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-6aaac61b78e928b7e1ae61d023b05cd03fc081899d5f9f4cfdb32b324d331341 | A snowplow (x) leaves the house at 1:00 PM with a speed inversely proportional to the amount of snow it is pushing, where the snow starts falling at a constant rate \( r \) at 12:00 PM. At 3:00 PM, a second snowplow (y) leaves the house with a speed also inversely proportional to the amount of snow it is clearing. When... | [
"7:389 PM"
] | [
"A snowplow (x) leaves the house at 1:00 PM with a speed inversely proportional to the amount of snow it is pushing, where the snow starts falling at a constant rate \\( r \\) at 12:00 PM. At 3:00 PM, a second snowplow (y) leaves the house with a speed also inversely proportional to the amount of snow it is clearin... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-794cbe11b9ababe9e17278886434e6c8ec273d44efb93e2bf1999eafef487809 | A snowplow can remove snow at a constant rate (in cubic feet per minute). One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, the plow began to remove snow. It had cleared 2 miles of snow between noon and 1 PM, and 1 more mile of snow between 1 PM... | [
"11 AM"
] | [
"A snowplow can remove snow at a constant rate (in cubic feet per minute). One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, the plow began to remove snow. It had cleared 2 miles of snow between noon and 1 PM, and 1 more mile of snow between... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-829d0ba76f14ee60a45cb6d2430ae49997defb53ac70462f95ceba67da89e1b6 | A snowplow can remove snow at a constant rate. One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, the plow began to remove snow. It had cleared 2 miles of snow between noon and 1 PM, and 1 more mile of snow between 1 PM and 2 PM. At what time did... | [
"11 AM"
] | [
"A snowplow can remove snow at a constant rate. One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, the plow began to remove snow. It had cleared 2 miles of snow between noon and 1 PM, and 1 more mile of snow between 1 PM and 2 PM. At what tim... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-ccc4fa122e7db334d2f3fe7aaee3474e2e93dbdb5bf084703c9c9366ae47f796 | A snowplow can remove snow at a constant rate. One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, the plow began to remove snow. It had cleared 2 miles of snow between noon and 1 PM, and 1 more mile of snow between 1 PM and 2 PM. What time did it... | [
"11:23 AM"
] | [
"A snowplow can remove snow at a constant rate. One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, the plow began to remove snow. It had cleared 2 miles of snow between noon and 1 PM, and 1 more mile of snow between 1 PM and 2 PM. What time d... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 22
} | 44 | 0 | 0 |
omr-ws-352c301b0ab0f505d26b8df66c028c46efed72a1940e97303e0d167214c093f3 | A soap film bubble blown from a circular hoop describes an undetermined region $\Omega \subseteq \{(x,y,z)\in \mathbb{R}^3: y\leq 0\}$ having a 3-dimensional volume equal to 10. Let $S$ denote the portion of $\partial \Omega$ comprised of the soap film (it does not include the unit disk $D$ in the $x,z-$plane). Suppose... | [
"30 - 5\\pi"
] | [
"A soap film bubble blown from a circular hoop describes an undetermined region $\\Omega \\subseteq \\{(x,y,z)\\in \\mathbb{R}^3: y\\leq 0\\}$ having a 3-dimensional volume equal to 10. Let $S$ denote the portion of $\\partial \\Omega$ comprised of the soap film (it does not include the unit disk $D$ in the $x,z-$p... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-6def3019971ed7c1de6013779533aa74e89c737ae614880ca1fb171d5aea8539 | A soccer ball is tiled with hexagons (6-gons) and pentagons (5-gons). Each pentagon is surrounded by 5 hexagons, and each hexagon is surrounded by 3 hexagons and 3 pentagons (alternating). How many pentagons and hexagons are there on the soccer ball? | [
"12 pentagons and 20 hexagons."
] | [
"A soccer ball is tiled with hexagons (6-gons) and pentagons (5-gons). Each pentagon is surrounded by 5 hexagons, and each hexagon is surrounded by 3 hexagons and 3 pentagons (alternating). How many pentagons and hexagons are there on the soccer ball?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 28
} | 29 | 0 | 0 |
omr-ws-7efd785126acbd3df3dc78cb2928b1846364f49042dc5e2e27cb1862a7b3ef45 | A soccer jersey must be numbered with a one- or two-digit integer. Ellen has 8 each of the digits 1-8. What is the maximum number of unique jerseys she can number? | [
"36"
] | [
"A soccer jersey must be numbered with a one- or two-digit integer. Ellen has 8 each of the digits 1-8. What is the maximum number of unique jerseys she can number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-f3229bd60d61e30905d22bffa88f34c57a2a7eaaeb8ace89424b968a28880d47 | A soccer match is played until a team has scored 10 goals. During the match, Little Johnny writes on a paper sheet the result at any point when a goal is scored. How many different paper sheets can Little Johnny obtain? (Two paper sheets are different if the results written on them differ in at least one position.) | [
"184756"
] | [
"A soccer match is played until a team has scored 10 goals. During the match, Little Johnny writes on a paper sheet the result at any point when a goal is scored. How many different paper sheets can Little Johnny obtain? (Two paper sheets are different if the results written on them differ in at least one position.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6226c212415c8ecb0950a2ee66c4f2b232ad3ea15543438ef672ed1de976a04a | A soccer team consists of 12 players. The players are to be paired in groups of two for the purpose of determining roommates. If the pairing is done at random, how many possible configurations are there? | [
"\\(\\frac{12!}{(2!)^6 \\cdot 6!}\\)"
] | [
"A soccer team consists of 12 players. The players are to be paired in groups of two for the purpose of determining roommates. If the pairing is done at random, how many possible configurations are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-251335066ed0a4f7a5ba32005cd79fa12d3ec041493d2a9b336b5df057b0309e | A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game... | [
"122"
] | [
"A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 5
} | 68 | 0 | 0 |
omr-ws-d5e4b405dff56593feee57b1c5230d3377a4971ddef8d3d13062458c75465897 | A soccer team has 22 available players, with 11 starting the game and 11 as substitutes. The coach can make up to 3 substitutions, where any one of the 11 players in the game is replaced by one of the substitutes. No player removed from the game may reenter, although a substitute can be replaced later. No two substitut... | [
"122"
] | [
"A soccer team has 22 available players, with 11 starting the game and 11 as substitutes. The coach can make up to 3 substitutions, where any one of the 11 players in the game is replaced by one of the substitutes. No player removed from the game may reenter, although a substitute can be replaced later. No two subs... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 7,
"tir": 8
} | 61 | 0 | 0 |
