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-6e5faade9746d947dc26a87624f7d9909753fc5ab03751ed3df397019a95c8aa | An ant is located at the point \((10,10)\) and needs to get to \((1,1)\). The ant's speed at any point \((x,y)\) is given by \(\lfloor x \rfloor\). Find the equation or describe the shortest route time-wise for the ant to travel from \((10,10)\) to \((1,1)\). | [
"(10,10) \\text{ to } (10,1) \\text{ to } (1,1)"
] | [
"An ant is located at the point \\((10,10)\\) and needs to get to \\((1,1)\\). The ant's speed at any point \\((x,y)\\) is given by \\(\\lfloor x \\rfloor\\). Find the equation or describe the shortest route time-wise for the ant to travel from \\((10,10)\\) to \\((1,1)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-8523271e2067a764b7a0b643e7572ab2fc30d81fc93ec98291f70832b503a3a8 | An ant is on 0 on a number line. Each turn, there is a 50% chance of the ant moving 1 unit forward and a 50% chance of the ant moving 1 unit backward. Find the probability that the ant ever reaches -1. | [
"1"
] | [
"An ant is on 0 on a number line. Each turn, there is a 50% chance of the ant moving 1 unit forward and a 50% chance of the ant moving 1 unit backward. Find the probability that the ant ever reaches -1."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c551e6ff691e96650ef6a4a15e0e599d82744e0dc7341c316ba0aaf2fb2bbaa3 | An ant is on a box with dimensions \( l \), \( h \), and \( d \) and needs to travel from point A to point B while staying on the surface. What is the minimum distance the ant must travel? | [
"\\min\\left\\{\\sqrt{(l + h)^2 + d^2}, \\sqrt{(l + d)^2 + h^2}, \\sqrt{(h + d)^2 + l^2}\\right\\}"
] | [
"An ant is on a box with dimensions \\( l \\), \\( h \\), and \\( d \\) and needs to travel from point A to point B while staying on the surface. What is the minimum distance the ant must travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-a244f5034c5eea669a627ca0ba4da209e18221a5abffdac18314ef9027b812ee | An ant is on a right circular cone with radius $4$ and height $8\sqrt{2}$. The ant starts at point $A$ on the edge of the base and wants to go to point $A'$ on the exact opposite side of the base. Find the square of the length of the shortest path across the lateral surface of the cone that the ant can take to get to $... | [
"144"
] | [
"An ant is on a right circular cone with radius $4$ and height $8\\sqrt{2}$. The ant starts at point $A$ on the edge of the base and wants to go to point $A'$ on the exact opposite side of the base. Find the square of the length of the shortest path across the lateral surface of the cone that the ant can take to ge... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-0c863bfbc532ab9617bcd6fc27c171a3ad020c285839d265f36ee3a17aaa89bb | An ant is on a vertex of a cube. After every minute, the ant randomly chooses one of the three adjacent vertices to move to. What is the probability that, after six minutes, the ant is back on the starting vertex? | [
"\\(\\frac{61}{243}\\)"
] | [
"An ant is on a vertex of a cube. After every minute, the ant randomly chooses one of the three adjacent vertices to move to. What is the probability that, after six minutes, the ant is back on the starting vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-17169e451498fb5f28d4f077de2a87efa5fab764707d7e3a50b035c2a018a132 | An ant is on an edge of a cube with side length 8. It must travel along the surface of the cube, visiting each of the six faces exactly once, and return to its starting point. Find the length of the ant’s shortest path. | [
"48"
] | [
"An ant is on an edge of a cube with side length 8. It must travel along the surface of the cube, visiting each of the six faces exactly once, and return to its starting point. Find the length of the ant’s shortest path."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 19,
"genselect": 4
} | 23 | 0 | 0 |
omr-ws-23d216cb049e7d6ad83f6fc51a16d3db50b71ff59887998e60221d9b6ca4bec5 | An ant is on an edge of a right rectangular prism one centimeter from a vertex. The prism has edges measuring 9 cm, 10 cm, and 12 cm. What is the smallest number of centimeters the ant can crawl to reach a point one centimeter from the opposite (furthest) corner? | [
"\\( \\sqrt{482} - 1 \\)"
] | [
