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-1611a05165098a62fc586b3ad5bbc1ed83217da65dded802ec0dd9ffe605b47f | A boy is traveling with a speed $s = 17\cos^2(t) + e^{17t}$, where $t$ is the time in seconds. How many units is the boy’s displacement in the first 10 seconds? | [
"\\frac{e^{170} - 1}{17} + \\frac{17}{4} \\sin(20) + 85"
] | [
"A boy is traveling with a speed $s = 17\\cos^2(t) + e^{17t}$, where $t$ is the time in seconds. How many units is the boy’s displacement in the first 10 seconds?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-7050e4b521d9e8a0bac5566e836adabf78fec1bb787da8e18cd4d1489adf5d68 | A boy lives on an island where every junction has three roads. He starts from his home and walks along the roads, turning right and left alternately (left, right, left, ...). If he takes exactly 2023 steps before returning to his home for the first time, what is the minimum number of junctions on the island? | [
"2024"
] | [
"A boy lives on an island where every junction has three roads. He starts from his home and walks along the roads, turning right and left alternately (left, right, left, ...). If he takes exactly 2023 steps before returning to his home for the first time, what is the minimum number of junctions on the island?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-697f50166953e3b92ae263d2357a6334f6a69742a27d907e0c472138d38381b9 | A boy moves 4 miles south, then turns 90 degrees to the left and moves forward 6 miles, then turns 90 degrees to the left again and moves forward 4 miles. How far is he now from his starting point? | [
"\\( 2\\sqrt{13} \\)"
] | [
"A boy moves 4 miles south, then turns 90 degrees to the left and moves forward 6 miles, then turns 90 degrees to the left again and moves forward 4 miles. How far is he now from his starting point?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a8ebabd2cc27505e8564df2f5b048794e3e1c4f4e4ca4e90bf2062b05e7c889 | A boy passing by the cinema on a bus could notice only the hours (but not the minutes!) when four (of the eight) shows began: $$\begin{array}{cccccc} 1\mbox{st} & \mbox{show} & - & 12\mbox{ (hours)} & \ldots & \mbox{(minutes)}\\ 2\mbox{nd} & \mbox{show} & - & 13\mbox{ (hours)} & \ldots & \mbox{(minutes)}\\ \ldots & \ld... | [
"On denoting by $12 + x$ the time of the beginning of the first performance and by $y$ the duration of the performance we can readily set a linear system of inequalities for the numbers $x$ and $y$"
] | [
"A boy passing by the cinema on a bus could notice only the hours (but not the minutes!) when four (of the eight) shows began: $$\\begin{array}{cccccc} 1\\mbox{st} & \\mbox{show} & - & 12\\mbox{ (hours)} & \\ldots & \\mbox{(minutes)}\\\\ 2\\mbox{nd} & \\mbox{show} & - & 13\\mbox{ (hours)} & \\ldots & \\mbox{(minute... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5a051946044a67c9f59312bf045c6d39b6d310c1a31a3299f359bca6fe26f45 | A boy plays 10 games, with the probability of winning any game being 0.25 (independent of other games). Let \( X \) be the number of games he wins. Find the probability \( P(X \leq 3) \). | [
"0.7759"
] | [
"A boy plays 10 games, with the probability of winning any game being 0.25 (independent of other games). Let \\( X \\) be the number of games he wins. Find the probability \\( P(X \\leq 3) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18833df2a62258fb68d3941db02422f07988df3b3a874d3f09c238833f2bd68e | A boy rides a scooter from one bus stop to another and looks in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change his direction. What is the maximum distance between the bus stops such that the boy is guaranteed not to miss the bus, given that he knows he is traveling at a... | [
"3"
] | [
"A boy rides a scooter from one bus stop to another and looks in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change his direction. What is the maximum distance between the bus stops such that the boy is guaranteed not to miss the bus, given that he knows he is traveling... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d02130e98c2c22ab4bc8c625689784177f1f0bc25f354d90ff2ee5311be7578e | A boy rolls 10 dice. What is the probability that five dice score '6' and the other five do not score '6'? | [
"\\dfrac{21875}{1679616}"
] | [
"A boy rolls 10 dice. What is the probability that five dice score '6' and the other five do not score '6'?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e476c407d3c0e9f4a8db9b965069737d97f7d46971a0b033652a429f0eedfeb4 | A boy stands on his tiptoes with his eyes at a height of 1 m above sea-level, and when seated on his father’s shoulders, his eyes are at a height of 2 m. What is the ratio of the distances visible to him in these two cases? (Assume the radius of the Earth is 6000 km and provide the answer to 0.1.) | [
"1.4"
] | [
"A boy stands on his tiptoes with his eyes at a height of 1 m above sea-level, and when seated on his father’s shoulders, his eyes are at a height of 2 m. What is the ratio of the distances visible to him in these two cases? (Assume the radius of the Earth is 6000 km and provide the answer to 0.1.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 5
} | 24 | 0 | 0 |
omr-ws-8fad9df3dd4ac13e5218253998b595b8b180edbd47190304887e9f3961d35e75 | A boy walks away from a vertical rope at a constant velocity \( v_{boy} \). The rope is attached to a food pack suspended from a tree limb at a height \( h \) above the boy's hands. If \( x \) is the distance the boy has walked away from the vertical rope, calculate the speed \( v \) of the food pack when the boy is \(... | [
"\\dfrac{v_{boy} x}{\\sqrt{x^2 + h^2}}"
] | [
"A boy walks away from a vertical rope at a constant velocity \\( v_{boy} \\). The rope is attached to a food pack suspended from a tree limb at a height \\( h \\) above the boy's hands. If \\( x \\) is the distance the boy has walked away from the vertical rope, calculate the speed \\( v \\) of the food pack when ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d89815575386db158d15ce565a6c520417b54a8d389f74f864fdf0713961102c | A boy walks from the incenter to the centroid and then to the circumcenter of a triangle with sides 9, 12, and 15. In simplest radical form, he walks $\frac{w-k\sqrt{p}}{2}$ farther than a girl who walks directly from the incenter to the circumcenter of the same triangle. Find the value of $k+p+w$. | [
"15"
] | [
"A boy walks from the incenter to the centroid and then to the circumcenter of a triangle with sides 9, 12, and 15. In simplest radical form, he walks $\\frac{w-k\\sqrt{p}}{2}$ farther than a girl who walks directly from the incenter to the circumcenter of the same triangle. Find the value of $k+p+w$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28eba142d650b32249c33982820d1e88384b2759cb08179cdd0453bbb10f1458 | A boy writes the natural number \( n \) three times on a blackboard. He then repeatedly performs the following operation: he erases one of the numbers and writes in its place the sum of the two remaining numbers minus 1. After several moves, one of the three numbers on the blackboard is 900. Find all possible values of... | [
"300"
] | [
"A boy writes the natural number \\( n \\) three times on a blackboard. He then repeatedly performs the following operation: he erases one of the numbers and writes in its place the sum of the two remaining numbers minus 1. After several moves, one of the three numbers on the blackboard is 900. Find all possible va... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f2c00ca73b540bad7e507e14fc530e4f6a1c913de75f27846879e617c0c2269 | A bridge is to be built across a small lake from B to C. The bearing from B to C is S $41^\circ$ W. From a point A, 100 meters from B, the bearings to B and C are S $74^\circ$ E and S $28^\circ$ E, respectively. Find the distance from B to C. | [