omr-ws-1268d65e0d9a2369861220a08048f4e3f2d993c77ae679bfcfe9ec9f81f38ef7 | A soccer team plays in 30 consecutive days in September, with at least one match each day and a total of no more than 45 matches. Let $N$ be the number of possible sequences of consecutive days during which the team played exactly 14 matches. Find the minimum possible value of $N$. | [
"2"
] | [
"A soccer team plays in 30 consecutive days in September, with at least one match each day and a total of no more than 45 matches. Let $N$ be the number of possible sequences of consecutive days during which the team played exactly 14 matches. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-9b0a904b87a47b23a4749ccc75e2f67f4e5a1420a3d42d0066d9defd4532da43 | A softball team played ten games, scoring $1, 2, 3, 4, 5, 6, 7, 8, 9,$ and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? | [
"45"
] | [
"A softball team played ten games, scoring $1, 2, 3, 4, 5, 6, 7, 8, 9,$ and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54888163a91844b9541baeffd2e47ce9c6ac9762ca247d545884e8cb2a0dd5bf | A software buyer is looking for a product to retail for $230. He wants to apply a markup of 38% (based on selling price) to the product. Allowing for this markup, what is the maximum amount the buyer can pay for the software? | [
"142.60"
] | [
"A software buyer is looking for a product to retail for $230. He wants to apply a markup of 38% (based on selling price) to the product. Allowing for this markup, what is the maximum amount the buyer can pay for the software?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2681f599baa28ed4ad8a1ef5465072adc33c75421b355c5f72c55f2b250b4b03 | A software package consists of 12 programs, five of which must be upgraded. If 4 programs are randomly chosen for testing, what is the expected number of programs, out of the chosen four, that must be upgraded? | [
"\\dfrac{5}{3}"
] | [
"A software package consists of 12 programs, five of which must be upgraded. If 4 programs are randomly chosen for testing, what is the expected number of programs, out of the chosen four, that must be upgraded?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86d3ecb5d8c76e01cab5186302418dff71a4f17dfbf3a2cdb58e0b28b9c0de99 | A soldier fires a rifle and hits a practice target 2600 meters away. The bullet travels at 800 meters per second and sound travels at 350 meters per second. How many seconds after firing the rifle does the soldier hear the bullet strike the target? Express your answer as a decimal rounded to the nearest tenth. | [
"10.7"
] | [
"A soldier fires a rifle and hits a practice target 2600 meters away. The bullet travels at 800 meters per second and sound travels at 350 meters per second. How many seconds after firing the rifle does the soldier hear the bullet strike the target? Express your answer as a decimal rounded to the nearest tenth."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-acdf185a823eccef205bb62fe0da66a23fa16a3c700ed56aba8435557ca51651 | A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dragon ... | [
"\\(\\frac{3}{5}\\)"
] | [
"A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-866d59e6f04f1af177fd09e91a341eca6054380c96c364e45deb7f1cfa0fed8b | A soldier is last in a 1/8 mile long line of troops marching forward at a steady 2 mph. He begins to run toward the front of the line at a steady 4 mph and, upon reaching the front, immediately runs back to the end of the line. Compute the total distance he has run (in miles). | [
"\\(\\frac{1}{3}\\)"
] | [
"A soldier is last in a 1/8 mile long line of troops marching forward at a steady 2 mph. He begins to run toward the front of the line at a steady 4 mph and, upon reaching the front, immediately runs back to the end of the line. Compute the total distance he has run (in miles)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3fe049058891b6d7cea2ecb1dedd933f82a848fd53b27fe91689c109e412a3db | A soldier needs to check for mines in the interior or on the sides of an equilateral triangle \(ABC\). His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier starts from one of the vertices of the triangle. What is the minimum distance the soldier needs to traverse to... | [
"\\( a \\frac{2\\sqrt{7} - \\sqrt{3}}{4} \\)"
] | [
"A soldier needs to check for mines in the interior or on the sides of an equilateral triangle \\(ABC\\). His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier starts from one of the vertices of the triangle. What is the minimum distance the soldier needs to trav... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-033f8dde7112ab98fbfc01a1f7f3f229c9f3bb4f2b771db1a0b1d6b47c72641f | A solid box has dimensions 11x1x12. If an insect lands on one corner of the box, what is the length of the shortest route to the corner diagonally opposite the starting corner? | [
"\\(12\\sqrt{2}\\)"
] | [
"A solid box has dimensions 11x1x12. If an insect lands on one corner of the box, what is the length of the shortest route to the corner diagonally opposite the starting corner?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-57c29c96a65333269cbdfd01831232412393f23d7c5d90ae4f5b43f7a7e75d2a | A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed? | [
"18\\%"
] | [
"A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-031b6cc59c848bd8a76b5537a4f78f09318e1f2c8264a066f93b80a1b5356fea | A solid circular cylinder with diameter $D$ cm is placed vertically. An inclined plane passing through a diameter of the base cuts the curved surface of the cylinder at a maximum height of $H$ cm. Find the volume of the cutout portion in cubic centimeters, in terms of $H$ and $D$. | [
"\\dfrac{1}{6} H D^2"
] | [
"A solid circular cylinder with diameter $D$ cm is placed vertically. An inclined plane passing through a diameter of the base cuts the curved surface of the cylinder at a maximum height of $H$ cm. Find the volume of the cutout portion in cubic centimeters, in terms of $H$ and $D$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0d1cf72fa16b880109d4c0081aa9ba6a2a9a2196946a71a4b1cffb72bf70177 | A solid consists of a cone, a cylinder, and a hemisphere with a base radius of 4 cm. The ratio of the volumes of the cone, the cylinder, and the hemisphere is 6:27:4. Find the height of the cone. | [
"12"
] | [
"A solid consists of a cone, a cylinder, and a hemisphere with a base radius of 4 cm. The ratio of the volumes of the cone, the cylinder, and the hemisphere is 6:27:4. Find the height of the cone."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-0e8f07da86f4cdc6889fd26be82e82ea7c03fd38d84f03135d7b1626fac46b42 | A solid consists of a cone, a cylinder, and a hemisphere with a base radius of 4 cm. The ratio of the volumes of the cone, the cylinder, and the hemisphere is 6:27:4. Find the height of the cylinder. | [
"18"
] | [
"A solid consists of a cone, a cylinder, and a hemisphere with a base radius of 4 cm. The ratio of the volumes of the cone, the cylinder, and the hemisphere is 6:27:4. Find the height of the cylinder."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-bea442763c8f479917496e675e4404222d552c121fe07b516ffa5376ee030707 | A solid consists of a cone, a cylinder, and a hemisphere with a base radius of 4 cm. The ratio of the volumes of the cone, the cylinder, and the hemisphere is 6:27:4. Find the volume of the solid. | [
