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-2f0f285142c5e8876f2541a587506080c4455fd88d9917f8189aae440a5531f6 | A dog buries a bone at $(0,1)$. He is at the end of a 1-unit long leash, and his master walks down the positive $x$-axis, dragging the dog along. The dog pulls the leash taut. Provide parametric equations for the dog's path in terms of $t$, where $t$ is the $x$-coordinate of the master. | [
"y(t) = \\text{sech}\\, t"
] | [
"A dog buries a bone at $(0,1)$. He is at the end of a 1-unit long leash, and his master walks down the positive $x$-axis, dragging the dog along. The dog pulls the leash taut. Provide parametric equations for the dog's path in terms of $t$, where $t$ is the $x$-coordinate of the master."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-893d9b02439566a3135c17b66c04e3863b4a73025bfe92af3908c957dede494b | A dog chases a cat. When the dog chases 4 times, the cat jumps 5 times. When the dog chases 5 times, the cat jumps 6 times. The velocity ratio of the dog to the cat can be expressed as \( x:y \), where \( x \) and \( y \) are co-prime numbers. Find \( x + y \). | [
"49"
] | [
"A dog chases a cat. When the dog chases 4 times, the cat jumps 5 times. When the dog chases 5 times, the cat jumps 6 times. The velocity ratio of the dog to the cat can be expressed as \\( x:y \\), where \\( x \\) and \\( y \\) are co-prime numbers. Find \\( x + y \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-cdedbcd5f086e3d5b225e733b18c8cd579e2f82c125b7ba7ca99787f5417053b | A dog has a leash that is \(2\pi\) meters long. The leash is attached to a circular shed with a radius of 1 meter. How much area can the dog cover? | [
"\\dfrac{4}{3}\\pi^3"
] | [
"A dog has a leash that is \\(2\\pi\\) meters long. The leash is attached to a circular shed with a radius of 1 meter. How much area can the dog cover?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-d9efe6a8788a0d788f99bd5f003a1ce1a496c654228dbbda8dde595083cf8c71 | A dog has infinitely many pieces of meat, each with a fly on it. At each move, the dog can either eat a piece of meat along with all flies on it or move a fly from one piece of meat to another. The dog does not want to eat more than one million flies. Let $N$ be the maximum number of pieces of meat the dog can ensure a... | [
"1000000"
] | [
"A dog has infinitely many pieces of meat, each with a fly on it. At each move, the dog can either eat a piece of meat along with all flies on it or move a fly from one piece of meat to another. The dog does not want to eat more than one million flies. Let $N$ be the maximum number of pieces of meat the dog can ens... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-0af6190ae2997238a47f025a9a7b9bc70af3e3fe35d9f5f9dff2fa89a99ce115 | A dog is attached to a $60$-meter leash anchored to the midpoint of one of the long sides of a $40$-meter by $20$-meter rectangular house. What is the total area of the yard that the dog can roam? | [
"2800\\pi"
] | [
"A dog is attached to a $60$-meter leash anchored to the midpoint of one of the long sides of a $40$-meter by $20$-meter rectangular house. What is the total area of the yard that the dog can roam?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-4f0386e9a6dad888f5c5e328ff38dd5ef2ab41c6b1e0ac792ade87c5b2abc612 | A dog is chasing a fox. Initially, the dog is 60 steps behind the fox. For every 9 steps the fox takes, the dog takes 6 steps. Additionally, 3 steps of the dog are equivalent to 7 steps of the fox. After how many steps of the dog will the dog catch the fox? | [
"72"
] | [
"A dog is chasing a fox. Initially, the dog is 60 steps behind the fox. For every 9 steps the fox takes, the dog takes 6 steps. Additionally, 3 steps of the dog are equivalent to 7 steps of the fox. After how many steps of the dog will the dog catch the fox?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-556eb43044a64033ab4c802c8625f4b7ffed34bd064dd6f347a792f72241775b | A dog is standing at the bottom left corner of a 46×46 grid and wants to reach the top right corner. The dog can only move up and to the right, with each move to the right consisting of at least 4 consecutive blocks and each move up consisting of at least 12 consecutive blocks. How many different intersections are unre... | [
"132"
] | [
"A dog is standing at the bottom left corner of a 46×46 grid and wants to reach the top right corner. The dog can only move up and to the right, with each move to the right consisting of at least 4 consecutive blocks and each move up consisting of at least 12 consecutive blocks. How many different intersections are... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b11fef2853b9d690354c05075ad301f44739cebcf70f650a90c3402fc77cbf7b | A dog is tied to a 15-foot leash at one corner of a rectangular garden measuring 12 by 8 feet. How many square yards of space does the dog have to roam in? Express your answer as a common fraction in terms of π. | [
"\\(\\frac{127 \\pi}{9}\\)"
] | [
"A dog is tied to a 15-foot leash at one corner of a rectangular garden measuring 12 by 8 feet. How many square yards of space does the dog have to roam in? Express your answer as a common fraction in terms of π."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2ab8776a5f7b86947edbd4a22ad5784fe57b8afbae14e869b379839b1de68aa8 | A dog is tied to a corner of a square house with a chain of 20 meters. The house has a square floor with each side measuring 10 meters. Calculate the area through which the dog can move. | [
"\\(350\\pi\\)"
] | [
"A dog is tied to a corner of a square house with a chain of 20 meters. The house has a square floor with each side measuring 10 meters. Calculate the area through which the dog can move."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-c8e5b1d4eef30da642f565bed2db392e9ac95f9b9b0c816d3a43487615ccf93e | A dog is tied to a perfectly smooth cylindrical column of radius \( r \) with a very fine rope that wraps around the column and has a loop. The dog has the free end of the rope around its neck. When the dog tries to reach someone and pulls the rope tight, it breaks. Determine the distance from the column to the knot at... | [
"r"
] | [
"A dog is tied to a perfectly smooth cylindrical column of radius \\( r \\) with a very fine rope that wraps around the column and has a loop. The dog has the free end of the rope around its neck. When the dog tries to reach someone and pulls the rope tight, it breaks. Determine the distance from the column to the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9f5a3bae4fa6bec859e26071b60ad3ec74e11d7503a123d5850d3359523cac9e | A dog is tied to a point on the inside of a circular patch of radius \( R \) using a rope of length \( L \). What is the value of \( L \) such that the dog can cover an area exactly \( \frac{1}{2} \) of the patch? | [
"\\dfrac{R}{\\sqrt{2}}"
] | [
"A dog is tied to a point on the inside of a circular patch of radius \\( R \\) using a rope of length \\( L \\). What is the value of \\( L \\) such that the dog can cover an area exactly \\( \\frac{1}{2} \\) of the patch?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-35a39d0ba6d0db21bbea222f7f87a96c9927a579a59fbd8c918f9423b10f9cf9 | A dog is tied to a point on the outside of a circular patch of radius \( R \) using a rope of length \( L \) such that \( L < \pi R \). What is the value of \( L \) such that the dog can cover an area exactly \( \frac{1}{2} \) of the patch? | [
"R"
] | [
"A dog is tied to a point on the outside of a circular patch of radius \\( R \\) using a rope of length \\( L \\) such that \\( L < \\pi R \\). What is the value of \\( L \\) such that the dog can cover an area exactly \\( \\frac{1}{2} \\) of the patch?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-55eeee438b79f7717594d6f75d9a0180673f3c6422cae20def387c3d94b9d251 | A dog multiplied three amounts instead of adding them, and the result was the same as if they were added, which is \$5.70. What were the three amounts? | [
"\\$2.85, \\$1.25, \\$1.60"
] | [
"A dog multiplied three amounts instead of adding them, and the result was the same as if they were added, which is \\$5.70. What were the three amounts?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-445e7fd58176c68c06c608295c6d70c9ad33db419d4787c79af916d013261354 | A dog named Nero is tied to a regular hexagonal house at one of its vertices with a rope that measures 2 meters. The side length of the hexagon is 1 meter. What is the area of the region outside the house that Nero can reach? | [
"\\(3\\pi\\)"
] | [
