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-63ccc1c4ccbc447e7bcf2940affd0c3512b04394f12b72326cf05f9327b1fcaa | An aviation study claims there were 24 fatalities and 38 accidents in commercial aviation among major (not commuter) airlines over 15.7 million flight hours, 8.2 billion miles flown, and 13.4 million departures. Compute the accident and fatality rates in units of accidents or deaths per 100,000 flight hours, per 1 bill... | [
"0.179"
] | [
"An aviation study claims there were 24 fatalities and 38 accidents in commercial aviation among major (not commuter) airlines over 15.7 million flight hours, 8.2 billion miles flown, and 13.4 million departures. Compute the accident and fatality rates in units of accidents or deaths per 100,000 flight hours, per 1... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-7755e41b02e8cc46215bb9e01b1c14b29d636e886a0bc354b8a8fc9b604d4c0f | An aviator heads his airplane due west. Due to a wind from the south, the course makes an angle of 20 degrees with the heading. If his airspeed is 100 m/h, what is his groundspeed and what is the speed of the wind? | [
"(32.20, 96.97)"
] | [
"An aviator heads his airplane due west. Due to a wind from the south, the course makes an angle of 20 degrees with the heading. If his airspeed is 100 m/h, what is his groundspeed and what is the speed of the wind?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37061d809c09a8b2223a0735a2b93878a0bed183e1a8fe17e1ad2a4709172f5b | An axis-aligned rectangle has vertices at $(0,0)$ and $(2, 2016)$. Let $f(x, y)$ be the maximum possible area of a circle with center at $(x, y)$ contained entirely within the rectangle. Compute the expected value of $f$ over the rectangle. | [
"\\dfrac{2015}{6048} \\pi"
] | [
"An axis-aligned rectangle has vertices at $(0,0)$ and $(2, 2016)$. Let $f(x, y)$ be the maximum possible area of a circle with center at $(x, y)$ contained entirely within the rectangle. Compute the expected value of $f$ over the rectangle."
] | [
"aops_c7_college_math"
] | [
"cot"
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"cot": 15
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omr-ws-75749277180325edc9b272dd3bddc791c9669574c046d5dd3f523cc8d778481d | An convex quadrilateral is drawn, and the 4 triangles obtained by choosing 3 of the 4 vertices of the quadrilateral are considered. If the area of the largest of these triangles is 1888 and the area of the smallest of these triangles is 123, determine the greatest value that the area of the quadrilateral can have. | [
"2011"
] | [
"An convex quadrilateral is drawn, and the 4 triangles obtained by choosing 3 of the 4 vertices of the quadrilateral are considered. If the area of the largest of these triangles is 1888 and the area of the smallest of these triangles is 123, determine the greatest value that the area of the quadrilateral can have.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
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"tir": 1
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omr-ws-6185e4a73df1221378a4e96a1ea70c789b01b305eccfb9de0f21fd946345f004 | An dice is thrown $n$ times. Let $X_n$ be the product of the outcomes and $Y_n$ be the sum of the outcomes. Find the probability that $X_n + Y_n$ is a multiple of 3. | [
"\\dfrac{1}{3}"
] | [
"An dice is thrown $n$ times. Let $X_n$ be the product of the outcomes and $Y_n$ be the sum of the outcomes. Find the probability that $X_n + Y_n$ is a multiple of 3."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
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omr-ws-693dbc4722cc924e46a0353f6a80c6762888e63bc8deceae09e7b3e38604a8a9 | An eastbound train traveling at $60$ meters per second passes through a railroad crossing. An observer stands $70$ meters south of the crossing. At what rate is the train moving away from the observer $4$ seconds after it passes through the intersection? | [
"57.6"
] | [
"An eastbound train traveling at $60$ meters per second passes through a railroad crossing. An observer stands $70$ meters south of the crossing. At what rate is the train moving away from the observer $4$ seconds after it passes through the intersection?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-4083671cfade0a4dbd75209e71f042d06d677b961d74b7ac9ad1ebdafdd32932 | An eccentric mathematician has a ladder with $n$ rungs. When ascending, each step covers $a$ rungs, and when descending, each step covers $b$ rungs, where $a$ and $b$ are fixed positive integers. The mathematician can climb from ground level to the top rung of the ladder and come back down to ground level again using a... | [
"\\( a + b - \\gcd(a, b) \\)"
] | [
"An eccentric mathematician has a ladder with $n$ rungs. When ascending, each step covers $a$ rungs, and when descending, each step covers $b$ rungs, where $a$ and $b$ are fixed positive integers. The mathematician can climb from ground level to the top rung of the ladder and come back down to ground level again us... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 7,
"tir": 12
} | 40 | 0 | 0 |
omr-ws-a1b0523763d9f63aae5cee258ca22e87dac1b71a3185aa990509bd8f19ef4e9a | An eccentric postman delivers mail to eight houses on Acacia Avenue, numbered $1, 3, 5, 7$ on one side and $2, 4, 6, 8$ on the opposite side. The postman alternates between odd and even-numbered houses and never crosses the road to a house directly opposite the one to which he has just delivered mail. He always starts ... | [
"12"
] | [
"An eccentric postman delivers mail to eight houses on Acacia Avenue, numbered $1, 3, 5, 7$ on one side and $2, 4, 6, 8$ on the opposite side. The postman alternates between odd and even-numbered houses and never crosses the road to a house directly opposite the one to which he has just delivered mail. He always st... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-e126bad29688cd2d7a24bfbb2b8958e39880f3dbff50628ac5067df0fd5ce423 | An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses numbered 1, 3, 5, 7 on one side of the street and 2, 4, 6, 8 on the opposite side, finally returning to house 1. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except house 1 which is visite... | [
"12"
] | [
"An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses numbered 1, 3, 5, 7 on one side of the street and 2, 4, 6, 8 on the opposite side, finally returning to house 1. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except house 1 which is v... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-5006ac98d90e51c05e8e9cda9c1ff297271759d03c397d8691b2591b4ead43d5 | An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses numbered 1, 3, 5, 7 on one side of the street and houses 2, 4, 6, 8 on the opposite side. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except house 1 which is visited twice, and never cros... | [
"12"
] | [
"An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses numbered 1, 3, 5, 7 on one side of the street and houses 2, 4, 6, 8 on the opposite side. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except house 1 which is visited twice, and never... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-e2b2148110927fa3ad449d8c261913b5c3c2aa3cfdfa408e2e7e2934699a0ffc | An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses on Hamilton Avenue, finally returning to house 1. The houses are numbered 1, 3, 5, 7 on one side and 2, 4, 6, 8 on the opposite side. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except ho... | [
"144"
] | [
"An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses on Hamilton Avenue, finally returning to house 1. The houses are numbered 1, 3, 5, 7 on one side and 2, 4, 6, 8 on the opposite side. The postman must follow an odd-even-odd-even pattern, visit each house exactly once exce... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c68b125da9a39fb6baab77958ea1c829459c803b6307bfb212f2512dfa1b007 | An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses on Hamilton Avenue, finally returning to house 1. The houses are numbered 1, 3, 5, 7 on one side and 2, 4, 6, 8 on the other side. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except house... | [
"12"
] | [
"An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses on Hamilton Avenue, finally returning to house 1. The houses are numbered 1, 3, 5, 7 on one side and 2, 4, 6, 8 on the other side. The postman must follow an odd-even-odd-even pattern, visit each house exactly once except ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-5b51b0c58b3f2b410e54aebe43de08dab996013ea79ded70ab406fb6a12d1179 | An econo-size bottle of perfume contains 14 ounces and consists of rose oil, oil of fermented prunes, and alcohol. The bottle contains 3 more ounces of alcohol than oil of fermented prunes, and the amount of alcohol is equal to the combined volume of the other two ingredients. How much of each ingredient is used in the... | [
"7 ounces of alcohol, 3 ounces of rose oil, and 4 ounces of oil of fermented prunes."
