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-96ef6ac6c5473f1e1a64262047dc72474954faa83fa3807ca198842f5e1f4b0b | A pile of \( n \) pebbles is placed in a vertical column. This configuration is modified according to the following rules: a pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right (if there are no pebbles to the right, consider this as a col... | [
"64"
] | [
"A pile of \\( n \\) pebbles is placed in a vertical column. This configuration is modified according to the following rules: a pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right (if there are no pebbles to the right, consider this a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-950ec9d63098dd60b9434415073cfc3955b58944f9f0c55952a74137e3530543 | A pile of cards, numbered from $1$ to $n$, is shuffled. The following operation is repeatedly performed: If the uppermost card has the number $k$, then the order of the $k$ uppermost cards is reversed. Let $f(n)$ be the maximum number of operations required for the card with the number $1$ to become the uppermost card ... | [
"88"
] | [
"A pile of cards, numbered from $1$ to $n$, is shuffled. The following operation is repeatedly performed: If the uppermost card has the number $k$, then the order of the $k$ uppermost cards is reversed. Let $f(n)$ be the maximum number of operations required for the card with the number $1$ to become the uppermost ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-e5dda8c74cc5c59cd1060206c40acb9f4379d340ebd73df4e29ed44b45c567c6 | A pile of gold and silver coins contains more silver coins than gold, but the total weight of each type is the same. When enough silver coins are removed to make the number of silver and gold coins equal, the removed silver coins weigh one-third of the weight of the remaining gold and silver coins. What is the ratio of... | [
"1 to 2"
] | [
"A pile of gold and silver coins contains more silver coins than gold, but the total weight of each type is the same. When enough silver coins are removed to make the number of silver and gold coins equal, the removed silver coins weigh one-third of the weight of the remaining gold and silver coins. What is the rat... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-a2e2e16e560dc8d1a3b2470eaf528cbfdb10394663580188c578d68a17f70e83 | A pilot flew from city A to city B, a distance of 300 miles, but took a course that was 12 degrees off course. What new average speed should the pilot maintain to ensure the total trip time is 100 minutes, given the original average speed was 225 miles per hour? | [
"184"
] | [
"A pilot flew from city A to city B, a distance of 300 miles, but took a course that was 12 degrees off course. What new average speed should the pilot maintain to ensure the total trip time is 100 minutes, given the original average speed was 225 miles per hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-617a6a76cbad9f167d730292ad4a430d54754dc18ee7a1819551d412321920f6 | A pilot flies from city A to city B, a distance of 330 miles, but takes a course that is $10^{\circ}$ in error. If the aircraft maintains an average speed of 220 miles per hour and the error is discovered after 15 minutes, through what angle should the pilot turn to head toward city B? | [
"168 degrees"
] | [
"A pilot flies from city A to city B, a distance of 330 miles, but takes a course that is $10^{\\circ}$ in error. If the aircraft maintains an average speed of 220 miles per hour and the error is discovered after 15 minutes, through what angle should the pilot turn to head toward city B?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31b3d1fa081ec95f7061f6a6198c652d4b0c28ccf2fb6712d211c4db250d0a76 | A pilot flies from city A to city B, a distance of 330 miles, but takes a course that is $10^{\circ}$ in error. If the error is discovered after 15 minutes, what new average speed should the pilot maintain so that the total time of the trip is 90 minutes? | [
"184 mph"
] | [
"A pilot flies from city A to city B, a distance of 330 miles, but takes a course that is $10^{\\circ}$ in error. If the error is discovered after 15 minutes, what new average speed should the pilot maintain so that the total time of the trip is 90 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-927ae39701f0274b6001f8c6ae8d65df8d93aec4606943cecfed7acc23041d47 | A pilot has just started on the glide path for landing at an airport where the length of the runway is 9000 feet. The angles of depression from the plane to the ends of the runway are $17.5^\circ$ and $18.8^\circ$. a) Find the air distance the plane must travel until touching down on the near end of the runway. b) Find... | [
"1473"
] | [
"A pilot has just started on the glide path for landing at an airport where the length of the runway is 9000 feet. The angles of depression from the plane to the ends of the runway are $17.5^\\circ$ and $18.8^\\circ$.\na) Find the air distance the plane must travel until touching down on the near end of the runway.... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b3a97937fc58322adad16b31c41524ad8eaa21f051aa67605878d032ace8004f | A ping-pong ball is dropped from a height of 16 ft and always rebounds $\frac{1}{4}$ of the distance of the previous fall. What distance does it rebound the 6th time? | [
"\\(\\frac{0.015625}{3}\\)"
] | [
"A ping-pong ball is dropped from a height of 16 ft and always rebounds $\\frac{1}{4}$ of the distance of the previous fall. What distance does it rebound the 6th time?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ae922fafe34d74ad1726ff654fb046d42b75f68ca4ff755de037e4fdbda9e32 | A ping-pong ball is dropped from a height of 16 ft and always rebounds $\frac{1}{4}$ of the distance of the previous fall. What is the total distance the ball has traveled after the 6th rebound? | [
"\\( 26.666666666 \\)"
] | [
"A ping-pong ball is dropped from a height of 16 ft and always rebounds $\\frac{1}{4}$ of the distance of the previous fall. What is the total distance the ball has traveled after the 6th rebound?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 19
} | 21 | 0 | 0 |
omr-ws-a93b1ced073fa13e659c2cc21cb87ad81948f67e56978adad3fb7300fb783bd8 | A pioneer detachment is lined up in a rectangle. In each rank, the tallest is noted, and from these, the shortest is selected. In each row, the shortest is noted, and from these, the tallest is selected. Which of these two pioneers is taller? | [
"A"
] | [
"A pioneer detachment is lined up in a rectangle. In each rank, the tallest is noted, and from these, the shortest is selected. In each row, the shortest is noted, and from these, the tallest is selected. Which of these two pioneers is taller?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-60177ce82ab937f7de38943ddef3b936f7c52e5adb5fc0ade7b9b8680f5adc05 | A pipe follows three sides of a square, giving it a total length of 300 meters. What is the area, in square meters, of the region consisting of all points (in the 2D plane of the square) that lie within 20 meters of the pipe? | [
"\\( 600\\pi + 8000 \\)"
] | [
"A pipe follows three sides of a square, giving it a total length of 300 meters. What is the area, in square meters, of the region consisting of all points (in the 2D plane of the square) that lie within 20 meters of the pipe?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-98875ed7b392977f17723a83cc3a352d369c2bd45f8eca771d7026a74d13e41c | A pipe has a flow rate of $15$ $m^3/hour$ but loses $0.25$ $m^3$ every $15$ minutes due to leaks. How long will it take for the pipe to deliver $350$ $m^3$ of water to its destination? | [
"25 hours"
] | [
"A pipe has a flow rate of $15$ $m^3/hour$ but loses $0.25$ $m^3$ every $15$ minutes due to leaks. How long will it take for the pipe to deliver $350$ $m^3$ of water to its destination?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-cb5ad0fc5a737686512e4b343a6e885d36165e6d76b830f02e831b23f6e74e39 | A pipe is being carried down a hallway that is 8 ft wide. At the end of the hallway, there is a right-angled turn into a narrower hallway 7 ft wide. Find the length of the longest pipe that can be carried around the corner. Round your answer to two decimal places. | [
