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-a280b12c5aaca2365c105d7502e12bdce203536801c8d2dde6ef08c5a601b7dc | A triangular pyramid with apex $O$ and base $\triangle{ABC}$ has the property that the perimeter of $\triangle{ABC}$ is $84$. Additionally, a cylinder of radius $4$ and height $10$ can be placed completely inside the pyramid such that one of its bases is in the same plane as $\triangle{ABC}$. Calculate the minimum poss... | [
"\\dfrac{70\\sqrt{3}}{7\\sqrt{3} - 6}"
] | [
"A triangular pyramid with apex $O$ and base $\\triangle{ABC}$ has the property that the perimeter of $\\triangle{ABC}$ is $84$. Additionally, a cylinder of radius $4$ and height $10$ can be placed completely inside the pyramid such that one of its bases is in the same plane as $\\triangle{ABC}$. Calculate the mini... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5167b47273431e513a4d05be17d66006b60459c35f54ee5deda8203b5754481d | A triangular region is bounded by the two coordinate axes and the line given by the equation $2x + y = 6$. What is the area of the region, in square units? | [
"9"
] | [
"A triangular region is bounded by the two coordinate axes and the line given by the equation $2x + y = 6$. What is the area of the region, in square units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7d745903075f452e5c1cf49d0486aa7b5459b4c07dabbe8e1043ccd105dd446 | A triangular region is enclosed by the lines with equations $y = \frac{1}{2} x + 3$, $y = -2x + 6$ and $y = 1$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth. | [
"8.45"
] | [
"A triangular region is enclosed by the lines with equations $y = \\frac{1}{2} x + 3$, $y = -2x + 6$ and $y = 1$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99e5a767f069ab08919c8e1a89ab9ec7ac64fa3c68c34a7e6fc8ad2882a77970 | A triangular sheet has an area of $1$. Fold the sheet once in any manner and place it on a table. Let $A$ be the area covered by the folded sheet. Find the maximum possible value of $A$. | [
"\\dfrac{3}{2}"
] | [
"A triangular sheet has an area of $1$. Fold the sheet once in any manner and place it on a table. Let $A$ be the area covered by the folded sheet. Find the maximum possible value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-6363295d49e99024f674d89176dfbe6184bf01c650b7440df9ddde8d7ff01710 | A triangular trough has sides meeting at angles of \( \theta \) degrees. A ball of radius \( R \) is placed in the trough. In terms of \( R \), what is the radius of the largest ball that will just fit in the trough beneath the bigger ball? | [
"\\( r = \\frac{R - R \\sin\\left(\\frac{1}{2}\\theta\\right)}{1 + \\sin\\left(\\frac{1}{2}\\theta\\right)} \\)"
] | [
"A triangular trough has sides meeting at angles of \\( \\theta \\) degrees. A ball of radius \\( R \\) is placed in the trough. In terms of \\( R \\), what is the radius of the largest ball that will just fit in the trough beneath the bigger ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-8525a5e448c6a0a5da6efb33395846c4ff0db9239a8d66f1ebc5e7b189e30303 | A triangulation of a polygon is a partitioning of the polygon into triangles all of whose vertices are vertices of the original polygon. Given an n-sided convex polygon, determine the number of different possible triangulations of it. | [
"\\frac{1}{n-1} \\dbinom{2n-4}{n-2}"
] | [
"A triangulation of a polygon is a partitioning of the polygon into triangles all of whose vertices are vertices of the original polygon. Given an n-sided convex polygon, determine the number of different possible triangulations of it."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9df88b3317cfcc8879d90ac6d5c88f7a4d42476b5256bcbc954d525b3c16ec55 | A triangulation of a polygon is a way to partition the polygon into triangles by adding diagonals that do not intersect each other (except possibly at the vertices of the polygon). Among all possible triangulations of a regular $2015$-gon, what is the maximum possible number of acute-angled triangles one can get? | [
"1005"
] | [
"A triangulation of a polygon is a way to partition the polygon into triangles by adding diagonals that do not intersect each other (except possibly at the vertices of the polygon). Among all possible triangulations of a regular $2015$-gon, what is the maximum possible number of acute-angled triangles one can get?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-1a026cf3c447dade73e83be842f6ed8ce270bc99fc0d15139f49ad5acead31a4 | A triathlete competes in a triathlon where the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. What is the triathlete's average speed, in kilometers per ho... | [
"6"
] | [
"A triathlete competes in a triathlon where the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. What is the triathlete's average speed, in kilometers p... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-589be3d3926acfa767fcb5900fe1b1a910c7d01c32d7fb9b8324de986679a192 | A tributary flows into a river. A boat leaves a landing-stage situated on the tributary and moves 60 km downstream to reach the junction. It then moves 65 km downstream along the river to reach another landing-stage $B$. Then following the same route the boat returns to landing-stage $A$. It takes the boat 10 hours to ... | [
"14 km/h"
] | [
"A tributary flows into a river. A boat leaves a landing-stage situated on the tributary and moves 60 km downstream to reach the junction. It then moves 65 km downstream along the river to reach another landing-stage $B$. Then following the same route the boat returns to landing-stage $A$. It takes the boat 10 hour... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df6eff8e4224e044c257f0ea4cd978ba598276b471844c8ef6f4882adac1739e | A trickster has a rigged die where the probability of rolling a 2, 3, 4, 5, and 6 are all equal, but the probability of rolling a 1 is higher. The trickster expects to make 50 cents per roll. What is the probability of rolling an odd number on this rigged die? | [
"\\dfrac{3}{5}"
] | [
"A trickster has a rigged die where the probability of rolling a 2, 3, 4, 5, and 6 are all equal, but the probability of rolling a 1 is higher. The trickster expects to make 50 cents per roll. What is the probability of rolling an odd number on this rigged die?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-bce628b88145dcedc50747e7fdfc8839b84fb36e43359ab319fe4dec1dc599fc | A triomino is an equilateral triangle with side length 1, with three integers (ranging from 1 to 10) on each side. Triominic is laying down triominoes on an equilateral triangle-shaped table with side length 5, such that adjacent triominoes must have the same label on the shared sides. Let \( S \) be the number of ways... | [
"2116"
] | [
"A triomino is an equilateral triangle with side length 1, with three integers (ranging from 1 to 10) on each side. Triominic is laying down triominoes on an equilateral triangle-shaped table with side length 5, such that adjacent triominoes must have the same label on the shared sides. Let \\( S \\) be the number ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-2af7ec9f86a368c130c02b3ce16f0bde859318e6522744223edd7dec2f648cb6 | A triple of integers \((a, b, c)\) is said to be \(\gamma\)-special if \(a \le \gamma(b+c)\), \(b \le \gamma(c+a)\), and \(c \le \gamma(a+b)\). For each integer triple \((a, b, c)\) such that \(1 \le a, b, c \le 20\), determine the smallest value of \(\gamma\) such that \((a, b, c)\) is \(\gamma\)-special. How many dis... | [
"256"
] | [
"A triple of integers \\((a, b, c)\\) is said to be \\(\\gamma\\)-special if \\(a \\le \\gamma(b+c)\\), \\(b \\le \\gamma(c+a)\\), and \\(c \\le \\gamma(a+b)\\). For each integer triple \\((a, b, c)\\) such that \\(1 \\le a, b, c \\le 20\\), determine the smallest value of \\(\\gamma\\) such that \\((a, b, c)\\) is... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52931f2a6068d3b7fc1c7629ad66b685bb85332d0f0fd8e0fb4c035d330b9c21 | A triple of positive integers \((a, b, c)\) is called "brazilian" if the following conditions are satisfied: \[ a \mid (bc + 1) \] \[ b \mid (ac + 1) \] \[ c \mid (ab + 1) \] Determine all the brazilian triples. | [
"\\((3,2,1)\\), \\((7,3,2)\\), \\((2,1,1)\\), \\((1,1,1)\\) and their permutations."