"A dog named Nero is tied to a regular hexagonal house at one of its vertices with a rope that measures 2 meters. The side length of the hexagon is 1 meter. What is the area of the region outside the house that Nero can reach?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-7aa485293a1da41162aabb457ed16b092df2d1a362f853ab116d886b22a36fd6 | A dog on a 10 meter long leash is tied to a 10 meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence? | [
"\\(\\frac{25}{4} \\pi\\)"
] | [
"A dog on a 10 meter long leash is tied to a 10 meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-6264f85be96bafdad7a35ccadc22cb10d5c35edb40d6a2e295f84ebd02c41be7 | A dog owns 4 different color shoes and 4 identical green socks. He can fit every shoe and sock on each of his four distinguishable paws. In how many different orders can he put on the shoes and socks, provided that on each paw he must put on the sock before the shoe? | [
"1680"
] | [
"A dog owns 4 different color shoes and 4 identical green socks. He can fit every shoe and sock on each of his four distinguishable paws. In how many different orders can he put on the shoes and socks, provided that on each paw he must put on the sock before the shoe?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7b6f07cc58db122516d23b3720bbe0f2fdcdb5a20dee367d133ae49996614c1 | A dog pen has the shape of a right-angled triangle with sides of length 30, 40, and 50. A straight fence goes from the corner with the right-angle to a point on the opposite side, dividing the dog pen into two sections which have the same perimeter. How long is the fence? | [
"\\( 12\\sqrt{5} \\)"
] | [
"A dog pen has the shape of a right-angled triangle with sides of length 30, 40, and 50. A straight fence goes from the corner with the right-angle to a point on the opposite side, dividing the dog pen into two sections which have the same perimeter. How long is the fence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e1702354e47e32f9412431a93bb0170bea1d30fe428c9f593ebec6e893940636 | A dog runs behind a flock of sheep that is 1000 meters long. The dog starts from the end of the flock, runs to the beginning, and then returns to the end. By the time the dog returns to the end, each sheep has traveled 1000 meters. What is the total distance the dog ran? | [
"\\( 1000\\sqrt{2} + 1000 \\) meters"
] | [
"A dog runs behind a flock of sheep that is 1000 meters long. The dog starts from the end of the flock, runs to the beginning, and then returns to the end. By the time the dog returns to the end, each sheep has traveled 1000 meters. What is the total distance the dog ran?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-077e6b982c453ce39107ee00c0c99d7144ca32050e79032c35954b3a76bcc6b8 | A dog runs once around a square garden with a side length of 10 meters, staying exactly 1 meter from the edge of the garden. To the nearest meter, how far does the dog run? | [
"46"
] | [
"A dog runs once around a square garden with a side length of 10 meters, staying exactly 1 meter from the edge of the garden. To the nearest meter, how far does the dog run?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0d011605fc8713c5d308d3546f56e30812dc41a39025cac7d59ffd44ab495893 | A dog standing at the center of a circular arena sees a rabbit at the wall. The rabbit runs around the wall, and the dog pursues it along a path determined by running at the same speed and staying on the radial line joining the center of the arena to the rabbit. If the arena has a radius of $R$ and the rabbit runs at a... | [
"\\dfrac{\\pi}{2\\omega}"
] | [
"A dog standing at the center of a circular arena sees a rabbit at the wall. The rabbit runs around the wall, and the dog pursues it along a path determined by running at the same speed and staying on the radial line joining the center of the arena to the rabbit. If the arena has a radius of $R$ and the rabbit runs... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-c1798fc27d9091e1b88e67dc7d63545e2aba9edc71f50f67e4ff26cb203b32f9 | A dog standing at the center of a circular arena sees a rabbit at the wall. The rabbit runs around the wall, and the dog pursues it along a path determined by running at the same speed and staying on the radial line joining the center of the arena to the rabbit. Let $C$ be the circumference of the circular arena. Calcu... | [
"\\dfrac{1}{4}"
] | [
"A dog standing at the center of a circular arena sees a rabbit at the wall. The rabbit runs around the wall, and the dog pursues it along a path determined by running at the same speed and staying on the radial line joining the center of the arena to the rabbit. Let $C$ be the circumference of the circular arena. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b4d7e603f044f8cb1ee409d32b23fdd93c334bb7f7abd661844cf1164da62ef1 | A dog standing at the center of a circular arena sees a rabbit at the wall. The rabbit runs around the wall, and the dog pursues it along a unique path by running at the same speed and always staying on the radial line joining the center of the arena and the rabbit. If the circumference of the arena is $400$ meters, ca... | [
"100"
] | [
"A dog standing at the center of a circular arena sees a rabbit at the wall. The rabbit runs around the wall, and the dog pursues it along a unique path by running at the same speed and always staying on the radial line joining the center of the arena and the rabbit. If the circumference of the arena is $400$ meter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bfd7d54527026f211a4bb84d801a039a0179f621cde1ce4eadc20ed3274c2eab | A dog trainer wants to buy eight dogs, all of which are either cocker spaniels, Irish setters, or Russian wolfhounds. In how many ways can she make the choice? | [
"45"
] | [
"A dog trainer wants to buy eight dogs, all of which are either cocker spaniels, Irish setters, or Russian wolfhounds. In how many ways can she make the choice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d961b94bdd28ef093482bc4628bf35877fc4030e3c763ba3582a1e1fc8461541 | A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to an adjacent segment and stains it red. After 100 minutes, what is the expected number of 1x1 squares which are colored completely red? | [
"1"
] | [
"A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to an adjacent segment and stains it red. After 100 minutes, what is the expected number of 1x1 squares which are colored completely red?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-5c971a207b975ea3e7c6c5af6a59ae2b7a7aafc1dbc33d40b817bfa9129bb6d6 | A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to an adjacent segment and stains it red. Can you make a formula for the expected number of 1x1 squares which are colored completely red after n minutes? | [
"0"
] | [
"A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to an adjacent segment and stains it red. Can you make a formula for the expected number of 1x1 squares which are colored completely red after n minutes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c9492703ab9a679d1501525bfff5896dae0ab1486476cc5aa0ac48a7d16d91b8 | A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to another adjacent segment (one of the six segments connected to the unit segment at either side), and stains it red. After 100 minutes, what is the expected number of ... | [
"1"
] | [
"A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to another adjacent segment (one of the six segments connected to the unit segment at either side), and stains it red. After 100 minutes, what is the expected numbe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-1ff793104c221100b280ac62a009f65c31dc5f70525dec10feb1a03b7e9c92fc | A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to another adjacent segment and stains it red. After 100 minutes, what is the expected number of 1x1 squares which are colored completely red? | [
"1"
] | [
"A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to another adjacent segment and stains it red. After 100 minutes, what is the expected number of 1x1 squares which are colored completely red?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-c2e0b86088961f161068a57bfe08deae04517555859e4635d4b32fe45be898be | A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to another adjacent segment and stains it red. After n minutes, what is the expected number of 1x1 squares which are colored completely red? | [
"0"
] | [
