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-83cb910407159e913f11ecf0190ba1ae33ee431585442424ceb1586fd6a65de7 | A ball of radius 14 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid. | [
"\\( 4 \\pi \\left(4 \\sqrt{33} + 49 \\arcsin\\left[\\frac{\\sqrt{33}}{7}\\right]\\right) \\)"
] | [
"A ball of radius 14 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43638132150491f4f2079d8b3ba127a44a69da8bed809fcad6969e17dd3d847a | A ball of radius 3 inches is rolled along the line $y=3x$ in the first quadrant until it strikes a vertical wall along the line $y=20-x$. Determine the $x$ coordinate of the point on the line $y=3x$ which lies directly below the point of contact of the ball with the wall when the ball strikes the wall. | [
"\\(\\frac{20+3\\sqrt{2}}{4}\\)"
] | [
"A ball of radius 3 inches is rolled along the line $y=3x$ in the first quadrant until it strikes a vertical wall along the line $y=20-x$. Determine the $x$ coordinate of the point on the line $y=3x$ which lies directly below the point of contact of the ball with the wall when the ball strikes the wall."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-d8ecdc6c4e4ddf246b56e47c6738769f8a97ad88d41292762128efdac5384ff4 | A ball of radius \( R \) is tangent to the floor and one wall of the room. Find, in terms of \( R \), the radius of the largest sphere that can be rolled through the space between the ball, the wall, and the floor. | [
"\\( R(\\sqrt{2} - 1) \\)"
] | [
"A ball of radius \\( R \\) is tangent to the floor and one wall of the room. Find, in terms of \\( R \\), the radius of the largest sphere that can be rolled through the space between the ball, the wall, and the floor."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ab90b51b348af1c5de2793fc157862f6eb8e69b061136f3b4ec18216d369297 | A ball of radius \( r \) is cut into two pieces by a horizontal plane \( a \) units above the center of the ball. Determine the volume of the upper piece using the shell method. | [
"\\(\\frac{2\\pi(r^\\frac{3}{2} - (r^2 - a^2)^\\frac{3}{2})}{3} - \\pi a^3\\)"
] | [
"A ball of radius \\( r \\) is cut into two pieces by a horizontal plane \\( a \\) units above the center of the ball. Determine the volume of the upper piece using the shell method."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-420ca1b7c1a267fc3fdde61e4564ca14742191f05b2b6137ad6b9d6eca63297d | A ball projected vertically upward with an initial speed \( v_0 \) ft/sec is at time \( t \) sec at a distance \( s \) ft from the point of projection as given by the formula \( s = v_0 t - 16 t^2 \). (i) If the ball is given an initial upward speed of 128 ft/sec, at what time will it be 100 ft above the point of proje... | [
"\\( t = \\frac{8 + \\sqrt{39}}{2} \\) or \\( t = \\frac{8 - \\sqrt{39}}{2} \\)"
] | [
"A ball projected vertically upward with an initial speed \\( v_0 \\) ft/sec is at time \\( t \\) sec at a distance \\( s \\) ft from the point of projection as given by the formula \\( s = v_0 t - 16 t^2 \\).\n(i) If the ball is given an initial upward speed of 128 ft/sec, at what time will it be 100 ft above the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 10
} | 17 | 0 | 0 |
omr-ws-190912ce92e15d9c02e043b5cc1f3faea1d7195ac285317000b8ec6785c986d6 | A ball rolls along a foot-ball field at right angles to its side line. Assume that moving with uniform deceleration the ball rolls 4 m in the first second and 0.75 m less than during the preceding second for every subsequent second. A player who is at a distance of 10 m from the ball runs in the same direction as the h... | [
"In 5 s 0.5 m short of the boundary of the field"
] | [
"A ball rolls along a foot-ball field at right angles to its side line. Assume that moving with uniform deceleration the ball rolls 4 m in the first second and 0.75 m less than during the preceding second for every subsequent second. A player who is at a distance of 10 m from the ball runs in the same direction as ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e38fc6d5639e8275c27b0887770244c9dc7252d98c23ed47e9a52857e0a43998 | A ball starts at a height of 1024 meters and bounces such that each bounce reaches half the height of the previous bounce. What is the total distance the ball travels before it reaches a height of 4 meters or less? | [
"3056"
] | [
"A ball starts at a height of 1024 meters and bounces such that each bounce reaches half the height of the previous bounce. What is the total distance the ball travels before it reaches a height of 4 meters or less?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-ee3992eb0509b130756f82c4d38160fcb66f5a3c7a1bc0e8cfe1ecbd9125a318 | A ball starts at a height of 124 meters and bounces such that each bounce reaches half the height of the previous bounce. What total distance will the ball cover before it reaches a height of 4 meters or less? | [
"356.5"
] | [
"A ball starts at a height of 124 meters and bounces such that each bounce reaches half the height of the previous bounce. What total distance will the ball cover before it reaches a height of 4 meters or less?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3bd7c6ee678d1db787264097b2acd191f0faa34c4b0ebfd212cd258f8dfafff0 | A ball starts in the middle of a 3x4 unit room and rolls at a 45-degree angle towards the wall. It continues straight until it hits a wall and bounces off at a 45-degree angle. After traveling 20 units, how many times has the ball hit the wall? | [
"10"
] | [
"A ball starts in the middle of a 3x4 unit room and rolls at a 45-degree angle towards the wall. It continues straight until it hits a wall and bounces off at a 45-degree angle. After traveling 20 units, how many times has the ball hit the wall?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1356f4a814ea12159cac7e0eb447a98bf6b89a998d4c6fb96ccd9c120160a376 | A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+32t+15$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | [
"31"
] | [
"A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+32t+15$, where $t$ is the time after launch. What is the maximum height of the ball, in feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96e67c416ea55bfdc93734d8bcd315fc1f7e6525c39333a3e16d7d35c6885365 | A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+64t+31$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | [
"95"
] | [