"21.20"
] | [
"A pipe is being carried down a hallway that is 8 ft wide. At the end of the hallway, there is a right-angled turn into a narrower hallway 7 ft wide. Find the length of the longest pipe that can be carried around the corner. Round your answer to two decimal places."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 24
} | 47 | 0 | 0 |
omr-ws-d9fd49a353a90d576877de0143795c855a67470e510ed52826928da73980402d | A pipe of negligible diameter is to be carried horizontally around a corner from a hallway 8 ft. wide into a hallway 4 ft. wide. What is the maximum possible length that the pipe can have? | [
"\\(\\sqrt{80+96\\sqrt[3]{2}+48\\sqrt[3]{4}}\\)"
] | [
"A pipe of negligible diameter is to be carried horizontally around a corner from a hallway 8 ft. wide into a hallway 4 ft. wide. What is the maximum possible length that the pipe can have?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4c09336a376ca724e897fcdf4c8727dbef75a942251c8b0a4593ce70e1f43a03 | A pipe that is 8 meters long and 3 cm in diameter must be transported down a corridor that is 3 meters wide and 3 meters high, which has a right-angled corner. The pipe must not be bent or twisted. Will the pipe fit around the corner? What is the longest pipe that will fit around the corner? | [
"4.16 meters"
] | [
"A pipe that is 8 meters long and 3 cm in diameter must be transported down a corridor that is 3 meters wide and 3 meters high, which has a right-angled corner. The pipe must not be bent or twisted. Will the pipe fit around the corner? What is the longest pipe that will fit around the corner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e5363ab4baeb809c3f4f104aeb9d8310b3afe885c5c3dc492058d7a6b6fcd25 | A pipe with inside diameter 10'' is to carry water from a reservoir to a small town in an arid land. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of 2''-inside-diameter pipes of the same length needed to carry the same volume of water to the arid town? | [
"25"
] | [
"A pipe with inside diameter 10'' is to carry water from a reservoir to a small town in an arid land. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of 2''-inside-diameter pipes of the same length needed to carry the same volume of water to the arid t... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6a9e6c38c4786f3059dcf7ea13bfdd275210f541cabacf72c5c740c10950d1d | A pipe-smoking mathematician carries two matchboxes, each initially containing \(N\) matches, one in his left-hand pocket and one in his right-hand pocket. Each time he needs a match, he is equally likely to take it from either pocket. What is the probability that, at the moment he first discovers one of the matchboxes... | [
"\\(\\frac{\\binom{2N-k}{N}}{2^{2N-k}}\\)"
] | [
"A pipe-smoking mathematician carries two matchboxes, each initially containing \\(N\\) matches, one in his left-hand pocket and one in his right-hand pocket. Each time he needs a match, he is equally likely to take it from either pocket. What is the probability that, at the moment he first discovers one of the mat... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-5a8aed5ce3674055575712ea2205a22af1e1c7b9166e02a865590339a4828484 | A pirate has five purses with 30 coins in each. One purse contains only gold coins, another contains only silver coins, the third contains only bronze coins, and the remaining two contain 10 gold, 10 silver, and 10 bronze coins each. What is the minimal number of coins that must be taken and examined to determine the c... | [
"5"
] | [
"A pirate has five purses with 30 coins in each. One purse contains only gold coins, another contains only silver coins, the third contains only bronze coins, and the remaining two contain 10 gold, 10 silver, and 10 bronze coins each. What is the minimal number of coins that must be taken and examined to determine ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e156e80de2df5c9e60f64bc33b3958d4331a41796555c1b476c7cd55b998239 | A pirate has five purses with 30 coins in each. One purse contains only gold coins, another only silver coins, and the third only bronze coins. The remaining two purses each contain 10 gold, 10 silver, and 10 bronze coins. The pirate can simultaneously take one or more coins from any of the purses and examine them. Wha... | [
"5"
] | [
"A pirate has five purses with 30 coins in each. One purse contains only gold coins, another only silver coins, and the third only bronze coins. The remaining two purses each contain 10 gold, 10 silver, and 10 bronze coins. The pirate can simultaneously take one or more coins from any of the purses and examine them... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f611e4c0b544220012d078a2fb8c92e1c910038a26eddf6c10a34265ba1a1a5 | A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_{5}$ dollars worth of gold coins, and $124_{5}$ dollars worth of rubbing alcohol. What is the total dollar amount that this pirate has accumulated? E... | [
"865"
] | [
"A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_{5}$ dollars worth of gold coins, and $124_{5}$ dollars worth of rubbing alcohol. What is the total dollar amount that this pirate has accumul... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-57b0878842903973ebbb55ef6113b4cb08a08602d972a343e0aa318d2ecc303f | A pirate is searching for buried treasure on 6 islands. On each island, there is a $\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\frac{1}{12}$ chance that the island has traps but no treasure, and a $\frac{2}{3}$ chance that the island has neither traps nor treasure. What is the probability ... | [
"\\frac{5}{54}"
] | [
"A pirate is searching for buried treasure on 6 islands. On each island, there is a $\\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\\frac{1}{12}$ chance that the island has traps but no treasure, and a $\\frac{2}{3}$ chance that the island has neither traps nor treasure. What is the prob... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26aa067f6c6dd929e596774e62fc1d37fbffce4a1a51990ce6a48e1d337f42b7 | A pirate ship spots an oblivious caravel 10 nautical miles to the east, sailing 60 degrees south of west at a steady 12 nm/hour. What is the minimum speed that the pirate ship must maintain to catch the caravel? | [
"12"
] | [
"A pirate ship spots an oblivious caravel 10 nautical miles to the east, sailing 60 degrees south of west at a steady 12 nm/hour. What is the minimum speed that the pirate ship must maintain to catch the caravel?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-94c1a37b42d433bb52a9d43e9d7d638c22aa66b0579b73b93264c9e04e62c72c | A pitcher has a $\frac{1}{2}$ chance of pitching a strike. If he pitches a strike, the batter will hit the ball $\frac{3}{4}$ of the time, ending the at bat. If he pitches 3 strikes, the batter strikes out, ending the at bat. Assuming there are no foul balls, what is the probability the batter gets a walk? A walk is 4 ... | [
"\\(\\frac{53}{512}\\)"
] | [
"A pitcher has a $\\frac{1}{2}$ chance of pitching a strike. If he pitches a strike, the batter will hit the ball $\\frac{3}{4}$ of the time, ending the at bat. If he pitches 3 strikes, the batter strikes out, ending the at bat. Assuming there are no foul balls, what is the probability the batter gets a walk? A wal... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-5409cbec8dd3fa60c7dfea83f6907fe6727c0fc9707bfb6836e30eaafd41614c | A pizza is divided into 7 regions of the same size. How many ways are there to flavor the pizza if there are 4 toppings to choose from, and no two adjacent slices can have the same flavor? | [
"2184"
] | [