"An ant is on an edge of a right rectangular prism one centimeter from a vertex. The prism has edges measuring 9 cm, 10 cm, and 12 cm. What is the smallest number of centimeters the ant can crawl to reach a point one centimeter from the opposite (furthest) corner?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23d1f6798ba95e4a45793602b86bf89b3328bb6359d20e59e26da53daf57d87f | An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face? | [
"6"
] | [
"An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8cb832ea348ee012a8ee0a439be91f7850325e44d1e55f06823395917aa3285 | An ant is on one of the eight corners of a toolbox with dimensions 2 decimeters in width, 3 decimeters in length, and 4 decimeters in height. The ant wants to crawl to the opposite corner of the toolbox, only moving on the surface. What is the shortest distance the ant has to crawl to accomplish this task, in decimeter... | [
"\\(\\sqrt{41}\\)"
] | [
"An ant is on one of the eight corners of a toolbox with dimensions 2 decimeters in width, 3 decimeters in length, and 4 decimeters in height. The ant wants to crawl to the opposite corner of the toolbox, only moving on the surface. What is the shortest distance the ant has to crawl to accomplish this task, in deci... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-9b26d72fe0f13f67e6b585ff3ccdb87f3d9c49eef4d0ce39cfe00c0b4abe839b | An ant is on the circumference of the top of a cylinder with radius 3 and height 3. The ant wants to crawl to a point diagonally across from its current position at the base of the cylinder. If the shortest path the ant can crawl is expressed as $\sqrt{a\pi^2 + b}$, find the value of $a + b$. | [
"18"
] | [
"An ant is on the circumference of the top of a cylinder with radius 3 and height 3. The ant wants to crawl to a point diagonally across from its current position at the base of the cylinder. If the shortest path the ant can crawl is expressed as $\\sqrt{a\\pi^2 + b}$, find the value of $a + b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43f6275b6166eb864dfb9f527c2a58ac97ea8a29a8022445924e6ae32feced0c | An ant is on top of a cylinder with a height of 8 inches and a diameter of 4 inches. If the ant wishes to walk to a point on the bottom of the cylinder directly opposite its current position, what is the least number of inches it may travel? | [
"2\\sqrt{\\pi^2 + 16}"
] | [
"An ant is on top of a cylinder with a height of 8 inches and a diameter of 4 inches. If the ant wishes to walk to a point on the bottom of the cylinder directly opposite its current position, what is the least number of inches it may travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1727497cdb88df527e6191f49338e79fb557052f2c635a20d083ed4c4538e131 | An ant is placed at one vertex of a cuboid with edge lengths $m$, $n$, and $p$ units, where $m$, $n$, and $p$ are prime numbers and $m < n < p$. The ant walks along a circuit crossing the four smallest faces of the cuboid, each time traveling along a diagonal of the face. By the time the ant returns to the starting ver... | [
"\\(4m + 2n + 2p - 8\\)"
] | [
"An ant is placed at one vertex of a cuboid with edge lengths $m$, $n$, and $p$ units, where $m$, $n$, and $p$ are prime numbers and $m < n < p$. The ant walks along a circuit crossing the four smallest faces of the cuboid, each time traveling along a diagonal of the face. By the time the ant returns to the startin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 5
} | 23 | 0 | 0 |
omr-ws-19e0afabeb745297cf544a16a874dc913e5f979723362b3d42c7106b6d752e41 | An ant is placed at point $A$ of a cube $ABCDA'B'C'D'$. The ant walks at a constant speed of $1$ side per minute. In how many ways can the ant move to point $D'$ (the farthest from $A$) in $9$ minutes? | [
"4920"
] | [
"An ant is placed at point $A$ of a cube $ABCDA'B'C'D'$. The ant walks at a constant speed of $1$ side per minute. In how many ways can the ant move to point $D'$ (the farthest from $A$) in $9$ minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-fd0bcd1c1476db9e124348ec876dded22982f7ef1942bcbd05784764d254fd28 | An ant is placed at vertex $A$ of a square $ABCD$ with $E$ as the center. Every minute, the ant randomly moves to one of the adjacent vertices ($B$, $E$, or $D$). Find the expected number of minutes until the ant reaches vertex $C$. | [
"\\( \\frac{15}{2} \\)"