"A dog with red stained paws is walking on an infinite lattice of 1x1 squares. Starting on a given (already stained red) segment, every minute he walks to another adjacent segment and stains it red. After n minutes, what is the expected number of 1x1 squares which are colored completely red?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-2dbad9623ba7050cfe808e78eba9b3574f929f960399016f4218ddb35238d3cb | A domain is bounded by the curves $x=0$, $x=\pi /2$, $y=\sin x$, $y=\sin a$. A point is thrown into a rectangle whose sides are the interval $[0, \pi /2]$ of the $x$-axis and the interval $[0,1]$ of the $y$-axis. For what value of $a$ is the probability of the point falling into the domain is the least? | [
"$\\frac{1}{2}$"
] | [
"A domain is bounded by the curves $x=0$, $x=\\pi /2$, $y=\\sin x$, $y=\\sin a$. A point is thrown into a rectangle whose sides are the interval $[0, \\pi /2]$ of the $x$-axis and the interval $[0,1]$ of the $y$-axis. For what value of $a$ is the probability of the point falling into the domain is the least?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ae5c586281af7826f2b4d5f0038cf21345453c90229c33d22be4f3fcf1d5eec | A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. There are $28$ dominos in total. We want to form a chain such that adjacent dominos have the same number at the adjacent ends, and after each domino is added, the total of all the numbers in the chain is odd. What is the largest number ... | [
"10"
] | [
"A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. There are $28$ dominos in total. We want to form a chain such that adjacent dominos have the same number at the adjacent ends, and after each domino is added, the total of all the numbers in the chain is odd. What is the largest nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-ea86813927ecfab2550a50e51916bef85887fc4a3579641d04dac36accf6cd85 | A domino is a rectangular tile composed of two squares. An integer is represented on both squares, and each integer 0-9 is paired with every integer 0-9 exactly once to form a complete set. A $\textit{double}$ is a domino that has the same integer on both of its squares. What is the probability that a domino randomly s... | [
"\\frac{2}{11}"
] | [
"A domino is a rectangular tile composed of two squares. An integer is represented on both squares, and each integer 0-9 is paired with every integer 0-9 exactly once to form a complete set. A $\\textit{double}$ is a domino that has the same integer on both of its squares. What is the probability that a domino rand... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f32c4fffeda59f2ed2a304ae31e3aa3b9324673d7ca59ef9fa90049ddf5dbc4 | A donkey hiccups regularly every 5 seconds, starting at 4:00 PM. At what time does the donkey’s 700th hiccup occur? | [
"15 seconds after 4:58"
] | [
"A donkey hiccups regularly every 5 seconds, starting at 4:00 PM. At what time does the donkey’s 700th hiccup occur?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-be11d1e882c1a53ef5adf588a252529e1fa60f251091ade0937a645f26a4c2aa | A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccups regularly every 5 seconds. At what time does the donkey’s 700th hiccup occur? | [
"15 seconds after 4:58"
] | [
"A donkey suffers an attack of hiccups and the first hiccup happens at 4:00 one afternoon. Suppose that the donkey hiccups regularly every 5 seconds. At what time does the donkey’s 700th hiccup occur?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4e56e0790ff999b5e435c42002f6eed9f1c1d851f04ce7d03894b2f45376557b | A donut shop sells 5 cream-filled and 7 jelly-filled donuts for $3.64, and 3 cream-filled and 9 jelly-filled donuts for $3.72. What is the cost of 1 cream-filled donut? | [
"28 cents"
] | [
"A donut shop sells 5 cream-filled and 7 jelly-filled donuts for $3.64, and 3 cream-filled and 9 jelly-filled donuts for $3.72. What is the cost of 1 cream-filled donut?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1c02fb31158f1f469393164992ec15ef3d029bd5339e0e55cc84e651d5e31a70 | A donut store has chocolate, plain, jelly, and powdered donuts. How many different ways can the bakery package 3 donuts into a pack? | [
"20"
] | [
"A donut store has chocolate, plain, jelly, and powdered donuts. How many different ways can the bakery package 3 donuts into a pack?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-4f626d0985ea542e31e921cf13342d8e4320543007d2cf0ce0256e3add5082fd | A door has the shape of an elliptical arch (half ellipse) that is 10 feet wide and 4 feet high at the center. A box 2 feet wide is to be pushed through the door. How high can the box be? | [
"\\(\\frac{8\\sqrt{6}}{5}\\)"
] | [
"A door has the shape of an elliptical arch (half ellipse) that is 10 feet wide and 4 feet high at the center. A box 2 feet wide is to be pushed through the door. How high can the box be?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-a62d731bad9366f39f732ce27d55072358446833fc626588b874f6ab718adb7e | A door is to be constructed inside an arch in the shape of the parabola given by the equation \( y = 9 - x^2 \), where \( x \) and \( y \) are in meters. If the area of the door is 20 square meters, find the dimensions of the door. The door is inscribed in the parabola, with one dimension on the x-axis. | [
"\\(4\\) meters by \\(5\\) meters"
] | [
"A door is to be constructed inside an arch in the shape of the parabola given by the equation \\( y = 9 - x^2 \\), where \\( x \\) and \\( y \\) are in meters. If the area of the door is 20 square meters, find the dimensions of the door. The door is inscribed in the parabola, with one dimension on the x-axis."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 20
} | 21 | 0 | 0 |
omr-ws-1a35a9f4125fc5033c0b623d5ccd1563f49c91b070b029d7ed320ddff2f06068 | A dormitory hall has 10 students, each with their own rooms. Two students can only switch rooms if they are friends. Let \( M \) be the least possible number of pairs of students that are friends such that every possible assignment of the 10 students to the 10 rooms could possibly be attained after a finite number of s... | [
"\\( 10^8 \\)"
] | [
"A dormitory hall has 10 students, each with their own rooms. Two students can only switch rooms if they are friends. Let \\( M \\) be the least possible number of pairs of students that are friends such that every possible assignment of the 10 students to the 10 rooms could possibly be attained after a finite numb... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-5914b7ab20e525215451c7884fdfbf0fd8692467a086109f9b155746eda40bfa | A dot is made on a circular sheet of paper with radius $R$. The paper is folded so that the edge passes through the dot, creating creases. If the creases all touch a fixed ellipse, find the sum of the lengths of the major and minor axes of this ellipse in terms of $R$. | [
"2R"
] | [
"A dot is made on a circular sheet of paper with radius $R$. The paper is folded so that the edge passes through the dot, creating creases. If the creases all touch a fixed ellipse, find the sum of the lengths of the major and minor axes of this ellipse in terms of $R$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-a1dc17fc5ef6f871f4fc1e005312cd58de93834af6dde8161c0a329b8b2afacd | A double cone with equation \(3z^2 = x^2 + y^2\) is sliced by a plane with equation \(2z - 5x = 4\). What is the eccentricity of the conic formed? | [
"\\dfrac{5\\sqrt{3}}{2}"
] | [
"A double cone with equation \\(3z^2 = x^2 + y^2\\) is sliced by a plane with equation \\(2z - 5x = 4\\). What is the eccentricity of the conic formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-f24e9c9d8bd2a4e5ff89c05a2b821454b4acae7347e439356416b323d5d9d91c | A double-decked bus can accommodate 13 passengers in the upper deck. If 5 passengers refuse to sit in the upper deck and 8 passengers refuse to sit in the lower deck, how many ways can the passengers be distributed between the two decks? | [
"1"
] | [
"A double-decked bus can accommodate 13 passengers in the upper deck. If 5 passengers refuse to sit in the upper deck and 8 passengers refuse to sit in the lower deck, how many ways can the passengers be distributed between the two decks?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-cefe7086e9fb643ca1b894b7361922877ff1c57000a0bb80acc090ecf973a13f | A doubly-indexed sequence \(a_{m,n}\) for nonnegative integers \(m\) and \(n\) is defined as follows: - \(a_{m,0} = 0\) for all \(m > 0\) and \(a_{0,0} = 1\) - \(a_{m,1} = 0\) for all \(m > 1\) and \(a_{1,1} = 1\), \(a_{0,1} = 0\) - \(a_{0,n} = a_{0,n-1} + a_{0,n-2}\) for all \(n \ge 2\) - \(a_{m,n} = a_{m,n-1} + a_{m,... | [
"250"
] | [