] | [
"An econo-size bottle of perfume contains 14 ounces and consists of rose oil, oil of fermented prunes, and alcohol. The bottle contains 3 more ounces of alcohol than oil of fermented prunes, and the amount of alcohol is equal to the combined volume of the other two ingredients. How much of each ingredient is used i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 25
} | 29 | 0 | 0 |
omr-ws-85afe3ece7024510df89fc2cf60fa5feb02d57e6d983200df64eb4687757ec1c | An economist and a statistician play a game on a calculator which displays only positive integers. The game consists of the following operation: if the calculator displays an integer $n$, a player can type an integer $m$ from the set $\{1, 2, \ldots, 99\}$, and if $m\%$ of $n$ is a positive integer, the calculator disp... | [
"951"
] | [
"An economist and a statistician play a game on a calculator which displays only positive integers. The game consists of the following operation: if the calculator displays an integer $n$, a player can type an integer $m$ from the set $\\{1, 2, \\ldots, 99\\}$, and if $m\\%$ of $n$ is a positive integer, the calcul... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b389061401818cadb780ad0d1c8e9c0dbd5027466c20407100ef6e242bab110 | An edge of a square can be expressed as $4x-15$ meters or as $20-3x$ meters. What is its area in square meters? | [
"25"
] | [
"An edge of a square can be expressed as $4x-15$ meters or as $20-3x$ meters. What is its area in square meters?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bc4707487efb311a06d8d6fbdb68591bc47f57534c8695da6a14a4e52eea658 | An egg is in a shape formed by vertically stretching the upper hemisphere of a sphere with radius 2 by a factor of 1.5. A horizontal plane cuts the egg into two regions of equal volume. How many units above the lowest point on the egg is the plane? | [
"6 \\cos \\dfrac{4\\pi}{9}"
] | [
"An egg is in a shape formed by vertically stretching the upper hemisphere of a sphere with radius 2 by a factor of 1.5. A horizontal plane cuts the egg into two regions of equal volume. How many units above the lowest point on the egg is the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-a5d213a88c94aa9537502831d18a396e48b04827361e28c439c717158a8b4b2e | An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form? | [
"10001"
] | [
"An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0acbb601141de0076e614747ceb73080cc8ead348a51f13ce12397b0228a4110 | An eight-digit number is formed using the digits 1, 2, 3, and 4 such that the product of all digits is 3072. The total number of ways to form such a number is given by $(23 \cdot \binom{8}{k})$. Determine the value of $k$. | [
"3"
] | [
"An eight-digit number is formed using the digits 1, 2, 3, and 4 such that the product of all digits is 3072. The total number of ways to form such a number is given by $(23 \\cdot \\binom{8}{k})$. Determine the value of $k$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-de002ba45619bb26232bf836306a4da12f047b843fa17e5864ffb90831a81e3e | An eight-foot stick and a twenty-two-foot stick are each randomly broken into two parts. What is the probability that the longer part of the eight-foot stick is longer than the shorter part of the twenty-two-foot stick? | [
"\\(\\frac{6}{11}\\)"
] | [
"An eight-foot stick and a twenty-two-foot stick are each randomly broken into two parts. What is the probability that the longer part of the eight-foot stick is longer than the shorter part of the twenty-two-foot stick?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-f74e48c9a776efbd4101bda84e740a62e8e9a94e1567ff2c408993708cedd571 | An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die? | [
"4.5"
] | [
"An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-44cb12c5d8d8ddeb3a05a351705f2353a0d1ac42282b71db9b243622531e64ca | An elastic ball of negligible dimensions rests at a point $A$ at a distance $a$ from the center of circular billiards of radius $R$. To what point $B$ of the cushion must the ball be directed so that it returns to the point $A$ after being reflected twice from the cushion? | [
"\\cos \\alpha =-\\frac{R}{4a}+\\sqrt {\\left(\\frac{R}{4a}\\right)^2+\\frac{1}{2}}.\n\nAdditionally, if \\(\\angle ABC = 0\\), then the points \\(B\\) and \\(C\\) are the two extremities of the diameter passing through the point \\(A\\). However, the specific coordinates of \\(B\\) are not provided in the solution... | [
"An elastic ball of negligible dimensions rests at a point $A$ at a distance $a$ from the center of circular billiards of radius $R$. To what point $B$ of the cushion must the ball be directed so that it returns to the point $A$ after being reflected twice from the cushion?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-083d3c4a30a18f815dfbd8886a9d2fa522410493a4e840d726837d96d77a09b3 | An elastic band is wound around a deck of playing cards to form seven horizontal stripes on the top. Ignoring the different ways the rubber band could overlap itself, how many different patterns are possible on the underside of the deck if two patterns are considered the same if one is a $180^\circ$ rotation of the oth... | [
"72"
] | [
"An elastic band is wound around a deck of playing cards to form seven horizontal stripes on the top. Ignoring the different ways the rubber band could overlap itself, how many different patterns are possible on the underside of the deck if two patterns are considered the same if one is a $180^\\circ$ rotation of t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-554907ba8ba9acac25fde0b51dcf90689196d1d1107a9b37b765425eb8665dc0 | An elastic strip of length 1 meter is attached to a wall and the back of a car. A fly starts at the wall and crawls on the strip towards the car, which drives away from the wall at a speed of 1 meter per second. Assuming the strip can be stretched infinitely long, will the fly ever reach the back of the car? If so, how... | [
"\\( t = e^{(1/x)} - 1 \\)"
] | [
"An elastic strip of length 1 meter is attached to a wall and the back of a car. A fly starts at the wall and crawls on the strip towards the car, which drives away from the wall at a speed of 1 meter per second. Assuming the strip can be stretched infinitely long, will the fly ever reach the back of the car? If so... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4d440ee722096e434d6b9b75724a0e13ebfb9183a5056fc92a940d0708573f19 | An electronics discount chain has a sale on a certain brand of stereo. The chain has stores in Santa Monica and El Toro and warehouses in Long Beach and Pasadena. To satisfy rush orders, 15 sets must be shipped from the warehouses to the Santa Monica store, and 19 must be shipped to the El Toro store. The cost of shipp... | [
