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-a3a739eedc1ca97fee39561c002235ddff72d9d6a4f4202eafa86fd7cdd535af | A printer printed a card with black ink on blue paper and an envelope with blue ink on blue paper. If the colors were chosen at random from a stock of 7 inks and 6 papers, what is the probability of this given result? | [
"\\(\\frac{1}{1764}\\)"
] | [
"A printer printed a card with black ink on blue paper and an envelope with blue ink on blue paper. If the colors were chosen at random from a stock of 7 inks and 6 papers, what is the probability of this given result?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-8f77f7ece4a87f9e8ea0c97ee2a9df544b265e5c686875e7fad2fa3668ced713 | A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number. | [
"12"
] | [
"A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6fee8f7765a7ffbb5b8b8cd5965577f29836c441970546cbee03d7e9fda04d0 | A prism has 15 edges. How many faces does the prism have? | [
"7"
] | [
"A prism has 15 edges. How many faces does the prism have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ccbf39138828697de595bc0a72c6e0cd8478f8c85a508de11f18d80fb5236052 | A prism has a regular hexagonal base with side length 1. The vertices of an oblique cutting plane are at distances 2, 3, \(x\), \(y\), 11, and \(z\) from the base. Calculate \(x + y + z\). | [
"26"
] | [
"A prism has a regular hexagonal base with side length 1. The vertices of an oblique cutting plane are at distances 2, 3, \\(x\\), \\(y\\), 11, and \\(z\\) from the base. Calculate \\(x + y + z\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 3
} | 23 | 0 | 0 |
omr-ws-e7474240ae496375c15f3fd9c1d8a85d86fb27b2476903f662a7d9b11396ab9e | A prism has its base and top as an \( n \)-sided regular polygon. Every vertex of the prism (all \( 2n \) of them) is labeled with the number \( 1 \) or \( -1 \). It is known that for every side (face) of the prism, the product of all the numbers which are labeled on all its vertices is \( -1 \). The sum of all natural... | [
"304"
] | [
"A prism has its base and top as an \\( n \\)-sided regular polygon. Every vertex of the prism (all \\( 2n \\) of them) is labeled with the number \\( 1 \\) or \\( -1 \\). It is known that for every side (face) of the prism, the product of all the numbers which are labeled on all its vertices is \\( -1 \\). The sum... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-d2e8c8aea37d227b52da266534a18e7915f5e445b9286bf1ed6590bc99cc5682 | A prism has more than 310 edges, fewer than 320 edges, and the number of edges is odd. How many edges does the prism have? | [
"315"
] | [
"A prism has more than 310 edges, fewer than 320 edges, and the number of edges is odd. How many edges does the prism have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-2c4cf2de65c9063bfc25ae7e784e0617b45d5ce26cb46553ee96d2dd6808eee4 | A prism is constructed so that its vertical edges are parallel to the $z$-axis. Its cross-section is a square of side length 10. [asy] import three; size(180); currentprojection = perspective(6,3,2); triple A, B, C, D, E, F, G, H; A = (1,1,0); B = (1,-1,0); C = (-1,-1,0); D = (-1,1,0); E = A + (0,0,1); F = B + (0,0,3);... | [
"225"
] | [
"A prism is constructed so that its vertical edges are parallel to the $z$-axis. Its cross-section is a square of side length 10.\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple A, B, C, D, E, F, G, H;\n\nA = (1,1,0);\nB = (1,-1,0);\nC = (-1,-1,0);\nD = (-1,1,0);\nE = A + (... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-2af6411266d3384c5fb209a3284a3474b6f55bdc8b6a907f34e7812ffb7f401b | A prism is right and has an equilateral triangle as its base. A plane cuts the prism but not cuts none of its bases, determining a triangular section of sides $a, b$, and $c$. Calculate the side of the prism base in terms of $a, b$ and $c$. | [
"\\sqrt{\\dfrac{a^2 + b^2 + c^2 - 2\\sqrt{a^4 + b^4 + c^4 - a^2b^2 - a^2c^2 - b^2c^2}}{3}}"
] | [
"A prism is right and has an equilateral triangle as its base. A plane cuts the prism but not cuts none of its bases, determining a triangular section of sides $a, b$, and $c$. Calculate the side of the prism base in terms of $a, b$ and $c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-daa55343813c3e93cba8531fabf81d0f4618fd2bac074a3fa81b13bc3e0302e5 | A prison consists of a square courtyard of side $b$ bounded by a perimeter wall and a square building of side $a$ placed centrally within the courtyard. The sides of the building are parallel to the perimeter walls. Guards can stand either at the middle of a perimeter wall or in a corner of the courtyard. Determine whi... | [
"The corner position is preferable when \\(2b \\le 9a\\)."
] | [
"A prison consists of a square courtyard of side $b$ bounded by a perimeter wall and a square building of side $a$ placed centrally within the courtyard. The sides of the building are parallel to the perimeter walls. Guards can stand either at the middle of a perimeter wall or in a corner of the courtyard. Determin... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19aa321789f866c3ce14f456c87d3af760525f3a65221424970a31fb26447fad | A prison guard tower is 30 meters from the nearest wall of the prison. The spotlight L on top of the tower rotates counterclockwise making one full revolution every six seconds, casting a moving beam of light onto the wall. Let N be the point on the wall nearest the spotlight, and M be the moving spot. Let \( d(t) \) b... | [
"30 \\tan\\left(\\frac{\\pi t}{3}\\right)"
] | [
"A prison guard tower is 30 meters from the nearest wall of the prison. The spotlight L on top of the tower rotates counterclockwise making one full revolution every six seconds, casting a moving beam of light onto the wall. Let N be the point on the wall nearest the spotlight, and M be the moving spot. Let \\( d(t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-4ab343b7d7b75598a3ff30580a8568ebf7f711226b0cf36b4eb05236eefb4ca6 | A prison has 2004 cells, numbered 1 through 2004. A jailer unlocked every cell, then on the nth trip, turned the key in every nth cell, locking the opened cells and unlocking the locked ones. After 2004 trips, how many prisoners were released? | [
"44"
] | [
"A prison has 2004 cells, numbered 1 through 2004. A jailer unlocked every cell, then on the nth trip, turned the key in every nth cell, locking the opened cells and unlocking the locked ones. After 2004 trips, how many prisoners were released?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 13
} | 14 | 0 | 0 |
omr-ws-1aa0ba887383ee57e15ffc2b57e2fcd0e08c210454ecc72422359970dce777c2 | A prisoner is trapped in a cell containing three doors. The first door leads back to the cell after 3 days, the second door leads back after 4 days, and the third door leads to freedom. The prisoner selects the doors with probabilities $\frac{1}{2}$, $\frac{7}{20}$, and $\frac{3}{20}$ respectively. Determine the expect... | [
"\\(\\frac{58}{3}\\)"
] | [
"A prisoner is trapped in a cell containing three doors. The first door leads back to the cell after 3 days, the second door leads back after 4 days, and the third door leads to freedom. The prisoner selects the doors with probabilities $\\frac{1}{2}$, $\\frac{7}{20}$, and $\\frac{3}{20}$ respectively. Determine th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-566d1c10454dac1b53b7d7416d7b0a49fd7199ba98622b8a1cf5892320e5cc3b | A privateer discovers a merchantman 10 miles away at 11:45 a.m. The privateer travels at 11 mph, while the merchantman travels at 8 mph. After a two-hour chase, the privateer's top sail is carried away, reducing her speed such that she can now travel 17 miles while the merchantman travels 15 miles. Determine the time a... | [
"5:30 p.m."