"77"
] | [
"A bridge is to be built across a small lake from B to C. The bearing from B to C is S $41^\\circ$ W. From a point A, 100 meters from B, the bearings to B and C are S $74^\\circ$ E and S $28^\\circ$ E, respectively. Find the distance from B to C."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-e743494a165a78118ad798a010b213e1f01eaa65f138b79c0616172042c8d337 | A bridge is to be built in the shape of a parabolic arch with a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is 10 feet. Find the height of the arch at its center. | [
"\\(\\frac{250}{9}\\)"
] | [
"A bridge is to be built in the shape of a parabolic arch with a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is 10 feet. Find the height of the arch at its center."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-3704601b525ce7c6842216de096c3d7fb91ecdf73b3839ef12a60592d95bbf05 | A bridge is to be built in the shape of a parabolic arch with a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center. | [
"\\(\\frac{250}{9}\\)"
] | [
"A bridge is to be built in the shape of a parabolic arch with a span of 100 feet. The height of the arch, a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6515cda3d9dc88c99b527e3b3f523a6fc2fa92a387648bea7b7b296a9a0e77ba | A bridge over a 16-meter wide valley stream is designed in the shape of a parabola with a height ranging between 4 meters and 6 meters at the middle. Determine the equation of the parabola that describes the bridge. | [
"\\( y = -\\frac{1}{16}x^2 + 4 \\) or \\( y = -\\frac{3}{32}x^2 + 6 \\)"
] | [
"A bridge over a 16-meter wide valley stream is designed in the shape of a parabola with a height ranging between 4 meters and 6 meters at the middle. Determine the equation of the parabola that describes the bridge."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 7
} | 24 | 0 | 0 |
omr-ws-86e3032fec3497f836311ba0ff418bac3968232e11f8e73b021d26b320814b0f | A bridge over a valley that is 16 meters wide is designed in the shape of a parabola. The height of the bridge ranges between 4 meters and 6 meters at the middle. Determine the equation of the parabola that describes the shape of the bridge. | [
"\\( y = \\frac{1}{32} x^2 + 4 \\)"
] | [
"A bridge over a valley that is 16 meters wide is designed in the shape of a parabola. The height of the bridge ranges between 4 meters and 6 meters at the middle. Determine the equation of the parabola that describes the shape of the bridge."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e5e58a03bfe90049abfdf1866258e700853d752989ccce47e228bca1b0322be0 | A broken line \( A_1A_2 \ldots A_n \) is drawn in a \( 50 \times 50 \) square such that the distance from any point in the square to the broken line is less than 1. Calculate the minimum possible length of the broken line. | [
"1250"
] | [
"A broken line \\( A_1A_2 \\ldots A_n \\) is drawn in a \\( 50 \\times 50 \\) square such that the distance from any point in the square to the broken line is less than 1. Calculate the minimum possible length of the broken line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-8f4bb2316217fa44dee8b3fcb0367649821ccc396b9c2d7f93a7c28afcf97fb8 | A broken line \( A_1A_2 \ldots A_n \) is drawn in a \( 50 \times 50 \) square, so that the distance from any point of the square to the broken line is less than \( 1 \). Let \( L \) be the total length of the broken line. Find the minimum integer value of \( L \). | [
"1250"
] | [
"A broken line \\( A_1A_2 \\ldots A_n \\) is drawn in a \\( 50 \\times 50 \\) square, so that the distance from any point of the square to the broken line is less than \\( 1 \\). Let \\( L \\) be the total length of the broken line. Find the minimum integer value of \\( L \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-ab8c08835de478a37942410e5e1e4aa3de85c082b682ae65a4ff843c3b05e105 | A broken line consists of 31 segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines. | [
"16",
"2"
] | [
"A broken line consists of 31 segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-404331a19be2fa6ddf6559d61f78e723284a5c7eb5624d9d30dc56198df63401 | A bug crawls from vertex to vertex on a cube along the edges. If the bug makes 7 moves, what is the probability that it visits every vertex exactly once? | [
"\\(\\frac{2}{243}\\)"
] | [
"A bug crawls from vertex to vertex on a cube along the edges. If the bug makes 7 moves, what is the probability that it visits every vertex exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 23
} | 34 | 0 | 0 |
omr-ws-0d73a006001ede1f6efd2caddfb6ef68bfbf1f34771954070c66cbc37652a94e | A bug either splits into two perfect copies of itself with probability \( p \) or dies. What is the probability that a bug's descendants die out? | [
"\\( P = \\begin{cases} \n1 & \\text{if } p \\leq \\frac{1}{2} \\\\\n\\frac{1 - p}{p} & \\text{if } p > \\frac{1}{2} \n\\end{cases} \\)"
] | [
"A bug either splits into two perfect copies of itself with probability \\( p \\) or dies. What is the probability that a bug's descendants die out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-4c50663e86c34e87d609a8576fde69ca47ec4bcd9b28ac4384da3c97551a0156 | A bug is at (3,5) and wants to get to (-17,-3). Its speed is 2 units per second in the 1st, 3rd, and 4th quadrants, and 1 unit per second in the 2nd quadrant. Find the path that gives the minimum time. | [
"(3,5) \\to (0,0) \\to (-17,-3)"
] | [
"A bug is at (3,5) and wants to get to (-17,-3). Its speed is 2 units per second in the 1st, 3rd, and 4th quadrants, and 1 unit per second in the 2nd quadrant. Find the path that gives the minimum time."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-54c88acf2a130036be1519037a54f62382333986629383246ecb339e5a5eb7c5 | A bug is crawling on a sheet of squared paper starting at the origin $(0,0)$. In each step, it can move two units to the right, four units to the left, three units up, or five units down, and then it makes a turn of exactly $90^\circ$. Determine the number of distinct lattice points $(x, y)$ that the bug can reach afte... | [
"36"
] | [
"A bug is crawling on a sheet of squared paper starting at the origin $(0,0)$. In each step, it can move two units to the right, four units to the left, three units up, or five units down, and then it makes a turn of exactly $90^\\circ$. Determine the number of distinct lattice points $(x, y)$ that the bug can reac... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31b5b426ff7f1cceaa955739ea298831b6f24ee859edb85efffdffe31fd55a7a | A bug is crawling on the coordinate plane from $(7,11)$ to $(-17,-3)$. The bug travels at a constant speed of one unit per second everywhere except in quadrant II, where it travels at $\frac{1}{2}$ unit per second. What path should the bug take to complete its journey in minimal time? | [
"(0,0)"
] | [
"A bug is crawling on the coordinate plane from $(7,11)$ to $(-17,-3)$. The bug travels at a constant speed of one unit per second everywhere except in quadrant II, where it travels at $\\frac{1}{2}$ unit per second. What path should the bug take to complete its journey in minimal time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-6184e42997b0b773478284103477a3fae2892e61deebfa345774597b10194b66 | A bug is crawling on the coordinate plane from $(7,11)$ to $(-17,-4)$. The bug travels at a constant speed of one unit per second everywhere except in quadrant II, where it travels at $.5$ units per second. What path should the bug take to complete its journey in minimal time? | [
"The bug should take a straight line to the origin \\((0,0)\\) and then a straight line to \\((-17,-4)\\)."