"\\(\\frac{1184 \\pi}{3}\\)"
] | [
"A solid consists of a cone, a cylinder, and a hemisphere with a base radius of 4 cm. The ratio of the volumes of the cone, the cylinder, and the hemisphere is 6:27:4. Find the volume of the solid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-56c18dacb9d20827f82a2b1ca8b5a579d86e8f0a8b5c45770d1d24699fa4b21e | A solid consists of a cylindrical middle with a conical cap at each end. The height of each conical cap is equal to the length of the cylindrical middle. Given a fixed surface area, what shape (i.e., dimensions) maximizes the volume of the solid? | [
"h = \\dfrac{2\\sqrt{5}}{5} r"
] | [
"A solid consists of a cylindrical middle with a conical cap at each end. The height of each conical cap is equal to the length of the cylindrical middle. Given a fixed surface area, what shape (i.e., dimensions) maximizes the volume of the solid?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 30,
"genselect": 3
} | 33 | 0 | 0 |
omr-ws-c7d00d9cd6776c730346d10ef621b611d902023b5a0984d780d40765ef20b691 | A solid cube has a side length of 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face, with the edges of each cut parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? | [
"7"
] | [
"A solid cube has a side length of 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face, with the edges of each cut parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d1ceaaf2af9077d2b2616ebc6315cb5c2c090c93dc2da7ac26327aecb4f0eeb | A solid cube has a side length of 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? | [
"7"
] | [
"A solid cube has a side length of 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5ad12bcfe69f1bc9e58615a98af5bc0aa6ca2f1d20fd699d0c1374e45489e676 | A solid cube has edges measuring 4 cm. A triangle is drawn with vertices at the midpoints of three connecting edges, and one "corner" of the cube is truncated so that this triangle is showing. The remaining figure is a solid with 7 faces. Find the surface area of this solid in square centimeters. | [
"90 + 2\\sqrt{3}",
"\\( 90 + 2\\sqrt{3} \\)"
] | [
"A solid cube has edges measuring 4 cm. A triangle is drawn with vertices at the midpoints of three connecting edges, and one \"corner\" of the cube is truncated so that this triangle is showing. The remaining figure is a solid with 7 faces. Find the surface area of this solid in square centimeters."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 26
} | 58 | 0 | 0 |
omr-ws-fe43da14c30b4925099c39bd9992d863dc8b7bc59b8f9ba37d4a8e7b48abd456 | A solid cube is divided into two pieces by a single rectangular cut. As a result, the total surface area increases by a fraction $f$ of the surface area of the original cube. What is the greatest possible value of $f$? | [
"\\(\\frac{\\sqrt{2}}{3}\\)"
] | [
"A solid cube is divided into two pieces by a single rectangular cut. As a result, the total surface area increases by a fraction $f$ of the surface area of the original cube. What is the greatest possible value of $f$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 24
} | 57 | 0 | 0 |
omr-ws-01164f693186c73203e1344345406fc7dba8cb93d9707e0cc06d40521e673c78 | A solid cube of side length $3$ has a solid cube of side length $1$ removed from each corner. How many edges does the remaining solid have? | [
"84"
] | [
"A solid cube of side length $3$ has a solid cube of side length $1$ removed from each corner. How many edges does the remaining solid have?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9ccdc4208ff669c9005b154f4371cd770dc97bb3d16c6962670dd193a30adf69 | A solid cube of side length $b$ is resting on level ground. Wet sand is piled up outside and on top of the cube to form a right circular cone that hides the cube from view. How high should the cone be in order to minimize the work done lifting the wet sand from ground level into the cone? | [
"\\( h = 2b \\)"
] | [
"A solid cube of side length $b$ is resting on level ground. Wet sand is piled up outside and on top of the cube to form a right circular cone that hides the cube from view. How high should the cone be in order to minimize the work done lifting the wet sand from ground level into the cone?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f6b4273ae288056a996738bf3d1b7fb3e713aa64927689d8297c5f6968527f76 | A solid cube with each side measuring 3 ft by 3 ft has three 1 ft by 1 ft square holes bored through the cube, connecting the center square on each face with the center square on the opposite face. Find the number of square feet in the surface area of the remaining solid. | [
"72"
] | [
"A solid cube with each side measuring 3 ft by 3 ft has three 1 ft by 1 ft square holes bored through the cube, connecting the center square on each face with the center square on the opposite face. Find the number of square feet in the surface area of the remaining solid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-f9db01f010e46a9e494c583e1fe235fb14a1cfe9f007eb3fe31ebb9c05927dc3 | A solid cylinder has a radius of 6 and a height of 5. It is sliced in half by a cutting plane that is tangent to both of its circular bases. What is the area of the cross section? | [
"\\( 39\\pi \\)"
] | [
"A solid cylinder has a radius of 6 and a height of 5. It is sliced in half by a cutting plane that is tangent to both of its circular bases. What is the area of the cross section?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-fb6bc9e9705e34babe65bdedfe35a34ddb40ee3124b14a0f5fa6e0f88d440934 | A solid in $\mathbb{R}^{2019}$ has the property that all of its angles are right angles. Let its edges have length $a_1, a_2, \dots, a_{2019}$, where each length is repeated the same number of times. Given that the maximum distance between any two vertices of the solid is $1$, what is the maximum value of the perimeter... | [
"\\(\\frac{2^{2018}}{\\sqrt{2019}}\\)"
] | [
"A solid in $\\mathbb{R}^{2019}$ has the property that all of its angles are right angles. Let its edges have length $a_1, a_2, \\dots, a_{2019}$, where each length is repeated the same number of times. Given that the maximum distance between any two vertices of the solid is $1$, what is the maximum value of the pe... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36d8293d4a475513a1fce8b4ffcf49f92474e84f92815d9a65661fef92c491e6 | A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum -shaped solid $F,$ in such a way that ... | [
"512"
] | [
"A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum -shaped solid $F,$ in such... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 8
} | 56 | 0 | 0 |
omr-ws-f7d79605bd23a5ae0503a5ee028006b10da99ad4abf136bdda37b1c536e0f44c | A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that t... | [
"512"
] | [
"A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way t... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b24899a6b011a6c55aed8b1e7fe907dd81a05b00f967a170b2f88d031b5c80c | A solid is constructed out of an infinite number of cones with height $h$. The bottom cone has base diameter $h$. Each successive cone has as its base the circular cross-section halfway up the previous cone. Find the volume of the solid. | [