"A pizza is divided into 7 regions of the same size. How many ways are there to flavor the pizza if there are 4 toppings to choose from, and no two adjacent slices can have the same flavor?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93334b6b9037fc69341a59efbfc813e2491bf6ada259104cd297d1a51080f7e9 | A pizza is in the shape of a regular octagon with side length 1. The pizza is to be cut into eight pieces of equal area using seven straight cuts, all parallel to one given pair of sides and no two cuts intersect. Find the length of each cut. | [
"\\( 2\\sqrt{2 + \\sqrt{2}} + 5(\\sqrt{2} + 1) \\)"
] | [
"A pizza is in the shape of a regular octagon with side length 1. The pizza is to be cut into eight pieces of equal area using seven straight cuts, all parallel to one given pair of sides and no two cuts intersect. Find the length of each cut."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d28cc3f3458018e97f938447473e7b776d595e71f447d7c5894367f8a01cc7f | A pizza parlor has six different toppings. How many different one- and two-topping pizzas can you order? | [
"21"
] | [
"A pizza parlor has six different toppings. How many different one- and two-topping pizzas can you order?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1fb8aa4e937463f3873e91f9f9593c5448b89c63f46927a0089365786064e58a | A pizza parlor offers six toppings. What is the greatest number of four-topping pizzas that can be made such that no two pizzas have the same topping combination? | [
"15"
] | [
"A pizza parlor offers six toppings. What is the greatest number of four-topping pizzas that can be made such that no two pizzas have the same topping combination?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88738721e6f6f314715f5240ca4b0e2f7be768a10a527355be21cebef8d2d712 | A pizza shop offers 30 different toppings, and a customer can choose any number of toppings. Let $M$ be the number of possible pizzas the customer can buy. What is the remainder when $M$ is divided by 1000? | [
"824"
] | [
"A pizza shop offers 30 different toppings, and a customer can choose any number of toppings. Let $M$ be the number of possible pizzas the customer can buy. What is the remainder when $M$ is divided by 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-95ada11d59c1c1f833f0968494c3efc12b1f98a41be98e7d3a044f99c1a75d89 | A pizza with a radius of 16 is cut into slices by making 4 horizontal and 4 vertical cuts. What is the area of the smallest slice? | [
"\\(\\frac{64}{3}\\pi - 64\\sqrt{3} + 64\\)"
] | [
"A pizza with a radius of 16 is cut into slices by making 4 horizontal and 4 vertical cuts. What is the area of the smallest slice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c24b3491ca7201a275936e25c4105de548e73b4f4d8fdc10b728efa62032072 | A placement exam is given to all entering students at a certain college. The scores are normally distributed with a mean of 80 and a standard deviation of 8. a) What percentage of the students scored between 74 and 86? b) What percentage of the students scored above 92? | [
"6.68\\%"
] | [
"A placement exam is given to all entering students at a certain college. The scores are normally distributed with a mean of 80 and a standard deviation of 8.\na) What percentage of the students scored between 74 and 86?\nb) What percentage of the students scored above 92?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9b0cc1f43fba246cadd4462edf25ef06267e7e6f9bd4fb693f3028a7caf3796 | A plane $(\alpha)$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. Define $p, q, r$ to be the signed distances of $P, Q, R$ from $(\alpha)$ measured along a directed normal to $(\alpha)$. If $a$ is the length of an edge of the tetrahedron, find the value of $p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p... | [
"2a^2"
] | [
"A plane $(\\alpha)$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. Define $p, q, r$ to be the signed distances of $P, Q, R$ from $(\\alpha)$ measured along a directed normal to $(\\alpha)$. If $a$ is the length of an edge of the tetrahedron, find the value of $p^2 + q^2 + r^2 + (q - r)^2 + (r - p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-c468163ad0550fe82f8ebcbffa539d8f8235c4c2aadff856b464b517a1f2a531 | A plane $\alpha$ intersects the edges $AB, BC, CD,$ and $DA$ of a triangular pyramid $ABCD$ at points $K, L, M,$ and $N$, respectively. It is given that the dihedral angles $\angle (KLA, KLM)$, $\angle (LMB, LMN)$, $\angle (MNC, MNK)$, and $\angle (NKD, NKL)$ are equal. Let $P, Q, R,$ and $S$ be the projections of the ... | [
"\\dfrac{9}{2}"
] | [
"A plane $\\alpha$ intersects the edges $AB, BC, CD,$ and $DA$ of a triangular pyramid $ABCD$ at points $K, L, M,$ and $N$, respectively. It is given that the dihedral angles $\\angle (KLA, KLM)$, $\\angle (LMB, LMN)$, $\\angle (MNC, MNK)$, and $\\angle (NKD, NKL)$ are equal. Let $P, Q, R,$ and $S$ be the projectio... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-6738569f608dd5215cdba504413123b2dadc04fda360a1b1085e9ff9fc3c9fa8 | A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. Define $p, q, r$ to be the signed distances of $P, Q, R$ from $\pi$ measured along a directed normal to $\pi$. If the length of an edge of the tetrahedron is $a$, find the value of $p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2$. | [
"2a^2"
] | [
"A plane $\\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. Define $p, q, r$ to be the signed distances of $P, Q, R$ from $\\pi$ measured along a directed normal to $\\pi$. If the length of an edge of the tetrahedron is $a$, find the value of $p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-efade79843ab8d77e3ef339e046cc0c7caa28a30dac017b1f46304a579d9008f | A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal. | [
"A"
] | [
"A plane $\\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\\rho$ for which $\\frac{AM}{AN}$ is minimal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-eed651464d15b3da4c81eb7f90d3cb4a048b61651496297d54de5e44ec84a594 | A plane 40 inches from the center of a sphere intersects the sphere in a circle with a circumference of $18\pi$ inches. Find the length of the sphere’s diameter in inches. | [
"82"
] | [
"A plane 40 inches from the center of a sphere intersects the sphere in a circle with a circumference of $18\\pi$ inches. Find the length of the sphere’s diameter in inches."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a532f30119ee90a7ddda5a5ff6424560c5fa632ef45cd91ac69b6db550c7942f | A plane \( p \) passes through a vertex of a cube so that the three edges at the vertex make equal angles with \( p \). Find the cosine of this angle. | [
"\\(\\frac{1}{\\sqrt{3}}\\)"
] | [
"A plane \\( p \\) passes through a vertex of a cube so that the three edges at the vertex make equal angles with \\( p \\). Find the cosine of this angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-79b7c113bf5ae9e4628e127ff204a8f3fb0882af94851ac4a4dd71dfd472bb57 | A plane \( p \) passes through a vertex of a cube so that the three edges at the vertex make equal angles with \( p \). Find the positions of the feet of the perpendiculars from the vertices of the cube onto \( p \). | [
"\\left(-\\frac{2}{3}, \\frac{1}{3}, \\frac{1}{3}\\right)"
] | [
"A plane \\( p \\) passes through a vertex of a cube so that the three edges at the vertex make equal angles with \\( p \\). Find the positions of the feet of the perpendiculars from the vertices of the cube onto \\( p \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-98d9f4486ae0e359a1e04f466c79d4444a47b55cdb55a6a56f92fd568c69fa91 | A plane \( p \) passes through a vertex of a cube so that the three edges at the vertex make equal angles with \( p \). There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane \( p \). | [
"\\text{Plane } p \\text{ is perpendicular to one of the 28 lines and parallel to two of the 20 planes.}"
] | [