] | [
"A bug is crawling on the coordinate plane from $(7,11)$ to $(-17,-4)$. The bug travels at a constant speed of one unit per second everywhere except in quadrant II, where it travels at $.5$ units per second. What path should the bug take to complete its journey in minimal time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 10
} | 21 | 0 | 0 |
omr-ws-23ad78da8ab97e1b137b9622df878fecfd991ecdca487640269031c5c6026370 | A bug is crawling on the coordinate plane from $(7,11)$ to $(-17,3)$. The bug travels at a constant speed of one unit per second everywhere except in quadrant II, where it travels at $\frac{1}{2}$ units per second. What path should the bug take to complete its journey in minimal time? | [
"The bug should travel in a straight line from $(7, 11)$ to $(-34, 6)$."
] | [
"A bug is crawling on the coordinate plane from $(7,11)$ to $(-17,3)$. The bug travels at a constant speed of one unit per second everywhere except in quadrant II, where it travels at $\\frac{1}{2}$ units per second. What path should the bug take to complete its journey in minimal time?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93a5c284753aa778c7a6ab17e956231b7a5b5f471baea865d44980325a6cad29 | A bug is crawling on the coordinate plane, starting at the origin. Each minute, the bug moves one space, either up or to the right. What is the probability that, after 6 minutes, the bug will have visited the point $(2, 2)$? | [
"\\(\\frac{3}{8}\\)"
] | [
"A bug is crawling on the coordinate plane, starting at the origin. Each minute, the bug moves one space, either up or to the right. What is the probability that, after 6 minutes, the bug will have visited the point $(2, 2)$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6ba8d2e74b51608240b78e0b8ba3b44ad3770d1f2ff32c907748522d39486ab4 | A bug is crawling on the edges of a cube. Every minute, it moves to a different vertex only by moving on the edges. What is the probability that after 6 minutes, the bug is back at its original vertex? | [
"\\dfrac{61}{243}"
] | [
"A bug is crawling on the edges of a cube. Every minute, it moves to a different vertex only by moving on the edges. What is the probability that after 6 minutes, the bug is back at its original vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f991800f80c99cdac05ce5b391ab81e6e033d1dec94d978bbccf6c837f0cb9ea | A bug is on a cube. Every minute, it moves to another vertex randomly. What is the probability that it winds up on the original vertex after $n$ moves? | [
"\\dfrac{(1 + (-1)^n)(1 + 3^{1 - n})}{8}"
] | [
"A bug is on a cube. Every minute, it moves to another vertex randomly. What is the probability that it winds up on the original vertex after $n$ moves?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-54180f616b9f581e848ae1fe01d78ed2e6845db765f05631fc799c773ab046f3 | A bug is on a vertex of a regular icosahedron. Every second, it can either move to one of the adjacent vertices or teleport to the opposite vertex, but it can teleport at most twice. Compute the last 3 nonzero digits of the number of ways the bug can return to the original vertex after exactly 7 seconds. | [
"265"
] | [
"A bug is on a vertex of a regular icosahedron. Every second, it can either move to one of the adjacent vertices or teleport to the opposite vertex, but it can teleport at most twice. Compute the last 3 nonzero digits of the number of ways the bug can return to the original vertex after exactly 7 seconds."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-9586c5c4aaab6c41fa4299c978154fadc1ba7990541f990fb6758028f0c8c201 | A bug is on the $0$ of a number line. At any point in time, it may jump up $1$ or down $1$ on the number line. What is the probability that the bug will at some point end up on $0$ again? | [
"1"
] | [
"A bug is on the $0$ of a number line. At any point in time, it may jump up $1$ or down $1$ on the number line. What is the probability that the bug will at some point end up on $0$ again?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0e890408151cadb9c1b1af8246ab0ba5c845ebe2763a11f10e82608c4ce806b | A bug is placed at the bottom left corner of a 9 by 7 grid and can only move one unit up or one unit to the right at a time. How many ways can the bug reach the top right corner of the grid? | [
"11440"
] | [
"A bug is placed at the bottom left corner of a 9 by 7 grid and can only move one unit up or one unit to the right at a time. How many ways can the bug reach the top right corner of the grid?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 26
} | 36 | 0 | 0 |
omr-ws-d4b43b622929dfd139f12b05ad2dfdf5ca873b983e4deaadf1ee33bad3b05c4e | A bug is placed on a 2-dimensional grid. Every second, the bug moves 1 unit up, down, left, or right, with each direction having an equal chance of being chosen. If the bug is allowed to move for 6 seconds (6 moves), what is the probability that the bug ends up where it started? | [
"\\(\\frac{25}{256}\\)"
] | [
"A bug is placed on a 2-dimensional grid. Every second, the bug moves 1 unit up, down, left, or right, with each direction having an equal chance of being chosen. If the bug is allowed to move for 6 seconds (6 moves), what is the probability that the bug ends up where it started?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 59
} | 91 | 0 | 0 |
omr-ws-b0a6d14cb4e359b427484df81311d3603079f3d530b18629cd97f3885fea4875 | A bug is sitting on one corner of a $3' \times 4' \times 5'$ block of wood. What is the minimum distance it needs to travel along the block’s surface to reach the opposite corner? | [
"\\sqrt{74}"
] | [
"A bug is sitting on one corner of a $3' \\times 4' \\times 5'$ block of wood. What is the minimum distance it needs to travel along the block’s surface to reach the opposite corner?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0b37f083e8ff78a6c3ae4a1662e4341761c134a776b86b6f3bf74c894af3344 | A bug is sitting on one corner of a 3' x 4' x 5' block of wood. What is the minimum distance it needs to travel along the block's surface to reach the opposite corner? Express the answer in the form \( m^{1/2} \), where \( m \) is a positive integer. Find \( m \). | [
"74"
] | [
"A bug is sitting on one corner of a 3' x 4' x 5' block of wood. What is the minimum distance it needs to travel along the block's surface to reach the opposite corner? Express the answer in the form \\( m^{1/2} \\), where \\( m \\) is a positive integer. Find \\( m \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bcb3de95f7ba3949020148377e6f9384bfd981c79009557b6f3f0555bfcba2c | A bug is standing on vertex $A$ of an octahedron $ABCDEF$, where $F$ and $A$ are the apexes of square pyramids $FBCDE$ and $ABCDE$, respectively. A move is counted when the bug travels to an adjacent vertex. What is the probability that the bug will be standing on vertex $F$ after 6 moves? | [
"\\(\\frac{11}{64}\\)"
] | [
"A bug is standing on vertex $A$ of an octahedron $ABCDEF$, where $F$ and $A$ are the apexes of square pyramids $FBCDE$ and $ABCDE$, respectively. A move is counted when the bug travels to an adjacent vertex. What is the probability that the bug will be standing on vertex $F$ after 6 moves?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d485c482b2f0999acbd48cfedcec52f8008fd1ad5a4099fdb633db883df242d4 | A bug jumps in a clockwise direction on a circle with five points numbered 1, 2, 3, 4, and 5. If the bug is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps, on which point will it be? | [
"4"
] | [
"A bug jumps in a clockwise direction on a circle with five points numbered 1, 2, 3, 4, and 5. If the bug is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps, on which point will it be?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-970b4ee5c9c3d968602bc879753a88da33d068d4ee3d74b4526b9c34b9188ef2 | A bug lives on a corner of a cube and is allowed to travel only on the edges of the cube. In how many ways can the bug visit each of the other seven corners exactly once and return to its home corner at the end of the trip? | [
"12"
] | [