] | [
"A triple of positive integers \\((a, b, c)\\) is called \"brazilian\" if the following conditions are satisfied:\n\\[ a \\mid (bc + 1) \\]\n\\[ b \\mid (ac + 1) \\]\n\\[ c \\mid (ab + 1) \\]\nDetermine all the brazilian triples."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-f1ed591f40adadfcf35726fcc22747d47cd29adb6f08cb7bd0b624f099e774b3 | A triple of real numbers \((c_1, c_2, c_3)\) is called flavorful if there exist real numbers \(k_1, k_2, k_3\) such that \[c_1 \sin(x + k_1) + c_2 \sin(x + k_2) + c_3 \sin(x + k_3) = 0\] for all \(x\). Compute the number of flavorful triples \((c_1, c_2, c_3)\) with \(c_1, c_2, c_3 \in \{1, \dots, 10\}\). | [
"640"
] | [
"A triple of real numbers \\((c_1, c_2, c_3)\\) is called flavorful if there exist real numbers \\(k_1, k_2, k_3\\) such that\n\\[c_1 \\sin(x + k_1) + c_2 \\sin(x + k_2) + c_3 \\sin(x + k_3) = 0\\]\nfor all \\(x\\). Compute the number of flavorful triples \\((c_1, c_2, c_3)\\) with \\(c_1, c_2, c_3 \\in \\{1, \\dot... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0ee6611ae87f4475ba995c05adf73ddd2281e30f210ac20aaba432fa6c628e8f | A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower $1$ foot of one leg breaks off. Let $h$ be the height in feet of the top... | [
"27"
] | [
"A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower $1$ foot of one leg breaks off. Let $h$ be the height in feet of th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-177914b8d68c5996bdbe7832e0ad879ca575616d7150f5c25bfcc4dfca05144e | A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top o... | [
"183"
] | [
"A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the... | [
"MATH_training_set"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-baf35befb44765f7b4e3c2017f5e38cab52720f57dc8c5b90196c681f89d9cbd | A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top o... | [
"183"
] | [
"A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the ... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d480be073b93c2a7b36c7f137053cdbcd712bb9c0ce81d4e6326aabc1449085 | A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal, and the top of the tripod is $4$ feet from the ground. If the lower $1$ foot of one leg breaks off, let $h$ be the new height of the top of the tripod from the ground. Then $h$ can be written in the ... | [
"7"
] | [
"A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal, and the top of the tripod is $4$ feet from the ground. If the lower $1$ foot of one leg breaks off, let $h$ be the new height of the top of the tripod from the ground. Then $h$ can be written in... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-065569a57a161a341290215b63c2c09a0ac7ddb68dfc691a19ba8cc3f95bbb87 | A trirectangular tetrahedron has the property that the areas of its four faces are in arithmetic progression, with the smallest face having an area of \(4 + 3\sqrt{2}\). Compute the volume of the tetrahedron. (A trirectangular tetrahedron is one in which the three edges emanating from one of the vertices are all perpen... | [
"\\(\\frac{577}{3} + 136\\sqrt{2}\\)"
] | [
"A trirectangular tetrahedron has the property that the areas of its four faces are in arithmetic progression, with the smallest face having an area of \\(4 + 3\\sqrt{2}\\). Compute the volume of the tetrahedron. (A trirectangular tetrahedron is one in which the three edges emanating from one of the vertices are al... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d75bb26f8d6b0607be1eda197d57e120f1d6f3a6e02becc42e3310b79d80c029 | A troop 5 meters long starts marching. A soldier at the end of the file steps out and starts marching forward at a higher speed. On reaching the head of the column, he immediately turns around and marches back at the same speed. As soon as he reaches the end of the file, the troop stops marching, and it is found that t... | [
"\\( 5(1 + \\sqrt{2}) \\)"
] | [
"A troop 5 meters long starts marching. A soldier at the end of the file steps out and starts marching forward at a higher speed. On reaching the head of the column, he immediately turns around and marches back at the same speed. As soon as he reaches the end of the file, the troop stops marching, and it is found t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-34720700397a1394c5fd2efe565a9ee3c9c6a537de3b8f7ae07caf50c8a9b6cb | A troop 5 metres long starts marching. A soldier at the end of the file starts marching forward at a higher speed. On reaching the head of the column, he immediately turns around and marches back at the same speed. As soon as he reaches the end of the file, the troop stops marching, and it is found that the troop has m... | [
"\\( 5(\\sqrt{2} + 1) \\) or approximately 12.07 meters"
] | [
"A troop 5 metres long starts marching. A soldier at the end of the file starts marching forward at a higher speed. On reaching the head of the column, he immediately turns around and marches back at the same speed. As soon as he reaches the end of the file, the troop stops marching, and it is found that the troop ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-5e7783cd5bb316bd8f92abe8b303354f643816c7e7279cc91802c433254fae21 | A trooper is hidden 50 feet from a highway. Two seconds after a truck passes, the angle \(\theta\) between the highway and the line of observation from the patrol car to the truck is measured. If the angle measures 20 degrees, how fast is the truck traveling? Express your answer in terms of miles per hour. | [
"6.2 mph"
] | [
"A trooper is hidden 50 feet from a highway. Two seconds after a truck passes, the angle \\(\\theta\\) between the highway and the line of observation from the patrol car to the truck is measured. If the angle measures 20 degrees, how fast is the truck traveling? Express your answer in terms of miles per hour."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 10
} | 18 | 0 | 0 |
omr-ws-62a23ebf76970d81730276696b9d7a65c6f360b5b5283dd5a7bd6debccdd9a64 | A trooper is hidden 50 feet from a highway. Two seconds after a truck passes, the angle \(\theta\) between the highway and the line of observation from the patrol car to the truck is measured. If the speed limit is 65 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour and more over the limit,... | [
"\\( \\theta \\geq 76.46^\\circ \\)"
] | [
"A trooper is hidden 50 feet from a highway. Two seconds after a truck passes, the angle \\(\\theta\\) between the highway and the line of observation from the patrol car to the truck is measured. If the speed limit is 65 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour and more over th... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-26bc04f11386aaf6a12c7e11f74f0564cb7bf705b4f2e160d1afc21ae8c8d869 | A trough has a cross section in the shape of an inverted isosceles triangle with base \(b\) and height \(h\). If the volume of the trough per unit length is given by \(V = \frac{1}{2} b h\), find the value of the vertex angle \(\theta\) that maximizes the volume \(V\) given that \(b = h \tan\left(\frac{\theta}{2}\right... | [
"2 \\arctan(2)"
] | [