"A plane \\( p \\) passes through a vertex of a cube so that the three edges at the vertex make equal angles with \\( p \\). There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane \\( p \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-f48987e947c16333503b93b622fc400459e5f2314bef4015ba28ba85e6c29975 | A plane \(P\) in \(\mathbb{R}^3\) is said to be a symmetry plane for a set \(S \subset \mathbb{R}^3\) if the image of \(S\) under the mirror reflection with respect to the plane \(P\) is equal to \(S\) (in other words, the mirror reflection preserves \(S\)). Assume that \(S\) is the union of three nonparallel lines. Wh... | [
"9"
] | [
"A plane \\(P\\) in \\(\\mathbb{R}^3\\) is said to be a symmetry plane for a set \\(S \\subset \\mathbb{R}^3\\) if the image of \\(S\\) under the mirror reflection with respect to the plane \\(P\\) is equal to \\(S\\) (in other words, the mirror reflection preserves \\(S\\)). Assume that \\(S\\) is the union of thr... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-f23ad1fd4d9bac39c2c4d158a718017a61769e437b075748312afe1a99d9dda4 | A plane angle is formed as the result of cutting a given dihedral angle by all possible planes intersecting its edge. Within what limits can the plane angle vary? | [
"From 0 to $\\pi $, irrespective of the size of the given dihedral angle"
] | [
"A plane angle is formed as the result of cutting a given dihedral angle by all possible planes intersecting its edge. Within what limits can the plane angle vary?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8d0a59f01b5ffac51758acdb51ee56a8587e6c231bd95007812550a0946c405 | A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Fin... | [
"607"
] | [
"A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-efb109a76e9fc57475f50af0fb53c8ccaa81ed99bb9d6700801ee24afdcb3864 | A plane contains $40$ lines, no $2$ of which are parallel. There are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, and $6$ points where exactly $6$ lines intersect. No points where more than $6$ lines intersect. Find the number ... | [
"607"
] | [
"A plane contains $40$ lines, no $2$ of which are parallel. There are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, and $6$ points where exactly $6$ lines intersect. No points where more than $6$ lines intersect. Find the nu... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-4d07e1825891a583ecbdfa27d28b68e1029d892c3f742ee300f077622d4dafed | A plane contains points $A$ and $B$ with $AB=1$. Let $R$ be the union of all disks of radius 1 in the plane that cover $\overline{AB}$. What is the area of $R$? Express your answer in terms of $\pi$ and simplest radical form. | [
"3\\pi - \\dfrac{\\sqrt{3}}{2}"
] | [
"A plane contains points $A$ and $B$ with $AB=1$. Let $R$ be the union of all disks of radius 1 in the plane that cover $\\overline{AB}$. What is the area of $R$? Express your answer in terms of $\\pi$ and simplest radical form."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-05dfa1c4dbccda2f60c83f6d8405da8835d3c28bd5847295782920891a22e2c0 | A plane contains points $A$ and $B$ with $AB=1$. Let $S$ be the union of all disks of radius 1 in the plane that cover $AB$. What is the area of $S$? | [
"\\( 3\\pi - \\frac{\\sqrt{3}}{2} \\)"
] | [
"A plane contains points $A$ and $B$ with $AB=1$. Let $S$ be the union of all disks of radius 1 in the plane that cover $AB$. What is the area of $S$?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-9c7ea3ef5ef6d528353c4d4f39039c30e8b650c8e19df4075d8a4656524be422 | A plane contains points $A$ and $B$ with the distance between them equal to $1$. Let $S$ be the union of all disks of radius $1$ in the plane that cover the segment $AB$. What is the area of $S$? | [
"3\\pi - \\dfrac{\\sqrt{3}}{2}"
] | [
"A plane contains points $A$ and $B$ with the distance between them equal to $1$. Let $S$ be the union of all disks of radius $1$ in the plane that cover the segment $AB$. What is the area of $S$?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-6ad3f9965edf0322d0c7a3d0272ff8cc9fa9189e5d91c544699b4373df387b2e | A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover AB. What is the area of S? | [
"3\\pi - \\dfrac{\\sqrt{3}}{2}"
] | [
"A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover AB. What is the area of S?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-1d00915f52b1a4853c1b1d751bb6fc2c558dcbbb38e808f25a47aeb4b47f38e1 | A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover point A. What is the area of S? | [
"\\( 3\\pi - \\frac{\\sqrt{3}}{2} \\)"
] | [
"A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover point A. What is the area of S?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ddac58e3b974a266d3428478794801a3a8e168a7f9297db89f426d9991c81ed | A plane contains points \( A \) and \( B \) with \( AB = 1 \). Let \( S \) be the union of all disks of radius \( 1 \) in the plane that cover \( \overline{AB} \). What is the area of \( S \)? | [
"\\( 3\\pi - \\frac{\\sqrt{3}}{2} \\)"
] | [
"A plane contains points \\( A \\) and \\( B \\) with \\( AB = 1 \\). Let \\( S \\) be the union of all disks of radius \\( 1 \\) in the plane that cover \\( \\overline{AB} \\). What is the area of \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-0ff18d87612648af4a9a6b14eff0b11997eb63c68a40a8c330933b558779be1a | A plane convex quadrilateral has an area of 32, and the sum of two opposite sides and a diagonal is 16. Determine all possible lengths for the other diagonal. | [
"8",
"\\( 8\\sqrt{2} \\)"
] | [
"A plane convex quadrilateral has an area of 32, and the sum of two opposite sides and a diagonal is 16. Determine all possible lengths for the other diagonal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 11,
"tir": 4
} | 33 | 0 | 0 |
omr-ws-03a6cf719a5774e43d825088842c5b88c31929614847e89aafc99155e66d9b06 | A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone. | [
"\\(\\frac{1}{3\\sqrt{3}}\\)"
] | [
"A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7fd5e59444f6e35a525719594998fe9f71f1491f7f01ef20555e0b64b4ed37ce | A plane cuts a right circular cone of volume \( V \) into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part. | [
"\\(\\frac{V}{3 \\sqrt 3}\\)"
] | [
"A plane cuts a right circular cone of volume \\( V \\) into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-4c4008b5bbe22827c2821eca0798de64fb95bc966c08d313d53d744e2c6b47e3 | A plane cuts a right circular cone with vertex \( V \) in an ellipse \( E \) and meets the axis of the cone at \( C \). \( A \) is an extremity of the major axis of \( E \). If the area of the curved surface of the slant cone with \( V \) as vertex and \( E \) as base is \( 120 \pi \) and the area of \( E \) is \( 60 \... | [
"2"
] | [
"A plane cuts a right circular cone with vertex \\( V \\) in an ellipse \\( E \\) and meets the axis of the cone at \\( C \\). \\( A \\) is an extremity of the major axis of \\( E \\). If the area of the curved surface of the slant cone with \\( V \\) as vertex and \\( E \\) as base is \\( 120 \\pi \\) and the area... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ea630a08fc8f54c8fd39de101a36d31c498ab58c9e77dcff9597e563d9179b1 | A plane cuts off from the lateral edges $SA$, $SB$ and $SC$ of a regular quadrangular pyramid $SABCD$ with vertex $S$ line segments $SK=2/3 SA$, $SL= 1/2 SB$, $SM=1/3SC$, respectively. The length of a lateral edge of the pyramid is equal to $a$. Find the length of line segment $SN$ cut off by this plane on edge $SD$. | [
"$2/5a$"
] | [
"A plane cuts off from the lateral edges $SA$, $SB$ and $SC$ of a regular quadrangular pyramid $SABCD$ with vertex $S$ line segments $SK=2/3 SA$, $SL= 1/2 SB$, $SM=1/3SC$, respectively. The length of a lateral edge of the pyramid is equal to $a$. Find the length of line segment $SN$ cut off by this plane on edge $S... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69a61b1454ad60bcb9a102df2f7d05f288a37aeeb1c7ff6f2d3475caf4fd38a7 | A plane cuts the lateral edges $SA$, $SB$ and $SC$ of the triangular pyramid $SABC$ at points $ K$, $L$ and $M$ respectively. In what ratio does the plane divide the volume of the pyramid if it is known that $$| SK | : | KA | = | SL | : | LB | = 2,$$ and the median $SN$ of the triangle $SBC$ is bisected by that plane? | [