] | [
"An ant is placed at vertex $A$ of a square $ABCD$ with $E$ as the center. Every minute, the ant randomly moves to one of the adjacent vertices ($B$, $E$, or $D$). Find the expected number of minutes until the ant reaches vertex $C$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8f24b3e32b4d5c5e48429c4d9ae597677ac8c64d700147ba549cc882d4923b5 | An ant is placed at vertex $A$ of a square $ABCD$ with center $E$. Every minute, the ant randomly moves to one of the adjacent vertices ($B$, $E$, or $D$). Find the expected number of minutes until the ant reaches vertex $C$. | [
"320"
] | [
"An ant is placed at vertex $A$ of a square $ABCD$ with center $E$. Every minute, the ant randomly moves to one of the adjacent vertices ($B$, $E$, or $D$). Find the expected number of minutes until the ant reaches vertex $C$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50744f74f63aff8f2425ea28b30fe624fe15032cc29dbff3d72780325e1d60dc | An ant is placed at vertex A of a square ABCD, with E being the center of the square. Every minute, the ant randomly moves to one of the adjacent vertices (B, E, or D). Find the expected number of minutes until the ant reaches vertex C. | [
"\\(\\frac{16}{3}\\)"
] | [
"An ant is placed at vertex A of a square ABCD, with E being the center of the square. Every minute, the ant randomly moves to one of the adjacent vertices (B, E, or D). Find the expected number of minutes until the ant reaches vertex C."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-a4882f61d58066c4d0f56b0df5fe71296c15c1bb5ac969e1e4c162cd73899466 | An ant is sitting at one of the vertices of a triangular prism with equilateral triangular faces of side length $10$. There are two piles of sugar on the opposite triangular face: one at the center and another at a vertex that does not share an edge with the ant's current vertex. The ant needs to walk the same distance... | [
"12"
] | [
"An ant is sitting at one of the vertices of a triangular prism with equilateral triangular faces of side length $10$. There are two piles of sugar on the opposite triangular face: one at the center and another at a vertex that does not share an edge with the ant's current vertex. The ant needs to walk the same dis... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51c74735f41902e9d9e94c3c687a1db3aa0f07e9df75daca85ca60cba7af8c40 | An ant is situated at vertex A of triangle ABC. Every movement of the ant consists of moving to one of the other two adjacent vertices from the previous vertex where it is situated. The probability of going to any of the other two adjacent vertices of the triangle is equal. If the ant will move 7 times, find the probab... | [
"\\(\\frac{21}{64}\\)"
] | [
"An ant is situated at vertex A of triangle ABC. Every movement of the ant consists of moving to one of the other two adjacent vertices from the previous vertex where it is situated. The probability of going to any of the other two adjacent vertices of the triangle is equal. If the ant will move 7 times, find the p... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-506c0251d55772cc09069d85ba9f1c4bf1a2763cdc61bef497e22a6cc43fd0c8 | An ant is standing 10 meters away from a pole. He walks 5 meters in a random direction, then walks 5 meters in another random direction. What is the probability that the ant is closer to the pole than he was when he started? | [
"\\(\\frac{1}{2}\\)"
] | [
"An ant is standing 10 meters away from a pole. He walks 5 meters in a random direction, then walks 5 meters in another random direction. What is the probability that the ant is closer to the pole than he was when he started?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5d39cae6aaf6f634de59424be53b1cf8f52595ae7b17551d02385026ff958bb | An ant is standing at the bottom left corner of a 3 by 3 grid. How many ways can it get to the top right corner if it can only move up, right, and left, and it is not allowed to cross the same edge twice? | [
"9"
] | [
"An ant is standing at the bottom left corner of a 3 by 3 grid. How many ways can it get to the top right corner if it can only move up, right, and left, and it is not allowed to cross the same edge twice?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-87170111ad2e8a580f7fb1cd4c60d2d180e79c16c0226f60e1ec95a2263f0efb | An ant is traveling from the point $(8, 15)$ to the point $(5, 12)$, but it wants to touch both axes along the way. Let $d$ be the minimum distance it has to travel. Find $d^{2}$. | [