"A trough has a cross section in the shape of an inverted isosceles triangle with base \\(b\\) and height \\(h\\). If the volume of the trough per unit length is given by \\(V = \\frac{1}{2} b h\\), find the value of the vertex angle \\(\\theta\\) that maximizes the volume \\(V\\) given that \\(b = h \\tan\\left(\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-f97093c0e676d5ad484f861ccd4190dded4d0fd5b6bba9d9327785dcd3553690 | A trough in the shape of the bottom half of a cylinder with radius $r$ and length $l$ is lying on its side and filled with a liquid of density $\rho$. Find the force exerted by the liquid on each of the two semi-circular ends of the tank and the outward force exerted by the liquid on the semi-cylindrical base of the ta... | [
"\\dfrac{1}{2} \\pi \\rho g r^2 l"
] | [
"A trough in the shape of the bottom half of a cylinder with radius $r$ and length $l$ is lying on its side and filled with a liquid of density $\\rho$. Find the force exerted by the liquid on each of the two semi-circular ends of the tank and the outward force exerted by the liquid on the semi-cylindrical base of ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-ac93a99fb2be0774c7ced6f0b98b5df8efb6a8eb38112e4b3b1817b0f572df9c | A trough is 10 feet long and its ends have the shape of isosceles triangles that are 3 feet across at the top and have a height of 1 foot. What is the depth, in feet, of water in the trough due to a 4-inch rainfall? | [
"\\dfrac{\\sqrt{6}}{3}"
] | [
"A trough is 10 feet long and its ends have the shape of isosceles triangles that are 3 feet across at the top and have a height of 1 foot. What is the depth, in feet, of water in the trough due to a 4-inch rainfall?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a87d1cce616547bc82538e4c31ecbcfcc4bbe8719675d81548ab4e389aebe002 | A truck can travel 500km on one load of petrol. There is an unlimited supply of petrol at the edge of an 800km wide desert, but no petrol is available within the desert. The truck can establish refueling stations at any point along the way. What is the minimum amount of petrol (in loads) the truck requires to cross the... | [
"4"
] | [
"A truck can travel 500km on one load of petrol. There is an unlimited supply of petrol at the edge of an 800km wide desert, but no petrol is available within the desert. The truck can establish refueling stations at any point along the way. What is the minimum amount of petrol (in loads) the truck requires to cros... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-1cc8e88b877fa8901cb7eb523bce855b4c4d8f749858ba677b63f3f2d3b74dfc | A truck develops engine trouble one mile down a straight highway from a service station. The truck continues to the service station at a constant speed, emitting smoke all the while. The smoke is carried to the service station by the wind at a steady rate of 20 miles per hour. The smoke envelops the service station for... | [
"48"
] | [
"A truck develops engine trouble one mile down a straight highway from a service station. The truck continues to the service station at a constant speed, emitting smoke all the while. The smoke is carried to the service station by the wind at a steady rate of 20 miles per hour. The smoke envelops the service statio... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 8
} | 53 | 0 | 0 |
omr-ws-5649b387c141506deb442732edd56f17f62c3fdc62acee400093fd117d147a9c | A truck driver has a truck that can transport 1000 apples at a time and eats 1 apple per km driven. The driver needs to transport 10,000 apples from one city to another, which are 500 km apart. The driver can use any strategy, including unloading and reloading apples at any point along the way. Determine the maximum nu... | [
"6300"
] | [
"A truck driver has a truck that can transport 1000 apples at a time and eats 1 apple per km driven. The driver needs to transport 10,000 apples from one city to another, which are 500 km apart. The driver can use any strategy, including unloading and reloading apples at any point along the way. Determine the maxim... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-daf8a852004b7efb67946d695c1c569f19a0c907c215bcab91ef0f71f36a2792 | A truck has a capacity of $c$ liters of fuel and one liter of fuel is enough to cross one kilometer. It goes on a trip $l$ kilometers long. On its way, it is possible to form any number of magazines with an arbitrary amount of fuel in each of them. It can go back to the starting position for a new fuel any number of ti... | [
"c \\sum_{k=1}^{n} \\frac{1}{2k - 1}"
] | [
"A truck has a capacity of $c$ liters of fuel and one liter of fuel is enough to cross one kilometer. It goes on a trip $l$ kilometers long. On its way, it is possible to form any number of magazines with an arbitrary amount of fuel in each of them. It can go back to the starting position for a new fuel any number ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-b10fb955303692cfec8ef6b48a6c01c7a81aae743b623a4c8b3f405d179a03af | A truck is traveling due north at 30 km/hr approaching a crossroad. On a perpendicular road, a police car is traveling west toward the intersection at 40 km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car... | [
"\\( (40, 30) \\)"
] | [
"A truck is traveling due north at 30 km/hr approaching a crossroad. On a perpendicular road, a police car is traveling west toward the intersection at 40 km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the polic... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c7ba181c5a2b0fd3e15b661e5cdaeae1a35c798ea497370efe19b7b2eb4b7a2 | A truck traveled the first 140 miles of a trip at one speed and the last 260 miles at an average speed of 5 miles per hour less. The entire trip took 6 hours. What was the average speed for the first part of the trip? | [
"70"
] | [
"A truck traveled the first 140 miles of a trip at one speed and the last 260 miles at an average speed of 5 miles per hour less. The entire trip took 6 hours. What was the average speed for the first part of the trip?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-916b2ce4feffc85827c93354b29ec2c37d887d37074030032fb25e53458dddab | A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes? | [
"\\(\\frac{10b}{t}\\)"
] | [
"A truck travels $\\frac{b}{6}$ feet every $t$ seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-85b343ce65034da8ec84aaefaa7df0a05b549c380031c83b0e7bc8b8a37404a5 | A truck travels 40 miles from point A to point B in exactly 1 hour. When the truck is halfway between point A and point B, a car starts from point A and travels at 50 miles per hour. How many miles has the car traveled when the truck reaches point B? | [
"25"
] | [
"A truck travels 40 miles from point A to point B in exactly 1 hour. When the truck is halfway between point A and point B, a car starts from point A and travels at 50 miles per hour. How many miles has the car traveled when the truck reaches point B?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11b533b258f7a0f91df7e6fe885f38b22dcc9ba7b0f75fe19a8e36d044aa4fe1 | A truck travels due west on a road that passes a famous landmark. At a given time, the bearing to the landmark is N 63 degrees W, and after the truck travels 6 miles farther, the bearing is N 45 degrees W. What is the closest the truck will come to the landmark while on the road? | [
"\\(\\dfrac{3\\sqrt{2}\\sin 27^\\circ}{\\sin 18^\\circ}\\) or approximately \\(6.2\\) miles."