"$\\frac{8}{37}$"
] | [
"A plane cuts the lateral edges $SA$, $SB$ and $SC$ of the triangular pyramid $SABC$ at points $ K$, $L$ and $M$ respectively. In what ratio does the plane divide the volume of the pyramid if it is known that $$| SK | : | KA | = | SL | : | LB | = 2,$$ and the median $SN$ of the triangle $SBC$ is bisected by that pl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29fc7e176df653733ca8e702a0af1a8d3fa5a0c06fdd1cd11128c869285ded6e | A plane cuts the lateral edges $SA$, $SB$ and $SC$ of the triangular pyramid $SABC$ at points $K$, $L$ and $M$ respectively. In what ratio does the plane divide the volume of the pyramid if it is known that $$| SK | : | KA | = | SL | : | LB | = 2,$$ and the median $SN$ of the triangle $SBC$ is bisected by that plane? | [
"$8:37$"
] | [
"A plane cuts the lateral edges $SA$, $SB$ and $SC$ of the triangular pyramid $SABC$ at points $K$, $L$ and $M$ respectively. In what ratio does the plane divide the volume of the pyramid if it is known that $$| SK | : | KA | = | SL | : | LB | = 2,$$ and the median $SN$ of the triangle $SBC$ is bisected by that pla... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ee3427dfab9c0d46bd6ed8d7a48d079b88b3224756d972da515136cd9368327 | A plane cuts through exactly three vertices of a cube with a side length of 2. What is the area of the plane shape that is the intersection between the plane and the cube? | [
"\\(2\\sqrt{3}\\)"
] | [
"A plane cuts through exactly three vertices of a cube with a side length of 2. What is the area of the plane shape that is the intersection between the plane and the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-84a1f0818dceb94419fc0408796dfa10a93683345d19880166936ed9043a19e7 | A plane departs from City $\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\mathbb{B}$. It then changes its course to $S 50^\circ E$ at 180 miles per hour for 5 hours to City $\mathbb{C}$. After reaching City $\mathbb{C}$, it returns to City $\mathbb{A}$.... | [
"178030.39"
] | [
"A plane departs from City $\\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\\mathbb{B}$. It then changes its course to $S 50^\\circ E$ at 180 miles per hour for 5 hours to City $\\mathbb{C}$. After reaching City $\\mathbb{C}$, it returns to City $\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-490bf78a13245c05a20db268dc581eb7a6c1372f07a10fe4f7f59b1169112ae6 | A plane departs from City $\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\mathbb{B}$. It then changes its course to $S 50^\circ E$ at 180 miles per hour for 5 hours to City $\mathbb{C}$. After reaching City $\mathbb{C}$, it returns to City $\mathbb{A}$.... | [
"$N 16.77^\\circ W$"
] | [
"A plane departs from City $\\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\\mathbb{B}$. It then changes its course to $S 50^\\circ E$ at 180 miles per hour for 5 hours to City $\\mathbb{C}$. After reaching City $\\mathbb{C}$, it returns to City $\... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-da7a0c74816cb733385481869d06ca63676cee13a4818e0cd156a4fdd85b83fe | A plane departs from City $\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\mathbb{B}$. It then changes its course to $S 50^\circ E$ at 180 miles per hour for 5 hours to City $\mathbb{C}$. After reaching City $\mathbb{C}$, it returns to City $\mathbb{A}$.... | [
"998"
] | [
"A plane departs from City $\\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\\mathbb{B}$. It then changes its course to $S 50^\\circ E$ at 180 miles per hour for 5 hours to City $\\mathbb{C}$. After reaching City $\\mathbb{C}$, it returns to City $\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 21
} | 33 | 0 | 0 |
omr-ws-3fca5af1fa03c425d0146765cd207b3b733b7d8956f65a6bc6e1fc77363f8c4b | A plane departs from City $\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\mathbb{B}$. It then changes its course to $S 50^\circ E$ at 180 miles per hour for 5 hours to City $\mathbb{C}$. After reaching City $\mathbb{C}$, it returns to City $\mathbb{A}$.... | [
"7.984"
] | [
"A plane departs from City $\\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\\mathbb{B}$. It then changes its course to $S 50^\\circ E$ at 180 miles per hour for 5 hours to City $\\mathbb{C}$. After reaching City $\\mathbb{C}$, it returns to City $\... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 13
} | 22 | 0 | 0 |
omr-ws-a1c9d2e2722b965f2f5469f385b797c7d6601acb90222fd901a4245ba25f229a | A plane departs from City $\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\mathbb{B}$. It then changes its course to $S 50^\circ E$ at 180 miles per hour for 5 hours to City $\mathbb{C}$. After reaching City $\mathbb{C}$, it returns to City $\mathbb{A}$.... | [
"11:35 PM"
] | [
"A plane departs from City $\\mathbb{A}$ at 7:00 AM with a bearing of $N 42.5^\\circ E$ and flies at 110 miles per hour for 3.6 hours to City $\\mathbb{B}$. It then changes its course to $S 50^\\circ E$ at 180 miles per hour for 5 hours to City $\\mathbb{C}$. After reaching City $\\mathbb{C}$, it returns to City $\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-429f45c4166c7def80bb9577cfac66f0ab60b62f7e7ef29c7de029f41bcb45d7 | A plane divides the medians of the faces $ABC$, $ACD$, and $ABD$ of a tetrahedron $ABCD$ coming from the vertex $A$ into ratios $1:2$, $1:1$, and $1:2$ (counting from point $A$). Find the ratio of the volumes of the parts into which this plane divides the given tetrahedron. | [
"\\dfrac{1}{15}"
] | [
"A plane divides the medians of the faces $ABC$, $ACD$, and $ABD$ of a tetrahedron $ABCD$ coming from the vertex $A$ into ratios $1:2$, $1:1$, and $1:2$ (counting from point $A$). Find the ratio of the volumes of the parts into which this plane divides the given tetrahedron."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-d4d881c6c3b5578d4ecab4a55b94810958cf2445bb352047d899e433ff3f0ea9 | A plane dividing a cube into two parts passes through two opposite vertices of the cube and the midpoints of the six edges not containing these vertices. Into each part of the cube a sphere is placed so that it is tangent to three faces of the cube and the cutting plane. Find the ratios of the volume of the cube to the... | [
"\\(\\frac{2\\left(3+\\sqrt{3}\\right)^3}{9\\pi}\\)"
] | [
"A plane dividing a cube into two parts passes through two opposite vertices of the cube and the midpoints of the six edges not containing these vertices. Into each part of the cube a sphere is placed so that it is tangent to three faces of the cube and the cutting plane. Find the ratios of the volume of the cube t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2427ed07fdd78399d0c02ff353f7103d76d95e07b91d8944269329800e97c4d2 | A plane first flew with a speed of 220 km/h. When it still had to fly 385 km less than the covered distance, its speed became equal to 330 km/h. The average speed of the plane during the flight was equal to 250 km/h. What distance was covered by the plane? | [
"1375 km"
] | [
"A plane first flew with a speed of 220 km/h. When it still had to fly 385 km less than the covered distance, its speed became equal to 330 km/h. The average speed of the plane during the flight was equal to 250 km/h. What distance was covered by the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3697c7064026ec25e3b88065efd39fa93e8519f0454293b2da3475fdba585d8b | A plane flew 720 miles in 3 hours with the wind and required 4 hours to fly the same distance against the wind. Find the rate of the plane in calm air and the rate of the wind. | [
"The rate of the plane in calm air is 210 mph and the rate of the wind is 30 mph."