"\\dfrac{\\pi h^3}{9}"
] | [
"A solid is constructed out of an infinite number of cones with height $h$. The bottom cone has base diameter $h$. Each successive cone has as its base the circular cross-section halfway up the previous cone. Find the volume of the solid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-080b408a81fd6674336bd9f010799dfc178a7bfd3eeda9b0d6030f9e57ecaaaf | A solid is formed by adjoining a hemisphere to each end of a right circular cylinder. The total volume of the figure is 12 cubic inches. Find the radius of the cylinder that produces the minimum surface area. | [
"\\sqrt[3]{\\dfrac{9}{\\pi}}"
] | [
"A solid is formed by adjoining a hemisphere to each end of a right circular cylinder. The total volume of the figure is 12 cubic inches. Find the radius of the cylinder that produces the minimum surface area."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-49fb8806601a1c98935f75609ffc600a6416a685e88509dfa7bf2303957005ed | A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 20 cubic cm. Find the radius of the cylinder that produces the minimum surface area. | [
"\\sqrt[3]{\\dfrac{15}{\\pi}}"
] | [
"A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 20 cubic cm. Find the radius of the cylinder that produces the minimum surface area."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ec12728eec59600e648d70f4fdf90e8164e1221d2474d3c9cc012c68455d183 | A solid is formed by joining semicircular cross sections (whose bases lie in the $xy$-plane and are perpendicular to the $x$-axis) across the ellipse given by the equation $9x^2 + 4y^2 = 36$. What is the volume of this solid? | [
"12\\pi"
] | [
"A solid is formed by joining semicircular cross sections (whose bases lie in the $xy$-plane and are perpendicular to the $x$-axis) across the ellipse given by the equation $9x^2 + 4y^2 = 36$. What is the volume of this solid?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7dbfdf6bc00e02d6250a4a0e0897d82928be72bafb3b282044523f65a6799042 | A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Calculate the maximum value of $\frac{a}{b}$ for which this solid can rest in stable equilibrium on its vertex. | [
"\\dfrac{2\\sqrt{10}}{5}"
] | [
"A solid is formed by rotating the first quadrant of the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Calculate the maximum value of $\\frac{a}{b}$ for which this solid can rest in stable equilibrium on its vertex."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6e0d9f8a334e0721b7ed251741d9e15988f452433a8327109c17d499a1bb2caa | A solid is generated by revolving the region bounded by \( y = \frac{1}{2}x^2 \) and \( y = 2 \) about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one quarter of the volume is removed. Find the diameter of the hole. | [
"\\sqrt{2}"
] | [
"A solid is generated by revolving the region bounded by \\( y = \\frac{1}{2}x^2 \\) and \\( y = 2 \\) about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one quarter of the volume is removed. Find the diameter of the hole."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-2a35bd858f296a1b5befe9f3495e27c2baa06a8ae966c1038217263aa80fe823 | A solid is generated by revolving the triangle in the $XY$-plane with vertices $(4,0)$, $(4,2)$, and $(6,0)$ around the $Y$-axis. What is the volume of the resulting solid? | [
"$\\frac{56\\pi}{3}$"
] | [
"A solid is generated by revolving the triangle in the $XY$-plane with vertices $(4,0)$, $(4,2)$, and $(6,0)$ around the $Y$-axis. What is the volume of the resulting solid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-c43ed8143dab81fef7db18a5e8fd09f2b82b61fe629a9879fd9a4a68d68d552a | A solid pyramid $TABCD$ with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, how many ways are there to colour the pyramid? | [
"420"
] | [
"A solid pyramid $TABCD$ with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, how many ways are there to colour the pyramid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-fbd223b6653bb3c514d093f8171adb1ae877aded15c1520e4b77838fd080eaea | A solid rectangular block is formed by gluing together $N$ congruent $1$-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the $1$-cm cubes cannot be seen. Find the smallest possible value of $N$. | [
"384"
] | [
"A solid rectangular block is formed by gluing together $N$ congruent $1$-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the $1$-cm cubes cannot be seen. Find the smallest possible value of $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-68ce1646f49a73635ab9b18709515a9c2ed26dc2520bcd6a8ece5048502ee0a1 | A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$ | [
"384"
] | [
"A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-870c2257ea6d69af3ca1906e368f9ae2e34f0299725e63bbcce54b5b61d6e727 | A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$. | [
"384"
] | [
"A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d4814009e99d5f19791fed624ef4ed0b593f119c5875b82cd3ab9cc7927610f6 | A solid rectangular block is formed by gluing together \( N \) congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes can be seen. Find the smallest possible value of \( N \). | [
"231"
] | [
"A solid rectangular block is formed by gluing together \\( N \\) congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes can be seen. Find the smallest possible value of \\( N \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 22,
"genselect": 8
} | 30 | 0 | 0 |
omr-ws-5510dcc3287d19e9552a7a93aa46de9be2707b00ec9826bd99764c777c5329dc | A solid rectangular block is formed by gluing together \(N\) congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of \(N\). | [
"384"
] | [
"A solid rectangular block is formed by gluing together \\(N\\) congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of \\(N\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-a9530e9e1ffab4228cd56a4e0d69ce50b3dbfc6e00c41b9458f742faf664dbf8 | A solid right circular cylinder with height \( h \) and base-radius \( r \) has a solid hemisphere of radius \( r \) resting upon it. The center of the hemisphere \( O \) is on the axis of the cylinder. Let \( P \) be any point on the surface of the hemisphere and \( Q \) the point on the base circle of the cylinder th... | [
"no_valid_answer_found"
] | [
"A solid right circular cylinder with height \\( h \\) and base-radius \\( r \\) has a solid hemisphere of radius \\( r \\) resting upon it. The center of the hemisphere \\( O \\) is on the axis of the cylinder. Let \\( P \\) be any point on the surface of the hemisphere and \\( Q \\) the point on the base circle o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f7fe24f6bd5ae16d95a65e182bdcb4182ce82323417aa3990d16e53af6e4689 | A solid sculpture consists of a $4 \times 4 \times 4$ cube with a $3 \times 3 \times 3$ cube sticking out. Three vertices of the smaller cube lie on edges of the larger cube, the same distance along each. Calculate the value of $n$ if the total volume of the sculpture is expressed as $n^2 + 1$. | [
"9"
] | [