] | [
"A privateer discovers a merchantman 10 miles away at 11:45 a.m. The privateer travels at 11 mph, while the merchantman travels at 8 mph. After a two-hour chase, the privateer's top sail is carried away, reducing her speed such that she can now travel 17 miles while the merchantman travels 15 miles. Determine the t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-983926f43a37318466d5305292d301d5a12291fc2723877c33781efc1c77e996 | A privateer discovers a merchantman 10 miles away at 11:45 a.m. The privateer travels at 11 mph, while the merchantman travels at 8 mph. After a two-hour chase, the privateer's top sail is carried away, reducing her speed such that she can now travel 17 miles while the merchantman travels 15 miles. Determine the time a... | [
"5:30 p.m."
] | [
"A privateer discovers a merchantman 10 miles away at 11:45 a.m. The privateer travels at 11 mph, while the merchantman travels at 8 mph. After a two-hour chase, the privateer's top sail is carried away, reducing her speed such that she can now travel 17 miles while the merchantman travels 15 miles. Determine the t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c6e746eaeb415b8c45f607db9dcf0c48b0faa4d16d433fb26aee187b4a554669 | A producer of fluorescent tubes has learned from experience that for tubes of a certain type, A, the number of hours this tube can burn before it should be renewed is a normal distributed random variable with a mean, μ, of 9,600 hours and a standard deviation, σ, of 1,200 hours. A chain of shops has recently opened a n... | [
"0.0175"
] | [
"A producer of fluorescent tubes has learned from experience that for tubes of a certain type, A, the number of hours this tube can burn before it should be renewed is a normal distributed random variable with a mean, μ, of 9,600 hours and a standard deviation, σ, of 1,200 hours. A chain of shops has recently opene... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-270a5af2bd1d8666e2661692bdb83fa0e14240f9f7de349461617bb59ef4d4dc | A professor grades an exam that has 20 questions worth five points each. The professor awards zero, two, four, or five points on each problem. Find a concise ordinary generating function that can be used to determine the number of ways to obtain an exam score of \( k \) points. | [
"(1 + x^2 + x^4 + x^5)^{20}"
] | [
"A professor grades an exam that has 20 questions worth five points each. The professor awards zero, two, four, or five points on each problem. Find a concise ordinary generating function that can be used to determine the number of ways to obtain an exam score of \\( k \\) points."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f650fe8588f04aa6d41b5e3bbd3d37a587a0dd31d404e75cdd3b8145f0636da5 | A professor has 10 students and wants to give each student an individualized exam with three problems. Each pair of students must have no more than one problem in common. What is the minimum number of distinct problems the professor must prepare to accomplish this? | [
"9"
] | [
"A professor has 10 students and wants to give each student an individualized exam with three problems. Each pair of students must have no more than one problem in common. What is the minimum number of distinct problems the professor must prepare to accomplish this?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 7,
"tir": 8
} | 56 | 0 | 0 |
omr-ws-22eae91232563cafc7d7a94a442dc28f883a347c859a879162c397db80359a7c | A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students? | [
"59049"
] | [
"A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b9e93747558c1f0d32344edd8f817aa243e7c05575ac350bd53186a535c4104 | A professor lectures in city A and has a project in city B every Monday to Wednesday for five weeks. A round trip ticket bought on Sunday to return on Wednesday is 20% more expensive than a ticket bought before Saturday. A one-way ticket costs 75% of the standard round trip ticket price, which is $900. Determine the be... | [
"4500"
] | [
"A professor lectures in city A and has a project in city B every Monday to Wednesday for five weeks. A round trip ticket bought on Sunday to return on Wednesday is 20% more expensive than a ticket bought before Saturday. A one-way ticket costs 75% of the standard round trip ticket price, which is $900. Determine t... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-48d0a220232d0d4bc6e4b3980ae423a626286c1dcb5f2aa22b2ce7616103d704 | A professor tells 3 jokes in his class each year. To ensure that he does not repeat the same triple of jokes over a period of 12 years, what is the minimum number of jokes he needs to have? | [
"6"
] | [
"A professor tells 3 jokes in his class each year. To ensure that he does not repeat the same triple of jokes over a period of 12 years, what is the minimum number of jokes he needs to have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a7b3d96cfd017ff9fc80c0efb86b590af9efad8efe8e4f13048797bafc9166f4 | A professor tries not to be late for class too often. If he is late some day, he is 90% sure to be on time the next time. If he is on time, the next day there is a 25% chance of his being late. In the long run, how often is he late for class? | [
"\\(\\frac{5}{23}\\)"
] | [
"A professor tries not to be late for class too often. If he is late some day, he is 90% sure to be on time the next time. If he is on time, the next day there is a 25% chance of his being late. In the long run, how often is he late for class?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-dc2e898e0acee763cbefe29d33a5d53643905480a440e96466edd37165c72312 | A project team consists of 100 people. Every member is appointed to an equal number of projects, and no two people are appointed to the exact same projects. This situation cannot be achieved if every member is appointed to 3 projects, but can be achieved if every member is appointed to 4 projects. How many projects doe... | [
"9"
] | [
"A project team consists of 100 people. Every member is appointed to an equal number of projects, and no two people are appointed to the exact same projects. This situation cannot be achieved if every member is appointed to 3 projects, but can be achieved if every member is appointed to 4 projects. How many project... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 15
} | 63 | 0 | 0 |
omr-ws-1adf36fbf433bf02476c6b2ec3688273cc14deea753ed43411157725d284d85c | A project team of 100 people has a certain number of projects, with each person assigned to an equal number of projects. The goal is to ensure that no two people are involved in exactly the same set of projects. This situation cannot be achieved if everyone takes part in 3 projects, but can be achieved if everyone take... | [
"9"
] | [
"A project team of 100 people has a certain number of projects, with each person assigned to an equal number of projects. The goal is to ensure that no two people are involved in exactly the same set of projects. This situation cannot be achieved if everyone takes part in 3 projects, but can be achieved if everyone... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-d0d7e4e57b322dd81b9949a1459d5857672a7f1a06f1e047f57c4da6c04e3983 | A projectile \( P \) is launched from point \( A \) in a perfect semicircle \( O \) of radius 1 mile. After traveling at a constant velocity for 1 minute, \( P \) returns to the ground at point \( B \). Find a function \( f(t) \) that describes the distance \( d \) from \( A \) to \( P \) as a function of time \( t \) ... | [