"184"
] | [
"An electronics discount chain has a sale on a certain brand of stereo. The chain has stores in Santa Monica and El Toro and warehouses in Long Beach and Pasadena. To satisfy rush orders, 15 sets must be shipped from the warehouses to the Santa Monica store, and 19 must be shipped to the El Toro store. The cost of ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3aff7d4193be28ba6451947d1ef082c804b96fa236e35adf6b80c76019d85409 | An element \( n = y(a, b) \) (where \( a \le b \)) in \( Y \) is called excellent if its representation is not unique (for instance, \( 20 = y(1, 5) = y(2, 3) \)). Find the number of excellent elements in \( Y \). | [
"17"
] | [
"An element \\( n = y(a, b) \\) (where \\( a \\le b \\)) in \\( Y \\) is called excellent if its representation is not unique (for instance, \\( 20 = y(1, 5) = y(2, 3) \\)). Find the number of excellent elements in \\( Y \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a0bc09a2acd2c728f5b008059a01e4a03a70057af433517570f5f64445ef027 | An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$? Note: The 1 at the top is often labelled the "zeroth" row of Pascal's Triangle, by convention. So to count a total of 15 rows, use rows 0 through 14. | [
"\\frac{29}{120}"
] | [
"An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?\n\nNote: The 1 at the top is often labelled the \"zeroth\" row of Pascal's Triangle, by convention. So to count a total of 15 rows, use rows 0 through 14."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2941a78d964eee246f3911de1281103b72cee7a28dfeeecb7f27ae1a10f6982 | An element of the alternating group $A_5$ is picked at random (each element is equally likely to be picked). What is the expected value of its order? | [
"\\(\\frac{211}{60}\\)"
] | [
"An element of the alternating group $A_5$ is picked at random (each element is equally likely to be picked). What is the expected value of its order?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d337cc1f560bd9e0d8a3f62bdd07a7dd9a65ebcd2f24a577b597152ef8607aa1 | An elephant is drinking from a water tank at a constant rate. A hose refills the tank at 2 liters per minute. If the hose rate is increased to 4 liters per minute, the elephant takes 40 extra minutes to empty the tank. If the hose rate is decreased to 1 liter per minute, the elephant takes 10 minutes less to empty the ... | [
"300 litres"
] | [
"An elephant is drinking from a water tank at a constant rate. A hose refills the tank at 2 liters per minute. If the hose rate is increased to 4 liters per minute, the elephant takes 40 extra minutes to empty the tank. If the hose rate is decreased to 1 liter per minute, the elephant takes 10 minutes less to empty... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-bb064feacaf6d42c7a7e7ba955291dfa088371409630825325f0666bd06cb8b5 | An elevator ascends from the ground with uniform speed. At time \( T_1 \), a boy drops a marble through the floor. The marble falls with uniform acceleration \( g = 9.8 \frac{m}{s^2} \) and hits the ground \( T_2 \) seconds later. Find the height of the elevator at time \( T_1 \). | [
"\\(\\frac{g T_2^2 T_1}{2 (T_1 - T_2)}\\)"
] | [
"An elevator ascends from the ground with uniform speed. At time \\( T_1 \\), a boy drops a marble through the floor. The marble falls with uniform acceleration \\( g = 9.8 \\frac{m}{s^2} \\) and hits the ground \\( T_2 \\) seconds later. Find the height of the elevator at time \\( T_1 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-098b8c6e29fc02232f938414ed3627a5dda076f51f290161951e2c73a3ce02f6 | An elevator starts on the top floor of a 100-floor building and stops at at least 40 floors, including both the top and bottom floors. Let \( S \) be the set of all possible floor stops. Define \( d(S) \) as the minimum difference between any two consecutive stops in \( S \). Find the maximum possible value of \( d(S) ... | [
"2"
] | [
"An elevator starts on the top floor of a 100-floor building and stops at at least 40 floors, including both the top and bottom floors. Let \\( S \\) be the set of all possible floor stops. Define \\( d(S) \\) as the minimum difference between any two consecutive stops in \\( S \\). Find the maximum possible value ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac1a7d9f4044b4029396bad2509f765303b615b6e92b99795edfcfd863d65349 | An elf and a dwarf are dividing a chest containing 100 tokens and 100 bags with gold coins, where the $i$-th bag contains $i$ coins. During each turn, the elf chooses a bag, and the dwarf can either take one token and give the bag to the elf or give up one token and take the bag for himself. What is the maximum number ... | [
"1275"
] | [
"An elf and a dwarf are dividing a chest containing 100 tokens and 100 bags with gold coins, where the $i$-th bag contains $i$ coins. During each turn, the elf chooses a bag, and the dwarf can either take one token and give the bag to the elf or give up one token and take the bag for himself. What is the maximum nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-e177ecf1c5b6214ba240eeba29801aa1c2221aaae2e8e0f04b2930a513416f5a | An ellipse $\epsilon$ has foci $F_1$ and $F_2$. Vertices $A$ and $C$ of a square $ABCD$ with side length $5$ lie on $\epsilon$, and $F_1$ and $F_2$ lie on the perimeter of $ABCD$ such that $BF_1=DF_2=1$. The square of the area of $\epsilon$ can be expressed as $(a+\sqrt{b})\pi^2$ for positive integers $a$ and $b$. Comp... | [
"750"
] | [
"An ellipse $\\epsilon$ has foci $F_1$ and $F_2$. Vertices $A$ and $C$ of a square $ABCD$ with side length $5$ lie on $\\epsilon$, and $F_1$ and $F_2$ lie on the perimeter of $ABCD$ such that $BF_1=DF_2=1$. The square of the area of $\\epsilon$ can be expressed as $(a+\\sqrt{b})\\pi^2$ for positive integers $a$ and... | [
"aops_c4_high_school_math"
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"additional_problems"
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omr-ws-ac4ba8d2bb487cdb355efd5f86452652d90a9cd1532f29ce1449cea2eff164ba | An ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is tangent to each of the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 + y^2 = 1$ at two points. Find the ordered pair $(a, b)$ that minimizes the area of the ellipse. | [
"\\(\\left( \\frac{3\\sqrt{2}}{2}, \\frac{\\sqrt{6}}{2} \\right)\\)"
] | [
"An ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ is tangent to each of the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 + y^2 = 1$ at two points. Find the ordered pair $(a, b)$ that minimizes the area of the ellipse."