"A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+64t+31$, where $t$ is the time after launch. What is the maximum height of the ball, in feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c909a017bbf0ec298b26e0ec7b4b7f08803578de2d03704d84cfba74f9981ea | A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+80t+21$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | [
"121"
] | [
"A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+80t+21$, where $t$ is the time after launch. What is the maximum height of the ball, in feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb8013dd6b6457fd132b299d39956d0c58a2f074d85700c32781bdafecbb4eec | A ball was floating in a lake when the lake froze. The ball was removed, leaving a hole 24 cm across at the top and 8 cm deep. What was the radius of the ball in centimeters? | [
"13"
] | [
"A ball was floating in a lake when the lake froze. The ball was removed, leaving a hole 24 cm across at the top and 8 cm deep. What was the radius of the ball in centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-dbda960eb44b2c70014fddb078adcd1d8c7e74a7c0677e10c01ebedc9747a411 | A ball was launched on a rectangular billiard table at an angle of $45^\circ$ to one of the sides. It reflected from all sides (the angle of incidence is equal to the angle of reflection) and returned to its original position. It is known that one of the sides of the table has a length of one meter. Find the length of ... | [
"1 meter"
] | [
"A ball was launched on a rectangular billiard table at an angle of $45^\\circ$ to one of the sides. It reflected from all sides (the angle of incidence is equal to the angle of reflection) and returned to its original position. It is known that one of the sides of the table has a length of one meter. Find the leng... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-3b3be2d601dd2f07c3a70ddfde0b348bdcf22b3685d2eec5f4ad534b85534091 | A ball was removed from a frozen lake, leaving a hole 34 cm across at the top and 8 cm deep. Find the radius of the ball. | [
"\\(\\frac{353}{16}\\)"
] | [
"A ball was removed from a frozen lake, leaving a hole 34 cm across at the top and 8 cm deep. Find the radius of the ball."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-42d941a36450a3f4a8f026e42e1fd46a0f65cc1d9842638dbb88abb417805f7c | A ball with a diameter of 4 inches starts at point A and rolls along a track comprised of three semicircular arcs with radii \( R_1 = 100 \) inches, \( R_2 = 60 \) inches, and \( R_3 = 80 \) inches, respectively. The ball remains in contact with the track and does not slip. What is the distance the center of the ball t... | [
"\\( 238\\pi \\)"
] | [
"A ball with a diameter of 4 inches starts at point A and rolls along a track comprised of three semicircular arcs with radii \\( R_1 = 100 \\) inches, \\( R_2 = 60 \\) inches, and \\( R_3 = 80 \\) inches, respectively. The ball remains in contact with the track and does not slip. What is the distance the center of... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7eeff4d1ffc0ef35f47a3a2e910afe1d8df695dfaf6542a68e4909225fdc184 | A ball with a diameter of 4 inches starts at point A and rolls along a track consisting of three semicircular arcs with radii \( R_1 = 100 \) inches, \( R_2 = 60 \) inches, and \( R_3 = 80 \) inches, respectively. The ball remains in contact with the track and does not slip. What is the distance traveled by the center ... | [
"\\( 238\\pi \\)"
] | [
"A ball with a diameter of 4 inches starts at point A and rolls along a track consisting of three semicircular arcs with radii \\( R_1 = 100 \\) inches, \\( R_2 = 60 \\) inches, and \\( R_3 = 80 \\) inches, respectively. The ball remains in contact with the track and does not slip. What is the distance traveled by ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22c048d09159eed8c1646dae93267259ad42557c59094b5182103aea0eafb440 | A ball with radius 1 is placed tangent to a wall and the floor. Another ball, just large enough to fit under the first ball, is placed under the first ball, so that it is tangent to the ball, the wall, and the floor. This process is continued indefinitely with an infinite number of balls. What is the area (two-dimensio... | [
"\\(\\frac{4\\sqrt{2}+8+(6-7\\sqrt{2})\\pi}{32}\\)"
] | [
"A ball with radius 1 is placed tangent to a wall and the floor. Another ball, just large enough to fit under the first ball, is placed under the first ball, so that it is tangent to the ball, the wall, and the floor. This process is continued indefinitely with an infinite number of balls. What is the area (two-dim... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6417eaeea81c0455d36773360a42a6628431ee6253fc0548eedd73dd3e285675 | A ball with radius \( R \) is filled with water at a constant rate of \( X \) litres per second. Determine how the water height \( H \) in the ball varies with time. | [
"R H^2 - \\frac{1}{3} H^3 = \\frac{X}{\\pi} t"
] | [
"A ball with radius \\( R \\) is filled with water at a constant rate of \\( X \\) litres per second. Determine how the water height \\( H \\) in the ball varies with time."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-5b3e5fdf952a6d4f8c48705d1ff6eca80b6298745957f4d594fcc05165d4c8da | A balloon initially moving at 5 m/sec has a total mass of 210 kg. After a body mass of 70 kg falls from the balloon, find the distance the balloon travels in 5 seconds after the body falls. | [
"\\(\\frac{100}{3}\\) meters"
] | [
"A balloon initially moving at 5 m/sec has a total mass of 210 kg. After a body mass of 70 kg falls from the balloon, find the distance the balloon travels in 5 seconds after the body falls."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bd9d8d1bb39690024b0ca9df07c892e54ca260fb72053022f7271249da4cbef | A balloon is observed simultaneously from three points A, B, and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then what is the height of the balloon? | [
"100\\sqrt{3}",
"\\( 100\\sqrt{3} \\)"
] | [
"A balloon is observed simultaneously from three points A, B, and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then what is the height of the balloon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 7
} | 37 | 0 | 0 |
omr-ws-3c0b27145a2aefd2009bb514eb31dfa7043a66f03b37d0e0dd75226a571a103e | A balloon is rising at a constant speed of 2 m/s. A boy is cycling along a straight road at a speed of 5 m/s. When he passes under the balloon, it is 15 m above him. How fast is the distance between the boy and the balloon increasing 3 seconds later? | [
"\\(\\frac{39}{\\sqrt{74}}\\) metres per second or approximately \\(4.53\\) metres per second."