"A doubly-indexed sequence \\(a_{m,n}\\) for nonnegative integers \\(m\\) and \\(n\\) is defined as follows:\n- \\(a_{m,0} = 0\\) for all \\(m > 0\\) and \\(a_{0,0} = 1\\)\n- \\(a_{m,1} = 0\\) for all \\(m > 1\\) and \\(a_{1,1} = 1\\), \\(a_{0,1} = 0\\)\n- \\(a_{0,n} = a_{0,n-1} + a_{0,n-2}\\) for all \\(n \\ge 2\\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-2bc25d2ce17dfdeb3ed27fe96025d82bfba99edb46f6b8ce08a2b077659aea3c | A dragon in the center of a circular forest with a diameter of 546 meters can breathe fire into a 45-degree sector all the way to the edge. What is the probability that the dragon's fire will hit a straight 273-meter long net held by two people entering the forest? Hitting at least one of the people counts as hitting t... | [
"\\(\\frac{5}{12}\\)"
] | [
"A dragon in the center of a circular forest with a diameter of 546 meters can breathe fire into a 45-degree sector all the way to the edge. What is the probability that the dragon's fire will hit a straight 273-meter long net held by two people entering the forest? Hitting at least one of the people counts as hitt... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-609f5928a70749a188d252918c8546d5379cce73c7e7349ba212becea2cf0ee5 | A dragon is flying above a Cartesian plane, starting from the origin. He wishes to return to his home, which is the point $H=(5,4)$. Each minute, he flies from his position $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$ with probability $\frac{2}{3}$ and $\frac{1}{3}$, respectively. However, there is a magical wizard standin... | [
"352"
] | [
"A dragon is flying above a Cartesian plane, starting from the origin. He wishes to return to his home, which is the point $H=(5,4)$. Each minute, he flies from his position $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$ with probability $\\frac{2}{3}$ and $\\frac{1}{3}$, respectively. However, there is a magical wizard ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-a70ceb1fd2115645a940d01ad0b8bae0f88871938ec50f8ddebc6c73a62ffe17 | A drain should be constructed using a 10 cm diameter pipe. However, only small pipes with a diameter of 3 cm are available. So that the channel capacity is not smaller than desired, how many 3 cm pipes need to be used instead of one 10 cm pipe? | [
"12"
] | [
"A drain should be constructed using a 10 cm diameter pipe. However, only small pipes with a diameter of 3 cm are available. So that the channel capacity is not smaller than desired, how many 3 cm pipes need to be used instead of one 10 cm pipe?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36c4a69650598d6facc1f90185e521082bda498ac094a7f1259193659bb97b17 | A drawer contains $10$ baubles, $8$ bangles, and $11$ beads. Assuming that all of each type of jewelry are indistinguishable, and the order of selection is irrelevant, determine the number of ways to select $16$ items from the drawer, if Victoria must select an even number of baubles, an odd number of bangles, and a pr... | [
"14"
] | [
"A drawer contains $10$ baubles, $8$ bangles, and $11$ beads. Assuming that all of each type of jewelry are indistinguishable, and the order of selection is irrelevant, determine the number of ways to select $16$ items from the drawer, if Victoria must select an even number of baubles, an odd number of bangles, and... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dbe27a569b51cfd6e0071f7eb7ad62a8e8177f628c6192ebd0bf4149f0c77db5 | A drawer contains 100 black, 80 blue, 60 red, and 40 white socks. What is the smallest number of socks that must be taken out randomly to guarantee that 10 pairs of socks (where each pair consists of two socks of the same color) have been taken out? | [
"23"
] | [
"A drawer contains 100 black, 80 blue, 60 red, and 40 white socks. What is the smallest number of socks that must be taken out randomly to guarantee that 10 pairs of socks (where each pair consists of two socks of the same color) have been taken out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-53849944345899fb93c548d61d088879c604d953845729122a27d64f526ed4a0 | A drawer contains 100 red socks, 80 green socks, 60 blue socks, and 40 black socks. What is the smallest number of socks that must be selected to guarantee that the selection contains at least 10 pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.) | [
"23"
] | [
"A drawer contains 100 red socks, 80 green socks, 60 blue socks, and 40 black socks. What is the smallest number of socks that must be selected to guarantee that the selection contains at least 10 pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-8e62ac73e314c37d0ea2bdc4319a289f5276540dcdf5e66f1d108e00a85930fe | A drawer contains 8 distinct pairs of gloves. If you pull out four individual gloves at random, what is the probability that you have at least one pair of matching gloves? | [
"\\(\\frac{5}{13}\\)"
] | [
"A drawer contains 8 distinct pairs of gloves. If you pull out four individual gloves at random, what is the probability that you have at least one pair of matching gloves?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5361c75314717f0c7aa02988373d1cb37085ae85fbe792909c06b71c83126ea7 | A drawer contains 8 distinct pairs of gloves. If you pull out four individual gloves at random, what is the probability that you have two pairs of matching gloves? | [
"\\(\\frac{1}{65}\\)"
] | [
"A drawer contains 8 distinct pairs of gloves. If you pull out four individual gloves at random, what is the probability that you have two pairs of matching gloves?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-5f007eccd215f5cc864aeeb8875ad9677609b3b19274b41a3b5231689f6eac7c | A drawer contains \( n \) socks. If you randomly draw 2 socks without replacement, the probability that they are both blue is \(\dfrac{5}{26}\). What is the smallest possible value of the number of blue socks? | [
"6"
] | [
"A drawer contains \\( n \\) socks. If you randomly draw 2 socks without replacement, the probability that they are both blue is \\(\\dfrac{5}{26}\\). What is the smallest possible value of the number of blue socks?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-725eec98643a5164c91f393dceaeb481da3dc0d64b3810b789bd07a8cdc86286 | A drawer contains a large number of white, red, and black socks, with equal numbers of each color. Three socks are drawn at random without replacement. Let \( p \) be the probability that at least two of the socks drawn are the same color. Calculate the value of \( 9p - 7 \). | [
"0"
] | [
"A drawer contains a large number of white, red, and black socks, with equal numbers of each color. Three socks are drawn at random without replacement. Let \\( p \\) be the probability that at least two of the socks drawn are the same color. Calculate the value of \\( 9p - 7 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8910cb275a19e6dff466285b91512b5d16715fd9275316d5e011a7cfb80aab82 | A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consist... | [
"990"
] | [
"A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is c... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 24
} | 56 | 0 | 0 |
omr-ws-9fac0aa7a0886aa0a2bdfee3a86aa02c65861c5776e52ab40c23b565d95c8b67 | A drawer contains red and black socks. The probability that two randomly drawn socks are both red is $\frac{1}{2}$. How small can the total number of socks in the drawer be if the number of black socks is even? | [
"21"
] | [
"A drawer contains red and black socks. The probability that two randomly drawn socks are both red is $\\frac{1}{2}$. How small can the total number of socks in the drawer be if the number of black socks is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c33496c20e81de08c0b5fb2b2b6cb56dc6356ab773544e41647c000e1777b713 | A drawer contains white and blue socks, with at least two of each color. If two socks are drawn at random without replacement, the probability that they are the same color is exactly $\frac{1}{2}$. What is the smallest possible number of socks in the drawer? | [
"9"
] | [
"A drawer contains white and blue socks, with at least two of each color. If two socks are drawn at random without replacement, the probability that they are the same color is exactly $\\frac{1}{2}$. What is the smallest possible number of socks in the drawer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-1ee92aa27a07617ecb3008031b0fd7c3a873d79c27d568da55305d388973e459 | A drill team initially formed a perfect square. They then changed formation to a rectangle with 5 more columns than before. How many columns were in the rectangular formation? | [
"e. none of the above."