] | [
"aops_c4_high_school_math"
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"cot",
"tir"
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"cot": 15,
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omr-ws-b245129a485c3b9c45b3eab3d0e95325c01ceeaffd6d13c5de49acc185ceee39 | An ellipse $\mathcal{A}$ has foci at $(-6, 0)$ and $(6, 0)$ and intersects the x-axis at $(-9, 0)$ and $(9, 0)$. Another ellipse $\mathcal{B}$ has foci at $(0, -5)$ and $(0, 5)$ and has a major axis of length $20$. What is the maximum area of a triangle constructed in the overlap of $\mathcal{A}$ and $\mathcal{B}$? | [
"15\\sqrt{15}"
] | [
"An ellipse $\\mathcal{A}$ has foci at $(-6, 0)$ and $(6, 0)$ and intersects the x-axis at $(-9, 0)$ and $(9, 0)$. Another ellipse $\\mathcal{B}$ has foci at $(0, -5)$ and $(0, 5)$ and has a major axis of length $20$. What is the maximum area of a triangle constructed in the overlap of $\\mathcal{A}$ and $\\mathcal... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
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"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-9321e0dd91009b24bbd4f9f68420fed1a8026a63364ef1424c8fba201874d6f5 | An ellipse and a circle are drawn with equal areas. The ellipse has axes whose lengths sum to five times the radius of the circle. The ratio of the radius of the circle to the major axis of the ellipse equals $\frac{m}{n}$ for positive integers $m$ and $n$ with a greatest common divisor of $1$. Find $m+n$. | [
"5"
] | [
"An ellipse and a circle are drawn with equal areas. The ellipse has axes whose lengths sum to five times the radius of the circle. The ratio of the radius of the circle to the major axis of the ellipse equals $\\frac{m}{n}$ for positive integers $m$ and $n$ with a greatest common divisor of $1$. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
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"cot": 16
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omr-ws-9116886de6e4e1ba61faae3cfd039fb405defd310eb0001719d07408164d7512 | An ellipse and a hyperbola share common foci $F_1$ and $F_2$. The ellipse has vertices at the origin and $(10, 24)$, and a semi-minor axis of length $11$. The hyperbola has a conjugate axis of length $4\sqrt{3}$. The ellipse and hyperbola intersect at four points. Let $P$ be one of the points of intersection. Compute $... | [
"410"
] | [
"An ellipse and a hyperbola share common foci $F_1$ and $F_2$. The ellipse has vertices at the origin and $(10, 24)$, and a semi-minor axis of length $11$. The hyperbola has a conjugate axis of length $4\\sqrt{3}$. The ellipse and hyperbola intersect at four points. Let $P$ be one of the points of intersection. Com... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-dcb60450f976273d7cc056965623ac9ddd60029401d92009ee8f84c5027197aa | An ellipse has a major axis of 7 feet and a minor axis of 6 feet. The major axis is growing at a rate of 1 foot/min and the minor axis is shrinking at a rate of 0.5 feet/min. At what rate is the area decreasing or increasing when the major axis is 11 feet? | [
"\\(\\frac{-3\\pi}{2}\\) ft/min"
] | [
"An ellipse has a major axis of 7 feet and a minor axis of 6 feet. The major axis is growing at a rate of 1 foot/min and the minor axis is shrinking at a rate of 0.5 feet/min. At what rate is the area decreasing or increasing when the major axis is 11 feet?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a90e69858adfe2d9af2cd8004e24c27656a9a52986af78e4e6119584d597500 | An ellipse has a major axis of 7 feet and a minor axis of 6 feet. The major axis is growing at a rate of 1 foot/min and the minor axis is shrinking at a rate of 0.5 feet/min. At what time does the ellipse have maximum area? What is that area? | [
"\\( t = \\frac{5}{2} \\) minutes, Area = \\( \\frac{361\\pi}{8} \\)"
] | [
"An ellipse has a major axis of 7 feet and a minor axis of 6 feet. The major axis is growing at a rate of 1 foot/min and the minor axis is shrinking at a rate of 0.5 feet/min. At what time does the ellipse have maximum area? What is that area?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-a95ba0ec305d87da18ef9aa960f086dc851eec29f8cf970441cd00d8c055bf06 | An ellipse has a major axis of 7 feet and a minor axis of 6 feet. The major axis is growing at a rate of 1 foot/min and the minor axis is shrinking at a rate of 0.5 feet/min. When, if at all, is the area accelerating (either growing or shrinking)? | [
"2.5"
] | [
"An ellipse has a major axis of 7 feet and a minor axis of 6 feet. The major axis is growing at a rate of 1 foot/min and the minor axis is shrinking at a rate of 0.5 feet/min. When, if at all, is the area accelerating (either growing or shrinking)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
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"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-1d643caae593fd0204c4e2f513ecf0f24eacee33156c29741a505d70aef17d8e | An ellipse has a major axis of length 10 and a minor axis of length 8. A circle is centered at one of its foci. What is the radius of the largest circle that is entirely within the ellipse? | [
"2"
] | [
"An ellipse has a major axis of length 10 and a minor axis of length 8. A circle is centered at one of its foci. What is the radius of the largest circle that is entirely within the ellipse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41283cc2f7e164dcd2ca3cf98f56da8a593ece0ebc5f2209e3c4ccea9a065f24 | An ellipse has a major axis of length 122 and a minor axis of length 120. A circle centered at a focus of the ellipse is tangent to it. What is the sum of all possible diameters of the circle? | [
"244"
] | [
"An ellipse has a major axis of length 122 and a minor axis of length 120. A circle centered at a focus of the ellipse is tangent to it. What is the sum of all possible diameters of the circle?"
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
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"cot": 31,
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} | 39 | 0 | 0 |
omr-ws-c4e5b09f15304e3258e202cdf568bfafb5cf413b7515feff49777fbe63270caa | An ellipse has a vertex at $(8,22)$ and a corresponding focus at $(8,12)$. One endpoint of the latus rectum that goes through this focus is $(-8,12)$. Find the coordinates of the center of this ellipse. | [
"(8, 17)"
] | [
"An ellipse has a vertex at $(8,22)$ and a corresponding focus at $(8,12)$. One endpoint of the latus rectum that goes through this focus is $(-8,12)$. Find the coordinates of the center of this ellipse."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4569fc6d342a9def4b494ebb0dea13d6c3437be2080d51dc9d7b5f158cfd913a | An ellipse has center O and the ratio of the lengths of the axes is \(2 + \sqrt{3}\). If P is a point on the ellipse, calculate the minimum value of the cosine of the acute angle between the tangent to the ellipse at P and the radius vector PO. | [
"0"
] | [
"An ellipse has center O and the ratio of the lengths of the axes is \\(2 + \\sqrt{3}\\). If P is a point on the ellipse, calculate the minimum value of the cosine of the acute angle between the tangent to the ellipse at P and the radius vector PO."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3f888d2095f02b88be1e28f882cb843674cbfa623854495ace7e522556210dd8 | An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find the ordered quadruple $(a, b, h, k)$. (En... | [
" (8\\sqrt3, 14, 2, 4)"
] | [
"An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1,\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find the ordered quadruple $(a, b, h,... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-687e79049004802c4d4e89f1fab72c1b08bcb62817c98f366c1891ac23414b71 | An ellipse has foci at $(0, 2)$ and $(3, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair. | [
"\\left(\\tfrac{15}{4},0\\right)"
] | [
"An ellipse has foci at $(0, 2)$ and $(3, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6a5351cfeaf926bb62e5678bb3847306e9ec89695e9a96c787375e976f2a68fe | An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? | [