"\\( 2 \\sin \\left( \\frac{\\pi t}{120} \\right) \\)"
] | [
"A projectile \\( P \\) is launched from point \\( A \\) in a perfect semicircle \\( O \\) of radius 1 mile. After traveling at a constant velocity for 1 minute, \\( P \\) returns to the ground at point \\( B \\). Find a function \\( f(t) \\) that describes the distance \\( d \\) from \\( A \\) to \\( P \\) as a fu... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1b0b64409f3f7aa554e11c06336a10b402f248c6aa08c92daee5828903ed84ed | A projectile is fired with an initial speed of 160 ft/s at a target 600 ft away. There is a hill 300 ft high midway between the starting point and the target. At what initial angle of elevation should the projectile be fired? | [
"65.7^\\circ"
] | [
"A projectile is fired with an initial speed of 160 ft/s at a target 600 ft away. There is a hill 300 ft high midway between the starting point and the target. At what initial angle of elevation should the projectile be fired?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-49e348206a0879095f07fd0a500127d6ed788902bd5bc7353d1a459fdc5488ac | A projectile is fired with an initial velocity of $v$ at an angle of $\theta$ from the ground. Then its trajectory can modeled by the parametric equations \begin{align*} x &= vt \cos \theta, \\ y &= vt \sin \theta - \frac{1}{2} gt^2, \end{align*}where $t$ denotes time and $g$ denotes acceleration due to gravity, formin... | [
"\\frac{\\pi}{8}"
] | [
"A projectile is fired with an initial velocity of $v$ at an angle of $\\theta$ from the ground. Then its trajectory can modeled by the parametric equations\n\\begin{align*}\nx &= vt \\cos \\theta, \\\\\ny &= vt \\sin \\theta - \\frac{1}{2} gt^2,\n\\end{align*}where $t$ denotes time and $g$ denotes acceleration du... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9be7f0b5c12ee94c0ae5f61311875855e845c77af851df54407de49468640187 | A projectile is launched from a height of 4 meters with an initial velocity of 90 meters/second. Determine the amount of time the projectile stays in the air. | [
"\\( t \\approx 18.43 \\text{ seconds} \\)"
] | [
"A projectile is launched from a height of 4 meters with an initial velocity of 90 meters/second. Determine the amount of time the projectile stays in the air."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-64a9cafb1d34323cae5cc2726f0184429b3715e384adb7328afcddd16c46a332 | A projectile is launched from a height of 4 meters with an initial velocity of 90 meters/second. Determine the velocity of the projectile on impact. | [
"90.43"
] | [
"A projectile is launched from a height of 4 meters with an initial velocity of 90 meters/second. Determine the velocity of the projectile on impact."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 14
} | 23 | 0 | 0 |
omr-ws-291dcbdc8e20249ed08495bfcd208029decbff5973868b032a37522e3188e8d2 | A projectile is launched from a point such that it just clears two walls of height \( H \) meters located at distances \( B \) meters and \( C \) meters from the launch point. Given \( H = 7 \) meters, \( B = 7 \) meters, and \( C = 14 \) meters, calculate the value of \( \tan A \). | [
"\\dfrac{3}{2}"
] | [
"A projectile is launched from a point such that it just clears two walls of height \\( H \\) meters located at distances \\( B \\) meters and \\( C \\) meters from the launch point. Given \\( H = 7 \\) meters, \\( B = 7 \\) meters, and \\( C = 14 \\) meters, calculate the value of \\( \\tan A \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e90b6d90dcd21dd3a13fc7557fbedafb05ef8399e0faeef793fa60093bd1c09 | A projectile is launched from the center of a rectangular room that is 40 feet wide and has a very high ceiling. The projectile hits the wall at a height of 10 feet above the floor, reflects off this wall, hits the opposite wall, and reflects again, finally landing back exactly where it was launched, without hitting th... | [
"13.333"
] | [
"A projectile is launched from the center of a rectangular room that is 40 feet wide and has a very high ceiling. The projectile hits the wall at a height of 10 feet above the floor, reflects off this wall, hits the opposite wall, and reflects again, finally landing back exactly where it was launched, without hitti... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-f976b045bb6116b0c8aae10be236b8bca841a17ec9c98680c1f14644b4df8c8b | A projectile is launched from the center of a rectangular room that is 40 feet wide. It hits the wall at a height of 10 feet above the floor, reflects off this wall, hits the opposite wall, and reflects again, finally landing back at the launch point without hitting the ceiling. The projectile travels along parabolic s... | [
"\\(\\frac{40}{3}\\)"
] | [
"A projectile is launched from the center of a rectangular room that is 40 feet wide. It hits the wall at a height of 10 feet above the floor, reflects off this wall, hits the opposite wall, and reflects again, finally landing back at the launch point without hitting the ceiling. The projectile travels along parabo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 14
} | 36 | 0 | 0 |
omr-ws-f7bb1cf4928fff83df61222bb2c86564705c0e99df96529a3dff60c6f6357ca1 | A projectile is launched straight up into the air with an initial velocity such that it returns to the same elevation with a velocity of 20 meters per second. Given that the acceleration due to gravity is 9.90 meters per second squared and there is no air resistance, determine the maximum height, in centimeters, that t... | [
"2020 cm"
] | [
"A projectile is launched straight up into the air with an initial velocity such that it returns to the same elevation with a velocity of 20 meters per second. Given that the acceleration due to gravity is 9.90 meters per second squared and there is no air resistance, determine the maximum height, in centimeters, t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-2ec8615046da5c9c58b4eeb4e44d94a4369c52403e23b4779591ff76a6541399 | A projectile is observed from four radio stations situated at the corners of a square of side \( b \). If the distances from the projectile to the stations are \( R_1, R_2, R_3, \) and \( R_4 \), calculate the value of \( R_{1}^{2} + R_{3}^{2} - R_{2}^{2} - R_{4}^{2} \). | [
"0"
] | [
"A projectile is observed from four radio stations situated at the corners of a square of side \\( b \\). If the distances from the projectile to the stations are \\( R_1, R_2, R_3, \\) and \\( R_4 \\), calculate the value of \\( R_{1}^{2} + R_{3}^{2} - R_{2}^{2} - R_{4}^{2} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-55ef5c77f8b240d192b18bf4fa762655ec3f385c6a5048c41bd4a4185eec626e | A projectile moves in a resisting medium with the horizontal distance given by $x=f(t)$. Given that the vertical distance $y$ is expressed as $$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt + Af(t) + B$$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity, find the expression f... | [
"y = -4.9 t^2 \\ln t + A t^2 + B"
] | [
"A projectile moves in a resisting medium with the horizontal distance given by $x=f(t)$. Given that the vertical distance $y$ is expressed as\n$$y=-gf(t) \\int \\frac{dt}{f'(t)} + g \\int \\frac{f(t)}{f'(t)} \\, dt + Af(t) + B$$\nwhere $A$ and $B$ are constants and $g$ is the acceleration due to gravity, find the ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-4d5898c8ff5aad089372c85425b5d8e5e36a58060ef5a66b41128815951cc3b4 | A projectile, thrown with initial velocity \( v_0 \) in a direction making angle \( \alpha \) with the horizontal, is acted on by no force except gravity. Find the length of its path until it strikes a horizontal plane through the starting point. | [