] | [
"A plane flew 720 miles in 3 hours with the wind and required 4 hours to fly the same distance against the wind. Find the rate of the plane in calm air and the rate of the wind."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-9ea94f4b2a6093ff8e7369e98e8b8465e95f71ea9338ce5e5076129fa4830cd1 | A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was. | [
"63"
] | [
"A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ea47c7903f0afbefe436110ba8532deadaadf299a386f3db032cf760a9b9d76f | A plane flies 11 km above the surface of the Earth and travels westward following the line of the equator at 510 km per hour. How long will it take for the plane to return to the same place above the Earth’s surface from where it started? | [
"78.7"
] | [
"A plane flies 11 km above the surface of the Earth and travels westward following the line of the equator at 510 km per hour. How long will it take for the plane to return to the same place above the Earth’s surface from where it started?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-ba7e3a3122341c97a92d587ed758c8d93378df7f16f8276a623dc3b84174bfeb | A plane flies from city A to city B, a distance of 150 miles, and then turns through an angle of 30 degrees and heads toward city C. If the distance from city A to city C is 400 miles, how far is it from city B to city C? | [
"303.32"
] | [
"A plane flies from city A to city B, a distance of 150 miles, and then turns through an angle of 30 degrees and heads toward city C. If the distance from city A to city C is 400 miles, how far is it from city B to city C?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33f6ee9a34b7e871f88864988b697cf620cf5f6fb63fec5e15bbe40c098611e7 | A plane flies over a car at an altitude of 2,000 meters. The plane is flying North-East at 600 kilometers per hour, while the car is driving North at 100 kilometers per hour. How fast are the car and plane separating 20 minutes later? | [
"533.95 kmph"
] | [
"A plane flies over a car at an altitude of 2,000 meters. The plane is flying North-East at 600 kilometers per hour, while the car is driving North at 100 kilometers per hour. How fast are the car and plane separating 20 minutes later?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-25f521d34d34d4a3f4b3037e4ca662869f15cceaa8afc75bbf206e8f864e3e57 | A plane has been partitioned into $n$ regions by 3 sets of parallel lines such that no three lines intersect at a single point. What is the least number of lines required such that $n > 2001$? | [
"36"
] | [
"A plane has been partitioned into $n$ regions by 3 sets of parallel lines such that no three lines intersect at a single point. What is the least number of lines required such that $n > 2001$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-110d2b7b26f2c71f3e4833490ab575398eb6d7e51cee761efe4addd99b10ff29 | A plane has one hundred seats and one hundred people boarding in the order of their seat numbers from $1$ to $100$. The first person, Grandma, picks a random seat instead of her assigned seat. Each subsequent person takes their assigned seat if it is available, or a random free seat otherwise. What is the probability t... | [
"\\(\\frac{1}{2}\\)"
] | [
"A plane has one hundred seats and one hundred people boarding in the order of their seat numbers from $1$ to $100$. The first person, Grandma, picks a random seat instead of her assigned seat. Each subsequent person takes their assigned seat if it is available, or a random free seat otherwise. What is the probabil... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-1dcd56355591b704f751a4e3edd0c928aee41af296f47a3f1e9e8f9e91e20e36 | A plane has one hundred seats and one hundred people boarding in the order of their seat numbers from $1$ to $100$. The first two people, Grandma and Grandpa, each pick a random seat independently. Each subsequent person takes their assigned seat if it is available, or a random free seat otherwise. What is the probabil... | [
"\\(\\frac{1}{2}\\)"
] | [
"A plane has one hundred seats and one hundred people boarding in the order of their seat numbers from $1$ to $100$. The first two people, Grandma and Grandpa, each pick a random seat independently. Each subsequent person takes their assigned seat if it is available, or a random free seat otherwise. What is the pro... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-f3744f37281d52894102f33ba79107a564af5970168d8dcae8c2aa556b2bb951 | A plane intersects a polyhedron without passing through any of its vertices. Let $E$ be the number of edges of the polyhedron and $I$ be the number of edges intercepted by the plane. Find the minimum possible value of $E - I$. | [
"2"
] | [
"A plane intersects a polyhedron without passing through any of its vertices. Let $E$ be the number of edges of the polyhedron and $I$ be the number of edges intercepted by the plane. Find the minimum possible value of $E - I$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-98610bea27c2cd9484c9dc6acd4a191ac171c97b0a604f3144bbbe5250908f58 | A plane intersects a sphere h units from the center of the sphere. The sphere has an area of 9π square units and h increases at a rate of 1/5 unit per second. Circle Q is the circle of intersection for the plane and the sphere. Find the rate at which the area of circle Q changes when h is 1/2 unit. Give your answer in ... | [
"-\\dfrac{\\pi}{5}",
"\\(-\\frac{\\pi}{5}\\)"
] | [
"A plane intersects a sphere h units from the center of the sphere. The sphere has an area of 9π square units and h increases at a rate of 1/5 unit per second. Circle Q is the circle of intersection for the plane and the sphere. Find the rate at which the area of circle Q changes when h is 1/2 unit. Give your answe... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 27
} | 58 | 0 | 0 |
omr-ws-cb65b1c3e968716df5a9d568c4178a4a346aade4bee28837278550c1eecbeb15 | A plane intersects a sphere of radius $R$ at a distance $x$ from the center of the sphere, cutting a cap-shaped piece off from the sphere. A cone is inscribed in this cap such that the circular base of the sphere cap is the base of the cone. What is the limit as $x$ tends to $R$ from below of the volume of the cone div... | [
"\\(\\frac{2}{3}\\)"
] | [
"A plane intersects a sphere of radius $R$ at a distance $x$ from the center of the sphere, cutting a cap-shaped piece off from the sphere. A cone is inscribed in this cap such that the circular base of the sphere cap is the base of the cone. What is the limit as $x$ tends to $R$ from below of the volume of the con... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ec1d8e70a97f69633661b80b8d7a2498bbce7e9997f8315a1064fa01ec260a92 | A plane intersects a unit cube and divides it into two polyhedrons. It is known that the distance between any two points of one polyhedron does not exceed $3/2$. What value can the area of this section take? | [
"\\dfrac{3\\sqrt{3}}{4}"
] | [
"A plane intersects a unit cube and divides it into two polyhedrons. It is known that the distance between any two points of one polyhedron does not exceed $3/2$. What value can the area of this section take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-9577c3c0429900135aeb953f23c774174594a6ac13749059f699ea7b7b42eeeb | A plane intersects a unit cube, dividing it into two polyhedra. It is known that for each polyhedron, the distance between any two points does not exceed $\frac{3}{2}$ meters. What can be the cross-sectional area of the cube cut by this plane? | [
"1"
] | [