"85"
] | [
"An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b55c293015649a9acac87f3541039561ff74d6009e8a96df90be6b76fb8498d1 | An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$ -plane and is tangent to the $x$ -axis. What is the length of its major axis? | [
"085"
] | [
"An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$ -plane and is tangent to the $x$ -axis. What is the length of its major axis?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-69c882b82f5fc07d12239d7c20a589302f2042de2c51ee8bf0e425f1563f4b51 | An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? | [
"85"
] | [
"An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 57,
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"tir": 22
} | 93 | 0 | 0 |
omr-ws-944117a0fee43c5d1d068e9f3edcefc17065b1c89461ae29c600f18b73dd7c56 | An ellipse has foci at $(9,20)$ and $(49,55)$. Find the length of its major axis if the ellipse is tangent to the x-axis. | [
"85"
] | [
"An ellipse has foci at $(9,20)$ and $(49,55)$. Find the length of its major axis if the ellipse is tangent to the x-axis."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-11cce5cdfb986c5c34d549c56a85953218e0f32d4c71bb43a35ad85ac2414342 | An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? | [
"\\left( \\frac{15}{4}, 0 \\right)"
] | [
"An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53b1d71c91da6b722a7db6c4b56593d7e15f351c0791594d87918ba0f8167f64 | An ellipse has foci at (0,0) and (4,0) and is tangent to the line $x + 2y = 5$. Find the length of the major axis of the ellipse. | [
"\\(2\\sqrt{5}\\)"
] | [
"An ellipse has foci at (0,0) and (4,0) and is tangent to the line $x + 2y = 5$. Find the length of the major axis of the ellipse."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 23
} | 71 | 0 | 0 |
omr-ws-1c0332d62e73cf7448a82e254c12d8c1cd688d06f393518f744636b62e943df3 | An ellipse has foci at \( (9, 20) \) and \( (49, 55) \) in the xy-plane and is tangent to the x-axis. What is the length of its major axis? | [
"85"
] | [
"An ellipse has foci at \\( (9, 20) \\) and \\( (49, 55) \\) in the xy-plane and is tangent to the x-axis. What is the length of its major axis?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 11
} | 34 | 0 | 0 |
omr-ws-d6f5f7865ad8acc8a408b5cec32891380259a8c3576ae727a6e98ea5f76ea120 | An ellipse has foci that are 24 feet apart and intersects a line on which the foci lie at points 40 feet apart. What is the shortest possible distance one can run from a point on the ellipse to one focus, then to the other focus, and back to the starting point on the ellipse? | [
"64"
] | [
"An ellipse has foci that are 24 feet apart and intersects a line on which the foci lie at points 40 feet apart. What is the shortest possible distance one can run from a point on the ellipse to one focus, then to the other focus, and back to the starting point on the ellipse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-60d4cccf816714b3233f5941778dacd0d2661ca94bf22ab5ba674fb0782a471d | An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ | [
"3"
] | [
"An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58aa26c961b57482af65f438ba1d9aff5bcccfa0d17f23733eba7d4bab0a68f7 | An ellipse has major and minor axes of lengths $10$ and $8$ respectively. Using one focus as the center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle. | [
"2"
] | [
"An ellipse has major and minor axes of lengths $10$ and $8$ respectively. Using one focus as the center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d68d9f91200a4a81b4dc1f837ce70ade9908a8d0d3c60a841db6e869c3beeac | An ellipse has major and minor axes of lengths $a$ and $b$ respectively. A chord $\overline{AB}$ passes through a focus $F$ of the ellipse. If $AF = x$ and $BF = y$, find the value of $\frac{1}{x} + \frac{1}{y}$ in terms of $a$ and $b$. | [
"\\dfrac{2a}{b^2}"
] | [
"An ellipse has major and minor axes of lengths $a$ and $b$ respectively. A chord $\\overline{AB}$ passes through a focus $F$ of the ellipse. If $AF = x$ and $BF = y$, find the value of $\\frac{1}{x} + \\frac{1}{y}$ in terms of $a$ and $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-beb18641e472b22171bdef78875d52cefd965d8b98096a80138c501330275434 | An ellipse has major and minor axes of lengths 10 and 8, respectively. Using one focus as the center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle. | [
"2"
] | [
"An ellipse has major and minor axes of lengths 10 and 8, respectively. Using one focus as the center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-1c65df1511493da0a00d31d0391791b8f33bf73e6c12cbf7ab55834043cdde6d | An ellipse has the equation $\frac{x^2}{16} + \frac{y^2}{4} = 1$ with foci $A$ on the left and $B$ on the right. A point $P$ lies on the line $y = \frac{x}{\sqrt{3}} + \frac{8 + 2\sqrt{3}}{\sqrt{3}}$. When $\angle APB$ is at its maximum value, what is the value of $\frac{PA}{PB}$? | [
"$\\sqrt{3} - 1$"
] | [
"An ellipse has the equation $\\frac{x^2}{16} + \\frac{y^2}{4} = 1$ with foci $A$ on the left and $B$ on the right. A point $P$ lies on the line $y = \\frac{x}{\\sqrt{3}} + \\frac{8 + 2\\sqrt{3}}{\\sqrt{3}}$. When $\\angle APB$ is at its maximum value, what is the value of $\\frac{PA}{PB}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-4d50bcfc4014ff16b0e1291a7850db24c1d4d84b0f9b78593e7e6128ff486192 | An ellipse has two foci $F_1$ and $F_2$. A point $A$ is outside the ellipse, and the lines $AF_1$ and $AF_2$ intersect the ellipse at points $B_1$ and $B_2$ respectively. Let $I_1$ and $I_2$ be the incenters of triangles $\triangle AB_1F_2$ and $\triangle AB_2F_1$. If $A$ has coordinates $(a, b)$ and $F_1$ and $F_2$ ha... | [
"\\left( \\dfrac{8 + 2\\sqrt{13} + 2\\sqrt{5}}{2 + \\sqrt{13} + \\sqrt{5}}, \\dfrac{6 + 2\\sqrt{13}}{2 + \\sqrt{13} + \\sqrt{5}} \\right)"
] | [
"An ellipse has two foci $F_1$ and $F_2$. A point $A$ is outside the ellipse, and the lines $AF_1$ and $AF_2$ intersect the ellipse at points $B_1$ and $B_2$ respectively. Let $I_1$ and $I_2$ be the incenters of triangles $\\triangle AB_1F_2$ and $\\triangle AB_2F_1$. If $A$ has coordinates $(a, b)$ and $F_1$ and $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-497f82446297a2f548b840ad2123ae4e26f443932557380961c198293fe43859 | An ellipse in the Cartesian plane is given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a > b > 0 \). Let \( F_1 \) and \( F_2 \) be the foci, \( O \) be the origin, and \( A \) be a point on the ellipse such that \( \overline{AF_2} \) is perpendicular to the \( x \)-axis. Let \( B \) be a point... | [
"\\(\\sqrt{2}\\)"
] | [
"An ellipse in the Cartesian plane is given by the equation \\( \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\) where \\( a > b > 0 \\). Let \\( F_1 \\) and \\( F_2 \\) be the foci, \\( O \\) be the origin, and \\( A \\) be a point on the ellipse such that \\( \\overline{AF_2} \\) is perpendicular to the \\( x \\)-axis... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-de00b415e79afb03b71641731d531f6435f0ac10d5c8e2cfa03d48703d42eb7d | An ellipse in the coordinate plane has a focus at $(2014, 2015)$. The average distance from a point on the ellipse to this focus is $75$, and the range of distances from a point on the ellipse to this focus is $[30,120]$. If $y$ is the $y$-coordinate of the point on the ellipse furthest away from the $x$-axis, find the... | [
"60"
] | [
"An ellipse in the coordinate plane has a focus at $(2014, 2015)$. The average distance from a point on the ellipse to this focus is $75$, and the range of distances from a point on the ellipse to this focus is $[30,120]$. If $y$ is the $y$-coordinate of the point on the ellipse furthest away from the $x$-axis, fin... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b74f8e354d6682de95762c44fe88e18ccaf33cf16fbb206ed2f47b73a102abd4 | An ellipse in the coordinate plane has a focus at (2014, 2015). The average distance from a point on the ellipse to this focus is 75, and the range of distances from a point on the ellipse to this focus is [30, 120]. If \( y \) is the y-coordinate of the point on the ellipse furthest away from the x-axis, find the leas... | [