" \\dfrac{v_0^2}{g} \\left( \\sin \\alpha + \\cos^2 \\alpha \\ln \\left( \\sec \\alpha + \\tan \\alpha \\right) \\right) "
] | [
"A projectile, thrown with initial velocity \\( v_0 \\) in a direction making angle \\( \\alpha \\) with the horizontal, is acted on by no force except gravity. Find the length of its path until it strikes a horizontal plane through the starting point."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-c0b557170cbd3afdaf6daffe4efb5b1e4102c34036ed77e874258a15e9440a9c | A projection takes $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ to $\begin{pmatrix} \frac{3}{2} \\ -\frac{3}{2} \end{pmatrix}.$ Which vector does the projection take $\begin{pmatrix} -4 \\ 1 \end{pmatrix}$ to? | [
"\\begin{pmatrix} -5/2 \\\\ 5/2 \\end{pmatrix}"
] | [
"A projection takes $\\begin{pmatrix} 1 \\\\ -2 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{3}{2} \\\\ -\\frac{3}{2} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -4 \\\\ 1 \\end{pmatrix}$ to?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09c6559532f6324394d3fb5635857df9242727470beba9ea9eb36fc8435434ad | A projection takes $\begin{pmatrix} 4 \\ 4 \end{pmatrix}$ to $\begin{pmatrix} \frac{60}{13} \\ \frac{12}{13} \end{pmatrix}.$ Which vector does the projection take $\begin{pmatrix} -2 \\ 2 \end{pmatrix}$ to? | [
"\\begin{pmatrix} -20/13 \\\\ -4/13 \\end{pmatrix}"
] | [
"A projection takes $\\begin{pmatrix} 4 \\\\ 4 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{60}{13} \\\\ \\frac{12}{13} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -2 \\\\ 2 \\end{pmatrix}$ to?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36659fd737c15aba006b0cf308c7352bfe63a4b83227d32e8b08bbab6c02a645 | A proposal will make years that end in double zeroes a leap year only if the year leaves a remainder of 200 or 600 when divided by 900. Under this proposal, how many leap years will there be that end in double zeroes between 1996 and 4096? | [
"5"
] | [
"A proposal will make years that end in double zeroes a leap year only if the year leaves a remainder of 200 or 600 when divided by 900. Under this proposal, how many leap years will there be that end in double zeroes between 1996 and 4096?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d08269acba62552a9818817fb670e3f7a4c178d4df54c31f290bd33ff23b2564 | A prospective doctor is required to intern in a hospital for five days in July 2011, with the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Determine the number of possible schedul... | [
"42504"
] | [
"A prospective doctor is required to intern in a hospital for five days in July 2011, with the following rules:\na) Internships may not be conducted on two consecutive days.\nb) The fifth day of internship can only be done after four days counted since the fourth day of internship.\nDetermine the number of possible... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 10
} | 15 | 0 | 0 |
omr-ws-2be334fbf9fca731b38436147b37c9a623ea26bf8c8d6d82eb820aec46433cd4 | A protractor has its scale edge 7.5 cm from the center point. If the scale is marked in degrees, how far apart are the marks along the edge? | [
"\\(\\frac{\\pi}{24}\\) cm"
] | [
"A protractor has its scale edge 7.5 cm from the center point. If the scale is marked in degrees, how far apart are the marks along the edge?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-c4da1cca015e7d9168166f48d0876cf9d55a8817a749cf3dc5c22810a2da27bc | A province consists of nine cities named $1, 2, 3, 4, 5, 6, 7, 8, 9$. From city $a$ there is a direct road to city $b$ if and only if $\overline{ab}$ and $\overline{ba}$ are two-digit numbers that are divisible by $3$. Two distinct cities $a_1$ and $a_n$ are said to be connected if there is a sequence of cities $a_1, a... | [
"6"
] | [
"A province consists of nine cities named $1, 2, 3, 4, 5, 6, 7, 8, 9$. From city $a$ there is a direct road to city $b$ if and only if $\\overline{ab}$ and $\\overline{ba}$ are two-digit numbers that are divisible by $3$. Two distinct cities $a_1$ and $a_n$ are said to be connected if there is a sequence of cities ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-77cd6daf6ee9890ac48b3f8b6790516390c5fb5958da240bdea253ec8f0500ad | A pulley is wrapped around two circles with diameters of 8 and 5 respectively. If the larger circle spins at 80 revolutions per minute, how many revolutions does the smaller circle make in one second? | [
"\\(\\frac{32}{15}\\)"
] | [
"A pulley is wrapped around two circles with diameters of 8 and 5 respectively. If the larger circle spins at 80 revolutions per minute, how many revolutions does the smaller circle make in one second?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8c505e96471963c66eb2a8ca70b1c396fac15f990a95bd1a3a6ce6255d5058be | A pupil is swimming at the center of a circular pond with a radius of 100 meters. At the edge of the pond, there is a teacher who wishes to catch the pupil but cannot swim. The teacher can run four times as fast as the pupil can swim, but not as fast as the pupil can run. If the pupil swims directly towards the edge of... | [
"100(4 - \\pi)"
] | [
"A pupil is swimming at the center of a circular pond with a radius of 100 meters. At the edge of the pond, there is a teacher who wishes to catch the pupil but cannot swim. The teacher can run four times as fast as the pupil can swim, but not as fast as the pupil can run. If the pupil swims directly towards the ed... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-d568ed1e207a76241e0f9fd6fdc31aed2d9766857875afc2e9acefd95a480093 | A puppy and two cats together weigh 24 pounds. The puppy and the larger cat together weigh exactly twice as much as the smaller cat, and the puppy and the smaller cat together weigh exactly the same as the larger cat. How many pounds does the puppy weigh? | [
"4"
] | [
"A puppy and two cats together weigh 24 pounds. The puppy and the larger cat together weigh exactly twice as much as the smaller cat, and the puppy and the smaller cat together weigh exactly the same as the larger cat. How many pounds does the puppy weigh?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64e9c89c3cc5d3446e6dd881e51f727968702f81aaf5d4344d5d567c6556a69e | A purchaser buys electrical components in lots of size 40. The purchaser inspects 5 components randomly and rejects the lot if a defective component is found. If 30 percent of the lots have 4 defective components and 70 percent have only 1 defective component, what proportion of lots does the purchaser reject? | [
"\\(\\frac{31529}{146224}\\)"
] | [
"A purchaser buys electrical components in lots of size 40. The purchaser inspects 5 components randomly and rejects the lot if a defective component is found. If 30 percent of the lots have 4 defective components and 70 percent have only 1 defective component, what proportion of lots does the purchaser reject?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 31
} | 42 | 0 | 0 |
omr-ws-6c9a556ee5ae0eacef538525d24ad1ab49c30b3d80b48196d6290e329b4f8f94 | A purse contains 4 coins which are either sovereigns or shillings. Two coins are drawn and both are found to be shillings. If these are replaced, what is the probability that another drawing will give a sovereign? | [
"\\dfrac{1}{8}"
] | [