] | [
"A drill team initially formed a perfect square. They then changed formation to a rectangle with 5 more columns than before. How many columns were in the rectangular formation?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c824064492dcc07bb16498e599091173603641cf92ad8b0616b6a82a9e53dc2c | A driver in a car is in the desert, 10 km from the nearest point, $N$, on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 20 km from $N$? | [
"40 minutes"
] | [
"A driver in a car is in the desert, 10 km from the nearest point, $N$, on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 20 km from $N$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-79a7f80f1dc53ef92106399487f0cc6a26333c748021d0498f51c2d49430cf37 | A driver in a car is in the desert, 10 km from the nearest point, $N$, on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 30 km from $N$? | [
"\\(\\frac{13}{15}\\) hours or 52 minutes."
] | [
"A driver in a car is in the desert, 10 km from the nearest point, $N$, on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 30 km from $N$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-48d2805364a2b4ce6f3f48a8119765ff3b39a86a551c4d051f9ab93e8892bb1e | A driver in a car is in the desert, 10 km from the nearest point, \( N \), on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 20 km from \( N \)? | [
"\\dfrac{2}{3}"
] | [
"A driver in a car is in the desert, 10 km from the nearest point, \\( N \\), on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 20 km from \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75d2ef4339f34f96f93182874b0aaebe9494188f29de03e87e16a23d99e9c144 | A driver in a car is in the desert, 10 km from the nearest point, \( N \), on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 30 km from \( N \)? | [
"\\dfrac{13}{15}"
] | [
"A driver in a car is in the desert, 10 km from the nearest point, \\( N \\), on a long, straight road. The driver can cover 30 km/h in the desert and 50 km/h on the road. What is the shortest time for the driver to reach a point on the road that is 30 km from \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5e3cbdc70ed569b7a59038ab8ef0fd4ce1832e033ca5eae91aaed7f1a94a2ab | A driver is at point A, 10 km from a long, straight road and 45 km from her base camp at point B. The base camp is 10 km from the road, on the same side of the road as the driver. On the road, the driver can travel at 50 kph, but in the desert, the speed is only 30 kph. The driver wants to return to base camp as quickl... | [
"1.5 hours"
] | [
"A driver is at point A, 10 km from a long, straight road and 45 km from her base camp at point B. The base camp is 10 km from the road, on the same side of the road as the driver. On the road, the driver can travel at 50 kph, but in the desert, the speed is only 30 kph. The driver wants to return to base camp as q... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 6
} | 13 | 0 | 0 |
omr-ws-3eec4c8fe46ab7b9ecaa03242d8d82435617284ade9439a0fa71fc4cce883fb3 | A driver is at point A, 10 km from a paved road, and wishes to travel to point B, which is 45 km from A and 10 km from the road. The driver's speed on the unpaved dirt is 30 km/hr and on the paved road is 50 km/hr. What is the minimum time needed to travel from A to B? | [
"1.43 hours"
] | [
"A driver is at point A, 10 km from a paved road, and wishes to travel to point B, which is 45 km from A and 10 km from the road. The driver's speed on the unpaved dirt is 30 km/hr and on the paved road is 50 km/hr. What is the minimum time needed to travel from A to B?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-ace0dd1c6ef8f4f4038da95f583fe7874b2042e50b516091711db975e585a4c3 | A driver is on a highway with three lanes: the fast lane, the middle lane, and an exit lane. The driver starts in the middle lane. Every minute, if in the fast lane, they have a 1/2 chance of moving to the middle lane and a 1/2 chance of staying in the fast lane. If in the middle lane, they have a 1/2 chance of moving ... | [
"4"
] | [
"A driver is on a highway with three lanes: the fast lane, the middle lane, and an exit lane. The driver starts in the middle lane. Every minute, if in the fast lane, they have a 1/2 chance of moving to the middle lane and a 1/2 chance of staying in the fast lane. If in the middle lane, they have a 1/2 chance of mo... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-e7f3b969a24c4ae4bb95f8fb0ec3bd7f0a8260dc50c2db2373b51971c2787487 | A driving instructor purchases new traffic cones when his old ones are cracked, dented, and losing their reflective coating. In his current stock, \( 65\% \) of the cones are cracked, \( 80\% \) are dented, and \( 70\% \) have lost their coating. What is the smallest possible percentage of traffic cones in this bunch t... | [
"15%"
] | [
"A driving instructor purchases new traffic cones when his old ones are cracked, dented, and losing their reflective coating. In his current stock, \\( 65\\% \\) of the cones are cracked, \\( 80\\% \\) are dented, and \\( 70\\% \\) have lost their coating. What is the smallest possible percentage of traffic cones i... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 14
} | 17 | 0 | 0 |
omr-ws-0149e58607f468046c1a663367722dd9c8d67dc7ca56d884d309364109efbe72 | A droplet of mist is a perfect sphere and picks up moisture at a rate proportional to its surface area. If the initial radius of the droplet is $r_0$ and after $t$ seconds the radius is $r(t)$, find the expression for $r(t)$ assuming the rate of increase of the radius is constant. Express your answer in terms of $r_0$,... | [
"r(t) = r_0 + kt"
] | [
"A droplet of mist is a perfect sphere and picks up moisture at a rate proportional to its surface area. If the initial radius of the droplet is $r_0$ and after $t$ seconds the radius is $r(t)$, find the expression for $r(t)$ assuming the rate of increase of the radius is constant. Express your answer in terms of $... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8140a88308eb65017ba8e5446ea77eddc2c6c42311fa9e113a68ad1ac17ff510 | A drug is injected into a patient at a rate given by \( r(t) = a t e^{-bt} \) ml/sec, where \( t \) is in seconds since the injection started, \( 0 \leq t \leq 5 \), and \( a \) and \( b \) are constants. The maximum rate of 0.3 ml/sec occurs half a second after the injection starts. Find the values of \( a \) and \( b... | [
"2"
] | [
"A drug is injected into a patient at a rate given by \\( r(t) = a t e^{-bt} \\) ml/sec, where \\( t \\) is in seconds since the injection started, \\( 0 \\leq t \\leq 5 \\), and \\( a \\) and \\( b \\) are constants. The maximum rate of 0.3 ml/sec occurs half a second after the injection starts. Find the values of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-5311ae50784fe0294886de657e85728795f93d3aafc585a13f068acfc342f481 | A drug's concentration in the blood must be between \( L \) and \( H \) for it to be safe and effective. Given that the concentration decays exponentially over time with a decay constant \( k \), determine the time interval \( T \) between doses to maintain the concentration within this range. If \( L = 2 \) and \( H =... | [