"898"
] | [
"An ant is traveling from the point $(8, 15)$ to the point $(5, 12)$, but it wants to touch both axes along the way. Let $d$ be the minimum distance it has to travel. Find $d^{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-2f954dd6ea82770be31597674f7f920e4a10fb0d80c7e7e80b2c68441d1450c2 | An ant is walking on the edges of a cube. Every time it arrives at a vertex, it randomly chooses another edge and continues walking. It takes 1 minute to travel one edge. What is the average amount of time it takes for the ant to go from one vertex to the opposite vertex? | [
"6"
] | [
"An ant is walking on the edges of a cube. Every time it arrives at a vertex, it randomly chooses another edge and continues walking. It takes 1 minute to travel one edge. What is the average amount of time it takes for the ant to go from one vertex to the opposite vertex?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-9c38009d50ab1951ad296ef8b8dc4cfc08ea9c555a0de17c1898f7568c72b842 | An ant is walking on the second hand of a clock at a rate of 6 cm per minute. The second hand measures 6 cm. The ant starts from the tip of the second hand and walks towards the center. How long (in distance, not time) does the ant travel? | [
"6 cm"
] | [
"An ant is walking on the second hand of a clock at a rate of 6 cm per minute. The second hand measures 6 cm. The ant starts from the tip of the second hand and walks towards the center. How long (in distance, not time) does the ant travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-b94ab48fa3c019fa5e3260be8bbaa0c495029bbe9f75b5684bff2ebb414877df | An ant is walking on the vertices of a tetrahedron $ABCD$. It starts at its home at vertex $A$ and chooses a random vertex to walk to every second until it returns home. What is the expected number of distinct vertices it will visit, including $A$, during its walk? | [
"3"
] | [
"An ant is walking on the vertices of a tetrahedron $ABCD$. It starts at its home at vertex $A$ and chooses a random vertex to walk to every second until it returns home. What is the expected number of distinct vertices it will visit, including $A$, during its walk?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0bf7cf452f59fadf449c24fe73c1f7ea2555d09499bf0eecac5d5f7d5d174be | An ant is walking up a soda can that is 6 inches tall at a 45-degree angle to the can. How far will the ant have walked when it reaches the top of the soda can? | [
"\\(6\\sqrt{2}\\)"
] | [
"An ant is walking up a soda can that is 6 inches tall at a 45-degree angle to the can. How far will the ant have walked when it reaches the top of the soda can?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-cf0983ef584aa6e0bf93dd011a540ed9140e28cb8853613d6a5eb99314499d9e | An ant is wandering on the edges of a cube, moving randomly to one of the three adjacent vertices every second. What is the probability that the ant is at its starting vertex after exactly 6 seconds? | [
"\\dfrac{61}{243}"
] | [
"An ant is wandering on the edges of a cube, moving randomly to one of the three adjacent vertices every second. What is the probability that the ant is at its starting vertex after exactly 6 seconds?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fcf7cbb51e2f8590ff7d65bfcd2bb40a075bfd8c8a338aa87ec94dbc1813130 | An ant leaves his anthill for its morning walk. It walks 5 feet east and then makes a 160-degree turn to the right and walks 5 more feet. It then makes another 160-degree turn to the right and walks 5 more feet. If the ant continues this pattern until it reaches the anthill again, how far will it have walked? | [
"45"
] | [
"An ant leaves his anthill for its morning walk. It walks 5 feet east and then makes a 160-degree turn to the right and walks 5 more feet. It then makes another 160-degree turn to the right and walks 5 more feet. If the ant continues this pattern until it reaches the anthill again, how far will it have walked?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-bf93467dbe29f5a3144ff076fb778ac71657bd3887fd729d48413cbb21040eeb | An ant leaves the anthill and walks 4 feet east, then makes a $160^\circ$ turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is the total distance in feet that the ant would have walked? | [