"A solid sculpture consists of a $4 \\times 4 \\times 4$ cube with a $3 \\times 3 \\times 3$ cube sticking out. Three vertices of the smaller cube lie on edges of the larger cube, the same distance along each. Calculate the value of $n$ if the total volume of the sculpture is expressed as $n^2 + 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f1b5a7540c93510fb401a300f6691664f3bc346abe5df5b6c924bc5a2b66b66c | A solid sphere of radius \( r \) has a cylindrical hole drilled through its center, leaving a hollow cylinder core of height \( h \). Find the volume of the remaining solid in terms of \( h \). | [
"\\dfrac{\\pi h^3}{6}"
] | [
"A solid sphere of radius \\( r \\) has a cylindrical hole drilled through its center, leaving a hollow cylinder core of height \\( h \\). Find the volume of the remaining solid in terms of \\( h \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-12927e57d35d1d2513c85872e0ce2903b284ac39395ab82c7d3c4eb4a4deca07 | A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface fa... | [
"\\(\\frac{2\\sqrt{3}}{3}\\)",
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surfa... | [
"aops_c5_contests_amp_programs"
] | [
"genselect",
"tir"
] | {
"genselect": 10,
"tir": 10
} | 20 | 0 | 0 |
omr-ws-9a6c1d40c7e2247ae8806acb4bd323866c0c7d2fdd4aac01d825ad5981f73ac4 | A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face dow... | [
"\\(\\sqrt{\\frac{2}{3}}\\)",
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface fac... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 12,
"tir": 1
} | 29 | 0 | 0 |
omr-ws-ca4842d3ba13bf770e8ab6305a5c009d438a994db89adab81e045d0d78b1f88a | A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of... | [
"394"
] | [
"A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, n... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-9838cdaee96c80cac0d3cd058e1b2820b8e984097453f71600ac7699d1226615 | A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of... | [
"394"
] | [
"A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no t... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 7,
"tir": 9
} | 41 | 0 | 0 |
omr-ws-49a7dddc51e572cb23a74706085b72b62ad0cca98980c1617ddb7f1ce8c36179 | A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of... | [
"\\dfrac{9}{385}"
] | [
"A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-440ca7640cf0dda0f044bc6883ffcaec5f6097686f58be17fce72cb9c8cb4f64 | A solitaire game is played with 8 red, 9 green, and 10 blue cards. Totoro plays each card exactly once in some order, one at a time. When he plays a card of color \( c \), he gains a number of points equal to the number of cards that are not of color \( c \) in his hand. Find the maximum number of points that he can ob... | [
"242"
] | [
"A solitaire game is played with 8 red, 9 green, and 10 blue cards. Totoro plays each card exactly once in some order, one at a time. When he plays a card of color \\( c \\), he gains a number of points equal to the number of cards that are not of color \\( c \\) in his hand. Find the maximum number of points that ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-45b1bc5904929addfd18dca3c10cdeb9e98f2085626616d95c6899e57152b5f7 | A solitaire game is played with six distinct pairs of matched tiles placed in a bag. The player draws tiles one at a time and retains them unless they form a match, which are then put aside. The game ends if the player ever holds three tiles, no two of which match; otherwise, the drawing continues until the bag is empt... | [
"394"
] | [
"A solitaire game is played with six distinct pairs of matched tiles placed in a bag. The player draws tiles one at a time and retains them unless they form a match, which are then put aside. The game ends if the player ever holds three tiles, no two of which match; otherwise, the drawing continues until the bag is... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-b3c7b481f724ddd1006d936712dd9f07f783b45336312b27fc4fc09b772ffed9 | A solved $3 \times 3$ Rubik’s cube is scrambled randomly. If the expected number of white squares that are on the white side of the cube after scrambling can be expressed as $\frac{a}{b}$, then find $a + b$. | [
"5"
] | [
"A solved $3 \\times 3$ Rubik’s cube is scrambled randomly. If the expected number of white squares that are on the white side of the cube after scrambling can be expressed as $\\frac{a}{b}$, then find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58c1df4de22e8317842a5ef1027087a037c18376df848a0ef2f49fd2721c4025 | A sorcerer is concocting a healing potion. In order to do so, he must have one of three roots, and one of five minerals. However, one of the minerals is incompatible with two of the roots. Otherwise, he does not foresee any complications with other combinations of roots and minerals. In how many ways can he brew his po... | [
"13"
] | [
"A sorcerer is concocting a healing potion. In order to do so, he must have one of three roots, and one of five minerals. However, one of the minerals is incompatible with two of the roots. Otherwise, he does not foresee any complications with other combinations of roots and minerals. In how many ways can he brew h... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72995b013eb69d4bd1d092ab2147656990708cd9dc43bf8fc504057ebe8dc5aa | A sorting method with time complexity \(O(n \log n)\) spends exactly 1 millisecond to sort 1,000 data items. Assuming that the time \(T(n)\) of sorting \(n\) items is directly proportional to \(n \log n\), derive a formula for \(T(n)\) given the time \(T(N)\) for sorting \(N\) items, and estimate how long this method w... | [
"2000"
] | [
"A sorting method with time complexity \\(O(n \\log n)\\) spends exactly 1 millisecond to sort 1,000 data items. Assuming that the time \\(T(n)\\) of sorting \\(n\\) items is directly proportional to \\(n \\log n\\), derive a formula for \\(T(n)\\) given the time \\(T(N)\\) for sorting \\(N\\) items, and estimate h... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-52a8b05f3a9b219d0ae9697969cf9a61b2042fad3f17c4a16c85f09fcb4df172 | A space city consists of 99 space stations, with every two stations connected by a one-way space highway, except for 99 highways which are two-way. A group is defined as a set of four stations such that we can travel from any one to any other of the four along the highways. Determine the maximum number of such groups i... | [
"1001"
] | [
"A space city consists of 99 space stations, with every two stations connected by a one-way space highway, except for 99 highways which are two-way. A group is defined as a set of four stations such that we can travel from any one to any other of the four along the highways. Determine the maximum number of such gro... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-cbaa7c4ae71bf45c36a3e1ddba8a59424a10121db3c0085819078f4b652c245c | A space shuttle is spotted flying at an angle of elevation of 17° when sighted from Cape Canaveral. At the moment it is sighted, it is directly overhead a ship located 12 miles from the Cape. What is the height of the shuttle at this moment? | [
"\\( 12 \\tan(17^\\circ) \\)"
] | [