"A plane intersects a unit cube, dividing it into two polyhedra. It is known that for each polyhedron, the distance between any two points does not exceed $\\frac{3}{2}$ meters. What can be the cross-sectional area of the cube cut by this plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2ae63c48faf439919ce8218952bb2e813e8f8b501985a50bcba8b1fa9178dd02 | A plane is colored into black and white squares in a chessboard pattern. Then, all the white squares are colored red and blue such that any two initially white squares that share a corner are different colors (one is red and the other is blue). Let $\ell$ be a line not parallel to the sides of any squares. For every li... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"A plane is colored into black and white squares in a chessboard pattern. Then, all the white squares are colored red and blue such that any two initially white squares that share a corner are different colors (one is red and the other is blue). Let $\\ell$ be a line not parallel to the sides of any squares. For ev... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-07c6e6cfa70bf60246b42c6881d09320427ecb2d905b3783a027281577274434 | A plane is divided by \( n \) straight lines, with no three lines concurrent and no two lines parallel. Find the maximum number of angular domains in this configuration. | [
"\\( 2n \\)"
] | [
"A plane is divided by \\( n \\) straight lines, with no three lines concurrent and no two lines parallel. Find the maximum number of angular domains in this configuration."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f007f53485bfbed385e659c2f09062f6a3c5ab5b729334b3a8b165451f73cd11 | A plane is divided by \( n \) straight lines, with no three lines concurrent and no two lines parallel. Find the minimum number of angular domains in this configuration. | [
"\\( 2n \\)"
] | [
"A plane is divided by \\( n \\) straight lines, with no three lines concurrent and no two lines parallel. Find the minimum number of angular domains in this configuration."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e4df4df1a586733adc3c800deb763ffe39f7dc5887010aca98bfa931c90e39c | A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$? | [
"35"
] | [
"A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e9a19a5b9e47ef5ff50c683c77d56821a9fc29ddfb0c31b7b4f2f42f971b6bfd | A plane is divided into unit squares and colored using $n$ colors. A polygon $P$ is formed by connecting $n$ unit squares along their sides. It is given that any translation of $P$ covers exactly $n$ unit squares, each of a different color. Determine the minimum number of distinct colors that must be used to color the ... | [
"n"
] | [
"A plane is divided into unit squares and colored using $n$ colors. A polygon $P$ is formed by connecting $n$ unit squares along their sides. It is given that any translation of $P$ covers exactly $n$ unit squares, each of a different color. Determine the minimum number of distinct colors that must be used to color... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-dbf3e326f7effde9081b26336a508eaa1ac3a563eef0fa1ca06be37618d79e5c | A plane is divided into unit squares, each colored either black or white. It is known that each rectangle of dimensions 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can the plane be colored? | [
"6"
] | [
"A plane is divided into unit squares, each colored either black or white. It is known that each rectangle of dimensions 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can the plane be colored?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-faf4fedabe7d285b20e42d7661bbc0c0cadce1576587f8b86dfc031dc0404ed5 | A plane is divided like a chessboard, and a real number is written in each square such that the number in each square is equal to the arithmetic mean of the numbers in the adjacent squares (up, down, left, and right). If the number in the square at the origin (0,0) is 10, and the number in the square at (1,0) is 8, fin... | [
"6"
] | [
"A plane is divided like a chessboard, and a real number is written in each square such that the number in each square is equal to the arithmetic mean of the numbers in the adjacent squares (up, down, left, and right). If the number in the square at the origin (0,0) is 10, and the number in the square at (1,0) is 8... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44ea4da765d0548fcbc0d578b50e92192c5378dafb30ba1d0838755922a6368d | A plane is drawn in a triangular truncated pyramid through a side of the upper base parallel to the opposite lateral edge. In what ratio does the plane divide the volume of the pyramid if the respective sides of the bases are related as $1 : 2$? | [
"$3:4$"
] | [
"A plane is drawn in a triangular truncated pyramid through a side of the upper base parallel to the opposite lateral edge. In what ratio does the plane divide the volume of the pyramid if the respective sides of the bases are related as $1 : 2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-283a8dbb5c7ba7f1ac1567a6c43c11a876fa7395f5e3990c59e3903132dd5ce7 | A plane is drawn through a side of the base of a regular quadrangular pyramid perpendicular to the opposite face. Find the area of the resulting section if the altitude of the pyramid is equal to $h$ and the ratio of the lateral surface area to the area of the base is equal to $\sqrt 3$. | [
"$4\\sqrt 6 h^2/9$"
] | [
"A plane is drawn through a side of the base of a regular quadrangular pyramid perpendicular to the opposite face. Find the area of the resulting section if the altitude of the pyramid is equal to $h$ and the ratio of the lateral surface area to the area of the base is equal to $\\sqrt 3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71d8a43f113954c9229e4b2862d1120034fa7246230590e547cdb3f3c6574032 | A plane is drawn through a vertex of a regular quadrangular prism so that a rhombus with an acute angle $\alpha $ results in the section. Find the angle of inclination of that plane to the plane of the base of the prism. | [
"$\\arccos \\left(\\tan\\frac{\\alpha }{2}\\right)$"
] | [
"A plane is drawn through a vertex of a regular quadrangular prism so that a rhombus with an acute angle $\\alpha $ results in the section. Find the angle of inclination of that plane to the plane of the base of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e393eea766af2a7cd54f9f0601b0067c44eb599d2791971a562257d6c81fdb85 | A plane is drawn through one of the sides of the base of a regular right triangular prism at an angle $\alpha $ to the base and cuts from the prism a pyramid of volume $V$. Determine the area of the section. | [
"$\\sqrt 3V^{2/3}\\cos ^{-1/3}\\alpha \\sin ^{-2/3}\\alpha $"
] | [
"A plane is drawn through one of the sides of the base of a regular right triangular prism at an angle $\\alpha $ to the base and cuts from the prism a pyramid of volume $V$. Determine the area of the section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98edf010d1c29a5c78d09bb87ccf723cc2efd63bc0faef29ff776b578a63febe | A plane is drawn through the center of a cube with side length 2. If the intersection of the plane with the cube forms a polygon, what is the maximum number of sides this polygon can have? Calculate the area of this polygon when it has the maximum number of sides. | [
"3\\sqrt{3}"
] | [
"A plane is drawn through the center of a cube with side length 2. If the intersection of the plane with the cube forms a polygon, what is the maximum number of sides this polygon can have? Calculate the area of this polygon when it has the maximum number of sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d75cfdf26a4f6b3bb434cea62ff6494f7657c82511a6c5133069763505387cdb | A plane is drawn through the midpoint of a diagonal of a cube perpendicular to the diagonal. Determine the area of the figure resulting from the section of the cube cut by this plane if the edge of the cube is equal to $a$. | [