"36"
] | [
"An ant leaves the anthill and walks 4 feet east, then makes a $160^\\circ$ turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is the total distance in feet that the ant would have walked?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36a41bcd6b059baa52af146f83f1761dfb3a650c5331db5d72c266c54fc5a1d8 | An ant lies on a vertex of a \( 1 \times 1 \times 2 \) box. What point on the box is farthest (along the surface) from the ant? And, exactly how far is it from the ant? | [
"\\(\\sqrt{10}\\)"
] | [
"An ant lies on a vertex of a \\( 1 \\times 1 \\times 2 \\) box. What point on the box is farthest (along the surface) from the ant? And, exactly how far is it from the ant?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-16d3bc8bcbab4e50382324ae0e6e7bdce4c228d6d84ed96cb82af4a9a1b53a50 | An ant lies on a vertex of a unit cube. If the ant has to move 5 steps, in how many ways can it get to the opposite vertex by traveling only on the edges? | [
"60"
] | [
"An ant lies on a vertex of a unit cube. If the ant has to move 5 steps, in how many ways can it get to the opposite vertex by traveling only on the edges?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 27
} | 55 | 0 | 0 |
omr-ws-bef8285634d9ebc1e22e3be0e2060937ab80b9dac9c77b88fe417bcc42d0f8d8 | An ant lies on a vertex of a unit cube. What is the length of its shortest path to the opposite vertex? | [
"$\\sqrt{5}$"
] | [
"An ant lies on a vertex of a unit cube. What is the length of its shortest path to the opposite vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-22679fe32b5060cdb3c36156ebe59abbfc77f8114c4cfb538dd23606b236b6c2 | An ant lies on one corner of an $a$ by $b$ rectangular grid divided into $ab$ unit squares. The ant can only take horizontal and vertical unit steps. How many ways are there for the ant to reach the other corner of the grid if it takes $N$ steps, where $N \geq a + b - 2 + 2k$ for $k \in \mathbb{Z}^+$? | [
"\\dbinom{N}{\\frac{N + a + b}{2}} \\dbinom{\\frac{N + a + b}{2}}{\\frac{N + a - b}{2}}"
] | [
"An ant lies on one corner of an $a$ by $b$ rectangular grid divided into $ab$ unit squares. The ant can only take horizontal and vertical unit steps. How many ways are there for the ant to reach the other corner of the grid if it takes $N$ steps, where $N \\geq a + b - 2 + 2k$ for $k \\in \\mathbb{Z}^+$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-2c9c7e4b7b36c7a1ad7b5e330d9f5b5a936212e9a5f73c2a79e14c36ed5d6bf3 | An ant living on a solid tetrahedron with side length 1 unit wants to build a road system to go from any corner to any other corner on the surface of the tetrahedron. What is the minimum total length of the road system required? | [
"\\(1 + \\sqrt{3}\\)"
] | [
"An ant living on a solid tetrahedron with side length 1 unit wants to build a road system to go from any corner to any other corner on the surface of the tetrahedron. What is the minimum total length of the road system required?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
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"tir": 1
} | 13 | 0 | 0 |
omr-ws-df7e59c34d48ebd0940d5e73c5b38c206f6449ef89cfd402b85b3924b7c17993 | An ant located at a corner of a $2 \text{ in} \times 3 \text{ in} \times 5 \text{ in}$ rectangular block of wood wants to crawl along the surface to the opposite corner of the block. What is the length of the shortest such path? | [
"\\(\\sqrt{50}\\) or \\(5\\sqrt{2}\\)"
] | [
"An ant located at a corner of a $2 \\text{ in} \\times 3 \\text{ in} \\times 5 \\text{ in}$ rectangular block of wood wants to crawl along the surface to the opposite corner of the block. What is the length of the shortest such path?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
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} | 44 | 0 | 0 |
omr-ws-13de6fbae2cdfa21796466c435e8dafd5fd6ba292e09d84124aef166f21811ee | An ant located at point $A$ on a cube with side length $3$ moves along the grid to point $B$ and then returns to $A$. The ant uses only the fastest route, does not pass through any point it has already walked through, and avoids the interior of face $Y$ and point $P$ on the way to $B$. On the return trip, the ant avoid... | [