"A space shuttle is spotted flying at an angle of elevation of 17° when sighted from Cape Canaveral. At the moment it is sighted, it is directly overhead a ship located 12 miles from the Cape. What is the height of the shuttle at this moment?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-73247ea346fae0f26f53abd65bf591a0f409ae58662142d1c231e0f1db39f97e | A spacecraft orbits a perfectly spherical planet with a circumference of 40,000 km, photographing the surface. The spacecraft must return to its starting point after completing its mission. Determine the minimum total distance, in kilometers, the spacecraft must travel to ensure the entire surface is photographed, assu... | [
"40000"
] | [
"A spacecraft orbits a perfectly spherical planet with a circumference of 40,000 km, photographing the surface. The spacecraft must return to its starting point after completing its mission. Determine the minimum total distance, in kilometers, the spacecraft must travel to ensure the entire surface is photographed,... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-a2ea51b7d5128b280cb44c44bfb87c958ea410968270727d87fa4be3fee91f58 | A spaceship is at $ (0,0,0)$. Every time it teleports, it moves to a lattice point one unit away. Let $ m$ be the number of paths the spaceship can take so that it takes 12 teleportations to reach $ (4,4,4)$, while avoiding the black holes at $ (1,1,1)$ and $ (3,3,3)$. Compute the sum of the digits of $ m$. | [
"18"
] | [
"A spaceship is at $ (0,0,0)$. Every time it teleports, it moves to a lattice point one unit away. Let $ m$ be the number of paths the spaceship can take so that it takes 12 teleportations to reach $ (4,4,4)$, while avoiding the black holes at $ (1,1,1)$ and $ (3,3,3)$. Compute the sum of the digits of $ m$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1f7cc2ce0c6be1e548ef8d2c38b76a81fc487980f6c5f550d22a1ba5bafd2db8 | A spaceship is at \((0,0,0)\). Every time it teleports, it moves to a lattice point one unit away. Let \( m \) be the number of paths the spaceship can take so that it takes 12 teleportations to reach \((4,4,4)\), while avoiding the black holes at \((1,1,1)\), \((2,2,2)\), and \((3,3,3)\). Compute the sum of the digits... | [
"18"
] | [
"A spaceship is at \\((0,0,0)\\). Every time it teleports, it moves to a lattice point one unit away. Let \\( m \\) be the number of paths the spaceship can take so that it takes 12 teleportations to reach \\((4,4,4)\\), while avoiding the black holes at \\((1,1,1)\\), \\((2,2,2)\\), and \\((3,3,3)\\). Compute the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb2699191dbf9e2fd3d3aad06415d4e3015c375e3235d73d7e3051568d3ff79f | A spaceship is at the point (0, 0, 0). Every time it teleports, it moves to a lattice point one unit away. Calculate the number of paths the spaceship can take to reach (4, 4, 4) in exactly 12 teleportations while avoiding the black holes at (1, 1, 1), (2, 2, 2), and (3, 3, 3). Compute the sum of the digits of this num... | [
"15"
] | [
"A spaceship is at the point (0, 0, 0). Every time it teleports, it moves to a lattice point one unit away. Calculate the number of paths the spaceship can take to reach (4, 4, 4) in exactly 12 teleportations while avoiding the black holes at (1, 1, 1), (2, 2, 2), and (3, 3, 3). Compute the sum of the digits of thi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9abf5feeed21b67fe1c91951ae748f6ea04c31373af2a478a8d4ffeb0874c68 | A spaceship is navigating along the path \( r(t) = (t^2 - 2)i + (t - 2)j + (t - t^2)k \) for \( t > 0 \). An astronaut leaves the ship at \( t = 2 \) and is not tethered. Assuming no other forces are applied to the astronaut after he is separated from the ship, how far is he from the ship at \( t = 4 \)? | [
"\\(4\\sqrt{2}\\)"
] | [
"A spaceship is navigating along the path \\( r(t) = (t^2 - 2)i + (t - 2)j + (t - t^2)k \\) for \\( t > 0 \\). An astronaut leaves the ship at \\( t = 2 \\) and is not tethered. Assuming no other forces are applied to the astronaut after he is separated from the ship, how far is he from the ship at \\( t = 4 \\)?"
... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4e06206b866a0ad627952f0684bfdf106cb9c807fbeb309b20e13444e3422694 | A spammer needs to place 1,000,000 smiley faces ( :D ) under a "click to reveal hidden content" thingie. The spammer can either press the Mr. Green button to add one smiley face or use copy and paste operations. Each press of Mr. Green, each copy, and each paste is considered a move. What is the quickest way to achieve... | [
"38 clicks"
] | [
"A spammer needs to place 1,000,000 smiley faces ( :D ) under a \"click to reveal hidden content\" thingie. The spammer can either press the Mr. Green button to add one smiley face or use copy and paste operations. Each press of Mr. Green, each copy, and each paste is considered a move. What is the quickest way to ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e392e27cef90221ff5b67d8bee10ae1eb1db7efb76b55ec033ee92a1498aecfd | A sparrow, flying horizontally in a straight line, is 50 feet directly below an eagle and 100 feet directly above a hawk. Both the hawk and the eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. At what rate does the eagle fly? | [
"\\( v \\)"
] | [
"A sparrow, flying horizontally in a straight line, is 50 feet directly below an eagle and 100 feet directly above a hawk. Both the hawk and the eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. At what rate does the eagle fly?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58c50d2e430f5b04a02c053f65950653d19b797e0e5edea79a0e003e5cacc42f | A sparrow, flying horizontally in a straight line, is 50 feet directly below an eagle and 100 feet directly above a hawk. Both the hawk and the eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far does each bird fly? | [
"- Sparrow: 50 feet\n- Hawk: 100 feet\n- Eagle: 50 feet"
] | [
"A sparrow, flying horizontally in a straight line, is 50 feet directly below an eagle and 100 feet directly above a hawk. Both the hawk and the eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far does each bird fly?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef73d159c3d9f60ac00cf2f02606bbadf256bca22d494e0b39e734aeaa4cfd66 | A speaks the truth 2 out of 3 times, and B speaks the truth 4 out of 5 times. They both assert that a red ball has been drawn from a bag containing 6 balls of different colors. Find the probability that their statement is true. | [
"\\(\\frac{40}{41}\\)"
] | [
"A speaks the truth 2 out of 3 times, and B speaks the truth 4 out of 5 times. They both assert that a red ball has been drawn from a bag containing 6 balls of different colors. Find the probability that their statement is true."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-a04dc9270e6eba931fb16fe5c91632c48702b8bf7484e6f1d57bce600b592dcc | A special calculator starts with the number 1 and has two buttons: @ and #. The @ button multiplies the current number by 5 and then adds 2, while the # button multiplies the current number by 7 and then subtracts 5. How many different three-digit numbers can be displayed by pressing these buttons any number of times? | [
"13"
] | [
"A special calculator starts with the number 1 and has two buttons: @ and #. The @ button multiplies the current number by 5 and then adds 2, while the # button multiplies the current number by 7 and then subtracts 5. How many different three-digit numbers can be displayed by pressing these buttons any number of ti... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 12
