways allowing movement between adjacent cubes. A mouse starts at a corner and a piece of cheese is placed in the center cube. Each minute, the mouse moves randomly to an adjacent cube. Find the expected number of minutes before the mouse reaches the chee... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-c19f82afda5357c85a93705b80f803e6c58004b2aaac32850acc562f9e632d7f | A 3×3 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° clockwise about its center, and every white square in a position formerly occupied by a black square is painted blac... | [
"\\dfrac{49}{512}"
] | [
"A 3×3 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° clockwise about its center, and every white square in a position formerly occupied by a black square is painted... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-32fe630ad8a5461f2c453c5cfda66301916eac47948454786f29279681e730be | A 3×3×3 cube is made of 1×1×1 cubes glued together. What is the maximal number of small cubes one can remove so that the remaining solid has the following features: 1) The projection of this solid on each face of the original cube is a 3 × 3 square. 2) The resulting solid remains face-connected (from each small cube on... | [
"24"
] | [
"A 3×3×3 cube is made of 1×1×1 cubes glued together. What is the maximal number of small cubes one can remove so that the remaining solid has the following features:\n1) The projection of this solid on each face of the original cube is a 3 × 3 square.\n2) The resulting solid remains face-connected (from each small ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a1cf71e1c4fb6ab0f131877f6c3ab0f0ba97334b4e48199b4c107828c03e13a | A 3×5 board can toggle its cells between black and white. Each day after Day 1, the board changes its pattern, but it dies if any two patterns differ by fewer than 3 cells or if it becomes all white. If the board starts with all cells black on Day 1, compute the maximum number of days it can stay alive. | [
"2047"
] | [
"A 3×5 board can toggle its cells between black and white. Each day after Day 1, the board changes its pattern, but it dies if any two patterns differ by fewer than 3 cells or if it becomes all white. If the board starts with all cells black on Day 1, compute the maximum number of days it can stay alive."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-12e98fe216c60fac0c717b467eb08e564d577d112bb3ecad7b6f4f3cc718bfa5 | A 4 by 4 square has its opposite corners cut off so that in total, one unit square is removed. Determine the length of the cut needed to form two pieces which can be rearranged to make a 3 by 5 rectangle. | [
"\\sqrt{10}"
] | [
"A 4 by 4 square has its opposite corners cut off so that in total, one unit square is removed. Determine the length of the cut needed to form two pieces which can be rearranged to make a 3 by 5 rectangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-97d12b16df78fd79760095c69ae48e013fb11a51a01ea2c6740aac89266a3747 | A 4-digit number $abcd$ is "cool" if and only if: - $a$ is divisible by 4, - $ab$ is divisible by 5, - $abc$ is divisible by 6, - $abcd$ is divisible by 7, - and the digit 8 cannot appear in the number. How many "cool" numbers are there? | [
"3"
] | [
"A 4-digit number $abcd$ is \"cool\" if and only if:\n- $a$ is divisible by 4,\n- $ab$ is divisible by 5,\n- $abc$ is divisible by 6,\n- $abcd$ is divisible by 7,\n- and the digit 8 cannot appear in the number.\nHow many \"cool\" numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e9c9979d23f0fe2c80b3aa6077321195fc691c074c4f336b3d303ad95b51740 | A 4-digit number $n$ exists such that the sum of the factors of $n$ equals $101^2-1$. Find the remainder when $n$ is divided by $1000$. | [
"983"
] | [
"A 4-digit number $n$ exists such that the sum of the factors of $n$ equals $101^2-1$. Find the remainder when $n$ is divided by $1000$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-4e3d961c455299b3f5dd4334daf3a47a0b8d4ac1688b91d19540774844d672e0 | A 4-digit number is randomly picked from all 4-digit numbers. What is the probability that the product of its digits is divisible by 3? | [
"\\(\\frac{107}{125}\\)"
] | [
"A 4-digit number is randomly picked from all 4-digit numbers. What is the probability that the product of its digits is divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-11650420725549b88876f5072e066e69f90dfe9c7f8fad5a2dd4e877481dae53 | A 4-digit number is subtracted by a 3-digit number resulting in a 3-digit number. If $abcd - efg = hij$ where all 10 digits used in the equation are different, what is the smallest possible value of $hij$? | [