"1.833"
] | [
"A drug's concentration in the blood must be between \\( L \\) and \\( H \\) for it to be safe and effective. Given that the concentration decays exponentially over time with a decay constant \\( k \\), determine the time interval \\( T \\) between doses to maintain the concentration within this range. If \\( L = 2... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9ff3f415dcb9e7a218c93edd8d0ef215d5a96f9b1603867bf08841a8b522bf2 | A drunk person walks along the number line starting from \( x = 0 \). At each step, they move forward or backward with equal probability. What is the probability that they will reach \( x = 3 \) before reaching \( x = -2 \)? | [
"\\(\\frac{2}{5}\\)"
] | [
"A drunk person walks along the number line starting from \\( x = 0 \\). At each step, they move forward or backward with equal probability. What is the probability that they will reach \\( x = 3 \\) before reaching \\( x = -2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-fd665213431d30515630ff6db09c5ed8a2fb026c7534b59676d5f5311b4d0b2f | A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as 3 square tiles, labeled 1, 2, and 3. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. Assu... | [
"\\(\\frac{7}{3}\\)"
] | [
"A drunkard is randomly walking through a city when he stumbles upon a $2 \\times 2$ sliding tile puzzle. The puzzle consists of a $2 \\times 2$ grid filled with a blank square, as well as 3 square tiles, labeled 1, 2, and 3. During each turn you may fill the empty square by sliding one of the adjacent tiles into i... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b909088f748e1eacb2af97031a11f6f85632b96597c030df3d28d3569d7c26dd | A drunkard performs an infinitely long random walk on the two-dimensional coordinate plane starting at $(0, 0)$. Every point he arrives at is colored red. What is the probability that at some point during his random walk, only the points $(0, 0)$, $(1, 0)$, $(2, 0)$, $(3, 0)$ are colored red? | [
"\\(\\frac{1}{56}\\)"
] | [
"A drunkard performs an infinitely long random walk on the two-dimensional coordinate plane starting at $(0, 0)$. Every point he arrives at is colored red. What is the probability that at some point during his random walk, only the points $(0, 0)$, $(1, 0)$, $(2, 0)$, $(3, 0)$ are colored red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-3022af17661c59cc762a6f1c161f6777ebc502ef3bd1a11e0796c8fb684b2487 | A drunkard starts at $0$ on the number line. Each minute, he steps to the left (decreasing his position by $1$) with probability $\frac{1}{3}$, or to the right (increasing his position by $1$) with probability $\frac{2}{3}$. What is the probability that the drunkard eventually reaches $-1$ ("falls off" the number line)... | [
"\\(\\frac{1}{2}\\)"
] | [
"A drunkard starts at $0$ on the number line. Each minute, he steps to the left (decreasing his position by $1$) with probability $\\frac{1}{3}$, or to the right (increasing his position by $1$) with probability $\\frac{2}{3}$. What is the probability that the drunkard eventually reaches $-1$ (\"falls off\" the num... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c7fd5dd11e448b56f7bf7f335b2d1b6867fa75408e092999e04917262aec54fd | A drunkard starts at pub A and moves to any directly connected pub with equal probability. The pubs are connected as follows: A-B, A-C, B-C, B-D, C-D (A and D are not connected). What is the probability that the drunkard is at pub C after five drinks? | [
"\\(\\frac{1}{3}\\)"
] | [
"A drunkard starts at pub A and moves to any directly connected pub with equal probability. The pubs are connected as follows: A-B, A-C, B-C, B-D, C-D (A and D are not connected). What is the probability that the drunkard is at pub C after five drinks?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8d284b6fbdae6ffe555165a77fbffee62ab4db451717abdef19feebc4c838b3 | A drunkard starts at pub A and moves to any directly connected pub with equal probability. The pubs are connected as follows: A-B, A-C, B-C, B-D, C-D (A and D are not connected). Where is the drunkard most likely to be after n drinks (n > 5)? | [
"C"
] | [
"A drunkard starts at pub A and moves to any directly connected pub with equal probability. The pubs are connected as follows: A-B, A-C, B-C, B-D, C-D (A and D are not connected). Where is the drunkard most likely to be after n drinks (n > 5)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-05d020d088cc29a1dd60b482245c5b2aa22be32249e73dd2f1e720076570c5c0 | A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the horse starts at $(0, 0)$, calculate the probability th... | [
"\\(\\dfrac{1}{32} - \\dfrac{1}{2^{2027}}\\)"
] | [
"A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \\in \\mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \\rightarrow (a \\pm 1, b \\pm 2),$$ $$(a, b) \\rightarrow (a \\pm 2, b \\pm 1)$$ are equally likely. Knowing that the horse starts at $(0, 0)$, calculate the p... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-d6e8100f32269b6b51f64ca6141cc06799463aa8adb4cc0981bb86cb8999d3e3 | A duck drew a square \(ABCD\), then reflected point \(C\) across point \(B\) to obtain point \(E\). The center of the square is \(F\). Point \(G\) is on ray \(EF\) beyond \(F\) such that \(\angle AGC = 135^\circ\). Calculate the measure of \(\angle CGD\). | [
"135^\\circ"
] | [
"A duck drew a square \\(ABCD\\), then reflected point \\(C\\) across point \\(B\\) to obtain point \\(E\\). The center of the square is \\(F\\). Point \\(G\\) is on ray \\(EF\\) beyond \\(F\\) such that \\(\\angle AGC = 135^\\circ\\). Calculate the measure of \\(\\angle CGD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75528a868dacfd4a2697f111cb62ea0408430cfd274feaee32523ec2cdb3dc73 | A duck walked up to a lemonade stand, and it said to the man running the stand, "Hey, if you pour half of the lemonade in your pitcher into my jug of grape juice, the mixture will be 40% grape juice." The lemonade man did so. The duck then poured 1/3 of his new mixture back into the lemonade pitcher. What fraction of t... | [
"\\(\\frac{6}{7}\\)"
] | [
"A duck walked up to a lemonade stand, and it said to the man running the stand, \"Hey, if you pour half of the lemonade in your pitcher into my jug of grape juice, the mixture will be 40% grape juice.\" The lemonade man did so. The duck then poured 1/3 of his new mixture back into the lemonade pitcher. What fracti... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-d75a7de3c41f9662f7b23ec1e72067f462011492d7c7c33dd4d4b9072484ab64 | A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\pi$. | [
"32 \\pi"
] | [
"A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2cd2c57741fbe6657a93c98f26a25547e14d5258207c78df91b5e81a2758366 | A fabric uses tools with a useful life distributed as $\text{N}(300,100)$ (normal distribution) in hours. The fabric uses these tools one at a time, replacing them immediately after they are broken. Find the probability that the fabric needs exactly 4 tools to function for 1000 hours. | [