] | [
"A truck travels due west on a road that passes a famous landmark. At a given time, the bearing to the landmark is N 63 degrees W, and after the truck travels 6 miles farther, the bearing is N 45 degrees W. What is the closest the truck will come to the landmark while on the road?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 19
} | 32 | 0 | 0 |
omr-ws-e28fa6fd25562496f406d832509c286c5173614634b82bfa3e9e625c72ae2e8b | A truck weighing 30,000 pounds is parked on a slope that is \( d \) degrees from the horizontal. Find the force required to keep the truck from rolling down the hill in terms of \( d \). | [
"\\( 30000 \\sin(d) \\)"
] | [
"A truck weighing 30,000 pounds is parked on a slope that is \\( d \\) degrees from the horizontal. Find the force required to keep the truck from rolling down the hill in terms of \\( d \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-d3e649e5a117eef9e8f05f4ac0e107a1055fe7baa3342c2a99a5ab7155c15f6e | A truffle pig either finds a truffle or is fooled by a decoy with equal and independent probabilities on each attempt. The pig will stop after finding 8 truffles or being fooled by 8 decoys. What is the probability that the pig makes exactly 14 attempts before going home? | [
"\\(\\frac{429}{2048}\\)"
] | [
"A truffle pig either finds a truffle or is fooled by a decoy with equal and independent probabilities on each attempt. The pig will stop after finding 8 truffles or being fooled by 8 decoys. What is the probability that the pig makes exactly 14 attempts before going home?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f15e3dca4fc2b94376c2117615275d28cd21be5cd75410113573afbc734fc97f | A truncated cone has a height of $h$, a lower base radius of $R$, and an upper base radius of $r$. Without using calculus, derive an expression for the volume of this truncated cone. Simplify your answer and express it in terms of $h$, $R$, and $r$. | [
"\\dfrac{1}{3} \\pi h (R^2 + R r + r^2)"
] | [
"A truncated cone has a height of $h$, a lower base radius of $R$, and an upper base radius of $r$. Without using calculus, derive an expression for the volume of this truncated cone. Simplify your answer and express it in terms of $h$, $R$, and $r$."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e8e15f5c54b9a5c7910868066780b732bd3e1619e505c5754d77cf03c97670e | A truncated cone has a larger base radius of \( r \) cm and the generatrix makes an angle with the base such that the tangent of that angle is \( m \). The truncated cone is made of a material with density \( d \) (g/cm\(^3\)), and the smaller base is covered by a special material with density \( p \) (g/cm\(^2\)). Det... | [
"r m - \\dfrac{2p}{d}"
] | [
"A truncated cone has a larger base radius of \\( r \\) cm and the generatrix makes an angle with the base such that the tangent of that angle is \\( m \\). The truncated cone is made of a material with density \\( d \\) (g/cm\\(^3\\)), and the smaller base is covered by a special material with density \\( p \\) (g... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e43dcd183d1fd229fe2f73c3e4caef212b33d11d335a65f98e2c80f01e711bf | A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? | [
"6"
] | [
"A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-9b86b4fdb5c3829b56363295fa289bb43f547aac6db479d1d4abcabce3bc9d2f | A truncated cone has two different radii. A sphere is positioned inside the truncated cone so that it touches the top, bottom, and lateral surfaces of the cone. If one of the radii is 2008, what is the minimum length of the other radius so that the sphere has an integer radius? | [
"B, 502.[/hide]\n\nThe answer provided is \\( r_1 = 502 \\)."
] | [
"A truncated cone has two different radii. A sphere is positioned inside the truncated cone so that it touches the top, bottom, and lateral surfaces of the cone. If one of the radii is 2008, what is the minimum length of the other radius so that the sphere has an integer radius?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-5c3deb99c4928c65cb8795a241f5005cffc6abde8380aef264159ab4e8c7d21e | A truncated triangular pyramid has bases with surface areas \( B_1 \) and \( B_2 \). It can be cut with a plane parallel to the bases so that a sphere can be inscribed in each of the obtained parts. If the lateral surface area of the given pyramid is given by \( (\sqrt{B_1} + \sqrt{B_2})(\sqrt[4]{B_1} + \sqrt[4]{B_2})^... | [
"325"
] | [
"A truncated triangular pyramid has bases with surface areas \\( B_1 \\) and \\( B_2 \\). It can be cut with a plane parallel to the bases so that a sphere can be inscribed in each of the obtained parts. If the lateral surface area of the given pyramid is given by \\( (\\sqrt{B_1} + \\sqrt{B_2})(\\sqrt[4]{B_1} + \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8dcaf405de5acf594912bcfa22d626e59a850f770a4ed096b727df778a3d8292 | A tube of ice cream with height 10 inches melts or freezes based on the weather. If the probability that the ending height of the ice cream after 2023 days is 6 inches can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \), find \( m + n \). | [
"1"
] | [
"A tube of ice cream with height 10 inches melts or freezes based on the weather. If the probability that the ending height of the ice cream after 2023 days is 6 inches can be expressed as \\( \\frac{m}{n} \\) for relatively prime positive integers \\( m \\) and \\( n \\), find \\( m + n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-572584aed6ee73e84b315077be4f677ea7b54016b65e6407be6f20f693a0ea29 | A tug-boat has a minimum time to tow two pontoons down the river a distance of 1 km. It was decided that one pontoon would be sent down the river by itself and the boat would tow the other pontoon for some time, then would leave it and return to the first one and tow it to the point of destination. For how many km must... | [
"$\\frac{3}{3u+v}$"
] | [
"A tug-boat has a minimum time to tow two pontoons down the river a distance of 1 km. It was decided that one pontoon would be sent down the river by itself and the boat would tow the other pontoon for some time, then would leave it and return to the first one and tow it to the point of destination. For how many km... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4391aa18e8937b4ad28989ed76b3fa051f4f0fc5bb208a2b289a312933566877 | A tunnel has a cross section consisting of two half-discs representing traffic lanes and a smaller disc representing a gas discharge duct. The two half-discs and the small disc are tangent to each other and internally tangent to a large disc with a diameter of 12 meters. What is the diameter of the gas discharge duct? | [
"4"
] | [
"A tunnel has a cross section consisting of two half-discs representing traffic lanes and a smaller disc representing a gas discharge duct. The two half-discs and the small disc are tangent to each other and internally tangent to a large disc with a diameter of 12 meters. What is the diameter of the gas discharge d... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-5978b09bbe021c18bb9f0dfa6df807352e3bfbe41537e8c3a60661000e60a298 | A tunnel in the shape of a semi-ellipse is sixty feet wide and twelve yards high in the center. Find its height (in feet) six feet from the edge. | [
"21.6"
] | [
"A tunnel in the shape of a semi-ellipse is sixty feet wide and twelve yards high in the center. Find its height (in feet) six feet from the edge."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13ff998202283d94d367c97026c569147f96cec2153e248c143aeba13d579d54 | A tuple \((z_1, z_2, \ldots, z_n) \in \mathbb{C}^n\) is said to be an MTRP tuple if for all \(\alpha, \beta, \gamma \in \mathbb{C}\) satisfying \(\alpha \overline{\beta} + \beta \overline{\gamma} + \gamma \overline{\alpha} = 0\), the following holds: \[ \prod_{i=1}^n (\alpha \overline{\beta} - z_i) + \prod_{i=1}^n (\be... | [