"A bug lives on a corner of a cube and is allowed to travel only on the edges of the cube. In how many ways can the bug visit each of the other seven corners exactly once and return to its home corner at the end of the trip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-5e3aeee1eaa0bdb31b115c82317b9061592b919fa5c7a26ddf2dbf7052c38ba4 | A bug moves around a triangle wire. At each vertex, it has a 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as \( \frac{p}{q} \) for positive relatively prime integers \( p \) and \( q \). Determine \( p + q ... | [
"683"
] | [
"A bug moves around a triangle wire. At each vertex, it has a 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as \\( \\frac{p}{q} \\) for positive relatively prime integers \\( p \\) and \\( q \\). Determi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c59a484c6b0aaa6d4e5dd5580e440ceb64dde612c41ad68de792040f7c98279b | A bug moves around a wire triangle. At each vertex, it has a 1/2 chance of moving towards each of the other two vertices. What is the probability that after crawling along 10 edges, it reaches its starting point? | [
"171/512"
] | [
"A bug moves around a wire triangle. At each vertex, it has a 1/2 chance of moving towards each of the other two vertices. What is the probability that after crawling along 10 edges, it reaches its starting point?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 60
} | 92 | 0 | 0 |
omr-ws-bf9c05ccdbffcbb0e44a749f37294d7af6ba678e6fdbdd8ed4e72a9a52eed577 | A bug sits at a corner of a rectangular box with dimensions $1 \times 1 \times 2$. The bug can only travel on the faces and edges of the box. Find the shortest path for the bug to travel from its corner to the diagonally opposed corner. | [
"$2\\sqrt{2}$"
] | [
"A bug sits at a corner of a rectangular box with dimensions $1 \\times 1 \\times 2$. The bug can only travel on the faces and edges of the box. Find the shortest path for the bug to travel from its corner to the diagonally opposed corner."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 28
} | 51 | 0 | 0 |
omr-ws-a45362ed6ae28b3c6e835cd2dc376d9b0c931a80d61b33a515003bc687b00078 | A bug sits on one corner of a unit cube and wants to crawl to the diagonally opposite corner, staying on the surface of the cube. What is the length of the shortest path? | [
"\\(\\sqrt{5}\\)"
] | [
"A bug sits on one corner of a unit cube and wants to crawl to the diagonally opposite corner, staying on the surface of the cube. What is the length of the shortest path?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-6a1403fd420fbf23e590132a7fcf9ef65ef7f430c46858aac9ecf21d46e28dd3 | A bug starts a neverending journey at the origin in a coordinate plane. He travels $1$ unit up, $2$ units right, $3$ units up, $4$ units right, and so on. A point $P$ is randomly chosen in the first quadrant. What is the probability that at some point on its journey, the bug is located directly above point $P$? | [
"\\dfrac{1}{2}"
] | [
"A bug starts a neverending journey at the origin in a coordinate plane. He travels $1$ unit up, $2$ units right, $3$ units up, $4$ units right, and so on. A point $P$ is randomly chosen in the first quadrant. What is the probability that at some point on its journey, the bug is located directly above point $P$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-860ba18160dc7f91fababd95d3eb1528e9be31549a098fe95e691cc8af270092 | A bug starts at $(0,0)$ on the x-axis and each of its moves is either a step right of length one, a step left of length one, or staying put. In how many ways can it land on $(i,0)$ after $n$ moves? | [
" \\sum_{k=0}^{\\left\\lfloor \\frac{n - i}{2} \\right\\rfloor} \\binom{n}{k} \\binom{n - k}{k + i} "
] | [
"A bug starts at $(0,0)$ on the x-axis and each of its moves is either a step right of length one, a step left of length one, or staying put. In how many ways can it land on $(i,0)$ after $n$ moves?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-28d1c004a02cc07c5dc05d5bbc0c4c25ae8660159f1e77ad1801656321dac698 | A bug starts at $(0,0)$ on the x-axis and each of its moves is either a step right of length one, a step left of length one, or staying put. It's equally likely for the bug to go either left or right at each move, but it's twice as likely that the bug stays put than that it chooses to go right or left at each move. Com... | [
"\\dfrac{\\dbinom{2n}{n+i}}{4^n}"
] | [
"A bug starts at $(0,0)$ on the x-axis and each of its moves is either a step right of length one, a step left of length one, or staying put. It's equally likely for the bug to go either left or right at each move, but it's twice as likely that the bug stays put than that it chooses to go right or left at each move... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0efa151832d45789da6c5ad8c5e18db4bb062295daac7394ba89722f1d42aa46 | A bug starts at $(0,0)$. It can make three moves: a step right of length 1, a step left of length 1, or stay put. The probability of staying put is twice the probability of going left, and the probability of going left is the same as going right. Compute the probability that the bug lands on $(i,0)$ after $n$ moves. | [
"\\dfrac{\\dbinom{2n}{n+i}}{4^n}"
] | [
"A bug starts at $(0,0)$. It can make three moves: a step right of length 1, a step left of length 1, or stay put. The probability of staying put is twice the probability of going left, and the probability of going left is the same as going right. Compute the probability that the bug lands on $(i,0)$ after $n$ move... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-62b4af785baa981cc1093db8e2a073c8f2b57dff8852123cfb350039f425c37b | A bug starts at (0, 0) and each of its moves is either a step right of length one or a step left of length one. In how many ways can it land on (i,0) after n moves? | [
"\\(\\binom{n}{\\frac{n + i}{2}}\\)"
] | [
"A bug starts at (0, 0) and each of its moves is either a step right of length one or a step left of length one. In how many ways can it land on (i,0) after n moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-530ab8ee18967c24112c95e1358e515aca2fe37aa46c187a1badaf881c65cbe3 | A bug starts at (0, 0) and each of its moves is either a step right of length one, a step left of length one, or staying put. In how many ways can it land on (i,0) after n moves? | [
" \\sum_{k=0}^{\\left\\lfloor \\frac{n - |i|}{2} \\right\\rfloor} \\frac{n!}{k! \\, (k + |i|)! \\, (n - 2k - |i|)!} "
] | [
"A bug starts at (0, 0) and each of its moves is either a step right of length one, a step left of length one, or staying put. In how many ways can it land on (i,0) after n moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-72bc2e2d9de92b38edb9d572a3836f6b3ab2a8bb0bbabf8b1fc6c4e4fefcbb16 | A bug starts at (0, 0) and each of its moves is either a step right of length one, a step left of length one, or staying put. The probability of staying put is twice the probability of moving right or left. Compute the probability that the bug lands on (i, 0) after n moves. | [
" \\dfrac{ \\dbinom{2n}{n + i} }{4^n} ",
" \\dfrac{1}{4^n} \\dbinom{2n}{n + i} "
] | [
"A bug starts at (0, 0) and each of its moves is either a step right of length one, a step left of length one, or staying put. The probability of staying put is twice the probability of moving right or left. Compute the probability that the bug lands on (i, 0) after n moves."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 19,
"genselect": 6
} | 25 | 0 | 0 |
omr-ws-232224417da232dd2960f0853c80a4546244ac931716e1db0a73b0612e317af2 | A bug starts at (0,0) in the Cartesian plane and moves one unit up or one unit to the right each minute. Define $S=\{ (1,1), (2,2), (3,3), \ldots, (8,8), (9,9)\}$. How many paths pass through exactly 5 points in $S$ and end at $(10,10)$? | [
"27456"
] | [