"2075"
] | [
"An ellipse in the coordinate plane has a focus at (2014, 2015). The average distance from a point on the ellipse to this focus is 75, and the range of distances from a point on the ellipse to this focus is [30, 120]. If \\( y \\) is the y-coordinate of the point on the ellipse furthest away from the x-axis, find t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2d79c8b1d426ac5670967c125dd73ca3546f347e157ea5337178e8e5ba23dcb | An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$ and the other focus is at $(d,7).$ Compute $d.$ | [
"\\frac{49}{3}"
] | [
"An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$ and the other focus is at $(d,7).$ Compute $d.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abef147f76780cf0d6fb15aba71f7bf718919a813d1178373a741b76d4bb981f | An ellipse intersects the hyperbola \(2x^2 - 2y^2 = 1\) orthogonally. The eccentricity of the ellipse is the reciprocal of the eccentricity of the hyperbola. If the axes of the ellipse are along the coordinate axes, find the equation of the ellipse. | [
"\\(\\frac{x^2}{2} + y^2 = 1\\)"
] | [
"An ellipse intersects the hyperbola \\(2x^2 - 2y^2 = 1\\) orthogonally. The eccentricity of the ellipse is the reciprocal of the eccentricity of the hyperbola. If the axes of the ellipse are along the coordinate axes, find the equation of the ellipse."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-a07c3e3ba81a208adce0ca3bf19875c341e4485628350075d3cf6833c05f64af | An ellipse intersects the sides of a triangle \(ABC\) at points \(A_{1}\), \(A_{2}\), \(B_{1}\), \(B_{2}\), \(C_{1}\), \(C_{2}\) such that \(A_{1}, A_{2} \in BC\), \(B_{1}, B_{2} \in CA\), and \(C_{1}, C_{2} \in AB\). Define points \(A'\), \(B'\), and \(C'\) as follows: \[ \{A'\} = B_{1}C_{1} \cap B_{2}C_{2}, \quad \{B... | [
"(1, 1)"
] | [
"An ellipse intersects the sides of a triangle \\(ABC\\) at points \\(A_{1}\\), \\(A_{2}\\), \\(B_{1}\\), \\(B_{2}\\), \\(C_{1}\\), \\(C_{2}\\) such that \\(A_{1}, A_{2} \\in BC\\), \\(B_{1}, B_{2} \\in CA\\), and \\(C_{1}, C_{2} \\in AB\\). Define points \\(A'\\), \\(B'\\), and \\(C'\\) as follows:\n\\[\n\\{A'\\} ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b163f69d900819a0a6b95091b81d74693089ed81923e885b6c44d5d2231417ba | An ellipse is defined parametrically by \[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right).\]Then the equation of the ellipse can be written in the form \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\gcd(|A|,|B|,|C|,|D|,|E|,|... | [
"1381"
] | [
"An ellipse is defined parametrically by\n\\[(x,y) = \\left( \\frac{2 (\\sin t - 1)}{2 - \\cos t}, \\frac{3 (\\cos t - 5)}{2 - \\cos t} \\right).\\]Then the equation of the ellipse can be written in the form\n\\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\\gcd(|... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4339581df2830705ac3872044aa95289b79eac83376cc71812d0fa80f661e541 | An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as a center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle. | [
"2"
] | [
"An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as a center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a69a24d84c2d3c95fcd034d5f3e7737cb1d06cfbd682b2b022936cf94df8ae3 | An ellipse is given by the equation $ \frac{x^2}{3} + \frac{y^2}{2} = 1 $. Line $ BD $ passes through focus $ F_1 $ and intersects the ellipse at points $ B $ and $ D $. Line $ AC $ passes through focus $ F_2 $ and intersects the ellipse at points $ A $ and $ C $. Line $ BD $ is perpendicular to $ AC $ at point $ P $. ... | [
"\\(\\frac{96}{25}\\)"
] | [
"An ellipse is given by the equation $ \\frac{x^2}{3} + \\frac{y^2}{2} = 1 $. Line $ BD $ passes through focus $ F_1 $ and intersects the ellipse at points $ B $ and $ D $. Line $ AC $ passes through focus $ F_2 $ and intersects the ellipse at points $ A $ and $ C $. Line $ BD $ is perpendicular to $ AC $ at point ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-5f42b704ca705cf2c983c745fe77ceecf29c0773cd4ea321c0de678cab23c947 | An ellipse is inscribed in triangle \(ABC\) and touches the sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Let \(P\), \(Q\), and \(R\) be the midpoints of segments \(EF\), \(FD\), and \(DE\) respectively. If the coordinates of \(A\), \(B\), and \(C\) are \((0,0)\), \((4,0)\), and \((0,... | [
"\\left( \\dfrac{4}{3}, 1 \\right)"
] | [
"An ellipse is inscribed in triangle \\(ABC\\) and touches the sides \\(BC\\), \\(CA\\), and \\(AB\\) at points \\(D\\), \\(E\\), and \\(F\\) respectively. Let \\(P\\), \\(Q\\), and \\(R\\) be the midpoints of segments \\(EF\\), \\(FD\\), and \\(DE\\) respectively. If the coordinates of \\(A\\), \\(B\\), and \\(C\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-7ff538da0c2417ca457f0167986291b4f3d503bf90aef947d2a056cdbfdcbcae | An ellipse is reflected over one of its tangents. The reflected ellipse rotates around the original ellipse, which remains fixed. Determine the locus of the foci of the rotating ellipse. | [
"Two circles centered at the foci of the fixed ellipse with radius \\( a \\)."
] | [
"An ellipse is reflected over one of its tangents. The reflected ellipse rotates around the original ellipse, which remains fixed. Determine the locus of the foci of the rotating ellipse."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-1722d6c126f306cbc6d03f447ea26b4e3e7629344a847afa7929965b3fba36de | An ellipse is sliding in the coordinate plane such that it always touches the x-axis (OX) and the y-axis (OY). The lengths of the major and minor axes are given as \(2a\) and \(2b\), respectively. Find the locus of the center of the ellipse. | [
"x^2 + y^2 = a^2 + b^2"
] | [
"An ellipse is sliding in the coordinate plane such that it always touches the x-axis (OX) and the y-axis (OY). The lengths of the major and minor axes are given as \\(2a\\) and \\(2b\\), respectively. Find the locus of the center of the ellipse."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-46dcc662af571b97cdad8003535a64cc1aae8c4f480c102f7454df175eac0ae7 | An ellipse is sliding in the coordinate plane such that it always touches the x-axis (OX) and the y-axis (OY). The lengths of the major and minor axes are given as \(2a\) and \(2b\), respectively. Find the locus of the foci of the ellipse. | [
"\\left( x - \\frac{b^2}{x} \\right)^2 + \\left( y - \\frac{b^2}{y} \\right)^2 = 4(a^2 - b^2)"
] | [
"An ellipse is sliding in the coordinate plane such that it always touches the x-axis (OX) and the y-axis (OY). The lengths of the major and minor axes are given as \\(2a\\) and \\(2b\\), respectively. Find the locus of the foci of the ellipse."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-bbe8025e3c2346f33c6ecaaf16a3b48ddf7ed225cfb2b218e60aa804f5d1b991 | An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci. | [
"$\\frac{21}{4}$"
] | [
"An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-108360604df75b9e7d8031d31cba01e0295208e96c1a181c6423f781cb3b872e | An ellipse lies in the closed positive orthant, with foci at $(x_1, y_1)$ and $(x_2, y_2)$. The ellipse touches the coordinate axes at points with abscissa $p$ and ordinate $q$, respectively. Given that the point $(p, q)$ is collinear with the origin and the center of the ellipse, calculate the numerical eccentricity o... | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"An ellipse lies in the closed positive orthant, with foci at $(x_1, y_1)$ and $(x_2, y_2)$. The ellipse touches the coordinate axes at points with abscissa $p$ and ordinate $q$, respectively. Given that the point $(p, q)$ is collinear with the origin and the center of the ellipse, calculate the numerical eccentric... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0a87ea5b2100e5fc642d30866fe074651565d04c4e1574039fea7820338c42f9 | An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(4, 0)$ and tangent to the $y$-axis at $(0, 1).$ Find the distance between the foci of the ellipse. | [
"2\\sqrt{15}"
] | [