"A purse contains 4 coins which are either sovereigns or shillings. Two coins are drawn and both are found to be shillings. If these are replaced, what is the probability that another drawing will give a sovereign?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-ad2bb8b1f244de06fa990ed88b63445733d30d437159aed1c1240ba8516c3d9a | A pyramid $OABC$ has an equilateral triangle $ABC$ as its base with $AB = 4$. The lengths $OA$, $OB$, and $OC$ are equal, and the distance between line $OA$ and line $BC$ is $\sqrt{3}$. Find the length of $OA$. | [
"\\dfrac{8}{3}"
] | [
"A pyramid $OABC$ has an equilateral triangle $ABC$ as its base with $AB = 4$. The lengths $OA$, $OB$, and $OC$ are equal, and the distance between line $OA$ and line $BC$ is $\\sqrt{3}$. Find the length of $OA$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e3e7a1798972e802ca958e1782bf0194c20f36de34c1d71a422c7a543e43a02c | A pyramid $P$ has a square base and four slanted triangular faces, with all edges of equal length. A cube $C$ has an edge length of 6. Six pyramids similar to $P$ are constructed by using each face of $C$ as the base, with the apex points $P_i$ (all outside of $C$). The volume of the octahedron formed by the convex hul... | [
"74"
] | [
"A pyramid $P$ has a square base and four slanted triangular faces, with all edges of equal length. A cube $C$ has an edge length of 6. Six pyramids similar to $P$ are constructed by using each face of $C$ as the base, with the apex points $P_i$ (all outside of $C$). The volume of the octahedron formed by the conve... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7d3f7ebe5a022d156e070e3d447712139740a4cdb2a2ab4dc27f37905e48fa1 | A pyramid SABC has all angles with a common vertex S as right angles. The areas of the sides SAB, SAC, and SBC are respectively 3, 4, and 6. What is the volume of the pyramid? | [
"4"
] | [
"A pyramid SABC has all angles with a common vertex S as right angles. The areas of the sides SAB, SAC, and SBC are respectively 3, 4, and 6. What is the volume of the pyramid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-8b4b92f2c28d41e230f1a587c4b70cb3fb10a3c57cb21c6badfe20cbe15d3ca4 | A pyramid \( P \) has a square base and four slanted triangular faces such that all of its edges are equal in length. A cube \( C \) has an edge length of 6. Six pyramids similar to \( P \) are constructed by taking points \( P_i \) (all outside of \( C \)) where \( i = 1, 2, \dots, 6 \) and using the nearest face of \... | [
"434"
] | [
"A pyramid \\( P \\) has a square base and four slanted triangular faces such that all of its edges are equal in length. A cube \\( C \\) has an edge length of 6. Six pyramids similar to \\( P \\) are constructed by taking points \\( P_i \\) (all outside of \\( C \\)) where \\( i = 1, 2, \\dots, 6 \\) and using the... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-3e90aec70362c1ed407ae5d8cae6ef68f5eb68a5ef85049b0d3bce351eeea8b0 | A pyramid has 23 triangular faces. How many edges does this pyramid have? | [
"46"
] | [
"A pyramid has 23 triangular faces. How many edges does this pyramid have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-ed809b84943be650590c0d118ecc2de9f7846b7138687c08edc0cb443e274fb8 | A pyramid has a height of 10 and a square base with a perimeter of 24. If one gram of sand has a volume of 8, then how many grams of sand can this pyramid hold? | [
"15"
] | [
"A pyramid has a height of 10 and a square base with a perimeter of 24. If one gram of sand has a volume of 8, then how many grams of sand can this pyramid hold?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8f911c009107bdf7b90a574e6b7534f42229867874d515a92687328e535527c | A pyramid has a square base ABCD with vertex at E. The area of the base is 144, and the areas of triangles ADE and BCE are 60 and 48, respectively. Compute the volume of the pyramid. | [
"\\( 120\\sqrt{7} \\)"
] | [
"A pyramid has a square base ABCD with vertex at E. The area of the base is 144, and the areas of triangles ADE and BCE are 60 and 48, respectively. Compute the volume of the pyramid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-ef14498c2236b7bbac63aee96a56386990abf5f747129e403d94d7afda02a034 | A pyramid has a square base \(ABCD\) and vertex \(E\). The area of square \(ABCD\) is \(196\), and the areas of \(\triangle ABE\) and \(\triangle CDE\) are \(105\) and \(91\), respectively. What is the volume of the pyramid? | [
"784"
] | [
"A pyramid has a square base \\(ABCD\\) and vertex \\(E\\). The area of square \\(ABCD\\) is \\(196\\), and the areas of \\(\\triangle ABE\\) and \\(\\triangle CDE\\) are \\(105\\) and \\(91\\), respectively. What is the volume of the pyramid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 11
} | 43 | 0 | 0 |
omr-ws-d34bbd1edb1ec6b99a6d3cd49154d3204b60dbdd54df49291a83a1ae970a8781 | A pyramid has a square base \(ABCD\) with an area of 196. The areas of triangles \( \triangle ABE \) and \( \triangle CDE \) are 105 and 91, respectively. What is the volume of the pyramid? | [
"784"
] | [
"A pyramid has a square base \\(ABCD\\) with an area of 196. The areas of triangles \\( \\triangle ABE \\) and \\( \\triangle CDE \\) are 105 and 91, respectively. What is the volume of the pyramid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 8
} | 56 | 0 | 0 |
omr-ws-4054de52c36081a1200fe07f5d51c8235e65e30f21aee68a4fceda776263076d | A pyramid has a square base with an area of $1440 \text{ cm}^2$ and four identical triangular faces, each with an area of $880 \text{ cm}^2$. Find the height of the pyramid. | [
"\\(\\frac{2\\sqrt{4030}}{3}\\)"
] | [
"A pyramid has a square base with an area of $1440 \\text{ cm}^2$ and four identical triangular faces, each with an area of $880 \\text{ cm}^2$. Find the height of the pyramid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-00415d1825185d19093a35996e17ef5305921b8acbc1ca5fd442d7c480e12ffe | A pyramid has a square base with side length 1 and lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? | [
"\\( 5\\sqrt{2} - 7 \\)"
] | [
"A pyramid has a square base with side length 1 and lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 4,
"tir": 54
} | 88 | 0 | 0 |
omr-ws-0fe7a22a989c46a885d605c884d2425dd2d4204bb71b24328ebbc8345b9cc7d1 | A pyramid has a square base with side length 2 and height 1. A cube is placed inside the pyramid so that it sits on the square base and its top corners each touch the slanted edges of the pyramid. What is the volume of the cube? | [
"\\(\\frac{8}{27}\\)"
] | [
"A pyramid has a square base with side length 2 and height 1. A cube is placed inside the pyramid so that it sits on the square base and its top corners each touch the slanted edges of the pyramid. What is the volume of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d2185b8008450493302929a6449000ee07383f7d6bff0e2588aa2deed9a741bf | A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? | [
"\\(5\\sqrt{2}-7\\)"
] | [
"A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 2,
"tir": 26
} | 62 | 0 | 0 |
omr-ws-fbc40e17ddb2ca5b6694817851326b0e1af1ece3fdb11adc8887409b139ca65b | A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of... | [
"803"
] | [
"A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the squ... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 33
} | 65 | 0 | 0 |
omr-ws-ab09f50e65db85468b1adedec75b108c90df9ad82296541442f632b7ac734855 | A pyramid has a volume of $2\sqrt{3}$ and a vertex at $P(1,2,2)$. The base is a regular hexagon with its center at $(0,1,0)$ and one vertex at $(1,0,0)$. Find the foot of the altitude from $P$ to the base. | [