"A bug starts at (0,0) in the Cartesian plane and moves one unit up or one unit to the right each minute. Define $S=\\{ (1,1), (2,2), (3,3), \\ldots, (8,8), (9,9)\\}$. How many paths pass through exactly 5 points in $S$ and end at $(10,10)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-f90965ff6eb57fe614c2944f80b0abfbe2fb4868c7213549bc039ff22f08d5e8 | A bug starts at a corner vertex of a graph and at each step moves along an adjacent edge with equal probability for any adjacent edge. It continues until it gets to the center vertex. What is the expected (average) number of steps required until the first time it gets to the center vertex? | [
"6"
] | [
"A bug starts at a corner vertex of a graph and at each step moves along an adjacent edge with equal probability for any adjacent edge. It continues until it gets to the center vertex. What is the expected (average) number of steps required until the first time it gets to the center vertex?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2d6cd6b7589bf366959518660065513b6622ba3203b398e2b568d18370f0a64 | A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step, the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away... | [
"\\(\\frac{13}{108}\\)"
] | [
"A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step, the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 27
} | 79 | 0 | 0 |
omr-ws-aceddef150910e8dfd2ef7d12cb598b2ee1eb7e6b6100e46ef9362068e2374f9 | A bug starts at a vertex of a hexagon and moves to an adjacent vertex at each step. What is the probability that the bug returns to its starting vertex after 12 steps? | [
"\\(\\frac{683}{2048}\\)"
] | [
"A bug starts at a vertex of a hexagon and moves to an adjacent vertex at each step. What is the probability that the bug returns to its starting vertex after 12 steps?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-11b94562acc87875c9f5b9d1f1fc1c50912116cf533eeecfcc164b8d8629954e | A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | [
"683"
] | [
"A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-d2adfe00c97411cab3f78e4914f4a6dac873ef217879a3b51ba2b7a9ebb029b2 | A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is \( \frac{m}{n} \), where \( ... | [
"683"
] | [
"A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is \\( \\frac{m}{n} \\), w... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-c4f701b2469e15e09a91e8932df33c6c3eff89cc259f3ad2bcf553d140f8d5bd | A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug returns to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ a... | [
"683"
] | [
"A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug returns to its starting vertex on its tenth move is $m/n,$ where $m$ and ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-da1262d58d022cb932564e2f58666df750c1155275a6454872e4e0e48772ec52 | A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug returns to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ a... | [
"683"
] | [
"A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug returns to its starting vertex on its tenth move is $m/n,$ where $m$ and ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-454adcbfb65c4973e08f6fbf27a5bf69ade34a820bfe9f2e9bff638a5b1008cd | A bug starts at coordinate $x=1$ on the real axis. Each step, the bug can jump to either $x=a+2$ or $x=\frac{a}{2}$ from its current position $x=a$. Let $S(n)$ be the number of distinct positions (including the initial position) the bug can reach in at most $n$ steps. Calculate $S(10) - F_{14} + 14$, where $F_n$ is the... | [
"0"
] | [
"A bug starts at coordinate $x=1$ on the real axis. Each step, the bug can jump to either $x=a+2$ or $x=\\frac{a}{2}$ from its current position $x=a$. Let $S(n)$ be the number of distinct positions (including the initial position) the bug can reach in at most $n$ steps. Calculate $S(10) - F_{14} + 14$, where $F_n$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-7765e55a1c3f57cfda9c5a6975be1c4a892992c4281b6d916ac4c3fa4ab20de6 | A bug starts at coordinate \( x = 1 \) on the real axis. At each step, the bug can jump to either \( x = a + 2 \) or \( x = \frac{a}{2} \) from its current position \( x = a \). How many distinct positions can the bug reach in at most \( n \) steps, including the original position? | [
"F_{n+5} - n - 4"
] | [
"A bug starts at coordinate \\( x = 1 \\) on the real axis. At each step, the bug can jump to either \\( x = a + 2 \\) or \\( x = \\frac{a}{2} \\) from its current position \\( x = a \\). How many distinct positions can the bug reach in at most \\( n \\) steps, including the original position?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-4342728b84cf4d4c739c7ad066fbdb18bf1402644909af7cdfdf109d0139aa01 | A bug starts at one vertex of a cube and moves along the edges according to the rule that at each vertex, it will choose one of the three emanating edges with equal probability. What is the probability that after seven moves, the bug will have visited every vertex exactly once? | [
"\\dfrac{2}{243}"
] | [
"A bug starts at one vertex of a cube and moves along the edges according to the rule that at each vertex, it will choose one of the three emanating edges with equal probability. What is the probability that after seven moves, the bug will have visited every vertex exactly once?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-0d3837b7083e19bc8f6c0e06da0c53012a59b2f9f6806e83e61cc79866b0e7c8 | A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule: at each vertex, the bug will choose to travel along one of the three edges emanating from that vertex, each with equal probability, and all choices are independent. What is the probability that after seven moves ... | [
"\\dfrac{4}{243}"
] | [
"A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule: at each vertex, the bug will choose to travel along one of the three edges emanating from that vertex, each with equal probability, and all choices are independent. What is the probability that after seven m... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-371250d149af08ca9f9815ea7b15eb175d3493d70486352bae8bf1837dec681b | A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule: at each vertex, the bug will choose to travel along one of the three edges emanating from that vertex, with each edge having an equal probability of being chosen, and all choices are independent. What is the prob... | [
"\\(\\frac{2}{243}\\)",
"\\dfrac{2}{243}",
"\\dfrac{4}{243}"
] | [
"A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule: at each vertex, the bug will choose to travel along one of the three edges emanating from that vertex, with each edge having an equal probability of being chosen, and all choices are independent. What is the... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 5,
"tir": 54
} | 94 | 0 | 0 |
omr-ws-8365549fb6d88a593c0bfae77183616052ef3e3f484d5a50632e1cd1411864e8 | A bug starts at one vertex of a regular octahedron and moves along the edges according to the rule that at each vertex, it will choose one of the four emanating edges with equal probability. What is the probability that after five moves, the bug will have visited every vertex exactly once? | [
"\\dfrac{5}{128}"
] | [
"A bug starts at one vertex of a regular octahedron and moves along the edges according to the rule that at each vertex, it will choose one of the four emanating edges with equal probability. What is the probability that after five moves, the bug will have visited every vertex exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-30a6f3c11d3d1d8c57f910b51027d40c9e75193cec0f333c22c73f544a61c391 | A bug starts at point $A$ on a circle with points $A$, $B$, and $C$. At any moment, the bug can move to either of the other two points. If the bug's path includes the sequence $ABCA$, it is squashed. What is the expected waiting time for the bug to be squashed? | [
"24"
] | [
"A bug starts at point $A$ on a circle with points $A$, $B$, and $C$. At any moment, the bug can move to either of the other two points. If the bug's path includes the sequence $ABCA$, it is squashed. What is the expected waiting time for the bug to be squashed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-cef10bed81d113636ba36b1cae3e53e21cee5095f74f528d20ed5383976d4ed2 | A bug starts at point $O$, faces east, and walks a distance of 5 units due east on the first day. Each night, it rotates $60^\circ$ counterclockwise and each day it walks half the distance it walked the previous day in the new direction. The bug gets arbitrarily close to point $P$. Then $OP^2 = \frac{m}{n}$, where $m$ ... | [