"206"
] | [
"A 4-digit number is subtracted by a 3-digit number resulting in a 3-digit number. If $abcd - efg = hij$ where all 10 digits used in the equation are different, what is the smallest possible value of $hij$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9d21bc235779edcbb543178138722c512e36d15e0e08665063fd8f6648c8a01c | A 4-meter long stick is rotated 60° about a point on the stick 1 meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters. | [
"\\dfrac{2\\pi}{3}"
] | [
"A 4-meter long stick is rotated 60° about a point on the stick 1 meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1711a253d6285e2c9ed789604a390620c417a756e9c3d16e86c8cfeeea97cbe6 | A 4-tuple of real numbers is such that the sum formed by each of the four numbers and the product of the other three numbers is equal to 2. Find all such 4-tuples. | [
"Either all the numbers are equal to 1 or three of them are equal to $-1$ and the fourth one is equal to 3"
] | [
"A 4-tuple of real numbers is such that the sum formed by each of the four numbers and the product of the other three numbers is equal to 2. Find all such 4-tuples."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34ffdf75a9f4d113c5c0bbfbc1ceadf90fa64080497cb869ff2f72b7e9249cb6 | A 40 feet high screen is placed on a vertical wall 10 feet above eye level. How far should you stand from the wall to maximize the angle subtended by the screen (from top to bottom) at your eye? | [
"10\\sqrt{5}"
] | [
"A 40 feet high screen is placed on a vertical wall 10 feet above eye level. How far should you stand from the wall to maximize the angle subtended by the screen (from top to bottom) at your eye?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20c72e3f8f3253ec795ca1bd6a8343aea8a7bf8a5b6f176a1cecde67e9e7a52f | A 40 feet high screen is placed on a vertical wall 10 feet above eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye? | [
"10\\sqrt{5}"
] | [
"A 40 feet high screen is placed on a vertical wall 10 feet above eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f443b90da264f39e6cd666bb5d14a2056b7623e9a95561c816d8d0160dbba7a | A 40 feet high screen is placed on a vertical wall 10 feet above your eye level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye? | [
"\\( d = 10\\sqrt{5} \\) feet"
] | [
"A 40 feet high screen is placed on a vertical wall 10 feet above your eye level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-cc4ed8d3e2023aa1d2dc10ab4b7024b4550513afea4c47e7faeaa96982d17b3b | A 40 feet high screen is placed on a vertical wall 10 feet above your eye-level. How far should you stand from the wall to maximize the angle subtended by the screen (from top to bottom) at your eye? | [
"\\( 10\\sqrt{5} \\)"
] | [
"A 40 feet high screen is placed on a vertical wall 10 feet above your eye-level. How far should you stand from the wall to maximize the angle subtended by the screen (from top to bottom) at your eye?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-77b9151e8dbc0d6175fc82384e45d0c09dfb850824c5702cc73259c68818f67a | A 40 x 40 white square is divided into 1 x 1 squares by lines parallel to its sides. Some of these 1 x 1 squares are colored red such that each 1 x 1 square, regardless of whether it is colored or not, shares a side with at most one red square (not counting itself). What is the largest possible number of red squares? | [
"400"
] | [
"A 40 x 40 white square is divided into 1 x 1 squares by lines parallel to its sides. Some of these 1 x 1 squares are colored red such that each 1 x 1 square, regardless of whether it is colored or not, shares a side with at most one red square (not counting itself). What is the largest possible number of red squar... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-16369b11a4e589ae92ae348909ba56042728c22a3626dcb4e05d587335e2940e | A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a square. How many square feet larger than the old garden is the new garden? | [
"225"
] | [