] | [
"A balloon is rising at a constant speed of 2 m/s. A boy is cycling along a straight road at a speed of 5 m/s. When he passes under the balloon, it is 15 m above him. How fast is the distance between the boy and the balloon increasing 3 seconds later?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-49c61e6db48661dff3db0f0f112491b409e90ef6e34a2ba401f9cb714751bcdf | A balloon, when the volume is higher than 10 cm$^3$, will lower the rate of change of the volume to 8/$v$ cm$^3$/min. What is the rate of change of the radius when the radius is 2 cm? | [
"\\(\\frac{3}{64 \\pi^2}\\)"
] | [
"A balloon, when the volume is higher than 10 cm$^3$, will lower the rate of change of the volume to 8/$v$ cm$^3$/min. What is the rate of change of the radius when the radius is 2 cm?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e006feb99095026c354773d2bb25dbdcad4948e5e5f6c5148e4c99a3946a9cc4 | A band has 72 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 5 and 20 students per row, in how many possible row-lengths can the band be arranged? | [
"5"
] | [
"A band has 72 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 5 and 20 students per row, in how many possible row-lengths can the band be arranged?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c80417e8d09106a40469dec4d3f263c4ab583e7959f9c902165033763802e9e2 | A band of radius $1$ is placed in a corner of a room such that it touches the horizontal floor and two vertical walls perpendicular to each other. Find the locus of the center of the band for all of its possible positions. | [
"\\((x-1)^2 + (y-1)^2 = 1\\) for \\(x \\geq 1\\) and \\(y \\geq 1\\)"
] | [
"A band of radius $1$ is placed in a corner of a room such that it touches the horizontal floor and two vertical walls perpendicular to each other. Find the locus of the center of the band for all of its possible positions."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15a11ae0d81875f3ad35ede7864b12f7d1d3ff63e5e8bd4ef6f7891cc24a3750 | A bank PIN number has four digits where the leading digit cannot be zero, the four digits cannot be in sequence, and all four digits cannot be identical. What is the probability that a thief would guess the number in three attempts? | [
"\\dfrac{3}{8978}"
] | [
"A bank PIN number has four digits where the leading digit cannot be zero, the four digits cannot be in sequence, and all four digits cannot be identical. What is the probability that a thief would guess the number in three attempts?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c968e3cafe0f6b40be3785b9a20c3dd7987770ffd57e92d9f5ced082a8a9b5a | A bank charges $6 for a loan of $120. The borrower receives $114 and repays the loan in 12 installments of $10 a month. The interest rate is approximately how much? Explain how and where the value $59 is derived in the approximate interest calculation. | [
"10%"
] | [
"A bank charges $6 for a loan of $120. The borrower receives $114 and repays the loan in 12 installments of $10 a month. The interest rate is approximately how much? Explain how and where the value $59 is derived in the approximate interest calculation."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 7,
"tir": 15
} | 60 | 0 | 0 |
omr-ws-1659da09059b3ceaabe5e6da3736cf42d7df226d4b86fe24a45f9cb08381fce2 | A bank charges $6 for a loan of $120. The borrower receives $114 and repays the loan in 12 installments of $10 a month. What is the approximate interest rate? | [
"9%"
] | [
"A bank charges $6 for a loan of $120. The borrower receives $114 and repays the loan in 12 installments of $10 a month. What is the approximate interest rate?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-fefc18ccfb57d703e27bb5782b1604024c5c37755c1a1b184912f7ba513a2f69 | A bank compounds money annually such that the amount for each year is equal to the amount of the previous year minus the amount from the year before that plus a fixed interest of 2013 pesos. If today is the kth year and the current amount is ${2011}^2$ pesos, and the amount in year 2 was 4 pesos, and the sum of all ind... | [
"2013^{2012} + 2011"
] | [
"A bank compounds money annually such that the amount for each year is equal to the amount of the previous year minus the amount from the year before that plus a fixed interest of 2013 pesos. If today is the kth year and the current amount is ${2011}^2$ pesos, and the amount in year 2 was 4 pesos, and the sum of al... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-a5477b43d139384f8c4c0bf9173f468311a2bb561219d509c26d077bbfec67fa | A bank discounted a 3-month, \$2,200, noninterest-bearing note for Jenny Woods. The discount rate was 15.5%, and Jenny received \$2,114.75 as proceeds. What true rate of interest, to the nearest tenth percent, did Jenny pay on the note? | [
"D\". The options provided are:\n\na. 16.0\nb. 17.0\nc. 18.0\nd. 16.1\n\nSince \"D\" corresponds to 16.1, the true rate of interest Jenny paid on the note is 16.1%."
] | [
"A bank discounted a 3-month, \\$2,200, noninterest-bearing note for Jenny Woods. The discount rate was 15.5%, and Jenny received \\$2,114.75 as proceeds. What true rate of interest, to the nearest tenth percent, did Jenny pay on the note?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-a087d366297fe87569deed3f69c43ebfe2952e1689cf6a272a8b9fa2709a2033 | A bank plans to offer a savings account at 4.8% simple annual interest. If the bank compounds monthly, what is the annual yield for this savings account? (The annual yield is the amount of interest a given balance would earn when the effect of the compounding interest is included.) Do not round until the last step, and... | [
"4.9%"
] | [
"A bank plans to offer a savings account at 4.8% simple annual interest. If the bank compounds monthly, what is the annual yield for this savings account? (The annual yield is the amount of interest a given balance would earn when the effect of the compounding interest is included.) Do not round until the last step... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-d884f15d0d1aad875956e5cc261676c002705b2c6f1534f2875ddfa0e98f3c79 | A banker has a set of similarly looking golden coins, one of which is counterfeit and lighter in weight. Using a balance without weights, an expert must determine the counterfeit coin with the condition that each coin participates in no more than two weighings. What is the largest number of coins the banker could have ... | [
"12"
] | [
"A banker has a set of similarly looking golden coins, one of which is counterfeit and lighter in weight. Using a balance without weights, an expert must determine the counterfeit coin with the condition that each coin participates in no more than two weighings. What is the largest number of coins the banker could ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-70e33e9bd760c0f7bf53efb4c0f0f52f850ef8106d9d00e65c27765ec195aa2f | A bar code is composed of alternate strips of black and white, always beginning and ending with a black strip. Each strip (of either color) has a width of 1 or 2, and the total width of the bar code is 12. How many different codes are possible, always reading from left to right? | [
"116"
] | [
"A bar code is composed of alternate strips of black and white, always beginning and ending with a black strip. Each strip (of either color) has a width of 1 or 2, and the total width of the bar code is 12. How many different codes are possible, always reading from left to right?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-db78c9c8c6179513b611772d6fb9a8ce4fa9f6db8d64170475d7bef57859ea89 | A bar code is composed of alternating black and white stripes, with the first and last stripes always black. Each stripe can be either 1 or 2 units wide, and the entire bar code has a width of 12. How many different bar codes are possible? | [
"116"
] | [
"A bar code is composed of alternating black and white stripes, with the first and last stripes always black. Each stripe can be either 1 or 2 units wide, and the entire bar code has a width of 12. How many different bar codes are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-df759a32e02ecbf45888f679d014d44f273a79d13ae4a953e072d192b8a3b36a | A bar is broken at random in two places. Find the average size of the largest piece. | [
"\\(\\frac{11}{18}\\)"
] | [
"A bar is broken at random in two places. Find the average size of the largest piece."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 22
} | 32 | 0 | 0 |
omr-ws-01f1012dea574a2baa08093a17fec12264c2efc070be0b827ffa8f34bf14d5f1 | A bar is broken at random in two places. Find the average size of the middle-sized piece. | [
"\\(\\frac{5}{18}\\)"
] | [