"\\(\\left( -\\frac{1}{3}, \\frac{2}{3}, \\frac{4}{3} \\right)\\)"
] | [
"A pyramid has a volume of $2\\sqrt{3}$ and a vertex at $P(1,2,2)$. The base is a regular hexagon with its center at $(0,1,0)$ and one vertex at $(1,0,0)$. Find the foot of the altitude from $P$ to the base."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-1e84dd99e8917b9612dd2eae1d8ad57eb0b95c7befbad71d12263a55fcc550d9 | A pyramid has an equilateral triangle of side length \( a \) as its base. The lateral sides are each of length \( b \). Find the largest volume of the sphere that can be inscribed in this pyramid. | [
"\\(\\frac{4}{3} \\pi \\left( \\frac{-a^2 + a\\sqrt{12b^2 - 3a^2}}{12\\sqrt{b^2 - \\frac{1}{3}a^2}} \\right)^3\\)"
] | [
"A pyramid has an equilateral triangle of side length \\( a \\) as its base. The lateral sides are each of length \\( b \\). Find the largest volume of the sphere that can be inscribed in this pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1bc3a6158b3d3f6fb266b5deb4d1f49bf66c68c8f66ec935106c11890e8e46f6 | A pyramid has an equilateral triangle of side length \( a \) as its base. The lateral sides are each of length \( b \). Find the volume and total surface area of the pyramid. | [
"Total Surface Area: \\( \\frac{\\sqrt{3}}{4}a^2 + \\frac{3}{2}a\\sqrt{b^2 - \\frac{1}{4}a^2} \\)\nVolume: \\( \\frac{1}{4\\sqrt{3}}a^2\\sqrt{b^2 - \\frac{1}{3}a^2} \\)"
] | [
"A pyramid has an equilateral triangle of side length \\( a \\) as its base. The lateral sides are each of length \\( b \\). Find the volume and total surface area of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-a183d945a049fe23c9208f9cc1a675420d4d2980716cba64e9d7c70eb5cab496 | A pyramid has an equilateral triangle of side length \( a \) as its base. The lateral sides are each of length \( b \). Find the volume of the sphere circumscribing this pyramid. | [
"\\(\\frac{\\pi b^6}{6(b^2-\\tfrac13a^2)^\\frac32}\\)"
] | [
"A pyramid has an equilateral triangle of side length \\( a \\) as its base. The lateral sides are each of length \\( b \\). Find the volume of the sphere circumscribing this pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-055acff0ae6d1abaa307f5f353d3e513ac63c561d83f0adc6843dd903b6db636 | A pyramid is formed on a $6\times 8$ rectangular base. The four edges joining the apex to the corners of the rectangular base each have length $13$. What is the volume of the pyramid? | [
"192"
] | [
"A pyramid is formed on a $6\\times 8$ rectangular base. The four edges joining the apex to the corners of the rectangular base each have length $13$. What is the volume of the pyramid?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f253b7b60b32630fb54d7e382bdf9299269655bf36553b94c20c418502278314 | A pyramid of blocks is set up so that there are n blocks in the nth layer, with the 1st layer being on top, and each block is the sum of the two blocks below it. If the 8th layer is filled randomly with the numbers 0, 1, and 2, what are the odds that the top block will be divisible by 4? | [
"\\dfrac{547}{2187}"
] | [
"A pyramid of blocks is set up so that there are n blocks in the nth layer, with the 1st layer being on top, and each block is the sum of the two blocks below it. If the 8th layer is filled randomly with the numbers 0, 1, and 2, what are the odds that the top block will be divisible by 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d2929da8b84a71a950a2dfac279fcb6489784e5497cccc1990a689697961a8d | A pyramid of non-negative integers is constructed as follows: - The first row consists of only $0$. - The second row consists of $1$ and $1$. - The $n^{th}$ row (for $n > 2$) consists of $n$ integers where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of two adj... | [
"2"
] | [
"A pyramid of non-negative integers is constructed as follows:\n- The first row consists of only $0$.\n- The second row consists of $1$ and $1$.\n- The $n^{th}$ row (for $n > 2$) consists of $n$ integers where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61c63f9091c3bdc7cafe207a1ae368f5374b63882f6aad6888dcbb358e59d47a | A pyramid of total surface area $S$ is cut off from a regular quadrangular prism by a plane passing through a diagonal of the lower base and a vertex of the upper base. Find the total surface area of the prism if the vertex angle of the triangle in the section is equal to $\alpha $. | [
"S_{total}=4S\\frac{\\sin\\frac{\\alpha }{2}+\\sqrt {2\\cos\\alpha }} {\\cos\\displaystyle\\frac{\\alpha }{2}+\\sin\\displaystyle\\frac{\\alpha }{2}+\\sqrt {2\\cos\\alpha }}"
] | [
"A pyramid of total surface area $S$ is cut off from a regular quadrangular prism by a plane passing through a diagonal of the lower base and a vertex of the upper base. Find the total surface area of the prism if the vertex angle of the triangle in the section is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95795ed43a02db0aa96177c0ae848ea9fdcf2f54ecee60d026201b2b2d45d0ed | A pyramid originally measured 580 feet and 12 inches tall but has lost some height due to the loss of its topmost stones. Two angles of elevation were taken: 46.27 degrees from 100 feet away and 40.3 degrees from 200 feet away. Find the current height of the pyramid. | [
"137.25 feet"
] | [
"A pyramid originally measured 580 feet and 12 inches tall but has lost some height due to the loss of its topmost stones. Two angles of elevation were taken: 46.27 degrees from 100 feet away and 40.3 degrees from 200 feet away. Find the current height of the pyramid."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d05ed1501a7d3180db7697158911381d6411fc7b2735883df0ca8f8a8bd00fe8 | A pyramid whose base is a square is inscribed in a sphere of radius $R$. One of the lateral edges of the pyramid is perpendicular to the plane of the base and the larger lateral edge makes an angle $\alpha $ with the base. Find the lateral surface of the pyramid. | [
"$S=2\\sqrt 2R^2\\cos \\alpha \\left(\\sin \\alpha +\\sqrt {\\sin ^2 \\alpha +\\frac{1}{2}\\cos ^2 \\alpha }\\right)$"
] | [
"A pyramid whose base is a square is inscribed in a sphere of radius $R$. One of the lateral edges of the pyramid is perpendicular to the plane of the base and the larger lateral edge makes an angle $\\alpha $ with the base. Find the lateral surface of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-faed1394d6ac4c8aee58424abcd384c47078d20627be09c0203921d8dbac2cfd | A pyramid whose base is a square is inscribed in a sphere of radius $R$. Two lateral faces are perpendicular to the base. The larger lateral edge makes an angle $\alpha $ with the side of the base intersecting it. Find the lateral surface of the pyramid. | [
"$4R^2\\cos \\alpha \\left(\\sin \\alpha +\\sqrt {-\\cos 2\\alpha }\\right)$"
] | [
"A pyramid whose base is a square is inscribed in a sphere of radius $R$. Two lateral faces are perpendicular to the base. The larger lateral edge makes an angle $\\alpha $ with the side of the base intersecting it. Find the lateral surface of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0bd948c448b6a5ec743a3ebad7d6acc30fe8397534de4c0d856ff891ddc5dd28 | A pyramid whose lateral edges are equal to $c$ is inscribed into a sphere. Its base is a rectangle whose sides subtend arcs of $\alpha $ and $\beta $ radians in the sections of the ball by the planes of the lateral faces. Find the radius of the circumscribed sphere. | [
"$\\frac{c}{\\sqrt {2(\\cos \\alpha /2)+\\cos (\\beta /2)}}$"
] | [