"(0)"
] | [
"A tuple \\((z_1, z_2, \\ldots, z_n) \\in \\mathbb{C}^n\\) is said to be an MTRP tuple if for all \\(\\alpha, \\beta, \\gamma \\in \\mathbb{C}\\) satisfying \\(\\alpha \\overline{\\beta} + \\beta \\overline{\\gamma} + \\gamma \\overline{\\alpha} = 0\\), the following holds:\n\\[\n\\prod_{i=1}^n (\\alpha \\overline{... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9640d95027d82aef330718932124a2a879cbfd094c3c0fed66b576e6fe5051da | A turkey in a circus is launched out of a cannon at a target composed of four concentric circles of radii 1, 4, 5, and 6. The concentric circles are colored alternately red and white and the innermost circle is white. If the turkey has an equal chance of landing anywhere in the target, what is the probability that the ... | [
"\\dfrac{5}{18}"
] | [
"A turkey in a circus is launched out of a cannon at a target composed of four concentric circles of radii 1, 4, 5, and 6. The concentric circles are colored alternately red and white and the innermost circle is white. If the turkey has an equal chance of landing anywhere in the target, what is the probability that... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24131f164258cc1e117cb85813a082a45dd92107f67d0fb4e79528c2378213c8 | A turtle is on a field. It crawled forward for 6 hours (its speed can vary). The turtle's crawling was always watched by at least one turtle enthusiast at all times. Each enthusiast watched the turtle crawl for an hour before leaving, reporting that they saw the turtle crawl exactly 1 inch in that hour. What's the fart... | [
"6"
] | [
"A turtle is on a field. It crawled forward for 6 hours (its speed can vary). The turtle's crawling was always watched by at least one turtle enthusiast at all times. Each enthusiast watched the turtle crawl for an hour before leaving, reporting that they saw the turtle crawl exactly 1 inch in that hour. What's the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-5653a4fe57af9e854e5373c37b23107504be56f4bafe5993c16b8c86da46b87c | A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$.... | [
"10 + 100\\sqrt{69}"
] | [
"A turtle runs away from an UFO with a speed of $0.2 \\ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \\ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\\ell$ (in meters) it may turn through any acute angle $\\alpha$ such that $\\tan \\alpha < \\frac{\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-675d63c1b1c865c46f71efbbf290907e9f707498ad6bb36d63bb3af6a2b4e96a | A turtle walks one step east, then turns $90$ degrees counter-clockwise and walks two steps north. Then the turtle turns $90$ degrees counter-clockwise again, and walks three steps west. If the turtle continues this pattern, turning $90$ degrees counter-clockwise, and then walking $n$ steps on the $n^{\rm th}$ iteratio... | [
"\\(\\sqrt{7321}\\)"
] | [
"A turtle walks one step east, then turns $90$ degrees counter-clockwise and walks two steps north. Then the turtle turns $90$ degrees counter-clockwise again, and walks three steps west. If the turtle continues this pattern, turning $90$ degrees counter-clockwise, and then walking $n$ steps on the $n^{\\rm th}$ it... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 28
} | 76 | 0 | 0 |
omr-ws-aabf2708ef8b2c75120c65dddc3b05cedc064736574806d50cb628cb9bfd4f86 | A two-day mathematics conference has \( n \) participants. Some participants give a talk on Saturday, some on Sunday, and some do not give a talk at all. At the end of the conference, a few talks are selected to be included in a book. In how many different ways can the talks be selected for inclusion in the book, assum... | [
"\\( 5^n - 3^n \\)"
] | [
"A two-day mathematics conference has \\( n \\) participants. Some participants give a talk on Saturday, some on Sunday, and some do not give a talk at all. At the end of the conference, a few talks are selected to be included in a book. In how many different ways can the talks be selected for inclusion in the book... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 1,
"tir": 2
} | 43 | 0 | 0 |
omr-ws-ce3bf216e4fd318c16854b43ca8bb2392c20c43632cfb98d380c8ca423ca62c8 | A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$? | [
"225"
] | [
"A two-digit integer $AB$ equals $\\frac{1}{9}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d3e6ae54d8ff8f2bb8de9eccdd0e6c309fd71aa88bbc03d4696b2231b9023fe2 | A two-digit integer $\underline{a}\,\, \underline{b}$ is multiplied by $9$. The resulting three-digit integer is of the form $\underline{a} \,\,\underline{c} \,\,\underline{b}$ for some digit $c$. Evaluate the sum of all possible $\underline{a} \,\, \underline{b}$. | [
"120"
] | [
"A two-digit integer $\\underline{a}\\,\\, \\underline{b}$ is multiplied by $9$. The resulting three-digit integer is of the form $\\underline{a} \\,\\,\\underline{c} \\,\\,\\underline{b}$ for some digit $c$. Evaluate the sum of all possible $\\underline{a} \\,\\, \\underline{b}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-bd6ee5ef3ec5dad2380eac5c1ce0bb51a13d1fa86da21c4839665ccdffa8645c | A two-digit number \( ab \) is called almost prime if changing at most one of its digits \( a \) and \( b \) results in a two-digit prime number. How many almost prime two-digit numbers are there? | [
"90"
] | [
"A two-digit number \\( ab \\) is called almost prime if changing at most one of its digits \\( a \\) and \\( b \\) results in a two-digit prime number. How many almost prime two-digit numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 5
} | 17 | 0 | 0 |
omr-ws-676abd4470ec4f2236ab0ed81beb6afa3c3d259531521f856182d18f36aabe25 | A two-digit number \( n \) has the property that the sum of the digits of \( n \) is the same as the sum of the digits of \( 6n \). How many such numbers are there? | [
"10"
] | [
"A two-digit number \\( n \\) has the property that the sum of the digits of \\( n \\) is the same as the sum of the digits of \\( 6n \\). How many such numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba42f692ff0a1d878e996b791d5f0978db58d0bcd1c584a223754661a1377699 | A two-digit number \( s \) is special if \( s \) is the common two leading digits of the decimal expansion of \( 4^n \) and \( 5^n \), where \( n \) is a certain positive integer. Given that there are two special numbers, find these two special numbers. | [
"21 and 46"
] | [
"A two-digit number \\( s \\) is special if \\( s \\) is the common two leading digits of the decimal expansion of \\( 4^n \\) and \\( 5^n \\), where \\( n \\) is a certain positive integer. Given that there are two special numbers, find these two special numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-206ed22c76c7b737247d59157894e1e9a0c3d0fa0fb95056faa4753a01139c77 | A two-digit number has its digits separated by another digit to form a new number. The arithmetic mean of these two numbers is equal to the number obtained by reversing the digits of the original number. Find the original number. | [
"16"
] | [
"A two-digit number has its digits separated by another digit to form a new number. The arithmetic mean of these two numbers is equal to the number obtained by reversing the digits of the original number. Find the original number."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-78d22dfa6c683fae372d4701f3a09985d472874337b6f45f269a9cc60a0d2566 | A two-digit number is added to the number with its digits reversed. How many distinct pairs of these numbers sum to a perfect square? | [
"4"
] | [
"A two-digit number is added to the number with its digits reversed. How many distinct pairs of these numbers sum to a perfect square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a90ef1a3f39e8e2b7e103520a95280ffc6543c095bbbadc32fe07582a268506a | A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? | [