"A bar is broken at random in two places. Find the average size of the middle-sized piece."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-16bb671c105ca65a8c1fdab33956747457c3eefdb581d17d1860360d1217403b | A bar is broken at random in two places. Find the average size of the smallest piece. | [
"\\(\\frac{1}{9}\\)"
] | [
"A bar is broken at random in two places. Find the average size of the smallest piece."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-5caf1b0287db7aa0586bbbf0f3bed79d1682cb607d545711402238ac6fa3b58f | A bar is randomly broken into \( n \) pieces by breaking it at \( n-1 \) random points. Let \( L_i \) denote the expected length of the \( i \)-th smallest piece. Find the value of \( L_3 \) when \( n = 5 \). | [
"\\dfrac{47}{300}"
] | [
"A bar is randomly broken into \\( n \\) pieces by breaking it at \\( n-1 \\) random points. Let \\( L_i \\) denote the expected length of the \\( i \\)-th smallest piece. Find the value of \\( L_3 \\) when \\( n = 5 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9ba2be1c620b206ac68aa773623924d638b753a8b1bfc13c39bcf5c9b9b4c653 | A bar of length 10 meters is to be cut into 3 pieces. What is the probability that none of the pieces are greater than 4 meters in length? | [
"\\(\\frac{1}{25}\\)"
] | [
"A bar of length 10 meters is to be cut into 3 pieces. What is the probability that none of the pieces are greater than 4 meters in length?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-cc30703e79ac0a5bdc80f7329bc82749b07547230dc236d1298a9cbb749f05af | A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip of either color has a width of $1$ or $2$. The total width of the barcode is $12$. How many distinct barcodes are possible? | [
"116"
] | [
"A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip of either color has a width of $1$ or $2$. The total width of the barcode is $12$. How many distinct barcodes are possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-2863a16fe64039ce7b755b68a4702e9f295384a6bd69e87d9099656dd90674d6 | A barge with 600 tons of goods was unloaded in three days, $2/ 3$ of the goods being unloaded during the first and third days. The amount of goods unloaded during the second day was less than that unloaded on the first day, and the amount unloaded on the third day was less than that unloaded on the second day. The diff... | [
"250 tons was unloaded during the first day, 200 tons during the second, and 150 tons during the third"
] | [
"A barge with 600 tons of goods was unloaded in three days, $2/ 3$ of the goods being unloaded during the first and third days. The amount of goods unloaded during the second day was less than that unloaded on the first day, and the amount unloaded on the third day was less than that unloaded on the second day. The... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d63d3a9ee89fa25368ee28f4a97324adcbac77defbb8ed3f1cdbc5770a2a25a1 | A barn with a flat roof is rectangular in shape, 10 yd. wide, 13 yd. long, and 5 yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. What is the total number of square yards to be painted? | [
"590"
] | [
"A barn with a flat roof is rectangular in shape, 10 yd. wide, 13 yd. long, and 5 yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. What is the total number of square yards to be painted?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-a9ec4e803e39188c39bfb73eaee9daf449ceef7476eedde72bf5038c35834b28 | A base \( 3 \) number is called \(\text{fun}\) if it has no three consecutive \( 0 \)'s. Find the number of \( 7 \)-digit base \( 3 \) fun numbers. | [
"1296"
] | [
"A base \\( 3 \\) number is called \\(\\text{fun}\\) if it has no three consecutive \\( 0 \\)'s. Find the number of \\( 7 \\)-digit base \\( 3 \\) fun numbers."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-38e1173a6323bb4bc2398ce74f99cb1ce20490525b4a6e94073dfe2264c72079 | A base two numeral consists of 15 digits all of which are ones. This number, when tripled and written in base two, contains how many digits? | [
"17"
] | [
"A base two numeral consists of 15 digits all of which are ones. This number, when tripled and written in base two, contains how many digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-58603a5de01188fc1554f215857aab67c05583c418e2a0820439a4eef4028a6d | A base two numeral consists of 15 digits, all of which are ones. When this number is tripled and written in base two, how many digits does the resulting number contain? | [
"18"
] | [
"A base two numeral consists of 15 digits, all of which are ones. When this number is tripled and written in base two, how many digits does the resulting number contain?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a82953f1169414a83b3320b249293d881444fe1cfff02341aaecaedf0f9c4341 | A base-$10$ three-digit number $n$ is selected at random. What is the probability that the base-$9$ representation and the base-$11$ representation of $n$ are both three-digit numerals? | [
"0.7"
] | [
"A base-$10$ three-digit number $n$ is selected at random. What is the probability that the base-$9$ representation and the base-$11$ representation of $n$ are both three-digit numerals?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df13e0bfd84abb021becdf775e52a68a5d5786bdb9467e5817e785490dfa4c2a | A base-10 three-digit number \( n \) is selected at random. What is the probability that the base-9 representation and the base-11 representation of \( n \) are both three-digit numbers? | [
"0.7"
] | [
"A base-10 three-digit number \\( n \\) is selected at random. What is the probability that the base-9 representation and the base-11 representation of \\( n \\) are both three-digit numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8707a7fe3ba34bf55d0e04cbe76e57a8dd0d50bc524b69b5de271795af26d2f9 | A base-10 three-digit number \( n \) is selected at random. What is the probability that the base-9 representation and the base-11 representation of \( n \) are both three-digit numerals? | [
"0.7"
] | [
"A base-10 three-digit number \\( n \\) is selected at random. What is the probability that the base-9 representation and the base-11 representation of \\( n \\) are both three-digit numerals?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a1dfe1f435852be669606df948bde3a1cc986a346dc3a4abe8e782b18353beb | A base-two numeral consists of 15 digits, all of which are ones. When tripled and written in base two, how many digits does this number contain? | [
"17"
] | [
"A base-two numeral consists of 15 digits, all of which are ones. When tripled and written in base two, how many digits does this number contain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-c6734cda094e2b6d9df81a1b961bbc82a730eaee9212d6d286607e60bf1bf82c | A baseball diamond consists of a diamond with a side length of $30$ meters in a quarter-circle with a radius of $102$. What is the longest possible throw to second base (opposite vertex of the diamond from home base)? | [
"78"
] | [
"A baseball diamond consists of a diamond with a side length of $30$ meters in a quarter-circle with a radius of $102$. What is the longest possible throw to second base (opposite vertex of the diamond from home base)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-23923afb380465bf757585b99ab03e3ab9c5f44ae5e600d6d0a7cd2c797e23df | A baseball is cut in half through the center. If the area of the circular region formed by the cut is increased by 800%, then the volume of the baseball would be increased by K%. Compute K. | [
"2600"
] | [
"A baseball is cut in half through the center. If the area of the circular region formed by the cut is increased by 800%, then the volume of the baseball would be increased by K%. Compute K."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-c63614e1217978ecd689a4ccc708b1fff0d5e86863c51c0db6c086d173cee0b3 | A baseball is hit 4 feet above the ground at a speed of 80 feet per second at an angle of 60 degrees with respect to the horizontal. How high did the baseball go? | [
"79 feet"
] | [
"A baseball is hit 4 feet above the ground at a speed of 80 feet per second at an angle of 60 degrees with respect to the horizontal. How high did the baseball go?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-a66d455cc81b2e313c2934c5beb60735fa721b45d13d09dc42d1ae9b4d974494 | A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \( g = 32 \) ft/sec². Calculate the range and fligh... | [
"Range = 275.6 ft, Flight time = 2.652 sec"
] | [
"A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \\( g = 32 \\) ft/sec². Calculate the range an... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84f674f112f0bb80169dfb3c02b51f6b254935af116536b8c692b76d7f65575d | A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \( g = 32 \) ft/sec². Determine the maximum height ... | [