"A pyramid whose lateral edges are equal to $c$ is inscribed into a sphere. Its base is a rectangle whose sides subtend arcs of $\\alpha $ and $\\beta $ radians in the sections of the ball by the planes of the lateral faces. Find the radius of the circumscribed sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af75defef4bd105cf16ff71c2beac4686c06ce4ec9bf0e58d28c384817d4b3a3 | A pyramid with a square base and all edges of length $2$ is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid. | [
"22"
] | [
"A pyramid with a square base and all edges of length $2$ is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 47,
"genselect": 7
} | 54 | 0 | 0 |
omr-ws-2076c4d343cf1632116be37403289c43d0580818885dd874a724491a31232421 | A pyramid with a square base is cut by a plane that is parallel to its base and 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid? | [
"\\( 4 + 2\\sqrt{2} \\)"
] | [
"A pyramid with a square base is cut by a plane that is parallel to its base and 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-62eeb56ed55f27f196152c30298077e479ba712243e976553201f653e9af2f95 | A pyramid with a square base is cut by a plane that is parallel to its base and is 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid? | [
"\\( 4 + 2\\sqrt{2} \\)"
] | [
"A pyramid with a square base is cut by a plane that is parallel to its base and is 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 58
} | 89 | 0 | 0 |
omr-ws-00d26bcf1c72d2ee3f2539c79fe7627737e4c9585bff181e238d7aa99cd61a32 | A pyramid with apex \( T \) and base triangle \( \triangle ABC \) has three right angles at the apex. Determine the area of the base if the areas of the side faces are 8, 9, and 12. | [
"17"
] | [
"A pyramid with apex \\( T \\) and base triangle \\( \\triangle ABC \\) has three right angles at the apex. Determine the area of the base if the areas of the side faces are 8, 9, and 12."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9cadb6bf9274f7bd0fa5f8b6c56e47c9367eebac8c7a76f680a4e8620716b97b | A pyramid with base $ABCD$ and top $V$ is inscribed in a sphere. Given that $AD = 2BC$ and the rays $AB$ and $DC$ intersect at point $E$, compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$. | [
"\\(\\frac{1}{2}\\)"
] | [
"A pyramid with base $ABCD$ and top $V$ is inscribed in a sphere. Given that $AD = 2BC$ and the rays $AB$ and $DC$ intersect at point $E$, compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba6e7bccc1317aa23ac4b59fad2de23cb4cd78587cbc692dd6ed66de65223554 | A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\%$, what is the volume of the new pyramid, in cubic inches? | [
"360"
] | [
"A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\\%$, what is the volume of the new pyramid, in cubic inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92c61ce8f038fc16ecd2e76726a03e2630b75208d342e9fa591064013ed4d010 | A quadrangle $ABCD$ is inscribed in a circle with radius $\frac{65}{8}$. The perimeter of the quadrangle is $44$, and the side lengths of $BC$ and $CD$ are both $13$. Find the lengths of $AB$ and $DA$. | [
"4"
] | [
"A quadrangle $ABCD$ is inscribed in a circle with radius $\\frac{65}{8}$. The perimeter of the quadrangle is $44$, and the side lengths of $BC$ and $CD$ are both $13$. Find the lengths of $AB$ and $DA$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-9919ec9e372226c046161ced2e4990d78b7b879bb83b5f43d8f254ad6df6891c | A quadrangular pie was cut diagonally into four pieces. The weights of three consecutive triangular pieces are $120$, $200$, and $300$ grams. Find the weight of the fourth piece. | [
"180"
] | [
"A quadrangular pie was cut diagonally into four pieces. The weights of three consecutive triangular pieces are $120$, $200$, and $300$ grams. Find the weight of the fourth piece."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-487aa58dc8ab1197d936a26756ed249325162f3ad3beb14e8e7f0a564e7ac0b6 | A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of these two solutions? | [
"1"
] | [
"A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of these two solutions?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9777d94bb8acb9368551571d4e564fbb9ee76dadc75c81ce0400546f33bcb74b | A quadratic equation \( ax^2 - 2ax + b = 0 \) has two real solutions. What is the average of the solutions? | [
"1"
] | [
"A quadratic equation \\( ax^2 - 2ax + b = 0 \\) has two real solutions. What is the average of the solutions?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fbad59e02401ca0164196e15b01439b32b042edc155673cd2d5274f326943267 | A quadratic equation has the natural roots $a$ and $b$. Another quadratic equation has roots $b$ and $c$ with $a \neq c$. If $a$, $b$, and $c$ are prime numbers less than $15$, how many triplets $(a, b, c)$ meet these conditions (provided that the coefficient of the quadratic term is equal to $1$)? | [
"180"
] | [
"A quadratic equation has the natural roots $a$ and $b$. Another quadratic equation has roots $b$ and $c$ with $a \\neq c$. If $a$, $b$, and $c$ are prime numbers less than $15$, how many triplets $(a, b, c)$ meet these conditions (provided that the coefficient of the quadratic term is equal to $1$)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d73ce540e6a8ebe3372ba27462b6b3c008db4318ccba36bb3204192251be72fe | A quadratic function $y=px^2+qx+r$ satisfies the following conditions: - The graph of $y$ passes through the point $(1, 6)$. - $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ form an arithmetic progression. - $p, r, q$ form a geometric progression. Find the values of $p, q, r$. | [
"\\( (p, q, r) \\in \\{(2, 2, 2), (2, 8, -4)\\} \\)"
] | [
"A quadratic function $y=px^2+qx+r$ satisfies the following conditions:\n- The graph of $y$ passes through the point $(1, 6)$.\n- $\\frac{1}{p}, \\frac{1}{q}, \\frac{1}{r}$ form an arithmetic progression.\n- $p, r, q$ form a geometric progression.\n\nFind the values of $p, q, r$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-d91e5ae9296084572d0c47ed89d2c62cf9b3362036f143554606c658a9d8036f | A quadratic function \( f(x) = ax^2 + bx + c \) satisfies the conditions: 1. \( f(-1) = 0 \) 2. \( x \leq f(x) \leq \frac{x^2 + 1}{2} \) for all real numbers \( x \). Determine the value of \( abc \) in the form \( \frac{m}{n} \), where \( m \) and \( n \) are coprime integers, and find \( m + n \). | [
"33"
] | [
"A quadratic function \\( f(x) = ax^2 + bx + c \\) satisfies the conditions:\n1. \\( f(-1) = 0 \\)\n2. \\( x \\leq f(x) \\leq \\frac{x^2 + 1}{2} \\) for all real numbers \\( x \\).\n\nDetermine the value of \\( abc \\) in the form \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are coprime integers, and find \\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-213d980ed70123fc705c7cc5942758c11184d1d66beb6722188cff082cbe5726 | A quadratic function \( f(x) \) satisfies the following conditions: \[ f(3) = 3 \] \[ f(x+4) = 2f(x) + 2x + 3. \] Find the coefficient of \( x^3 \) in \( f(f(x)) \). | [
"0"
] | [
"A quadratic function \\( f(x) \\) satisfies the following conditions:\n\\[ f(3) = 3 \\]\n\\[ f(x+4) = 2f(x) + 2x + 3. \\]\nFind the coefficient of \\( x^3 \\) in \\( f(f(x)) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-7f1f6b4aa317423764ac8d37b945962ed93e15f46489633f8bf9219e54bebb10 | A quadratic function of the form \( f(x) = x^2 + bx + c \) intersects the x-axis and the y-axis in three different points. The circle through these three points intersects the graph of the function \( f \) in a fourth point. What are the coordinates of this fourth point of intersection? | [