"A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a square. How many square feet larger than the old garden is the new garden?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dee7463da28b2175bdb8293dea801ba49a125ac66f2377ddfefbce50dbc00866 | A 40-foot television antenna stands on top of a vertical building. From point A on the ground, the angle of elevation to the top of the antenna is $56^\circ$, and the angle of elevation to the bottom of the antenna is $42^\circ$. How tall is the building? (Assume the angle between the ground and the building is a right... | [
"61.9"
] | [
"A 40-foot television antenna stands on top of a vertical building. From point A on the ground, the angle of elevation to the top of the antenna is $56^\\circ$, and the angle of elevation to the bottom of the antenna is $42^\\circ$. How tall is the building? (Assume the angle between the ground and the building is ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5601b3a66b4fdaf9ae1cb8c60fc53c72d2e47d5d70c0c13e7b133f3e52b95d3 | A 40-sided polygon with each side of length 1 is divided into triangles, with each vertex of each triangle being a vertex of the original polygon. Let \( A \) be the area of the smallest triangle. What is the minimum number of square root signs needed to express the exact area of \( A \)? | [
"3"
] | [
"A 40-sided polygon with each side of length 1 is divided into triangles, with each vertex of each triangle being a vertex of the original polygon. Let \\( A \\) be the area of the smallest triangle. What is the minimum number of square root signs needed to express the exact area of \\( A \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-474b5cab17ce312d83d8d5476683cb4a9ef795174cc02ee5bb0d2e6ace793661 | A 40x40 square is divided into 1x1 squares by lines parallel to its sides. Some of these 1x1 squares are colored red so that each 1x1 square, regardless of whether it is colored red or not, shares a side with at most one red square (not counting itself). What is the largest possible number of red squares? | [
"800"
] | [
"A 40x40 square is divided into 1x1 squares by lines parallel to its sides. Some of these 1x1 squares are colored red so that each 1x1 square, regardless of whether it is colored red or not, shares a side with at most one red square (not counting itself). What is the largest possible number of red squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8c40dcfba2bd0899ce59ff054b1351b3060053fcc5854955c2018137dbee95dc | A 40x40 white square is divided into 1x1 squares by lines parallel to its sides. Some of these 1x1 squares are colored red such that each 1x1 square, regardless of its color, shares a side with at most one red square (excluding itself). What is the largest possible number of red squares? | [
"800"
] | [
"A 40x40 white square is divided into 1x1 squares by lines parallel to its sides. Some of these 1x1 squares are colored red such that each 1x1 square, regardless of its color, shares a side with at most one red square (excluding itself). What is the largest possible number of red squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-551a7f5d4450fabca7bc879832a35e9acf98370996bffe5332df2f8ff053d423 | A 4x3 grid needs to be filled with twelve different positive whole numbers such that any two neighboring numbers (sharing an edge) have a common divisor greater than 1. Denote the largest of these twelve numbers as G. What is the smallest possible value of G? | [
"16"
] | [
"A 4x3 grid needs to be filled with twelve different positive whole numbers such that any two neighboring numbers (sharing an edge) have a common divisor greater than 1. Denote the largest of these twelve numbers as G. What is the smallest possible value of G?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef1ddd872783dca3e33f712eaa96a7c2a2a0cf2c968071e12481aae3b99c138e | A 4x4 chessboard has 3 knights and 3 kings randomly placed on it. What is the expected number of checks that occur? | [
"\\dfrac{57}{20}"
] | [
"A 4x4 chessboard has 3 knights and 3 kings randomly placed on it. What is the expected number of checks that occur?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-ce5e8d4e5289bb4f806ffe3d55881de6e096e62f55ed31d07afdc7b36449be60 | A 4x4 table is divided into 16 white unit square cells. Two cells are called neighbors if they share a common side. A move consists of choosing a cell and changing the colors of its neighbors from white to black or from black to white. After exactly \( n \) moves, all the cells are black. Find all possible values of \(... | [
"\\( n \\geq 4 \\) and \\( n \\) is even."