"Maximum height = 90.988 ft, Time to reach maximum height = 1.32 sec"
] | [
"A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \\( g = 32 \\) ft/sec². Determine the maximum ... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-25dbf0a6a49e9a64e917d9bafae20ba4032881441f5aa927d5c7ae13ed41197d | A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \( g = 32 \) ft/sec². Determine when the baseball i... | [
"\\( t \\approx 0.323 \\text{ sec}, 2.323 \\text{ sec} \\) and \\( x \\approx 33.53 \\text{ ft}, 241.47 \\text{ ft} \\)"
] | [
"A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \\( g = 32 \\) ft/sec². Determine when the bas... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 19
} | 23 | 0 | 0 |
omr-ws-d38ae562d2f266b7490e4a24797a12bc4c0545549bae9b8857dd78b25c05fcaa | A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \( g = 32 \) ft/sec². Find a vector equation for th... | [
"\\(\\vec{r}(t) = \\left ( 103.917 t, -16 t^2 + 42.3239 t + 3 \\right )\\)"
] | [
"A baseball is hit when it is 3 ft above ground. It leaves the bat with an initial velocity of 130 ft/sec at a launch angle of 19 degrees. An instantaneous gust of wind adds a component of -19i (ft/sec) to the ball's initial velocity. The acceleration due to gravity is \\( g = 32 \\) ft/sec². Find a vector equation... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3cc3a6ed271824b6891be50a5e7c944ec3d478c4410cbc718bdde4db1e65252d | A baseball is hit when it is 3 ft above the ground. It leaves the bat with an initial speed of 127 ft/sec, making an angle of 13 degrees with the horizontal. Assume there is linear drag with a drag coefficient of \( k = 0.25 \) and an instantaneous gust of wind that adds a component of \(-13.7 \mathbf{i}\) (ft/sec). Th... | [
"\n\\left[ 549.78 \\left(1 - e^{-0.25t}\\right) - 13.7t \\right] \\mathbf{i} + \\left[ 3 + 626.28 \\left(1 - e^{-0.25t}\\right) - 128t \\right] \\mathbf{j}\n"
] | [
"A baseball is hit when it is 3 ft above the ground. It leaves the bat with an initial speed of 127 ft/sec, making an angle of 13 degrees with the horizontal. Assume there is linear drag with a drag coefficient of \\( k = 0.25 \\) and an instantaneous gust of wind that adds a component of \\(-13.7 \\mathbf{i}\\) (f... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-a0f0c25e5d6544c47ad4caab37e5b80ecc0350ba709ba9d995b2c3d5e550345e | A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games and every team in the other division $M$ games, with $N > 2M$ and $M > 4$. Each team plays a total of 76 games. How many games does a team play within its own division? | [
"16",
"48"
] | [
"A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games and every team in the other division $M$ games, with $N > 2M$ and $M > 4$. Each team plays a total of 76 games. How many games does a team play within its own division?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 8,
"tir": 6
} | 35 | 0 | 0 |
omr-ws-c2bdc356e31b25228c59b25059daa9bc4062abe5a43ec1d8b6a713eb8248d710 | A baseball league has nine teams. During the season, each of the nine teams plays exactly three games with each of the other teams. What is the total number of games played? | [
"108"
] | [
"A baseball league has nine teams. During the season, each of the nine teams plays exactly three games with each of the other teams. What is the total number of games played?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b65d36904f76e11eaf38f0a8effafdb5d1f48ad67b5b137bf23aa92f0972864 | A baseball team won 20% of its games in the first half of the season. To achieve a 50% winning average or better in the season, at least what percent of its remaining games must it win? | [
"80%"
] | [
"A baseball team won 20% of its games in the first half of the season. To achieve a 50% winning average or better in the season, at least what percent of its remaining games must it win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-38cf7debda3afdd8ab7f2d67c582660075eef2a1823661a17e6ef294ac72614a | A basement has a 24-foot by 32-foot rectangular floor. The basement is flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will it take to pump all of th... | [
"360"
] | [
"A basement has a 24-foot by 32-foot rectangular floor. The basement is flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will it take to pump ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a21fd63967e08c6ff25b3c511d048858b2767656c77b3138557104148fff157 | A basket contains \( n \) balls, each colored with a unique color. At each turn, a ball is picked randomly and painted with a new color (red). Find the expected number of turns until all the balls are painted red. | [
"\\(\\sum_{i=1}^n \\dfrac{n}{i}\\)"
] | [
"A basket contains \\( n \\) balls, each colored with a unique color. At each turn, a ball is picked randomly and painted with a new color (red). Find the expected number of turns until all the balls are painted red."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-2bf0c0118fec1ed1a7bb49dcd7e91f0cb062baff10cd6d7a28af6d0fa8eaac62 | A basket is called a "Stuff Basket" if it includes 10 kilograms of rice and 30 eggs. A market is to distribute 100 Stuff Baskets, with a total of 1000 kilograms of rice and 3000 eggs. However, some baskets may have more or less rice or eggs. In each step, market workers can select two baskets and move an arbitrary amou... | [
"99"
] | [
"A basket is called a \"Stuff Basket\" if it includes 10 kilograms of rice and 30 eggs. A market is to distribute 100 Stuff Baskets, with a total of 1000 kilograms of rice and 3000 eggs. However, some baskets may have more or less rice or eggs. In each step, market workers can select two baskets and move an arbitra... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c0ca87beb563ce8e31dc0a1f01aeb31e401c16641a11056fcc47f3c9ee215ce | A basket of peanuts is distributed into $n$ piles. These piles are then rearranged into $n + k$ new piles $(k > 0)$. Let $S$ be the total number of peanuts transferred to smaller piles than the respective original piles that contained them. Find the minimum possible value of $S$ if $n = 10$ and $k = 5$. | [
"6"
] | [
"A basket of peanuts is distributed into $n$ piles. These piles are then rearranged into $n + k$ new piles $(k > 0)$. Let $S$ be the total number of peanuts transferred to smaller piles than the respective original piles that contained them. Find the minimum possible value of $S$ if $n = 10$ and $k = 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-dc46dbec4e262fedd6685036d8e89a12fe0345cea9a61ab232c1dc6fc100b96c | A basketball club decided to label every basketball in the club. After labelling all $n$ of the balls, the labeller noticed that exactly half of the balls had the digit $1$. Find the sum of all possible three-digit integer values of $n$. | [
"702"
] | [
"A basketball club decided to label every basketball in the club. After labelling all $n$ of the balls, the labeller noticed that exactly half of the balls had the digit $1$. Find the sum of all possible three-digit integer values of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-69a2ef4748d1883b87ac60dae6bcabec4a28e2dc92fe66161fd37ba573e904f5 | A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations, and each location has five poles in a row. Flags are placed starting from the far right pole in sequence. If not all poles in each location must be fl... | [
"57600"
] | [
"A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations, and each location has five poles in a row. Flags are placed starting from the far right pole in sequence. If not all poles in each location must ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20ce535946f591eeeb87fab127a61136d18f3274f624fdea2a28a6b2ee3b412e | A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} =.4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ where $p$... | [
"660"
] | [
"A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} =.4$ and $a_n\\le.4$ for all $n$ such that $1\\le n\\le9$ is given to be $p^aq^br/\\left(s^c\\right)$... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
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} | 71 | 0 | 0 |
omr-ws-e7862af95167f502fe5a39440a39e39deda0edc6bcc1005e9372fdb596769741 | A basketball player has a constant probability of $0.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=0.4$ and $a_{n} \leq 0.4$ for all $n$ such that $1 \leq n \leq 9$ is given to be $\frac{p^a q^b r}{s^c}... | [
"660"
] | [
"A basketball player has a constant probability of $0.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=0.4$ and $a_{n} \\leq 0.4$ for all $n$ such that $1 \\leq n \\leq 9$ is given to be $\\frac{p^a q^... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