"\\((-b, c)\\)"
] | [
"A quadratic function of the form \\( f(x) = x^2 + bx + c \\) intersects the x-axis and the y-axis in three different points. The circle through these three points intersects the graph of the function \\( f \\) in a fourth point. What are the coordinates of this fourth point of intersection?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-fcc6d5bc32e9c96375c8670bfd6074011bbceb5e1fc0d15e364d79fef4daab5e | A quadratic has a root $\frac{\sqrt{p}}{\sqrt{p}+\sqrt{p-q}}$ with multiplicity 2. What is the quadratic in standard form? | [
"\\left(2p - q + 2\\sqrt{p(p - q)}\\right)x^2 - 2\\left(p + \\sqrt{p(p - q)}\\right)x + p = 0"
] | [
"A quadratic has a root $\\frac{\\sqrt{p}}{\\sqrt{p}+\\sqrt{p-q}}$ with multiplicity 2. What is the quadratic in standard form?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-1484043f4e2b02d7b4e5280f980f88895c5fb598a155d74943acbe29850e2fe7 | A quadratic in \( x \) with rational coefficients is of the form \( ax^2 - acx + c \). One root of the quadratic is \( \frac{3 + 2\sqrt{2}}{10} \). What is \( a \)? | [
"60"
] | [
"A quadratic in \\( x \\) with rational coefficients is of the form \\( ax^2 - acx + c \\). One root of the quadratic is \\( \\frac{3 + 2\\sqrt{2}}{10} \\). What is \\( a \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6beacc87284f4edd5864131b444097368b14616d1cc613ccd7ea8d52b8655ab8 | A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximiz... | [
"x^2 - \\frac{1}{2}x - \\frac{3}{16}"
] | [
"A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x)$ for which the sum of the roots is m... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d91234c240e69e6b61c7a1732dacc452395b82f323d55d5bd6b7b28115f2f529 | A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximiz... | [
"\\dfrac{5}{16}"
] | [
"A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\\tilde{p}(x)$ for which the sum of the roots is m... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ec6b6d0037ea1a79ab5e3ef03ce4a18b82309e772c1d75b216329f62b8cb0601 | A quadratic polynomial \( p(x) \) with real coefficients and leading coefficient 1 is called disrespectful if the equation \( p(p(x)) = 0 \) is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \( \tilde{p}(x) \) for which the sum of the roots ... | [
"\\(\\frac{5}{16}\\)"
] | [
"A quadratic polynomial \\( p(x) \\) with real coefficients and leading coefficient 1 is called disrespectful if the equation \\( p(p(x)) = 0 \\) is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \\( \\tilde{p}(x) \\) for which the sum o... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 27
} | 72 | 0 | 0 |
omr-ws-72c1f40a04ae7ac0cff7bf4f0b36b3b760c32dff0fd7cf8ef3ef66b0b44e1628 | A quadratic trinomial with integer coefficients intersects the $x$-axis at points $X$ and $Z$, and the $y$-axis at point $Y$. Given that $Y \neq X$ and $Y \neq Z$, what is the greatest possible value of the angle $XYZ$? | [
"$\\pi/2$"
] | [
"A quadratic trinomial with integer coefficients intersects the $x$-axis at points $X$ and $Z$, and the $y$-axis at point $Y$. Given that $Y \\neq X$ and $Y \\neq Z$, what is the greatest possible value of the angle $XYZ$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-2de45aa0a700de200727205852d7e5c87e98508ac4b189065387cc5ec7c4b0bb | A quadrilateral $A BCD$ is inscribed into a circle with the radius of 6 cm and center at a point $O$. Its diagonals $AC$ and $BD$ are mutually perpendicular and meet at a point $K$. Points $E$ and $F$ are the midpoints of $AC$ and $BD$ respectively. The segment $OK$ is 5 cm long, and the area of the quadrilateral $OEKF... | [
"$12\\sqrt {15}$"
] | [
"A quadrilateral $A BCD$ is inscribed into a circle with the radius of 6 cm and center at a point $O$. Its diagonals $AC$ and $BD$ are mutually perpendicular and meet at a point $K$. Points $E$ and $F$ are the midpoints of $AC$ and $BD$ respectively. The segment $OK$ is 5 cm long, and the area of the quadrilateral ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f872cf5ff04f372b3eb4fad5fe8effdb678f9580f3783ea2966ab4b86270ca8a | A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$. | [
"769"
] | [
"A quadrilateral $ABCD$ has a right angle at $\\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-6c71ac40364778891a14bd59a4537c0c27648e4e4de0afe55c7b0cf43f918017 | A quadrilateral $ABCD$ has an incircle $\Gamma$. Points $X$ and $Y$ are chosen such that $AX - CX = AB - BC$, $BX - DX = BC - CD$, $CY - AY = AD - DC$, and $DY - BY = AB - AD$. Given that the center of $\Gamma$ lies on $XY$, calculate the ratio of the area of triangle $ACX$ to the area of triangle $BDY$ if $AB = 13$, $... | [
"1"
] | [
"A quadrilateral $ABCD$ has an incircle $\\Gamma$. Points $X$ and $Y$ are chosen such that $AX - CX = AB - BC$, $BX - DX = BC - CD$, $CY - AY = AD - DC$, and $DY - BY = AB - AD$. Given that the center of $\\Gamma$ lies on $XY$, calculate the ratio of the area of triangle $ACX$ to the area of triangle $BDY$ if $AB =... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-d8cab8b08702634af295ae55d4daee15ed77fe426d4d0076892597874b60f195 | A quadrilateral $ABCD$ has an incircle $\omega$. The semi-lines $AB$ and $DC$ intersect at point $P$, and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point on $\omega$ closest to the line $PQ$. If $I$ is the incenter of triangle $PQC$, find the rat... | [
"1"
] | [
"A quadrilateral $ABCD$ has an incircle $\\omega$. The semi-lines $AB$ and $DC$ intersect at point $P$, and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point on $\\omega$ closest to the line $PQ$. If $I$ is the incenter of triangle $PQC$, find ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e3d8a9fc7b8d77dfbaa10619f4cee6363bd07ddaa0cb59bb08a2ae18d92eb098 | A quadrilateral $ABCD$ has an inscribed circle $\omega$. The rays $AB$ and $DC$ intersect at point $P$, and the rays $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point on $\omega$ that is closer to the line $PQ$. If the incenter of triangle $PQC$ is denoted as $I$... | [
"1"
] | [
"A quadrilateral $ABCD$ has an inscribed circle $\\omega$. The rays $AB$ and $DC$ intersect at point $P$, and the rays $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point on $\\omega$ that is closer to the line $PQ$. If the incenter of triangle $PQC$ is denoted... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5ae7010d227b033af55fd92c53596696895abdf766bed050f5c5e65866d091ce | A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The intersection point of $\omega$ and the diagonal $AC$, closest to $A$, is $E$. The point $F$ is diametrally opposite to the point $E$ on the circle $\omega$. The tangent to $\omega$ at the point $F$ intersects lines $AB$ and $BC$ in points $A_1$ and $C... | [
"1"
] | [
"A quadrilateral $ABCD$ is circumscribed about a circle $\\omega$. The intersection point of $\\omega$ and the diagonal $AC$, closest to $A$, is $E$. The point $F$ is diametrally opposite to the point $E$ on the circle $\\omega$. The tangent to $\\omega$ at the point $F$ intersects lines $AB$ and $BC$ in points $A_... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3749bf7696169befe434c9d046d3fe606f9de5210d96e8e0c92c6ccb8f68e7d7 | A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points ... | [
"25"
] | [
"A quadrilateral $ABCD$ is circumscribed about a circle $\\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
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