"1"
] | [
"A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72640b9aa95c4affb438955d2c1f6bb83ece56b3a8a1bf80e24e9435c6ae9ce1 | A two-dimensional object is bounded by $x = 0, y = 0,$ and $x + y = 1.$ It has density (mass per unit area) equal to $\frac{1}{1 + (x + y)^2}.$ Find its center of mass. | [
"\\(\\left(\\frac{4 - \\pi}{4 \\ln(2)}, \\frac{4 - \\pi}{4 \\ln(2)}\\right)\\)"
] | [
"A two-dimensional object is bounded by $x = 0, y = 0,$ and $x + y = 1.$ It has density (mass per unit area) equal to $\\frac{1}{1 + (x + y)^2}.$ Find its center of mass."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-bdbaca43d7d32f4fa8bb1b56ead40a37aab2db79a5acfc409f9472e1134ffbfb | A two-gallon container had all of its dimensions tripled. How many gallons does the new container hold? | [
"54"
] | [
"A two-gallon container had all of its dimensions tripled. How many gallons does the new container hold?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0486a55715fafbe8caf7312ba6ce182a35a4ab9b5726db5160d34c81f945779c | A two-inch cube ($2\times2\times2$) of silver weighs 3 pounds and is worth $\$200$. How much is a three-inch cube of silver worth? Round your answer to the nearest dollar. | [
"\\$675"
] | [
"A two-inch cube ($2\\times2\\times2$) of silver weighs 3 pounds and is worth $\\$200$. How much is a three-inch cube of silver worth? Round your answer to the nearest dollar."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0849bc2769e3215cfabf6f36a637927dec63e41a8bf5882c83611c1e40d1d4db | A type of digit-lock has 5 digits, each digit chosen from $\{1, 2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s? | [
"1441"
] | [
"A type of digit-lock has 5 digits, each digit chosen from $\\{1, 2, 3, 4, 5\\}$. How many different passwords are there that have an odd number of $1$'s?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-60bac8eda5228b807cd45b1b57b0468124263e78461066c933c09c7051a0bdab | A typist figures out that if she types 2 pages above her work quota daily, she will complete her work 3 days ahead of schedule, and if she makes 4 pages extra per day, she will finish 5 days ahead of time. How many pages does she have to type and in what time? | [
"120 pages, 15 days"
] | [
"A typist figures out that if she types 2 pages above her work quota daily, she will complete her work 3 days ahead of schedule, and if she makes 4 pages extra per day, she will finish 5 days ahead of time. How many pages does she have to type and in what time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae8e2ce82c565c2f6097e6bd374011e269ab1b6d272a0beb96a7e76aa66328f5 | A typist had to do a job in a specified time by typing a certain number of pages every day. She calculated that if she had typed 2 pages more than required per day, she would have completed the task 2 days ahead of time, but if she had turned out 60\% of her work quota, then she would have finished the job 4 days ahead... | [
"The quota was 10 pages per day, and the time limit, 12 days"
] | [
"A typist had to do a job in a specified time by typing a certain number of pages every day. She calculated that if she had typed 2 pages more than required per day, she would have completed the task 2 days ahead of time, but if she had turned out 60\\% of her work quota, then she would have finished the job 4 days... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcefbc46121400e12fa6f27ee928298425f87c6b5364ee0a140a5185a5b0e486 | A typist receives work according to a Poisson process with an average of four jobs per hour. The actual work time for each job is exponentially distributed with an average of 12 minutes. Determine the probability that a job will be completed in less than 45 minutes after arrival. | [
"0.528"
] | [
"A typist receives work according to a Poisson process with an average of four jobs per hour. The actual work time for each job is exponentially distributed with an average of 12 minutes. Determine the probability that a job will be completed in less than 45 minutes after arrival."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-3534f63ab1678c88e0e7a8cee515f65c3ddf7c64940e75605bd7c394004d2797 | A typist receives work according to a Poisson process with an average of four jobs per hour. The actual work time for each job is exponentially distributed with an average of 12 minutes. Determine the probability that the typist completes all work by the end of the day. | [
"0.2"
] | [
"A typist receives work according to a Poisson process with an average of four jobs per hour. The actual work time for each job is exponentially distributed with an average of 12 minutes. Determine the probability that the typist completes all work by the end of the day."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9d8f544102330f863c93de6c522bccd6fc338a9711ba15ca809e89bd643f6763 | A unicorn is tethered by a $20$ -foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the nearest point on the towe... | [
"813"
] | [
"A unicorn is tethered by a $20$ -foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the nearest point on t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41dd1f77df5ffedce507095e697cbbfb6eadf2cfb29bab5f7af0a3759ea56754 | A unicorn is tethered by a $20$-foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the nearest point on the tower... | [
"813"
] | [
"A unicorn is tethered by a $20$-foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the nearest point on the ... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78fa019f715c32b1aaad5c2463bc42603bbedfd678c8094a44cdbf9dda272ab4 | A unicorn is tethered by a 20-foot silver rope to the base of a cylindrical tower with a radius of 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, and the end of the rope is 4 feet from the nearest point on the tower. The length o... | [
"13"
] | [
"A unicorn is tethered by a 20-foot silver rope to the base of a cylindrical tower with a radius of 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, and the end of the rope is 4 feet from the nearest point on the tower. The len... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-1867e65e06fd62bf838acfa3955ff2a5d2a3e3f2da73361fd89a7d42a4ade065 | A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and th... | [
"813"
] | [
"A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-341322dcb107b00fa12fd059fd322df3ad26b43c1ed6c089e698dad1c18902f5 | A unicorn is tethered by a rope of length 20 to the base of a cylindrical tower. The rope is attached to the tower at ground level and to the unicorn at a height of 4, and pulled tight. The unicorn's end of the rope is a distance of 4 from the nearest point on the tower. Find the length of the rope which is in contact ... | [
"16"
] | [
"A unicorn is tethered by a rope of length 20 to the base of a cylindrical tower. The rope is attached to the tower at ground level and to the unicorn at a height of 4, and pulled tight. The unicorn's end of the rope is a distance of 4 from the nearest point on the tower. Find the length of the rope which is in con... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-105948e3d85f1ff947a6685dcf97e22f4cf9aca9277bb14a663224b864af1b7c | A unicorn is tethered by a rope of length 20 to the base of a cylindrical tower. The rope is attached to the tower at ground level and to the unicorn at a height of 4, and the unicorn's end of the rope is a distance of 4 from the nearest point on the tower. Find the length of the rope that is in contact with the tower. | [
"16"
] | [