} | 35 | 0 | 0 |
omr-ws-d2ac83d9c7bb12d29af399fc760ee3030c89680819bf12b58d5f71f8d0dce374 | A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one c... | [
"013"
] | [
"A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least ... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 7,
"tir": 8
} | 52 | 0 | 0 |
omr-ws-717944c22b6df3db2747dcd2606e8e5f110b0d558e0e28434f1abc30af7b7a87 | A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors, with each number-color combination appearing on exactly one card. John selects a set of eight cards from the deck at random. Given that he gets at least one card of each color and at least one card wi... | [
"13"
] | [
"A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors, with each number-color combination appearing on exactly one card. John selects a set of eight cards from the deck at random. Given that he gets at least one card of each color and at least one ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-dba534252aa7eae1fa3557873171e310f5c8fc3e9d2ea91f3f9ec422407676d3 | A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card wi... | [
"013"
] | [
"A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one ca... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 50,
"genselect": 16,
"tir": 15
} | 81 | 0 | 0 |
omr-ws-a83a2e84c6a097736f89d42ad52f2493d3fa077e1dc4d85c1053fa879f5aacfd | A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card wi... | [
"13"
] | [
"A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one ca... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 8,
"tir": 7
} | 51 | 0 | 0 |
omr-ws-c70cb7bbbba3d69d2cff4a737df8b2753e4af4523059958ba1569af381ba9f3c | A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to place these tiles in an $n \times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and no two tiles overlap. Let $A(n)$ be the minimum ... | [
"10"
] | [
"A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to place these tiles in an $n \\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and no two tiles overlap. Let $A(n)$ be the mi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-530ede1dfeb3ca8807ac48999aa04b7bb8cf11f8baa2952afb84a9443b91f080 | A special type of door lock has a panel with five buttons labeled with the digits 1 through 5. This lock is opened by a sequence of three actions. Each action consists of either pressing one of the buttons or pressing a pair of them simultaneously. How many combinations are possible if no digit is repeated in the combi... | [
"180"
] | [
"A special type of door lock has a panel with five buttons labeled with the digits 1 through 5. This lock is opened by a sequence of three actions. Each action consists of either pressing one of the buttons or pressing a pair of them simultaneously. How many combinations are possible if no digit is repeated in the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0085d7fdb3aeace9a6e2ca5cda55fdf378ea699c144665d9385ac9598a442616 | A special window has the shape of a rectangle surmounted by a semicircle, with the diameter of the semicircle equal to the width of the rectangle, labeled $x$. If the perimeter of the window is 16 feet, find the exact value of $x$ (in ft) so that the greatest possible amount of light is admitted. | [
"\\( \\frac{32}{8 + 3\\pi} \\)"
] | [
"A special window has the shape of a rectangle surmounted by a semicircle, with the diameter of the semicircle equal to the width of the rectangle, labeled $x$. If the perimeter of the window is 16 feet, find the exact value of $x$ (in ft) so that the greatest possible amount of light is admitted."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22b3bd7f75a396eac7eef6f270db3bf6e9ada6cf19353973cb48f61e9ef327c8 | A speeder going 75 mph passes a state trooper parked by the side of the thruway. The trooper gives chase; within 1.5 minutes he has reached a speed of 90 mph and has gone 0.15 miles. If he continues at this speed, how long does it take him to overtake the speeder? | [
"6.9"
] | [
"A speeder going 75 mph passes a state trooper parked by the side of the thruway. The trooper gives chase; within 1.5 minutes he has reached a speed of 90 mph and has gone 0.15 miles. If he continues at this speed, how long does it take him to overtake the speeder?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 10
} | 11 | 0 | 0 |
omr-ws-ece071b2e547d00c40881e40025e56f13fc86245c63fffae484b3609817584b6 | A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Let $r$ be the radius of the smallest sphere that is tangent to all the edges of $P$. If the length of each edge of $P$ is $3$, calculate the value of $r^2$. | [
"\\dfrac{9}{2}"
] | [
"A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Let $r$ be the radius of the smallest sphere that is tangent to all the edges of $P$. If the length of each edge of $P$ is $3$, calculate the value of $r^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-7837cea15b7f7e7ab048019ebc4c1b5397e8ea1869be311b4cc5d274b9823cd8 | A sphere $\mathcal{G}$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD,$ and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $\mathcal{G}$ touches planes $ABE, ACE,$ and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and le... | [
"1"
] | [
"A sphere $\\mathcal{G}$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD,$ and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $\\mathcal{G}$ touches planes $ABE, ACE,$ and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-bfe65bccca52d4d374656b6d3ab3eae0f89be4c2267aacb99cacd5e9539ea18a | A sphere \( S \) is tangent to the edges \( AB, BC, CD, DA \) of a tetrahedron \( ABCD \) at the points \( E, F, G, H \) respectively. If \( EFGH \) is a square, determine the ratio of the lengths \( AC \) to \( BD \) given that the sphere is tangent to both \( AC \) and \( BD \). | [
"\\sqrt{2}"
] | [
"A sphere \\( S \\) is tangent to the edges \\( AB, BC, CD, DA \\) of a tetrahedron \\( ABCD \\) at the points \\( E, F, G, H \\) respectively. If \\( EFGH \\) is a square, determine the ratio of the lengths \\( AC \\) to \\( BD \\) given that the sphere is tangent to both \\( AC \\) and \\( BD \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-93fa5bf652180a7d8125e58deea4e7e531ffa4d4b5574b1520d23569025061fc | A sphere \( S \) of radius \( r \) is positioned inside a right triangular prism \( P \) such that every face of the prism is tangent to the sphere. If the sum of the lengths of every edge of the triangular prism is equal to \( 22r \), what is the value of the ratio \( \frac{\text{Volume}(S)}{\text{Volume}(P)} \)? | [
"\\(\\frac{\\pi}{6}\\)"
] | [
"A sphere \\( S \\) of radius \\( r \\) is positioned inside a right triangular prism \\( P \\) such that every face of the prism is tangent to the sphere. If the sum of the lengths of every edge of the triangular prism is equal to \\( 22r \\), what is the value of the ratio \\( \\frac{\\text{Volume}(S)}{\\text{Vol... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-82ef424cf0a162dfc2d05385b0919b12fec52b446dfb45b383d1046dafdc0c30 | A sphere \( S \) with center \( O \) and radius \( R \) is given. Let \( P \) be a fixed point on this sphere. Points \( A, B, C \) move on the sphere \( S \) such that \( \angle APB = \angle BPC = \angle CPA = 90^\circ \). Let \( Q \) be the point where the plane of triangle \( ABC \) intersects the line segment \( OP... | [