"0.5603"
] | [
"A fabric uses tools with a useful life distributed as $\\text{N}(300,100)$ (normal distribution) in hours. The fabric uses these tools one at a time, replacing them immediately after they are broken. Find the probability that the fabric needs exactly 4 tools to function for 1000 hours."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-f11ac649e9447573c8001472bbde8210e4b2251c4f5681eab5d884adabeaebe3 | A face-down stack of 8 playing cards consisted of 4 Aces (A's) and 4 Kings (K's). After revealing and removing the top card, the new top card was moved to the bottom of the stack without revealing it. This procedure was repeated until only 1 card was left, which was then revealed. The cards revealed were AKAKAKAK, in t... | [
"AAKAAKKK"
] | [
"A face-down stack of 8 playing cards consisted of 4 Aces (A's) and 4 Kings (K's). After revealing and removing the top card, the new top card was moved to the bottom of the stack without revealing it. This procedure was repeated until only 1 card was left, which was then revealed. The cards revealed were AKAKAKAK,... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-f4b310485fe74b368fc3eb718c6d0bc0bd57a9e1b146bb185742795aed562ae6 | A factor of $12 \cdot 2023^{12}$ is chosen at random. What is the probability that the factor is divisible by 2023? | [
"\\(\\frac{276}{325}\\)"
] | [
"A factor of $12 \\cdot 2023^{12}$ is chosen at random. What is the probability that the factor is divisible by 2023?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-cc18d0c85dfbf96946a3c32991b49fa71c1e8dd7b2a615f5ea76efe98312e2e3 | A factory $P$ produces $10\%$ defective pens and another factory $Q$ produces $20\%$ defective pens. A bag contains $4$ pens from factory $P$ and $5$ pens from factory $Q$. If two pens are drawn at random from the bag, find the probability that both pens are non-defective. | [
"\\(\\frac{1283}{1800}\\)"
] | [
"A factory $P$ produces $10\\%$ defective pens and another factory $Q$ produces $20\\%$ defective pens. A bag contains $4$ pens from factory $P$ and $5$ pens from factory $Q$. If two pens are drawn at random from the bag, find the probability that both pens are non-defective."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-bea17b3ea0184d0cfd510c909cdaaea9ccf1522b0a6e89f78b1cec334e9bca07 | A factory A produces 10% defective valves and another factory B produces 20% defective valves. A bag contains 4 valves from factory A and 5 valves from factory B. If two valves are drawn at random from the bag, find the probability that at least one valve is defective. | [
"0.29"
] | [
"A factory A produces 10% defective valves and another factory B produces 20% defective valves. A bag contains 4 valves from factory A and 5 valves from factory B. If two valves are drawn at random from the bag, find the probability that at least one valve is defective."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 15
} | 28 | 0 | 0 |
omr-ws-1cd62b447b88f791fb423be495decaa1eb65e9549446ee9b91adfbae123d01c1 | A factory conveyor has an angle of depression of 18° and drops 10 feet. How long is the conveyor? | [
"32.5 feet"
] | [
"A factory conveyor has an angle of depression of 18° and drops 10 feet. How long is the conveyor?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-d47210ef6757d8f2cd9f0cae4f8d3a43002baa15b99c5f7ce19a2fc7ea140672 | A factory has 600 gears in stock and uses 20 gears daily. The stock must not fall below 100 at any time. It takes 7 days for a new delivery of gears to arrive. When does the factory need to order the next delivery of gears to maintain the stock above 100? | [
"18"
] | [
"A factory has 600 gears in stock and uses 20 gears daily. The stock must not fall below 100 at any time. It takes 7 days for a new delivery of gears to arrive. When does the factory need to order the next delivery of gears to maintain the stock above 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-c56ebe5c99ac2a34089d10ab256a6382782521a2b0fe66e21c71398370d6ab76 | A factory has to deliver 1100 parts to a client. The parts are packed in boxes of three types. One box of type One holds 70 parts, one of type Two holds 40, and one of type Three holds 25 parts. The cost of delivery in a type One box is 20 roubles, in a type Two box, 10 roubles, in a type Three box, 7 roubles. What kin... | [
"Four boxes of the third type and 25 boxes of the second type"
] | [
"A factory has to deliver 1100 parts to a client. The parts are packed in boxes of three types. One box of type One holds 70 parts, one of type Two holds 40, and one of type Three holds 25 parts. The cost of delivery in a type One box is 20 roubles, in a type Two box, 10 roubles, in a type Three box, 7 roubles. Wha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-432b4d76691296459668a933fb8468e3cc9b1aec683e6830fdb9ef048d8d49d8 | A factory has two plants, $T_1$ and $T_2$, which produce $20\%$ and $80\%$ of the total computers, respectively. The overall defect rate is $7\%$. The probability that a computer is defective given it is produced in plant $T_1$ is $10$ times the probability that a computer is defective given it is produced in plant $T_... | [
"\\(\\frac{26}{31}\\)"
] | [
"A factory has two plants, $T_1$ and $T_2$, which produce $20\\%$ and $80\\%$ of the total computers, respectively. The overall defect rate is $7\\%$. The probability that a computer is defective given it is produced in plant $T_1$ is $10$ times the probability that a computer is defective given it is produced in p... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-3809f01f8c46d7b8218c1615ded3589648fc2e315b71235f42900f4fc56cecac | A factory manufactures bolts where $10\%$ are brass and $90\%$ are steel. The weights of brass and steel bolts are normally distributed as $N(10,4)$ and $N(20,9)$ respectively. Given that a randomly selected bolt weighs $13$ grams, what is the conditional probability that the bolt is made of brass? | [
"\\(\\frac{\\frac{1}{\\sqrt{8 \\pi}} e^{\\left(-\\frac{9}{8}\\right)} \\times 0.1}{\\frac{1}{\\sqrt{8 \\pi}} e^{-\\left(-\\frac{9}{8}\\right)} \\times 0.1+\\frac{1}{\\sqrt{18 \\pi}} e^{\\left(-\\frac{49}{18}\\right)} \\times 0.9}\\)"
] | [
"A factory manufactures bolts where $10\\%$ are brass and $90\\%$ are steel. The weights of brass and steel bolts are normally distributed as $N(10,4)$ and $N(20,9)$ respectively. Given that a randomly selected bolt weighs $13$ grams, what is the conditional probability that the bolt is made of brass?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 7
} | 12 | 0 | 0 |
omr-ws-df14125b3cb4432107f464744439d50696b8ac3de6b0fc7fdc74b8f1083d5795 | A factory needs to produce 3000 toys of type A and 15000 toys of type B. Producing a type A toy takes twice as much time as producing a type B toy. The factory has 200 workers who need to be divided into two groups: one group to produce only type A toys and the other group to produce only type B toys. Determine how man... | [
"58 workers for type A toys and 142 workers for type B toys."