"An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(4, 0)$ and tangent to the $y$-axis at $(0, 1).$ Find the distance between the foci of the ellipse."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9cd60161a1944cf8483fd45482c85a9cc80ad2266fa60944892ef3f9b257a3d | An ellipse with a focus at $(6,2)$ is tangent to the positive $x$ axis at $(a,0)$. There is a maximum value of $b$ such that for all $0<a<b$, it is possible for this ellipse to be tangent to the $y$ axis. If this maximal value of $b$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, ... | [
"23"
] | [
"An ellipse with a focus at $(6,2)$ is tangent to the positive $x$ axis at $(a,0)$. There is a maximum value of $b$ such that for all $0<a<b$, it is possible for this ellipse to be tangent to the $y$ axis. If this maximal value of $b$ can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m$ and... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-5357befd58ba588267e1e5cc055bba2902182f755b429cf9388de0daa2372e1c | An ellipse with a major axis of \(20\sqrt{3}\) and a minor axis of 20 slides along the coordinate axes and always remains confined to the first quadrant. Determine the length of the arc described by the locus of the ellipse and the equation of the locus of the center of the ellipse. | [
"x^2 + y^2 = 400"
] | [
"An ellipse with a major axis of \\(20\\sqrt{3}\\) and a minor axis of 20 slides along the coordinate axes and always remains confined to the first quadrant. Determine the length of the arc described by the locus of the ellipse and the equation of the locus of the center of the ellipse."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-5df8c4d04c2524e21be32dc06701daadf35cf478bb7a4709698445c4dd0d7b3c | An ellipse with area $32\pi$ is inscribed in a $20$ by $18$ rectangle. Find the distance between the foci of the ellipse. | [
"\\( 2\\sqrt[4]{28665} \\)"
] | [
"An ellipse with area $32\\pi$ is inscribed in a $20$ by $18$ rectangle. Find the distance between the foci of the ellipse."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-3364333f4016f07752fd9abbe46020e485caac2720cc1827b45a91bad04d370d | An ellipse with equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 +y^2 = 1.$ Then the smallest possible area of the ellipse can be expressed in the form $k \pi.$ Find $k.$ | [
"\\frac{3 \\sqrt{3}}{2}"
] | [
"An ellipse with equation\n\\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 +y^2 = 1.$ Then the smallest possible area of the ellipse can be expressed in the form $k \\pi.$ Find $k.$"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-73b9b454ac704e78548f00c9be8681de127ecb5758c9f09d1fd1c1c20f82a563 | An ellipse with foci $B_0, B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0 = B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0 = C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1 = B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1 = C_0... | [
"2"
] | [
"An ellipse with foci $B_0, B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0 = B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0 = C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1 = B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1 ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-0d73129f239fbba80849bcee1be7f55339b847f246f39996a6e26e27b41c163e | An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Let $A$, $B$, $C$, and $D$ be the vertices of this quadrilateral. If the equation of the conic that circumsc... | [
"0"
] | [
"An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Let $A$, $B$, $C$, and $D$ be the vertices of this quadrilateral. If the equation of the conic that cir... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-4b8b08000efd56c9ed7ad7424b7241f16949273a9f4461a67b282a35bbb7c83d | An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square. | [
"\\dfrac{689}{2}"
] | [
"An ellipse with major and minor axes $20$ and $17$, respectively, is inscribed in a square whose diagonals coincide with the axes of the ellipse. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-e28e99559994dec6b39a163ab937be5d0907f86a87b08000f33807a955601479 | An ellipse with semi-axes lengths \(a\) and \(b\) rolls without slipping on the curve \(y = c \sin(x/a)\). The ellipse completes one revolution per period of the sine curve. Find the relationship between \(a\), \(b\), and \(c\). | [
"\\(b^2 = a^2 + c^2\\)"
] | [
"An ellipse with semi-axes lengths \\(a\\) and \\(b\\) rolls without slipping on the curve \\(y = c \\sin(x/a)\\). The ellipse completes one revolution per period of the sine curve. Find the relationship between \\(a\\), \\(b\\), and \\(c\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-8349cdebe50a61c7190a6e840bb2bc357ac58ac01fcd45ab2ed2a888e367067e | An ellipse with semi-axes of lengths $a$ and $b$ rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. Determine the relationship between $a$, $b$, and $c$ given that the ellipse completes one revolution when it traverses one period of the curve. | [
"\\(c^2 = b^2 - a^2\\)"
] | [
"An ellipse with semi-axes of lengths $a$ and $b$ rolls without slipping on the curve $y=c\\sin{\\left(\\frac{x}{a}\\right)}$. Determine the relationship between $a$, $b$, and $c$ given that the ellipse completes one revolution when it traverses one period of the curve."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-155959ef93e7a94e136da141544e35b1d922014d749291ce96df922c86b8be2a | An ellipsoid has 3 axes. Another ellipsoid has an axis that is half the axis of the first ellipsoid’s first axis, and has a second axis that is $\frac{2}{5}$ the axis of the first ellipsoid’s second axis. If the second ellipsoid has ten times the volume of the first ellipsoid, what is the ratio of the second ellipsoid’... | [
"50"
] | [
"An ellipsoid has 3 axes. Another ellipsoid has an axis that is half the axis of the first ellipsoid’s first axis, and has a second axis that is $\\frac{2}{5}$ the axis of the first ellipsoid’s second axis. If the second ellipsoid has ten times the volume of the first ellipsoid, what is the ratio of the second elli... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2779e58dee48e9ff723fd6dfc5898e0c8ceac91affe768edfc5970a7c7cd2407 | An elliptical garden has a perimeter of 30 meters and an area of 40 square meters. Calculate the semi-major and semi-minor axes of the garden. | [
"1.83"
] | [
"An elliptical garden has a perimeter of 30 meters and an area of 40 square meters. Calculate the semi-major and semi-minor axes of the garden."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-f973cdb5f4feeaff9a63e9c8b2ad3ebbf727b684575ad0ae0b58dd01fc097431 | An email will be sent between 1pm and 3pm. The recipient will check their email between 2pm and 3pm. What is the probability that the recipient will check their email after it has been sent? | [
"\\(\\frac{3}{4}\\)"
] | [
"An email will be sent between 1pm and 3pm. The recipient will check their email between 2pm and 3pm. What is the probability that the recipient will check their email after it has been sent?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-d0ef694905ce9adaf710e3d49d6c0f87168ad209dc647267853a1156be2b6855 | An employee contributes $200 into a superannuation fund at the end of each year. If the interest rate on this fund is 11.5% per annum, how much will the employee have at the end of 20 years? | [
"22299.21"
] | [
"An employee contributes $200 into a superannuation fund at the end of each year. If the interest rate on this fund is 11.5% per annum, how much will the employee have at the end of 20 years?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-251c066c432f26b49338777b46890c1cd22379597f38db152d5e342af011055c | An employee works 80 hours per pay period at a regular rate of $18.50 per hour, and any hours worked beyond 80 are paid at 1.5 times the regular rate. The goal is to adjust the regular hourly rate so that the total pay per pay period is reduced by exactly $200.00. Determine the new regular hourly rate that achieves thi... | [
"16.00"
] | [