"\\dfrac{R}{3}"
] | [
"A sphere \\( S \\) with center \\( O \\) and radius \\( R \\) is given. Let \\( P \\) be a fixed point on this sphere. Points \\( A, B, C \\) move on the sphere \\( S \\) such that \\( \\angle APB = \\angle BPC = \\angle CPA = 90^\\circ \\). Let \\( Q \\) be the point where the plane of triangle \\( ABC \\) inters... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d8de16db529a4d212c2a62dbb6ed2ceb1af5f950bdef9b63fee6e31f9ccbf98e | A sphere \( S_1 \) of radius \( R > 0 \) is inscribed in a right circular cone \( K \). Another sphere \( S_2 \) of radius \( r \), with \( 0 < r < R \), is contained within \( K \) and is simultaneously tangent to the sphere \( S_1 \) and the lateral surface of \( K \). Determine the volume of the cone \( K \). | [
"\\(\\frac{2\\pi R^5}{3r(R-r)}\\)"
] | [
"A sphere \\( S_1 \\) of radius \\( R > 0 \\) is inscribed in a right circular cone \\( K \\). Another sphere \\( S_2 \\) of radius \\( r \\), with \\( 0 < r < R \\), is contained within \\( K \\) and is simultaneously tangent to the sphere \\( S_1 \\) and the lateral surface of \\( K \\). Determine the volume of t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-b1dae00c274365bb436723affa6b59c8c993282dab4a08c2763741d110e5886d | A sphere and a right equilateral cone have the same surface area. What is the ratio between the volume of the sphere and the cone? | [
"\\(\\sqrt{2}\\)"
] | [
"A sphere and a right equilateral cone have the same surface area. What is the ratio between the volume of the sphere and the cone?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95e7a01317f187976f5b91cd62fa31d97eac11137e359a4d3f3e8204921baf24 | A sphere has a radius of 30. It is filled with liquid until the liquid reaches a height of 45. What is the volume of the liquid? | [
"\\(825\\pi\\)"
] | [
"A sphere has a radius of 30. It is filled with liquid until the liquid reaches a height of 45. What is the volume of the liquid?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fd168ec646baecb34afdd1ae82e952e82fda4bf303a4326f0ba4ae4b94b10a1 | A sphere has a surface area of \(676\pi\). What is its volume? Express your answer in terms of \(\pi\). | [
"\\(\\frac{8788 \\pi}{3}\\)"
] | [
"A sphere has a surface area of \\(676\\pi\\). What is its volume? Express your answer in terms of \\(\\pi\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-b863db3a70c8bcec1801f1d363f1b0f42f57ba294199e9cfec5e5c8cf7ca0124 | A sphere has an inside diameter of 4 feet and a uniform ice coating of thickness \(\frac{1}{10}\) inch. What will be the volume of water, to the nearest cubic inch, that results from melting all of this ice, given that water expands about 9% when it turns into ice? | [
"661"
] | [
"A sphere has an inside diameter of 4 feet and a uniform ice coating of thickness \\(\\frac{1}{10}\\) inch. What will be the volume of water, to the nearest cubic inch, that results from melting all of this ice, given that water expands about 9% when it turns into ice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-809b9979514110d684260de2c22b454b6b7ee5db063714de7671983395f67315 | A sphere has been partitioned into spherical triangles and quadrilaterals by drawing some great circles. No three of the great circles pass through the same point, and at least one quadrilateral was created. Let $T$ be the number of spherical triangles and $Q$ be the number of spherical quadrilaterals. Find the value o... | [
"20"
] | [
"A sphere has been partitioned into spherical triangles and quadrilaterals by drawing some great circles. No three of the great circles pass through the same point, and at least one quadrilateral was created. Let $T$ be the number of spherical triangles and $Q$ be the number of spherical quadrilaterals. Find the va... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-96036f784d1a08044db51e59a2cdd75be75482a3e727b6f120759e81c63d6cc7 | A sphere has radius \(3r\). Two circles of radii \(2r\) with centers \(A\) and \(B\) are drawn onto the surface of the sphere. The area of overlap is a tenth of the total surface area of the sphere. Find the length of the straight line \(AB\) in terms of \(r\). | [
"\\(2\\sqrt{2}r\\)"
] | [
"A sphere has radius \\(3r\\). Two circles of radii \\(2r\\) with centers \\(A\\) and \\(B\\) are drawn onto the surface of the sphere. The area of overlap is a tenth of the total surface area of the sphere. Find the length of the straight line \\(AB\\) in terms of \\(r\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a525cf0bab9fe99a4eee2e6c349a67c7fb876d8aa2ac3764b06fd194adb9c008 | A sphere intersects the $xy$-plane in a circle centered at $(2,4,0)$ with radius 1. The sphere also intersects the $yz$-plane in a circle centered at $(0,4,-7),$ with radius $r.$ Find $r.$ | [
"\\sqrt{46}"
] | [
"A sphere intersects the $xy$-plane in a circle centered at $(2,4,0)$ with radius 1. The sphere also intersects the $yz$-plane in a circle centered at $(0,4,-7),$ with radius $r.$ Find $r.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d52e7ffbf17489d63cf1b59c748c209eb8d5ea076495b51b68ae482750f69be | A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0)$, $(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^2$. | [
"51"
] | [
"A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0)$, $(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^2$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-99528cf500d37cb76da11bc655226ba93a9b17e0105abc4739a3c20dbc0c0fbe | A sphere is circumscribed about a cone with a base radius of 15 and a height of 25. What is the surface area of the sphere? | [
"\\( 1156\\pi \\)"
] | [
"A sphere is circumscribed about a cone with a base radius of 15 and a height of 25. What is the surface area of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-86bcc38fd6eb61316747107f0afaf34395c6221ea86c4fe3bd9188b7189082a8 | A sphere is circumscribed about a cube with edge length \(5\sqrt{2}\). What is the volume of the sphere? | [
"125\\sqrt{6}\\pi"
] | [
"A sphere is circumscribed about a cube with edge length \\(5\\sqrt{2}\\). What is the volume of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5307a3f3c422f268ac916e93c07599b29149c5306a604dd11aece95266c2dab7 | A sphere is contained within a cone, tangent to the center of the base and all the curved sides of the cone. The volume of the cone is three times the volume of the sphere, and the lateral side of the cone is 10. What is the surface area of the sphere? | [
"36\\pi"
] | [
"A sphere is contained within a cone, tangent to the center of the base and all the curved sides of the cone. The volume of the cone is three times the volume of the sphere, and the lateral side of the cone is 10. What is the surface area of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
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