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"tir": 30
} | 51 | 0 | 0 |
omr-ws-9423b1837b919b555f4bb69301d7eff0f287ebfa3ccdf489412af0caa7734e8c | A basketball player made the following number of free throws in 8 successive games: 6, 18, 15, 14, 19, 12, 19, and 15. What is the median number of successful free throws? | [
"15"
] | [
"A basketball player made the following number of free throws in 8 successive games: 6, 18, 15, 14, 19, 12, 19, and 15. What is the median number of successful free throws?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7da5c7233f5d196225189fef90836fb0d3de29d29419a804a9bda97d34e2cde7 | A basketball player makes a successful free throw with a probability of 90%. The probability of success is independent for each throw. If the player takes 100 free throws in a row, what is the probability that he hits at most 14 consecutive successful free throws? | [
"0.000237588120764901"
] | [
"A basketball player makes a successful free throw with a probability of 90%. The probability of success is independent for each throw. If the player takes 100 free throws in a row, what is the probability that he hits at most 14 consecutive successful free throws?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-799c9d4ced3f8e66fad2e6ebff64d35f8f574d16d882c96e82ec5e427c3d6e3b | A basketball player shoots $100$ times. The first shot is a success, and the second is a miss. For each subsequent shot, the probability of scoring is the number of points scored before the shot divided by the number of shots taken before the shot. What is the probability that he scores exactly $33$ points in the $100$... | [
"\\dfrac{1}{99}"
] | [
"A basketball player shoots $100$ times. The first shot is a success, and the second is a miss. For each subsequent shot, the probability of scoring is the number of points scored before the shot divided by the number of shots taken before the shot. What is the probability that he scores exactly $33$ points in the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08dd5a1356f1656d4a656a87c9d957007c4255e8ccc28bb7f5b8244d7107fd7b | A basketball player shoots $100$ times. The first shot is a success, and the second is a miss. For each subsequent shot, the probability of scoring is the number of points scored before the shot divided by the number of shots taken before the shot. What is the probability that he scores exactly $67$ points in the $100$... | [
"\\dfrac{1}{99}"
] | [
"A basketball player shoots $100$ times. The first shot is a success, and the second is a miss. For each subsequent shot, the probability of scoring is the number of points scored before the shot divided by the number of shots taken before the shot. What is the probability that he scores exactly $67$ points in the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a6febaa1a37f3040cd05c266b88f61951a72849e0f2feb5a1dd8f6338b25b985 | A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt? | [
"20"
] | [
"A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-19e25d0e202d7e37c3c5b49f10c3f2f56c846d4a0008ca96dbe18c7d777b0fa9 | A batch of cookies has 5 chocolate chip cookies, 5 red velvet cookies, and 5 snickerdoodles. If Tom chooses 3 cookies randomly, what is the probability that he is missing at least one flavor? | [
"\\(\\frac{66}{91}\\)"
] | [
"A batch of cookies has 5 chocolate chip cookies, 5 red velvet cookies, and 5 snickerdoodles. If Tom chooses 3 cookies randomly, what is the probability that he is missing at least one flavor?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7fecc5ae0f8c03e0112e86e7668643a6cd4f905205e85882f44149792ad96402 | A beach is a segment of length \( l \) uniformly occupied by bathers who wish to buy an ice-cream. The pleasure from eating an ice-cream is 10, and the cost of walking a unit length towards a seller is 1 (including the return walk). The sellers are situated at the edges of the segment and offer prices \( p_1 \) and \( ... | [
"\\( p_1 = 10 \\) and \\( p_2 = 10 \\)"
] | [
"A beach is a segment of length \\( l \\) uniformly occupied by bathers who wish to buy an ice-cream. The pleasure from eating an ice-cream is 10, and the cost of walking a unit length towards a seller is 1 (including the return walk). The sellers are situated at the edges of the segment and offer prices \\( p_1 \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62ff0889ad5bc9f8478399bf9e1ecd706c5c32b80a4f964f4d69868511a38f34 | A beacon of light 200 meters offshore rotates 2 times per minute, forming a spot of light that moves along a straight wall along the shore. How fast is the spot of light moving when it is 200 meters from point P? | [
"\\( 1600\\pi \\) meters per minute."
] | [
"A beacon of light 200 meters offshore rotates 2 times per minute, forming a spot of light that moves along a straight wall along the shore. How fast is the spot of light moving when it is 200 meters from point P?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-96463cbf822c1fcbaa86ce2a8744669ee6eb4c16e94d238a6fca65a96d3eae4e | A beacon of light 200 meters offshore rotates 2 times per minute, forming a spot of light that moves along a straight wall along the shore. How fast is the spot of light moving when the spot is at point P directly opposite the beam? | [
"\\(\\infty\\)"
] | [
"A beacon of light 200 meters offshore rotates 2 times per minute, forming a spot of light that moves along a straight wall along the shore. How fast is the spot of light moving when the spot is at point P directly opposite the beam?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be1367d36fda15789d9a36eab3fcbed9001b790a334666b51e94095889341191 | A beacon that makes one revolution every 10 seconds is located on a ship 4 km from a straight shoreline. How fast is the beam moving along the shoreline when it makes an angle of 45 degrees with the shore? | [
"\\( 1600 \\pi \\) m/s"
] | [
"A beacon that makes one revolution every 10 seconds is located on a ship 4 km from a straight shoreline. How fast is the beam moving along the shoreline when it makes an angle of 45 degrees with the shore?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-b583adeb6b90d3e17035e4b928561e6c001cd811b2593910adb047e2658ebd2a | A beam $ABCD.EFGH$ is formed by pasting two unit cubes $ABCD.PQRS$ and $PQRS.EFGH$. Point $K$ is the midpoint of edge $AB$, and point $L$ is the midpoint of edge $SH$. What is the length of the line segment $KL$? | [
"\\(\\frac{\\sqrt{14}}{2}\\)"
] | [
"A beam $ABCD.EFGH$ is formed by pasting two unit cubes $ABCD.PQRS$ and $PQRS.EFGH$. Point $K$ is the midpoint of edge $AB$, and point $L$ is the midpoint of edge $SH$. What is the length of the line segment $KL$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-219708856062d032d1f7e7cf8280a2ca2844640d0cc51bd33efe128f9c02050b | A beam can support a load that is jointly proportional to the beam's width and the square of its depth and inversely proportional to its length. A blue beam is half as wide, twice as long, and three times as deep as a red beam. Find the ratio of the load the blue beam can support to the load the red beam can support. | [
"\\(\\frac{4}{9}\\)"
] | [
"A beam can support a load that is jointly proportional to the beam's width and the square of its depth and inversely proportional to its length. A blue beam is half as wide, twice as long, and three times as deep as a red beam. Find the ratio of the load the blue beam can support to the load the red beam can suppo... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-621b21270ab2980b72d6cddb3c6f8e8b98519799e53a4cebb843b1f016cb19ed | A beam of light emanates from vertex \( A \) of a cube with side length 12 and reflects off face \( BCFG \) at point \( P \), which is 7 units from \( \overline{BG} \) and 5 units from \( \overline{BC} \). The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves... | [
"230"
] | [
"A beam of light emanates from vertex \\( A \\) of a cube with side length 12 and reflects off face \\( BCFG \\) at point \\( P \\), which is 7 units from \\( \\overline{BG} \\) and 5 units from \\( \\overline{BC} \\). The beam continues to be reflected off the faces of the cube. The length of the light path from t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-66752a0bcc8917141e7e4a82ec43702dbcf608e049b351f7456bdfe907892e12 | A beam of light enters a $10$ by $10$ square $ABCD$ at vertex $A$ and heads to point $P$ on side $CD$ where $CP = 3$ and $PD = 7$. The beam reflects off the sides of the square and can only leave via one of the vertices $A, B, C,$ or $D$. Determine the distance the light beam travels within the square before it leaves ... | [
"\\(10\\sqrt{149}\\) units, leaving via \\(B\\)."