"A unicorn is tethered by a rope of length 20 to the base of a cylindrical tower. The rope is attached to the tower at ground level and to the unicorn at a height of 4, and the unicorn's end of the rope is a distance of 4 from the nearest point on the tower. Find the length of the rope that is in contact with the t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-aa16c27061250083d789aad152af965c581c5dbb6610a9ec39bb2f496d152ed5 | A unicorn is tied to a cylindrical tower with a rope. The tower has a radius of $2$ and a height of $8$, and the rope is $10$ units long. The unicorn starts at the farthest point from the center of the tower and runs counterclockwise, causing the rope to wind around the tower. Find the area swept out by the shadow of t... | [
"\\( 18 + 9\\pi \\)"
] | [
"A unicorn is tied to a cylindrical tower with a rope. The tower has a radius of $2$ and a height of $8$, and the rope is $10$ units long. The unicorn starts at the farthest point from the center of the tower and runs counterclockwise, causing the rope to wind around the tower. Find the area swept out by the shadow... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8162a2d0a781300cd4332d29c5f4797d0959305c2833aab38c2967edb1868f14 | A unicorn is tied to a cylindrical wizard's magic tower with an elven rope stretching from the unicorn to the top of the tower. The tower has a radius of $2$ and a height of $8$, and the rope is of length $10$. The unicorn begins as far away from the center of the tower as possible. The unicorn runs counterclockwise, a... | [
"\\dfrac{250}{3}"
] | [
"A unicorn is tied to a cylindrical wizard's magic tower with an elven rope stretching from the unicorn to the top of the tower. The tower has a radius of $2$ and a height of $8$, and the rope is of length $10$. The unicorn begins as far away from the center of the tower as possible. The unicorn runs counterclockwi... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-c3639bda6d773f0c85b76b9ff66c3a72e3139b2b3ca0268e4c2047467f42fea9 | A unicycle travels 6 km. The diameter of the wheel is 1 meter. How many full revolutions did the wheel make during the trip? | [
"1909"
] | [
"A unicycle travels 6 km. The diameter of the wheel is 1 meter. How many full revolutions did the wheel make during the trip?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 40
} | 72 | 0 | 0 |
omr-ws-9b4f70494c14d40d6ce2acd3f3d27690dda37de6d181c529be3d4016e0d6f988 | A uniform rod of length $2k$ and weight $w$ rests with one end $A$ against a vertical wall, while the other end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A$. If the system is in equilibrium, determine the angle $ABC$. | [
"\\(\\arccos\\left(\\frac{b^2 + 3k^2}{4bk}\\right)\\)"
] | [
"A uniform rod of length $2k$ and weight $w$ rests with one end $A$ against a vertical wall, while the other end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A$. If the system is in equilibrium, determine the angle $ABC$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4baea0cf1f4dc7f87fc01584eec74748f1a12af2d8d13a57d50ed077eb34b956 | A unit circle is centered at the origin and a tangent line to the circle is constructed in the first quadrant such that it makes an angle \(\frac{5\pi}{6}\) with the \(y\)-axis. A series of circles centered on the \(x\)-axis are constructed such that each circle is both tangent to the previous circle and the original t... | [
"\\dfrac{(7\\sqrt{3} - 12)\\pi}{24}"
] | [
"A unit circle is centered at the origin and a tangent line to the circle is constructed in the first quadrant such that it makes an angle \\(\\frac{5\\pi}{6}\\) with the \\(y\\)-axis. A series of circles centered on the \\(x\\)-axis are constructed such that each circle is both tangent to the previous circle and t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-6107b01c39b47df84ec8f1325a74836cbeb272a45f9dde3f18501dfe2a9dc85f | A unit cube $C$ is rotated around one of its diagonals by an angle of $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$. | [
"\\(\\frac{5}{6}\\)"
] | [
"A unit cube $C$ is rotated around one of its diagonals by an angle of $\\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-ac7328747252b97a89f2be9e477c1cee3b186b3894b0489d36945e2097c8e13d | A unit cube $\mathcal{C}$ touches plane $\mathcal{P}$ at point $A$. Let $A \neq B, C, D$ be the three closest vertices to $\mathcal{P}$. Calculate the value of $d(B)^2 + d(C)^2 + d(D)^2$, where $d(X)$ is the distance from point $X$ to $\mathcal{P}$. | [
"1"
] | [
"A unit cube $\\mathcal{C}$ touches plane $\\mathcal{P}$ at point $A$. Let $A \\neq B, C, D$ be the three closest vertices to $\\mathcal{P}$. Calculate the value of $d(B)^2 + d(C)^2 + d(D)^2$, where $d(X)$ is the distance from point $X$ to $\\mathcal{P}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9de60eb6e983625a0ca3fa1ee40c9a4caa3eafa43f8710cdf9d57c08e2050e1 | A unit cube and a fair die are given. Upon tossing the die, the resulting number affects the dimensions of the cube as follows: - Rolling a 1 increases the cube's volume by 50 percent. - Rolling a prime number increases the length, width, and height by half the prime number obtained, in units. - Rolling a 4 or 6 decrea... | [
"14"
] | [
"A unit cube and a fair die are given. Upon tossing the die, the resulting number affects the dimensions of the cube as follows:\n- Rolling a 1 increases the cube's volume by 50 percent.\n- Rolling a prime number increases the length, width, and height by half the prime number obtained, in units.\n- Rolling a 4 or ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df7898a2d3295629bf51b5ebea862333ad2a57d72b5ee45c3cb5d6643950629e | A unit cube has its corners cut off such that the initial faces are now regular octagons. How many vertices does the resulting object have? | [
"32"
] | [
"A unit cube has its corners cut off such that the initial faces are now regular octagons. How many vertices does the resulting object have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e32c64346f7f6e0a0ce14e98c5f48f49aefa43d6d3d0d9dca50dc3b5caefc5c | A unit cube has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3',\) and \(P_4'\). Vertices \(P_2, P_3,\) and \(P_4\) are adjacent to \(P_1\), and for \(1 \leq i \leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_... | [
"\\(\\frac{3\\sqrt{2}}{4}\\)"
] | [
"A unit cube has vertices \\(P_1, P_2, P_3, P_4, P_1', P_2', P_3',\\) and \\(P_4'\\). Vertices \\(P_2, P_3,\\) and \\(P_4\\) are adjacent to \\(P_1\\), and for \\(1 \\leq i \\leq 4\\), vertices \\(P_i\\) and \\(P_i'\\) are opposite to each other. A regular octahedron has one vertex in each of the segments \\(P_1P_2... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 27,
"tir": 6
} | 33 | 0 | 0 |
omr-ws-d616bb0bef957f37aaecd654816eb2a1265ca59327fcd49e1f49404b2d0c7c3c | A unit cube is aligned in a 3D Cartesian coordinate system such that two of its opposite vertices (connected by a space diagonal) are on the z-axis. The cube is then rotated about the z-axis to form another solid. What is the surface area of the resulting solid? Is there a name for this solid? | [
"2\\pi\\sqrt{6}"
] | [
"A unit cube is aligned in a 3D Cartesian coordinate system such that two of its opposite vertices (connected by a space diagonal) are on the z-axis. The cube is then rotated about the z-axis to form another solid. What is the surface area of the resulting solid? Is there a name for this solid?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-bf92c9eca81b44de73a01cc212c744f9f30b1e6a1bd5e085f36ece885f2a335a | A unit cube is cut by a plane passing through two opposite vertices, dividing it into two congruent pieces. A sphere is inscribed in one of these pieces. What is the maximum possible volume of this inscribed sphere? | [
"\\(\\frac{\\pi}{3}(10 - 7\\sqrt{2})\\)"
] | [