"103"
] | [
"A bug starts at point $O$, faces east, and walks a distance of 5 units due east on the first day. Each night, it rotates $60^\\circ$ counterclockwise and each day it walks half the distance it walked the previous day in the new direction. The bug gets arbitrarily close to point $P$. Then $OP^2 = \\frac{m}{n}$, whe... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-518aede8bb1e3f10c8a0903369012e708f52618a748264f73d7e8daa61ab1fbc | A bug starts at position 0 on the number line. The probability that it moves up one space in one turn is \( \frac{1}{2} \), down one space is \( \frac{1}{3} \), and up two spaces is \( \frac{1}{6} \). What is the probability that the bug lands on the number \( n \)? | [
"1"
] | [
"A bug starts at position 0 on the number line. The probability that it moves up one space in one turn is \\( \\frac{1}{2} \\), down one space is \\( \\frac{1}{3} \\), and up two spaces is \\( \\frac{1}{6} \\). What is the probability that the bug lands on the number \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-ec1e70c969ae15952a8337c1608d650decd66d9b5bb9a08f52f5dc2c1cfa4b54 | A bug starts at position 0 on the number line. The probability that it moves up one space in one turn is \( p \), and the probability it moves down one space is \( 1-p \). What is the probability that the bug lands on the number \( n \)? | [
" \\begin{cases} \n1 & \\text{if } p \\geq \\dfrac{1}{2}, \\\\ \n\\left( \\dfrac{p}{1 - p} \\right)^n & \\text{if } p < \\dfrac{1}{2}. \n\\end{cases} "
] | [
"A bug starts at position 0 on the number line. The probability that it moves up one space in one turn is \\( p \\), and the probability it moves down one space is \\( 1-p \\). What is the probability that the bug lands on the number \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ef2a836673f03e186f9a9494e4bdb3b4d8e5882f8fc4f3174bb000aa12ce9ed | A bug starts at position \( x = 1 \) on the number line. At each step, the bug can jump to \( x = a + 2 \) or \( x = \frac{a}{2} \) from its current position \( x = a \). Calculate the number of different positions (including the initial location) the bug can reach in no more than 10 steps. Express your answer as \( F_... | [
"10"
] | [
"A bug starts at position \\( x = 1 \\) on the number line. At each step, the bug can jump to \\( x = a + 2 \\) or \\( x = \\frac{a}{2} \\) from its current position \\( x = a \\). Calculate the number of different positions (including the initial location) the bug can reach in no more than 10 steps. Express your a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c36b5f5bb895015f228077575f054f0ac16bbbd9676e5f6a8232be4979ab7e5 | A bug starts at the origin and crawls in the following pattern: 1 inch due east, 1/2 inch due north, 1/4 inch due west, 1/8 inch due south, and continues crawling like this indefinitely. a. Does the bug ever reach a specific point? b. If so, what are the coordinates of this point? c. What is the total distance that the... | [
"a. Yes, the bug tends towards a specific point.\nb. The coordinates of this point are \\(\\left(\\frac{4}{5}, \\frac{2}{5}\\right)\\).\nc. The total distance the bug crawls is 2 inches."
] | [
"A bug starts at the origin and crawls in the following pattern: 1 inch due east, 1/2 inch due north, 1/4 inch due west, 1/8 inch due south, and continues crawling like this indefinitely.\na. Does the bug ever reach a specific point?\nb. If so, what are the coordinates of this point?\nc. What is the total distance ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 26
} | 28 | 0 | 0 |
omr-ws-75f1bcc7488f7e5b3364a73b92b09951923cdb8475cc030652e1421d1497aea1 | A bug starts at the origin on a number line. Every second, it moves to the right 1 space with probability $\frac{1}{2}$ or to the left 1 space with probability $\frac{1}{2}$. Let $P_n$ be the probability that the bug visits the integer $n$ at least once. Calculate the value of $\sum_{n=1}^{\infty} P_n$. | [
"\\infty"
] | [
"A bug starts at the origin on a number line. Every second, it moves to the right 1 space with probability $\\frac{1}{2}$ or to the left 1 space with probability $\\frac{1}{2}$. Let $P_n$ be the probability that the bug visits the integer $n$ at least once. Calculate the value of $\\sum_{n=1}^{\\infty} P_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2292bdd5af0a6d15cf56f1ae78f19ced3bc1c64bb22465bbfabbec45e66c0ff7 | A bug starts at the point $(0,0)$ on the Cartesian plane. From any point $(x,y)$, the bug can jump to either $(x+1,y)$ or $(x,y+1)$. However, if the bug visits the point $(a,b) \neq (0,0)$, it cannot visit the point $(2a,2b)$. How many paths can the bug take from $(0,0)$ to $(6,6)$? | [
"224"
] | [
"A bug starts at the point $(0,0)$ on the Cartesian plane. From any point $(x,y)$, the bug can jump to either $(x+1,y)$ or $(x,y+1)$. However, if the bug visits the point $(a,b) \\neq (0,0)$, it cannot visit the point $(2a,2b)$. How many paths can the bug take from $(0,0)$ to $(6,6)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-385c3b450f470edab52f81e0c0fd3c1672846da2f3a0153764fd7c9383b09707 | A bug starts at the point $(0,0)$ on the coordinate plane and faces right. It moves one unit in the direction it is facing, then randomly turns $60$ degrees clockwise or counterclockwise. The bug then moves half the distance it previously moved in the direction it is currently facing, and again randomly turns $60$ degr... | [
"\\dfrac{2\\sqrt{7}}{3}"
] | [
"A bug starts at the point $(0,0)$ on the coordinate plane and faces right. It moves one unit in the direction it is facing, then randomly turns $60$ degrees clockwise or counterclockwise. The bug then moves half the distance it previously moved in the direction it is currently facing, and again randomly turns $60$... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-2ba7654e3c39052807cf9df3b9664a86b3fe85169282784961f3f64cae4f927f | A bug starts at the point $(0,0)$ on the coordinate plane and moves to point A. Each subsequent move is $k$ times the length of the previous move, where $k \in (0,1)$. After each move, the bug turns by an angle $\alpha$. Determine the final landing point of the bug. | [
"\\left( \\dfrac{a(1 - k \\cos \\alpha) - b k \\sin \\alpha}{1 - 2k \\cos \\alpha + k^2}, \\dfrac{a k \\sin \\alpha + b(1 - k \\cos \\alpha)}{1 - 2k \\cos \\alpha + k^2} \\right)"
] | [
"A bug starts at the point $(0,0)$ on the coordinate plane and moves to point A. Each subsequent move is $k$ times the length of the previous move, where $k \\in (0,1)$. After each move, the bug turns by an angle $\\alpha$. Determine the final landing point of the bug."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-556f1d042a842881ecfd4d080ca73c42632d371800b5ec90451013e1e9155267 | A bug starts at the point $(1,1)$ in a coordinate system and moves according to the following rules: - From any point $(a,b)$, the bug can move to $(2a,b)$ or $(2b,a)$. - From any point $(a,b)$, if $a > b$, the bug can move to $(a-b,b)$; if $a < b$, the bug can move to $(a,b-a)$. Determine for which points $(x,y)$ the ... | [
"\\(\\gcd(x, y) = 2^n\\) for some non-negative integer \\(n\\)"
] | [
"A bug starts at the point $(1,1)$ in a coordinate system and moves according to the following rules:\n- From any point $(a,b)$, the bug can move to $(2a,b)$ or $(2b,a)$.\n- From any point $(a,b)$, if $a > b$, the bug can move to $(a-b,b)$; if $a < b$, the bug can move to $(a,b-a)$.\nDetermine for which points $(x,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-44f2afd85120e99796b0f65cbe6a1a7827f502f512c1641c1b5e5f64d21c2601 | A bug starts at the point $(10, -19)$, travels to the line $y = -2x + 16$, and then travels to a point on the circle $x^2 - 6x + y^2 + 10y + 9 = 0$. Find the shortest possible total distance the bug could have traveled. | [