] | [
"A beam of light enters a $10$ by $10$ square $ABCD$ at vertex $A$ and heads to point $P$ on side $CD$ where $CP = 3$ and $PD = 7$. The beam reflects off the sides of the square and can only leave via one of the vertices $A, B, C,$ or $D$. Determine the distance the light beam travels within the square before it le... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-110e7f8d26c68bf39067de86e767ff838032793bfe3b5cce9409b366c6e0cbac | A beam of light exits from point $A$ on the diameter $AB$ of a perfectly reflective semicircle at an angle of $58$ degrees relative to $AB$. The beam reflects off the walls of the semicircle and eventually returns to point $A$. How many times does the beam reflect off the walls of the semicircle (not including the fina... | [
"290"
] | [
"A beam of light exits from point $A$ on the diameter $AB$ of a perfectly reflective semicircle at an angle of $58$ degrees relative to $AB$. The beam reflects off the walls of the semicircle and eventually returns to point $A$. How many times does the beam reflect off the walls of the semicircle (not including the... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-623c030f4600d9542560388b1eb7234dbb2314803ed3a10e8e1b29e6325fca11 | A beam of light exits from point $A$ on the diameter $AB$ of a perfectly reflective semicircle at an angle of $58^\circ$ relative to $AB$. The beam reflects off the walls of the semicircle and eventually returns to point $A$. How many times does the beam reflect off the walls of the semicircle (not including when it hi... | [
"59"
] | [
"A beam of light exits from point $A$ on the diameter $AB$ of a perfectly reflective semicircle at an angle of $58^\\circ$ relative to $AB$. The beam reflects off the walls of the semicircle and eventually returns to point $A$. How many times does the beam reflect off the walls of the semicircle (not including when... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4345f4bfbc5479cf3f4b51099c07e60950378747a8d57aac9bedf7843184ddc9 | A beam of light is shot into a space containing 4 parallel planes (windows) and bounces around $n$ times. Find the number of distinct paths the light can follow in terms of $n$, given that the light must enter between the panes of glass for it to count. | [
"\\( f(n) = C_1 r_1^n + C_2 r_2^n + C_3 r_3^n \\) where \\( r_1, r_2, r_3 \\) are the roots of \\( x^3 - 2x^2 - x + 1 = 0 \\)."
] | [
"A beam of light is shot into a space containing 4 parallel planes (windows) and bounces around $n$ times. Find the number of distinct paths the light can follow in terms of $n$, given that the light must enter between the panes of glass for it to count."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87f018aab37001374721a25f2b8bd07bdd8755770696df4589ee5151755523e4 | A beam of light is shot into a space containing 4 parallel planes (windows) and bounces around $n$ times. Find the number of distinct paths the light can follow in terms of $n$. (Note: The light must enter between the panes of glass for it to count.) | [
"\n\\begin{cases}\n3 \\cdot 2^{\\frac{n}{2}} & \\text{if } n \\text{ is even} \\\\\n2^{\\frac{n + 3}{2}} & \\text{if } n \\text{ is odd}\n\\end{cases}\n"
] | [
"A beam of light is shot into a space containing 4 parallel planes (windows) and bounces around $n$ times. Find the number of distinct paths the light can follow in terms of $n$. (Note: The light must enter between the panes of glass for it to count.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-a86a2e55fd96a4960602594a391c47479a651a0d7270f631ee45307f4eec9ddf | A beam of light strikes $\overline{BC}$ at point $C$ with an angle of incidence $\alpha = 19.94^\circ$ and reflects with an equal angle of reflection. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection.... | [
"71"
] | [
"A beam of light strikes $\\overline{BC}$ at point $C$ with an angle of incidence $\\alpha = 19.94^\\circ$ and reflects with an equal angle of reflection. The light beam continues its path, reflecting off line segments $\\overline{AB}$ and $\\overline{BC}$ according to the rule: angle of incidence equals angle of r... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c67dbdae2c1adce3deb27da5fcf6fb85c45c6f73163faff6da7419a2856747a3 | A beam of light strikes $\overline{BC}$ at point $C$ with an angle of incidence $\alpha = 19.94^\circ$ and reflects with an equal angle of reflection. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection.... | [
"10"
] | [
"A beam of light strikes $\\overline{BC}$ at point $C$ with an angle of incidence $\\alpha = 19.94^\\circ$ and reflects with an equal angle of reflection. The light beam continues its path, reflecting off line segments $\\overline{AB}$ and $\\overline{BC}$ according to the rule: angle of incidence equals angle of r... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d0f7bf22087106ba6aa2111f7c389758c9f2766e5acca3a4ef23bce027e91989 | A beam of light strikes line segment $\overline{BC}$ at point $C$ with an angle of incidence $\alpha = 19.94^\circ$ and reflects with an equal angle of reflection. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle o... | [
"71"
] | [
"A beam of light strikes line segment $\\overline{BC}$ at point $C$ with an angle of incidence $\\alpha = 19.94^\\circ$ and reflects with an equal angle of reflection. The light beam continues its path, reflecting off line segments $\\overline{AB}$ and $\\overline{BC}$ according to the rule: angle of incidence equa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e63d5663b41427fe52bce00a1e3acc5d9e5b030a57d30da6425b629385eb4bc | A beam of rectangular cross section must be sawn from a round log of diameter $d$. What should the width $x$ and the height $y$ of the cross section be for the beam to offer the greatest resistance: (a) to compression, (b) to bending? {\bf Remark.} The compressive strength of a beam is proportional to the area of the c... | [
"(a) $x=y=\\frac{d}{\\sqrt 2}$; (b) $x=\\frac{d}{\\sqrt 3}$, $y=d\\sqrt {\\frac{2}{3}}$"
] | [
"A beam of rectangular cross section must be sawn from a round log of diameter $d$. What should the width $x$ and the height $y$ of the cross section be for the beam to offer the greatest resistance: (a) to compression, (b) to bending? {\\bf Remark.} The compressive strength of a beam is proportional to the area of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f72da0c9fd9c1bbab144edeb76037b1c7cc33b9dde903358928e426e727d9a6 | A beam starts from point $A(-3,4)$, reaches point $B$ on the $x$-axis, and then reflects from the axis to reach point $C$ on the $y$-axis. If the ray reaches point $D(-1,6)$ after reflecting from the $y$-axis, determine the equation of $\overline{AB}$. | [
"\\( y = -\\frac{5}{2}x - \\frac{7}{2} \\)"
] | [