] | [
"A 4x4 table is divided into 16 white unit square cells. Two cells are called neighbors if they share a common side. A move consists of choosing a cell and changing the colors of its neighbors from white to black or from black to white. After exactly \\( n \\) moves, all the cells are black. Find all possible value... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-c0c0d0f543c2b5bb4dcfb476fb9bc1c337dc758e095a17d0fd1cfe8561c72a3a | A 4x4-square of integers contains 30 subsquares \(A_1, \cdots, A_{30}\). Let \(B_i\) be the sum of integers in \(A_i\). What is the maximum number \(i\) such that for any positive \(j < i\), we have \(j \in \{B_1, \cdots, B_{30}\}\)? | [
"25"
] | [
"A 4x4-square of integers contains 30 subsquares \\(A_1, \\cdots, A_{30}\\). Let \\(B_i\\) be the sum of integers in \\(A_i\\). What is the maximum number \\(i\\) such that for any positive \\(j < i\\), we have \\(j \\in \\{B_1, \\cdots, B_{30}\\}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-efd98bd211caaab0ac72f692b30f6453db68cb9fe0354268b98208c15df27aad | A 4x4x4 cube is made of 64 1x1x1 smaller cubes, each with 3 red sides and 3 yellow sides. If all 64 cubes are rolled like dice and combined to form the giant 4x4x4 cube based on their orientations, what is the expected number of yellow sides that show? | [
"48"
] | [
"A 4x4x4 cube is made of 64 1x1x1 smaller cubes, each with 3 red sides and 3 yellow sides. If all 64 cubes are rolled like dice and combined to form the giant 4x4x4 cube based on their orientations, what is the expected number of yellow sides that show?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60a9b8e2b2eecdfc560c0bd15eb8cc5072068bb995ab100e1d047f8323295a64 | A 4x4x4 cube is painted red. It is then divided into 8 2x2x2 cubes, and the faces not painted red are painted blue. The 8 cubes are then divided into 64 unit cubes, and the unpainted faces are painted yellow. What is the ratio of the number of red faces to blue faces to yellow faces? | [
"1:1:2"
] | [
"A 4x4x4 cube is painted red. It is then divided into 8 2x2x2 cubes, and the faces not painted red are painted blue. The 8 cubes are then divided into 64 unit cubes, and the unpainted faces are painted yellow. What is the ratio of the number of red faces to blue faces to yellow faces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-a8d3177c4cde4198f88ef9f22a1ac24c89ebf342ebee884c9f2f8dbbeb7f9811 | A 5 × 5 × 5 wooden cube is painted on exactly five of its six faces and then cut into 125 unit cubes. One unit cube is randomly selected and rolled. What is the probability that the top face of the unit cube that is rolled is painted? Express your answer as a common fraction. | [
"\\(\\frac{1}{6}\\)"
] | [
"A 5 × 5 × 5 wooden cube is painted on exactly five of its six faces and then cut into 125 unit cubes. One unit cube is randomly selected and rolled. What is the probability that the top face of the unit cube that is rolled is painted? Express your answer as a common fraction."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 26
} | 72 | 0 | 0 |
omr-ws-a1e938040e029694ea91b8ffbbedc250eccc87d7ab2dc973b07c03d5abb268ee | A 5,000 cubic foot capacity water tank with a constant cross section stands 10 feet tall. It drains from the bottom with a volume rate proportional to the instantaneous water level, and a full tank drains 50% in 6 hours. If the drain is opened and water is added to the tank at a rate of 100 cubic feet per hour, what is... | [
"\\(\\frac{6}{5 \\ln 2}\\)"
] | [
"A 5,000 cubic foot capacity water tank with a constant cross section stands 10 feet tall. It drains from the bottom with a volume rate proportional to the instantaneous water level, and a full tank drains 50% in 6 hours. If the drain is opened and water is added to the tank at a rate of 100 cubic feet per hour, wh... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 23
} | 26 | 0 | 0 |
omr-ws-793eafc6707aebefd7bdc310bb857cee75e57469ee53cb23397ae6041fa91dcb | A 5-digit number (in base 10) has digits \( k, k + 1, k + 2, 3k, k + 3 \) in that order, from left to right. If this number is \( m^2 \) for some natural number \( m \), find the sum of the digits of \( m \). | [
"15"
] | [
"A 5-digit number (in base 10) has digits \\( k, k + 1, k + 2, 3k, k + 3 \\) in that order, from left to right. If this number is \\( m^2 \\) for some natural number \\( m \\), find the sum of the digits of \\( m \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-2c2fa9d0206e9baf4540057c48de847384160ea09c94db55c8689c80a3498021 | A 5-digit number is a multiple of 41. If the highest order digit is removed and placed to the right of the units digit, the new number is a perfect cube. What was the original number? | [
"16892"
] | [
"A 5-digit number is a multiple of 41. If the highest order digit is removed and placed to the right of the units digit, the new number is a perfect cube. What was the original number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51c418ea77248e6534ebf8aba291e7086c739d5ffc38c2d6abda937fb5b78405 | A 5-digit number is called "leet" if the sum of the first 2 digits, the sum of the last 2 digits, and the middle digit are equal. How many 5-digit leet numbers exist? | [