"\\(5\\sqrt{17} - 5\\)"
] | [
"A bug starts at the point $(10, -19)$, travels to the line $y = -2x + 16$, and then travels to a point on the circle $x^2 - 6x + y^2 + 10y + 9 = 0$. Find the shortest possible total distance the bug could have traveled."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-fea4e7de68874900e4d10f96caa9cb86ad113c2e1149b882fcf066cdc4527893 | A bug starts at the point $A_0 = (1, 0)$ on the Cartesian plane and is facing rightwards. It turns $1^\circ$ counterclockwise and travels $1$ unit forward to the point $A_1$. It then again turns $1^\circ$ counterclockwise and travels $1$ unit forward to the point $A_2$. The bug continues to do repeat this infinitely ti... | [
"1078"
] | [
"A bug starts at the point $A_0 = (1, 0)$ on the Cartesian plane and is facing rightwards. It turns $1^\\circ$ counterclockwise and travels $1$ unit forward to the point $A_1$. It then again turns $1^\\circ$ counterclockwise and travels $1$ unit forward to the point $A_2$. The bug continues to do repeat this infini... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-b283a435a6c4f1c7dddb44c3cdb72c9c470063f0a6902e899c2f4d4897c5b404 | A bug starts at the point (0,0) on the plane and moves every 20 seconds by one of three vectors \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \) with integer entries. These vectors are not contained in any half-plane that includes (0,0). Every 40 seconds, a grid point is poisoned. If the bug steps on or will step on a po... | [
"4"
] | [
"A bug starts at the point (0,0) on the plane and moves every 20 seconds by one of three vectors \\( \\mathbf{v}_1, \\mathbf{v}_2, \\mathbf{v}_3 \\) with integer entries. These vectors are not contained in any half-plane that includes (0,0). Every 40 seconds, a grid point is poisoned. If the bug steps on or will st... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-53a5605f98f430a231a432014d7203db3d0ac761c8c1ccd8fbb4bbe138711edc | A bug starts at the point (10, -19), travels to the line whose equation is \( y = -2x + 16 \), and then travels to a point on the circle whose equation is \( x^2 - 6x + y^2 + 10y + 9 = 0 \). Find the shortest possible total distance the bug could have traveled, rounded to four significant figures. | [
"15.62"
] | [
"A bug starts at the point (10, -19), travels to the line whose equation is \\( y = -2x + 16 \\), and then travels to a point on the circle whose equation is \\( x^2 - 6x + y^2 + 10y + 9 = 0 \\). Find the shortest possible total distance the bug could have traveled, rounded to four significant figures."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-e221645c0bdfe3bc4205d7b4262b07b1a3f8444c0ef9fe704f294f4c92bad2e4 | A bug starts at the point (5,0) on the plane. Each second, it rotates 45 degrees clockwise and then moves 5 units to the right. Find its coordinates after 100 seconds. | [
"(0, 5(1+\\sqrt{2}))"
] | [
"A bug starts at the point (5,0) on the plane. Each second, it rotates 45 degrees clockwise and then moves 5 units to the right. Find its coordinates after 100 seconds."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-5626ac182ab157af5f446025f1bde98bff3184f81b4937fd5b684b251efc45ed | A bug starts at the point \((10, -19)\), travels to the line whose equation is \(y = -2x + 16\), and then travels to a point on the circle whose equation is \(x^2 - 6x + y^2 + 10y + 9 = 0\). Find the shortest possible total distance the bug could have traveled. | [
"\\(5\\sqrt{17} - 5\\)"
] | [
"A bug starts at the point \\((10, -19)\\), travels to the line whose equation is \\(y = -2x + 16\\), and then travels to a point on the circle whose equation is \\(x^2 - 6x + y^2 + 10y + 9 = 0\\). Find the shortest possible total distance the bug could have traveled."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-cf107c3937ea1a509244b8d20a71213d3a669391874dd4d2ca3f8c07837b1543 | A bug starts at the top left of a 2 by 10 grid of dots and must end at the bottom right. The bug can only move to an adjacent dot and cannot repeat a dot. How many ways can the bug travel from the top left to the bottom right? | [
"512"
] | [
"A bug starts at the top left of a 2 by 10 grid of dots and must end at the bottom right. The bug can only move to an adjacent dot and cannot repeat a dot. How many ways can the bug travel from the top left to the bottom right?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-48ddae850a3df4d331a54a84b2b6e605aecb4c0ea0d3e409c90763e1c810c724 | A bug starts at the vertex of a square. On each move, it moves to an adjacent vertex with probability $\frac14$ each and moves to the vertex opposite it with probability $\frac12$. The probability that after $9$ moves, it ends up at the same vertex it started at can be expressed as $\frac{m}{n}$ for relatively prime in... | [
"1279"
] | [
"A bug starts at the vertex of a square. On each move, it moves to an adjacent vertex with probability $\\frac14$ each and moves to the vertex opposite it with probability $\\frac12$. The probability that after $9$ moves, it ends up at the same vertex it started at can be expressed as $\\frac{m}{n}$ for relatively ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9489e2da9fa8430c833cefc9c29d9c5eb8caabb19d8185d68538343734ef8b65 | A bug starts at vertex $A$ on triangle $ABC$. Every time the bug is at any vertex, it flips a fair coin and walks to its right vertex if it lands heads and to its left vertex if it lands tails. After ten coin flips, what is the probability that the bug is back at vertex $A$? | [
"\\(\\frac{171}{512}\\)"
] | [
"A bug starts at vertex $A$ on triangle $ABC$. Every time the bug is at any vertex, it flips a fair coin and walks to its right vertex if it lands heads and to its left vertex if it lands tails. After ten coin flips, what is the probability that the bug is back at vertex $A$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-37f7d499dd52fb44dbf466d097e6687c4a3eeb9bdbcc1c0ca3d01a2b21f532f9 | A bug starts from the top of Pascal's triangle and moves down to the bottom, with equal probability of turning left or right at each step. Find the ratio of the probability of the bug reaching each of two specified colored numbers at the bottom of the triangle. | [
"\\(\\frac{10}{3}\\)"
] | [
"A bug starts from the top of Pascal's triangle and moves down to the bottom, with equal probability of turning left or right at each step. Find the ratio of the probability of the bug reaching each of two specified colored numbers at the bottom of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22560d15bd5fc815aa44420f9a397a992354d603a04287fd38ec96c02bb04fad | A bug starts to crawl from a point on the edge of the lower base of a cylinder with base radius 1 and height 10 at a $60^\circ$ angle of elevation. What is the total distance the bug travels to reach the top of the cylinder? | [
"\\(\\frac{20\\sqrt{3}}{3}\\)"
] | [
"A bug starts to crawl from a point on the edge of the lower base of a cylinder with base radius 1 and height 10 at a $60^\\circ$ angle of elevation. What is the total distance the bug travels to reach the top of the cylinder?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ab8430abeda3691640428478279b55767afae570c2b000d2d554ece2f072f66f | A bug travels from point A to point B along the segments in a hexagonal lattice. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there from point A to point B? | [
"20"
] | [
"A bug travels from point A to point B along the segments in a hexagonal lattice. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there from point A to point B?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-4ad0a6eac24c5566a3dafaf103e25c5b24827b416e0fea80bd390a71748b4a67 | A bug travels in the coordinate plane, moving only along lines parallel to the $X$ and $Y$ axes. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$. How many lattice points lie on at least one of these paths? | [
"87"
] | [