"A unit cube is cut by a plane passing through two opposite vertices, dividing it into two congruent pieces. A sphere is inscribed in one of these pieces. What is the maximum possible volume of this inscribed sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 23
} | 35 | 0 | 0 |
omr-ws-147768c1150ca1fd1779574373661bbb94ecdd1d71b79b18bdd6d80bcc1cc9ba | A unit cube is oriented such that its long diagonal is perpendicular to a tabletop and is then spun at a high speed around this diagonal. What is the volume of the solid captured by the spinning cube? | [
"\\dfrac{\\sqrt{3}}{3} \\pi"
] | [
"A unit cube is oriented such that its long diagonal is perpendicular to a tabletop and is then spun at a high speed around this diagonal. What is the volume of the solid captured by the spinning cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-f1d85a8b0dbc453f259de75bc4cb77e685f41132ede57e853acfe034894c5b69 | A unit cube moves on top of a $6 \times 6$ checkerboard whose squares are unit squares. Beginning in the bottom left corner, the cube is allowed to roll up or right, rolling about its bottom edges to travel from square to square, until it reaches the top right corner. Given that the side of the cube facing upwards in t... | [
"384"
] | [
"A unit cube moves on top of a $6 \\times 6$ checkerboard whose squares are unit squares. Beginning in the bottom left corner, the cube is allowed to roll up or right, rolling about its bottom edges to travel from square to square, until it reaches the top right corner. Given that the side of the cube facing upward... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-dbce1b5295824d154f1d6870548e286ff17eae7d7aac25ab244c3a6ec979fb8f | A unit cube of size $1 \times 1 \times 1$ is stacked to form a larger cube $ABCD-EFGH$ of size $n \times n \times n$. Let $P$ be the plane passing through the midpoints of $AB$, $FG$, and $DH$. How many of the $n^3$ unit cubes intersect with $P$? Express the number as a function of $n$. | [
"\\( 2n^2 \\)"
] | [
"A unit cube of size $1 \\times 1 \\times 1$ is stacked to form a larger cube $ABCD-EFGH$ of size $n \\times n \\times n$. Let $P$ be the plane passing through the midpoints of $AB$, $FG$, and $DH$. How many of the $n^3$ unit cubes intersect with $P$? Express the number as a function of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b922cc7213f187f26a5ed33b3467ede23d6573e9f2f43a74ed65871433b9a7a4 | A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps for in... | [
"\\(\\frac{\\sqrt{3}}{10}\\)"
] | [
"A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 20
} | 64 | 0 | 0 |
omr-ws-a9a1397acca900d7b65f6e400a8978a0885fc95e246e5f3d9d0672ce96baf32a | A unit of blood expires after $10!$ seconds. If Yasin donates a unit of blood at noon on January 1, on what day does his unit of blood expire? | [
"February 12"
] | [
"A unit of blood expires after $10!$ seconds. If Yasin donates a unit of blood at noon on January 1, on what day does his unit of blood expire?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0ea0f1bc9d3450b482e7cf9da959ad7e0efa46bce1cfe70f5f79f00c4a2fce59 | A unit of blood expires after \(10! = 10 \cdot 9 \cdot 8 \cdots 1\) seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire? | [
"February 12"
] | [
"A unit of blood expires after \\(10! = 10 \\cdot 9 \\cdot 8 \\cdots 1\\) seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7021b2d1469e88a2a4f71db78abf1a925413f91b4167ec8efae2269d91065215 | A unit sphere is centered at $(0,0,1)$. A point light source is located at $(1,0,4)$ and sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere's shadow on the $xy$ plane? | [
"\\dfrac{3\\sqrt{2}}{2} \\pi"
] | [
"A unit sphere is centered at $(0,0,1)$. A point light source is located at $(1,0,4)$ and sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere's shadow on the $xy$ plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6ccfb7e2a27f1af5549b2ee9c12f4934ab1cf4d978e64bb1bc12842091544544 | A unit sphere is filled to $\frac{1}{4}$ of its volume with water. Find the depth of the water in the sphere. | [
"\\(1 - 2\\cos80^\\circ\\)"
] | [
"A unit sphere is filled to $\\frac{1}{4}$ of its volume with water. Find the depth of the water in the sphere."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-b217cf421897f0ed794ed536b7144ddceea356e97ff593d4eb66da47bbe4c047 | A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\alpha$. Compute the common area of the original and the rotated squares. | [
"\\(\\frac{2}{\\sin \\alpha + \\cos \\alpha + 1}\\)"
] | [
"A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\\alpha$. Compute the common area of the original and the rotated squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-47fb69c15c191c1f37d85c8399d2576a3c2a56b7b3d9edf907f72334bb29f38c | A unit square from one of the corners of an $8 \times 8$ chessboard is cut and thrown away. At least how many triangles are necessary to divide the new board into triangles with equal areas? | [
"21"
] | [
"A unit square from one of the corners of an $8 \\times 8$ chessboard is cut and thrown away. At least how many triangles are necessary to divide the new board into triangles with equal areas?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-0f553e5540564f659e1d750858acb9ad72adb6955b751662d14a6dda1514089f | A unit square has a circle of radius \( r \) centered at its midpoint. Four quarter circles are centered at the vertices of the square and are tangent to the central circle. Find the maximum and minimum possible values of the area of the striped figure and the corresponding values of \( r \) at which these maximum and ... | [
"Maximum area \\( 1 - \\frac{\\pi}{4} \\) at \\( r = \\frac{\\sqrt{2}}{4} \\), minimum area \\( 1 - \\pi + \\frac{\\pi \\sqrt{2}}{2} \\) at \\( r = \\frac{1}{2} \\) or \\( r = \\frac{\\sqrt{2} - 1}{2} \\)."
] | [
"A unit square has a circle of radius \\( r \\) centered at its midpoint. Four quarter circles are centered at the vertices of the square and are tangent to the central circle. Find the maximum and minimum possible values of the area of the striped figure and the corresponding values of \\( r \\) at which these max... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-9528d61808f792123c6269e7bfef70634ea29008bbcec19d9e57e3d00768f61c | A unit square has an equilateral triangle inscribed inside, with one vertex of the triangle shared by the square. The other two vertices of the triangle are on sides of the square, not on a vertex. Find the area of the triangle. | [
"$2\\sqrt{3} - 3$"
] | [
"A unit square has an equilateral triangle inscribed inside, with one vertex of the triangle shared by the square. The other two vertices of the triangle are on sides of the square, not on a vertex. Find the area of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 29
} | 51 | 0 | 0 |
omr-ws-def0151f5cda013052f2acd48aeebb4bfa169127583ceefe609d2191fd120936 | A unit square has its corners chopped off to form a regular polygon with eight sides. What is the area of this polygon? | [
"\\(2\\sqrt{2} - 2\\)"
] | [
"A unit square has its corners chopped off to form a regular polygon with eight sides. What is the area of this polygon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f367ad2ddfc9708cf1d760e68cec82c9501023af2cf0dc56616964404404df17 | A unit square has two points randomly chosen within or on its borders. What is the probability that the distance between the two points is at least 0.5 units? | [
"\\(\\frac{125}{96} - \\frac{\\pi}{4}\\)"
] | [
"A unit square has two points randomly chosen within or on its borders. What is the probability that the distance between the two points is at least 0.5 units?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 9
} | 27 | 0 | 0 |
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