"An employee works 80 hours per pay period at a regular rate of $18.50 per hour, and any hours worked beyond 80 are paid at 1.5 times the regular rate. The goal is to adjust the regular hourly rate so that the total pay per pay period is reduced by exactly $200.00. Determine the new regular hourly rate that achieve... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0b0052f38c4c5db27595f4fa144faf0917a98952f94b4a6183a92da99de2cac | An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions: - The two $1 \times 1$ faces of each beam coi... | [
"2020^2",
"3030"
] | [
"An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\n- The two $1 \\times 1$ faces of e... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 14
} | 31 | 0 | 0 |
omr-ws-59aa46cb6372284869e60101d41ccd82d5abb590e6332e64060e6d7410ccc9d7 | An empty tank is being filled through two pipelines simultaneously with pure water and a constant concentration of an acid solution. When filled, the tank has a 5\% solution of the acid. If when the tank were half full, the water supply were cut off, the full tank would have a 10\% solution of the acid. Determine which... | [
"The water is delivered twice as fast"
] | [
"An empty tank is being filled through two pipelines simultaneously with pure water and a constant concentration of an acid solution. When filled, the tank has a 5\\% solution of the acid. If when the tank were half full, the water supply were cut off, the full tank would have a 10\\% solution of the acid. Determin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-018bd3817abaaa3607e92ef78787024ea0984e96b71e89c01991e651b0cd01d3 | An empty, inverted circular cone has a radius of $5$ meters and a height of $20$ meters. At time $t = 0$ seconds, the cone is empty, and at time $t \ge 0$ we fill the cone with water at a rate of $4t^2$ cubic meters per second. Compute the rate of change of the height of water with respect to time, at the point when th... | [
"\\(\\frac{1}{\\sqrt[3]{\\pi}}\\)"
] | [
"An empty, inverted circular cone has a radius of $5$ meters and a height of $20$ meters. At time $t = 0$ seconds, the cone is empty, and at time $t \\ge 0$ we fill the cone with water at a rate of $4t^2$ cubic meters per second. Compute the rate of change of the height of water with respect to time, at the point w... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-cc472d93abb727868b325f222687ae1976f983838a49340f37b86b8541e63754 | An encyclopedia consists of 2000 numbered volumes stacked in order with volume 1 on top and volume 2000 at the bottom. Two operations can be performed: (i) For even \( n \), take the top \( n \) volumes and place them at the bottom of the stack without changing their order. (ii) For odd \( n \), take the top \( n \) vo... | [
"(1000!)^2"
] | [
"An encyclopedia consists of 2000 numbered volumes stacked in order with volume 1 on top and volume 2000 at the bottom. Two operations can be performed:\n(i) For even \\( n \\), take the top \\( n \\) volumes and place them at the bottom of the stack without changing their order.\n(ii) For odd \\( n \\), take the t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-b5f4c3bf708f27fb46cb605009a5953eb5d0955d58cf350ccf5ce1cf6e344df7 | An enemy jet flying level passes a battleship 50 miles due east and travels at 1,000 miles per hour due north. How quickly (in radians per hour) is the anti-aircraft laser turning when the jet is closest to the ship? | [
"20 rad/hr"
] | [
"An enemy jet flying level passes a battleship 50 miles due east and travels at 1,000 miles per hour due north. How quickly (in radians per hour) is the anti-aircraft laser turning when the jet is closest to the ship?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e41ee7c81d233f703b941e3a28dfd9836094452903b72d629e9bff1836b06464 | An enemy submarine is submerged at an integer position $A$ on the number line and moves $B$ units to the right every hour, where $B$ is an integer. You can fire a torpedo to any integer position on the number line at the top of every hour. How can you guarantee eventually sinking the submarine? | [
"\\text{Enumerate all integer pairs } (A, B) \\text{ and fire at position } A + Bt \\text{ at time } t"
] | [
"An enemy submarine is submerged at an integer position $A$ on the number line and moves $B$ units to the right every hour, where $B$ is an integer. You can fire a torpedo to any integer position on the number line at the top of every hour. How can you guarantee eventually sinking the submarine?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-1ef2e2d43fd3fe465b364103f87f54e4cd18f78a129042bfd33c174f6a2233a6 | An engineer has two identical airbags to test on a 75-floor building to find the highest floor from which the airbag won't burst when dropped. What is the least number of drops required to determine this floor? | [
"12"
] | [
"An engineer has two identical airbags to test on a 75-floor building to find the highest floor from which the airbag won't burst when dropped. What is the least number of drops required to determine this floor?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1baa61c9e3b7cbea0aba172ece340ea2b6bf04035f4a6e1e78f4d894bf6bf048 | An engineer invested $\$10,\!000$ in a six-month savings certificate that paid a simple annual interest rate of $12\%$. After six months, she invested the total value of her investment in another six-month certificate. After six more months, the investment was worth $\$11,\!130$. If the annual interest rate of the seco... | [
"10"
] | [
"An engineer invested $\\$10,\\!000$ in a six-month savings certificate that paid a simple annual interest rate of $12\\%$. After six months, she invested the total value of her investment in another six-month certificate. After six more months, the investment was worth $\\$11,\\!130$. If the annual interest rate o... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd2d4b0c0a6654461f12cf33fe7528fbda8d9accebd0ce28b563d54afab6b2c3 | An engineer is building a straight road, 16 feet wide with negligible depth, and needs to measure the length to calculate a budget. From a 250-foot tall tower, she observes the following: 1. Turning 45° to the left and looking 60° downwards, her line of sight goes toward the origin city. 2. Turning 30° to the left and ... | [
"318100"
] | [
"An engineer is building a straight road, 16 feet wide with negligible depth, and needs to measure the length to calculate a budget. From a 250-foot tall tower, she observes the following:\n1. Turning 45° to the left and looking 60° downwards, her line of sight goes toward the origin city.\n2. Turning 30° to the le... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-4f859d1d933e50f0c2feadd44d40c7b905e52608d1e164418ee1828a5fa9e604 | An engraving measuring $18$ by $24$ will be placed on a wooden frame with the largest dimension being vertical. The width of the frame on the sides is half the width of the frame on the top and bottom. If the frame area is equal to the engraving area, what is the ratio of the smallest to the largest frame size? | [
"\\(\\frac{2}{3}\\)"
] | [
"An engraving measuring $18$ by $24$ will be placed on a wooden frame with the largest dimension being vertical. The width of the frame on the sides is half the width of the frame on the top and bottom. If the frame area is equal to the engraving area, what is the ratio of the smallest to the largest frame size?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-2577cabf3f4901fb9b4d64969082400d34fe28485efe7f08baaad545d9649369 | An ensemble has a violin, a viola, and a cello. A chord has 3 notes and each instrument can play $1$ or $2$ of them at a time. How many ways can they play the chord if every note in the chord must be played? (Octaves don’t matter.) | [
"343"
] | [
"An ensemble has a violin, a viola, and a cello. A chord has 3 notes and each instrument can play $1$ or $2$ of them at a time. How many ways can they play the chord if every note in the chord must be played? (Octaves don’t matter.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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