"$3\\sqrt 3a^{2/4}$"
] | [
"A plane is drawn through the midpoint of a diagonal of a cube perpendicular to the diagonal. Determine the area of the figure resulting from the section of the cube cut by this plane if the edge of the cube is equal to $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-634dd78ebb0aa9ebdc0030f9389fc093e53ae4688c3cd09b2486f6d97676e65b | A plane is drawn through the midpoint of the diagonal of a cube perpendicular to it. Find the area of the figure resulting from the section of the cube by the plane if the edge of the cube is equal to $a$. | [
"$\\frac{3a^2\\sqrt 3}{4}$"
] | [
"A plane is drawn through the midpoint of the diagonal of a cube perpendicular to it. Find the area of the figure resulting from the section of the cube by the plane if the edge of the cube is equal to $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc00b2d799046136d222c06215a7c724576d13f43e35166192cfdeb3df904d69 | A plane is drawn through the side of the base of a regular triangular pyramid and the center of the ball inscribed into it. In what ratio does the plane divide the volume of the pyramid if its lateral edge is 3.5 times as large as the side of the base? | [
"4"
] | [
"A plane is drawn through the side of the base of a regular triangular pyramid and the center of the ball inscribed into it. In what ratio does the plane divide the volume of the pyramid if its lateral edge is 3.5 times as large as the side of the base?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d15508abfcc71ac1d01dda350ef2e67786ba15f350dd9fc187a27512c11f6fb7 | A plane is drawn through the vertex of a regular triangular pyramid and the midpoints of two sides of the base. Determine the area of the section figure and volumes of the portions of the given pyramid into which it is divided by the cutting plane, given the side $a$ of the base and angle $\alpha $ formed by the cuttin... | [
"$S=\\frac{\\sqrt 3a^2}{48\\cos\\alpha };\\ V_1=\\displaystyle\\frac{a^3}{192}\\tan\\alpha ;\\ V_2=\\displaystyle\\frac{a^3}{64}\\tan\\alpha $"
] | [
"A plane is drawn through the vertex of a regular triangular pyramid and the midpoints of two sides of the base. Determine the area of the section figure and volumes of the portions of the given pyramid into which it is divided by the cutting plane, given the side $a$ of the base and angle $\\alpha $ formed by the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-540b30c2fcc093de24a7f2ef11c084b0f48fbdfc38b8378a095d5336117f5225 | A plane is drawn through the vertices $A$, $C$, and $D_1$ of a rectangular parallelepiped $ABCDA_1B_1C_1D_1$ which makes an angle of $60^\circ$ with the plane of the base. The sides of the base of the parallelepiped are equal to 4 and 3. Find the volume of the parallelepiped. | [
"$\\frac{144\\sqrt 3}{5}$"
] | [
"A plane is drawn through the vertices $A$, $C$, and $D_1$ of a rectangular parallelepiped $ABCDA_1B_1C_1D_1$ which makes an angle of $60^\\circ$ with the plane of the base. The sides of the base of the parallelepiped are equal to 4 and 3. Find the volume of the parallelepiped."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc14e1d4084fe09c7c94e59aca32f1ccaeef8174d009877bbd87087f0a45cd60 | A plane is drawn through two generatrices of a cone which make an angle $\alpha $. Find the ratio of the area of the section to the total surface of the cone if the angle between the generatrices of the cone and the plane of the base is $\beta $. | [
"$\\frac{\\sin \\alpha }{4\\pi \\cos \\beta \\cos ^2 \\frac{\\beta }{2}}$"
] | [
"A plane is drawn through two generatrices of a cone which make an angle $\\alpha $. Find the ratio of the area of the section to the total surface of the cone if the angle between the generatrices of the cone and the plane of the base is $\\beta $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e7d9189fc7c53cc62ddcce7b9d1c1176ab169ba8fa0e6ad9ab8a99949971699 | A plane is expressed parametrically by \[\mathbf{v} = \begin{pmatrix} 1 + s - t \\ 2 - s \\ 3 - 2s + 2t \end{pmatrix}.\]Find the equation of the plane. Enter your answer in the form \[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$ | [
"2x + z - 5 = 0"
] | [
"A plane is expressed parametrically by\n\\[\\mathbf{v} = \\begin{pmatrix} 1 + s - t \\\\ 2 - s \\\\ 3 - 2s + 2t \\end{pmatrix}.\\]Find the equation of the plane. Enter your answer in the form\n\\[Ax + By + Cz + D = 0,\\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\\gcd(|A|,|B|,|C|,|D|) = 1.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c971062bc63baca7172060aafbcb80cfe5c68ac19f7281f1477254b62e45a6d2 | A plane is flying at a constant altitude along a circle of radius 12 miles with center at point $A$. The speed of the aircraft is $v$. At some moment, a missile is fired from point $A$ with speed $v$ and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before col... | [
"12"
] | [
"A plane is flying at a constant altitude along a circle of radius 12 miles with center at point $A$. The speed of the aircraft is $v$. At some moment, a missile is fired from point $A$ with speed $v$ and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel befor... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdc72bb44765c1c10051f34cc20e15b1a3a9661011e511499e7b7db4ccfaab47 | A plane is flying at a speed and direction which in still air would be 500 mph due north. There is a wind of 100 mph blowing toward 30 degrees east of north. What is the resultant ground speed of the plane in mph? | [
"588.7"
] | [
"A plane is flying at a speed and direction which in still air would be 500 mph due north. There is a wind of 100 mph blowing toward 30 degrees east of north. What is the resultant ground speed of the plane in mph?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-874b1168e097b9a265b832ce3b80ec2f391e175d577beae79af642372be9ba2e | A plane is flying from San Francisco Airport to Los Angeles Airport, which is 540 km away at a bearing of 140 degrees. The plane's airspeed is 640 km/h, and there is a 60 km/h wind blowing from 290 degrees. Determine the compass heading the plane should take and the estimated time of arrival (ETA) to the nearest minute... | [
"Compass heading: 142.6867242 degrees, ETA: 2:56"
] | [
"A plane is flying from San Francisco Airport to Los Angeles Airport, which is 540 km away at a bearing of 140 degrees. The plane's airspeed is 640 km/h, and there is a 60 km/h wind blowing from 290 degrees. Determine the compass heading the plane should take and the estimated time of arrival (ETA) to the nearest m... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-7b4e2e5c76cbb3cc6ac5031f816d6b1dc0810bb132ba663279f5ff19208e2b81 | A plane is missing and is equally likely to have gone down in any of three possible regions. Let \(1 - A_i\) denote the probability that the plane will be found upon a search of the \(i\)th region when the plane is in fact in that region, where \(i = 1, 2, 3\). The constants \(A_i\) are called overlook probabilities. W... | [
"\\(\\frac{A_1}{A_1+2}\\) for the first region and \\(\\frac{1}{A_1+2}\\) for the second and third regions."
] | [
"A plane is missing and is equally likely to have gone down in any of three possible regions. Let \\(1 - A_i\\) denote the probability that the plane will be found upon a search of the \\(i\\)th region when the plane is in fact in that region, where \\(i = 1, 2, 3\\). The constants \\(A_i\\) are called overlook pro... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 13
} | 27 | 0 | 0 |
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