"330"
] | [
"A 5-digit number is called \"leet\" if the sum of the first 2 digits, the sum of the last 2 digits, and the middle digit are equal. How many 5-digit leet numbers exist?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b327fd3716647ffd72cd2d1a75d865fae99cc32723eaadf65dd7d0879761d8a9 | A 5-digit positive integer $\overline{abcde}$ is defined to be $\textit{wavy}$ if $a<b>c<d>e$ and $\textit{anti-wavy}$ if $a>b<c>d<e$. Find the remainder when the positive difference between the number of wavy integers and the number of anti-wavy integers is divided by $1000$. | [
"695"
] | [
"A 5-digit positive integer $\\overline{abcde}$ is defined to be $\\textit{wavy}$ if $a<b>c<d>e$ and $\\textit{anti-wavy}$ if $a>b<c>d<e$. Find the remainder when the positive difference between the number of wavy integers and the number of anti-wavy integers is divided by $1000$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-aef7c1955fbda044eb886bbb2374d9ce8ca9d7f3fdd8bb24a6fd4b7d5b3cf4a1 | A 5-ft ladder leans against a wall with its foot 4 ft from the wall. The ladder slides down 2 ft. How far is the ladder's foot from the wall after sliding? | [
"\\(2\\sqrt{6}\\)"
] | [
"A 5-ft ladder leans against a wall with its foot 4 ft from the wall. The ladder slides down 2 ft. How far is the ladder's foot from the wall after sliding?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-35f5aa077c4283c93a66bcdb14eb51ded5617c22c0fe1f2b34a95d550a3f6bbc | A 5-person committee consists of 2 red people and 3 blue people. The red people vote for red bills 75% of the time, and the blue people vote for red bills 25% of the time. Find the probability that a red bill will receive a majority vote. | [
"\\(\\frac{195}{512}\\)"
] | [
"A 5-person committee consists of 2 red people and 3 blue people. The red people vote for red bills 75% of the time, and the blue people vote for red bills 25% of the time. Find the probability that a red bill will receive a majority vote."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ce2a2d47c3b7989547620ac4d91746f1531b23b6e33c3a177ef03fd5fc61241 | A 5-question exam awards 5 points for a correct answer, 1 point for a blank, and 0 points for an incorrect answer. For each attainable score $ s $, let $ N(s) $ be the number of distinct answer patterns corresponding to that score. Over all attainable scores $ s $, the average value of $ N(s) $ can be written in the fo... | [
"263"
] | [
"A 5-question exam awards 5 points for a correct answer, 1 point for a blank, and 0 points for an incorrect answer. For each attainable score $ s $, let $ N(s) $ be the number of distinct answer patterns corresponding to that score. Over all attainable scores $ s $, the average value of $ N(s) $ can be written in t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-b6b4f2bbf575569c523437d2a59a1da872e2a058b930c059889adb323c0d62c7 | A 50-digit number N has all digits except the 26th digit (from the left) as 1. If N is divisible by 13, find the 26th digit. | [
"3"
] | [
"A 50-digit number N has all digits except the 26th digit (from the left) as 1. If N is divisible by 13, find the 26th digit."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-96246cfb96e20716365d8a605ce51a79cd44511d6fe210695f326b9cb3aad01e | A 50-digit prime number \( p \) is divided by 120, and the remainder \( r \) is not 1. What is \( r \)? | [
"119"
] | [
"A 50-digit prime number \\( p \\) is divided by 120, and the remainder \\( r \\) is not 1. What is \\( r \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd0ed4a3c08f92bad1580ff82b4a1cc7fed0ab8db06ce0adafae383e7ab99ffd | A 50-foot rope is tied at each end to two stakes that are 14 feet apart. A dog is tethered to the rope, and the tether slides freely along the rope between the stakes. What is the area of the region that the dog can reach? | [
"\\( 600\\pi \\)"
] | [
"A 50-foot rope is tied at each end to two stakes that are 14 feet apart. A dog is tethered to the rope, and the tether slides freely along the rope between the stakes. What is the area of the region that the dog can reach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 20
} | 38 | 0 | 0 |
omr-ws-2d989c1f73014b2e32b31a8a63ff3a5e5a3cd28c66cef7c5fbda2657ce8b0ffc | A 500 meter wide river is flowing downstream at 3 kph. A man swims across the river and reaches the other side 7.5 minutes later, directly across from the point where he started. If he had allowed the current to carry him instead of swimming partly against the current, how many minutes would he have taken to cross the ... | [
"6 minutes"
] | [
"A 500 meter wide river is flowing downstream at 3 kph. A man swims across the river and reaches the other side 7.5 minutes later, directly across from the point where he started. If he had allowed the current to carry him instead of swimming partly against the current, how many minutes would he have taken to cross... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
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