] | [
"A factory needs to produce 3000 toys of type A and 15000 toys of type B. Producing a type A toy takes twice as much time as producing a type B toy. The factory has 200 workers who need to be divided into two groups: one group to produce only type A toys and the other group to produce only type B toys. Determine ho... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-369e49108db1b001268dc0f07c434a59096cd65104f99b6839c73a5261d59b91 | A factory packs its products in cubic boxes and arranges them into a large $8 \times 8 \times 8$ cube, consisting of $512$ smaller cubic boxes. To ensure that each smaller box has at least one face not touching another box when the temperature exceeds a limit, what is the least number of horizontal and vertical plates ... | [
"4"
] | [
"A factory packs its products in cubic boxes and arranges them into a large $8 \\times 8 \\times 8$ cube, consisting of $512$ smaller cubic boxes. To ensure that each smaller box has at least one face not touching another box when the temperature exceeds a limit, what is the least number of horizontal and vertical ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-369a922689e99e670bb2e42a0661b36f3ce49662b8cbd1971f80fe2bf4314a1f | A factory produces cars, small cars, trucks, buses, and oil trucks. The demand for buses and oil trucks is 10 and 20, respectively, over a three-month period, while the demand for the other products is unlimited. The factory has the following department capacities in working hours: - Erecting shop: 100,000 hours - Carp... | [
"550"
] | [
"A factory produces cars, small cars, trucks, buses, and oil trucks. The demand for buses and oil trucks is 10 and 20, respectively, over a three-month period, while the demand for the other products is unlimited. The factory has the following department capacities in working hours:\n- Erecting shop: 100,000 hours\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e4c407e20fe4f8209bbd5f96b56a4216d33784a1011b297b25e53285fe2b3fa | A factory sold two used machines for $1200 each. One machine was sold at a 20% profit, and the other at a 20% loss. What is the total profit or loss from these sales? | [
"-100"
] | [
"A factory sold two used machines for $1200 each. One machine was sold at a 20% profit, and the other at a 20% loss. What is the total profit or loss from these sales?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eff6a8544c1fa86bc69e7ea4b2e1fb3001722f4f88e5a2ca490af7a31d45d339 | A fair $2015^{2015}$-sided die is labeled with the positive integers from $1$ to $2015^{2015}$, inclusive. Steve rolls the die until the sum of the numbers rolled is strictly greater than $2015^{2015}$. Let $E$ be the expected number of times Steve must roll the die for this to occur. Compute $\lfloor 100E\rfloor$. | [
"271"
] | [
"A fair $2015^{2015}$-sided die is labeled with the positive integers from $1$ to $2015^{2015}$, inclusive. Steve rolls the die until the sum of the numbers rolled is strictly greater than $2015^{2015}$. Let $E$ be the expected number of times Steve must roll the die for this to occur. Compute $\\lfloor 100E\\rfloo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-8d718f6f7845c9f324317bf5d994646711d9dc7446d460720cdde924339b45ae | A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? | [
"\\(\\frac{1}{20}\\)",
"\\dfrac{1}{20}"
] | [
"A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 110,
"genselect": 3,
"tir": 45
} | 158 | 0 | 0 |
omr-ws-54952f71a6ee4632880518666b62c0373c5a6a63a487606b4262f16d4fdef547 | A fair $6$-sided die is rolled repeatedly until a $6$ is obtained. Find the expected number of rolls conditioned on the event that none of the rolls yielded an odd number. | [
"\\dfrac{3}{2}"
] | [
"A fair $6$-sided die is rolled repeatedly until a $6$ is obtained. Find the expected number of rolls conditioned on the event that none of the rolls yielded an odd number."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-6781cb98fa18e7992ea8491048a2a0f3dc0f699003a59f63f9dad3108699ac05 | A fair 12-sided die has faces numbered 1 through 12. The die is rolled twice, and the results of the two rolls are \( x \) and \( y \), respectively. Given that \( \tan(2\theta) = \frac{x}{y} \) for some \( \theta \) with \( 0 < \theta < \frac{\pi}{2} \), compute the probability that \( \tan \theta \) is rational. | [
"\\(\\frac{1}{18}\\)"
] | [
"A fair 12-sided die has faces numbered 1 through 12. The die is rolled twice, and the results of the two rolls are \\( x \\) and \\( y \\), respectively. Given that \\( \\tan(2\\theta) = \\frac{x}{y} \\) for some \\( \\theta \\) with \\( 0 < \\theta < \\frac{\\pi}{2} \\), compute the probability that \\( \\tan \\t... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-438d312e187a7e38a3e8ccc6be0fa0c5780200ae3ffc74854f13296d08eb1705 | A fair 12-sided die with sides 0 through 11 is rolled 4 times to form a base 12 number. Find \(p + q\) where \(\frac{p}{q}\) is the probability that \(11 \cdot 9 \cdot 8\) (base 10) divides the resulting number, and \(p\) and \(q\) are relatively prime positive integers. | [
"769"
] | [
"A fair 12-sided die with sides 0 through 11 is rolled 4 times to form a base 12 number. Find \\(p + q\\) where \\(\\frac{p}{q}\\) is the probability that \\(11 \\cdot 9 \\cdot 8\\) (base 10) divides the resulting number, and \\(p\\) and \\(q\\) are relatively prime positive integers."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 8,
"tir": 16
} | 61 | 0 | 0 |
omr-ws-09d5547a5bf6c0b66546391351808473750dbb410d111ea3ce696218ec14e7bc | A fair 20-sided die has faces numbered 1 through 20. The die is rolled three times and the outcomes are recorded. If \(a\) and \(b\) are relatively prime integers such that \(a/b\) is the probability that the three recorded outcomes can be the sides of a triangle with positive area, find \(a+b\). | [
"1201"
] | [
"A fair 20-sided die has faces numbered 1 through 20. The die is rolled three times and the outcomes are recorded. If \\(a\\) and \\(b\\) are relatively prime integers such that \\(a/b\\) is the probability that the three recorded outcomes can be the sides of a triangle with positive area, find \\(a+b\\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-987c0fd65f3981a9bed464cff9fc730f90b4e80aa3f0eb5e736e3c5f97f522d2 | A fair 6-face die is thrown 3 times. Find the probability that the sum of the numbers obtained is a composite number. | [
"\\dfrac{143}{216}"
] | [
"A fair 6-face die is thrown 3 times. Find the probability that the sum of the numbers obtained is a composite number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-99b3a4edae549693f43fa141ac5ffda2ae6682a417cfeda3a44e75b5f1a7d170 | A fair 6-face die is thrown 3 times. Let \( \frac{x}{216} \) be the probability of getting one '3' or getting two '6's. Find the value of \( x \). | [
"29"
] | [
"A fair 6-face die is thrown 3 times. Let \\( \\frac{x}{216} \\) be the probability of getting one '3' or getting two '6's. Find the value of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25e6094e1844e9e8d5ca659253169b46b5fb7696af223f99fa420516f05977ee | A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number (2, 4, 6) appears at least once before the first occurrence of an odd number? | [
"\\(\\frac{1}{20}\\)"
] | [
"A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number (2, 4, 6) appears at least once before the first occurrence of an odd number?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 138,
"tir": 72
} | 210 | 0 | 0 |
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