"A bug travels in the coordinate plane, moving only along lines parallel to the $X$ and $Y$ axes. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$. How many lattice points lie on at least one of these paths?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf6dca4865586f81115685cbe389c8f8ac4e8a948a7ed3cc88902f0047ba108d | A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? | [
"195"
] | [
"A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b735e52b908de5fa3fd6a2ead25783863c6976ff6f1d5c3264adb8a4ea837234 | A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? | [
"195"
] | [
"A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 18
} | 39 | 0 | 0 |
omr-ws-fc336293c86dd58ebf9fec67d4b568bed72240bd6799848a6e9a50800ae1bcdf | A bug travels on an equilateral triangle and starts at vertex \( A \). During its traveling, it can only travel to either of the two adjacent vertices of the vertex the bug is currently standing at. Let \( P(n) \) denote the probability that the bug ends up back at vertex \( A \) after \( n \) moves. Which of these fra... | [
"\\(\\frac{1}{3}\\)"
] | [
"A bug travels on an equilateral triangle and starts at vertex \\( A \\). During its traveling, it can only travel to either of the two adjacent vertices of the vertex the bug is currently standing at. Let \\( P(n) \\) denote the probability that the bug ends up back at vertex \\( A \\) after \\( n \\) moves. Which... | [
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-63cbc5b071ef04e6da2c620697ec7a3ac1feb6486eb104b279d72c33dd24ddbb | A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the poi... | [
"103"
] | [
"A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to t... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-79c3fee23f3cf34ee02fc3dda3bf191d4bdc1baf993968e592ff66f50283ca0a | A bug walks along a wheel consisting of two concentric circles and five spokes, starting at point $A$. The bug can only move to adjacent labeled points, walking counterclockwise on the inner circle and clockwise on the outer circle. How many paths of 15 steps are there that begin and end at point $A$? Find the remainde... | [
"4"
] | [
"A bug walks along a wheel consisting of two concentric circles and five spokes, starting at point $A$. The bug can only move to adjacent labeled points, walking counterclockwise on the inner circle and clockwise on the outer circle. How many paths of 15 steps are there that begin and end at point $A$? Find the rem... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-6862ef38752ddded070890a749de44a7c6834b478df11dc90583dced67fc9cd4 | A bug walks along a wheel consisting of two concentric circles and five spokes, starting at point $A$. The bug can only move to adjacent labeled points. Along the inner circle, the bug walks counterclockwise, and along the outer circle, the bug walks clockwise. How many paths of 15 steps are there that begin and end at... | [
"004"
] | [
"A bug walks along a wheel consisting of two concentric circles and five spokes, starting at point $A$. The bug can only move to adjacent labeled points. Along the inner circle, the bug walks counterclockwise, and along the outer circle, the bug walks clockwise. How many paths of 15 steps are there that begin and e... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-ecdda5aeec69dae1ef05c505da14fbdb181e950df673ad008e95ba47c533a33d | A building has 120 apartments and 119 residents. An apartment is called "overpopulated" if it has at least 15 people living in it. Every day, the inhabitants of an overpopulated apartment move to different apartments to avoid each other. Determine the maximum number of days, \( n \), after which there are no overpopula... | [
"8"
] | [
"A building has 120 apartments and 119 residents. An apartment is called \"overpopulated\" if it has at least 15 people living in it. Every day, the inhabitants of an overpopulated apartment move to different apartments to avoid each other. Determine the maximum number of days, \\( n \\), after which there are no o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-bc2fd1316da0435a385ab0f30fae48087610df169b629aa1b06733c3f45cde3e | A building has 2 levels with 4 windows each. In level one, each window has a set of blue Christmas lights. In level two, each window has a Christmas wreath. Bill takes down the rightmost decoration from a level and can't take down more than 2 decorations in a level before switching floors. How many ways can Bill take d... | [
"34"
] | [
"A building has 2 levels with 4 windows each. In level one, each window has a set of blue Christmas lights. In level two, each window has a Christmas wreath. Bill takes down the rightmost decoration from a level and can't take down more than 2 decorations in a level before switching floors. How many ways can Bill t... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 48,
"genselect": 14,
"tir": 21
} | 83 | 0 | 0 |
omr-ws-ac703516aa4ca3659331c4c6813e587ce17a4a0b105a79ce267bdd926181e5e4 | A building has 7 lifts, and each lift stops at 6 floors. For any two floors, there is always a lift connecting them directly. Let $N$ be the maximum number of floors the building can have under these conditions. Calculate the value of $N^2 - 10N + 24$. | [
"80"
] | [
"A building has 7 lifts, and each lift stops at 6 floors. For any two floors, there is always a lift connecting them directly. Let $N$ be the maximum number of floors the building can have under these conditions. Calculate the value of $N^2 - 10N + 24$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-9406fd59616689d1745ffc0ade8a72e38a791e87b8ce05aec68fe4197c4ca923 | A building has a floor in the shape of a regular hexagon with each side equal to 60 feet. The roof is a portion of a sphere centered at the exact center of the floor, touching the floor at each vertex of the hexagon. Find the surface area of the roof. | [
"\\( 7200\\pi \\left(\\frac{3\\sqrt{3}}{2} - 2\\right) \\)"
] | [
"A building has a floor in the shape of a regular hexagon with each side equal to 60 feet. The roof is a portion of a sphere centered at the exact center of the floor, touching the floor at each vertex of the hexagon. Find the surface area of the roof."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51db7924449a979e2d95838643e69489f97c63ec4b26174f4bc7b954ef25c48e | A building is 1225 feet tall and situated about 1 mile inland from a shore. Ritter is in his pleasure boat directly in front of the building and measures an angle of elevation of 7 degrees. How far offshore is the boat? | [
"9999 feet"
] | [
"A building is 1225 feet tall and situated about 1 mile inland from a shore. Ritter is in his pleasure boat directly in front of the building and measures an angle of elevation of 7 degrees. How far offshore is the boat?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7283c841a20958596bc9529c136928913292982b23a4ef1e01798a142177588c | A building is in the shape of a hemisphere with a radius of 20 ft. A 24 ft. board is leaning against the building such that the base of the board is on the ground and the center of the base of the board is 5 ft. from the edge of the building. Find the distance from the center of the base of the board to the point where... | [
"15"
] | [
"A building is in the shape of a hemisphere with a radius of 20 ft. A 24 ft. board is leaning against the building such that the base of the board is on the ground and the center of the base of the board is 5 ft. from the edge of the building. Find the distance from the center of the base of the board to the point ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9935b99a2f329f12756c8a73f3f324288e94f912ace3a762335aafc4b4d4b063 | A building is to be braced by a beam that must rest on the ground and pass over a vertical wall 10 ft high and 8 ft from the building. Find the shortest length that the beam can be. | [
"25.40331527"
] | [
"A building is to be braced by a beam that must rest on the ground and pass over a vertical wall 10 ft high and 8 ft from the building. Find the shortest length that the beam can be."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
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