"A beam starts from point $A(-3,4)$, reaches point $B$ on the $x$-axis, and then reflects from the axis to reach point $C$ on the $y$-axis. If the ray reaches point $D(-1,6)$ after reflecting from the $y$-axis, determine the equation of $\\overline{AB}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-49608601c9fd47149115d4dbe6f5e1dcde28764f3c278af60dc6a1607b9dab5f | A beam starts from point $A(-3,4)$, reaches point $B$ on the $x$-axis, and then reflects from the axis to reach point $C$ on the $y$-axis. If the ray reaches point $D(-1,6)$ after reflecting from the $y$-axis, determine the equation of $\overline{CD}$. | [
"\\( y = 6 \\)"
] | [
"A beam starts from point $A(-3,4)$, reaches point $B$ on the $x$-axis, and then reflects from the axis to reach point $C$ on the $y$-axis. If the ray reaches point $D(-1,6)$ after reflecting from the $y$-axis, determine the equation of $\\overline{CD}$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-faf0a8efd3839de57b9f5c931f20e7496913443113cec0e0f39c98c4ca0666c7 | A bean packing plant has a machine that fills bags with beans in a sequence where the first $n$ bags are empty, the next $n$ bags each contain $1$ bean, the next $n$ bags each contain $2$ beans, and so on, until the last $n$ bags each contain $2006$ beans. The goal is to distribute these bags among $2007$ boxes such th... | [
"\\( n = 2007k \\) for any integer \\( k \\)."
] | [
"A bean packing plant has a machine that fills bags with beans in a sequence where the first $n$ bags are empty, the next $n$ bags each contain $1$ bean, the next $n$ bags each contain $2$ beans, and so on, until the last $n$ bags each contain $2006$ beans. The goal is to distribute these bags among $2007$ boxes su... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 9
} | 14 | 0 | 0 |
omr-ws-fe6f987f047a24d7f8432b6299c19bcd57dd26e41a23d3d39077e6a5e30bf187 | A bear is in the center of the left-down corner of a $100 \times 100$ grid. A bear cycle is defined as a cycle that visits each square exactly once and returns to the starting point. By removing a row or column, the bear cycle is divided into several paths. Determine the minimum value of $k$ such that, for any bear cyc... | [
"4999"
] | [
"A bear is in the center of the left-down corner of a $100 \\times 100$ grid. A bear cycle is defined as a cycle that visits each square exactly once and returns to the starting point. By removing a row or column, the bear cycle is divided into several paths. Determine the minimum value of $k$ such that, for any be... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-120e102206d7f64450856c5e1ed903ef43757f5aad6683f32c01fa53bb5e59b9 | A bee can fly or walk in a straight line between any two corners on the inside of a cubic box of length 1. The bee managed to move so that it visited every corner of the box without passing through the same point twice in the air or on the wall of the box. What is the largest possible length of such a path? | [
"\\( 6\\sqrt{2} + \\sqrt{3} \\)"
] | [
"A bee can fly or walk in a straight line between any two corners on the inside of a cubic box of length 1. The bee managed to move so that it visited every corner of the box without passing through the same point twice in the air or on the wall of the box. What is the largest possible length of such a path?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-1cdac10998a24e237dbf12d0ff78c7200be17e8bc67e4e773870cbee2e8429fe | A bee can fly or walk only in a straight line between any two corners on the inside of a cubic box with edge length 1. The bee managed to move so that it visited every corner of the box without passing through the same point twice in the air or on the wall of the box. What is the largest possible length of such a path? | [
"\\(6 \\sqrt{2} + \\sqrt{3}\\)"
] | [
"A bee can fly or walk only in a straight line between any two corners on the inside of a cubic box with edge length 1. The bee managed to move so that it visited every corner of the box without passing through the same point twice in the air or on the wall of the box. What is the largest possible length of such a ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-e11c1a589785e57bf437a6a76921c074f88e3562f94a4a6e05ceed9be3104fed | A bee flies 10√2 meters in the direction 45° clockwise of North (that is, in the NE direction). Then, the bee turns 135° clockwise, and flies 20 meters forward. It continues by turning 60° counterclockwise, and flies forward 14 meters. Finally, the bee turns 120° clockwise and flies another 14 meters forward before fin... | [
"26"
] | [
"A bee flies 10√2 meters in the direction 45° clockwise of North (that is, in the NE direction). Then, the bee turns 135° clockwise, and flies 20 meters forward. It continues by turning 60° counterclockwise, and flies forward 14 meters. Finally, the bee turns 120° clockwise and flies another 14 meters forward befor... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-56e7f3753be17499d91fba596f43f082da54262fd4d91aa525fff7069168e588 | A bee flies in a semicircle with radius \( r \). It then flies another semicircle with half the area of the previous semicircle, and repeats this process infinitely. How far from its starting point will the bee end up? | [
"\\( r (2 + \\sqrt{2}) \\)"
] | [
"A bee flies in a semicircle with radius \\( r \\). It then flies another semicircle with half the area of the previous semicircle, and repeats this process infinitely. How far from its starting point will the bee end up?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-86a4b4f2f798cd19a6ce1af00959c6a23be51d626e1f04140dfc0c5557e856a4 | A bee is moving in three-dimensional space starting at the point $(0, 0, 0)$. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled four times. The bee moves according to the die roll: $A^+$ moves the bee to $(a+1, b, c)$, $A^-$ to $(a-1, b, c)$, $B^+$ to $(a, b+1, c)$, $B^-$ to $(a, b-... | [
"\\(\\frac{7}{54}\\)"
] | [
"A bee is moving in three-dimensional space starting at the point $(0, 0, 0)$. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled four times. The bee moves according to the die roll: $A^+$ moves the bee to $(a+1, b, c)$, $A^-$ to $(a-1, b, c)$, $B^+$ to $(a, b+1, c)$, $B^-$ to $(... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-8df4d6096df61ca613dd45aa8e707e1f4c703c0635dccd9a6a248a3423dd4a9d | A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015},$ how far from $P_0$ is she, in inches? | [
"1008 \\sqrt{6} + 1008 \\sqrt{2}"
] | [
"A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $P_j$, she turns $30^{\\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015},$ how far from $P_0$ is she, in inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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