"384"
] | [
"An ant located at point $A$ on a cube with side length $3$ moves along the grid to point $B$ and then returns to $A$. The ant uses only the fastest route, does not pass through any point it has already walked through, and avoids the interior of face $Y$ and point $P$ on the way to $B$. On the return trip, the ant ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-16b344f94a43ec01492a437e19968807d07f7f03e921ae39e66a03d7d7d06981 | An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \le 2$. Let $N$ be the number of paths the ant can take. Find the remaind... | [
"442"
] | [
"An ant makes $4034$ moves on a coordinate plane, beginning at the point $(0, 0)$ and ending at $(2017, 2017)$. Each move consists of moving one unit in a direction parallel to one of the axes. Suppose that the ant stays within the region $|x - y| \\le 2$. Let $N$ be the number of paths the ant can take. Find the r... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-a97858dfbb0daf21e785d95b4a4525c2fb6cef974df3e5e506521480c1ec8cc2 | An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, th... | [
"049"
] | [
"An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first mov... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 17
} | 65 | 0 | 0 |
omr-ws-f73a7ab0bb11ffaa1aa7ef205f086751ade089b7f4414ce47c66f6bf8a228509 | An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, th... | [
"49"
] | [
"An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first mov... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-cf83aa94112b65a0c8aa1c183fc8bc0098f0482f1bb1e48e16eef6e80213d307 | An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially, the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, t... | [
"49"
] | [
"An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially, the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first mo... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 7,
"tir": 20
} | 70 | 0 | 0 |
omr-ws-6d9cd7e3eb6f0f88bf25a559c161a367ba03f00c582a1af28414694b67c28450 | An ant moves along the edges of a regular octahedron, starting at vertex $A$. Each move takes one minute, and the ant chooses a new random edge, different from the previous one, at each vertex. After 4 minutes, what is the probability that the ant is back at vertex $A$? | [
"\\(\\frac{5}{27}\\)",
"\\dfrac{5}{27}"
] | [
"An ant moves along the edges of a regular octahedron, starting at vertex $A$. Each move takes one minute, and the ant chooses a new random edge, different from the previous one, at each vertex. After 4 minutes, what is the probability that the ant is back at vertex $A$?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 5,
"tir": 8
} | 50 | 0 | 0 |
omr-ws-06ca937ea0fe58842214abf444aaeac63f3899f62e8b07885ad5f7be1354977e | An ant moves along the grid bars on the shortest path from $A_0$ to $K_2$ on a cubic space grid with edge length $2$. How many different shortest paths are there for the ant if it moves on any bars? | [
"90"
] | [
"An ant moves along the grid bars on the shortest path from $A_0$ to $K_2$ on a cubic space grid with edge length $2$. How many different shortest paths are there for the ant if it moves on any bars?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4385fe2d255d8c3d11c63a9b49ec0a1807fe1c819dacd9ee7cc56d246d32355a | An ant moves along the grid bars on the shortest path from $A_0$ to $K_2$ on a cubic space grid with edge length $2$. How many different shortest paths are there for the ant if it moves only along the bars on the surface of the cube? | [
"18"
] | [
"An ant moves along the grid bars on the shortest path from $A_0$ to $K_2$ on a cubic space grid with edge length $2$. How many different shortest paths are there for the ant if it moves only along the bars on the surface of the cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-950d160825d03a7e7fe4de0a05bfba38768bd9d4d0581acfae200a9331aebde7 | An ant moves along the segments of a cubic grid from $A_0$ to $K_2$. How many shortest ways are there for the ant if it moves along arbitrary segments? | [
"90"
] | [
"An ant moves along the segments of a cubic grid from $A_0$ to $K_2$. How many shortest ways are there for the ant if it moves along arbitrary segments?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 29,
"genselect": 3
} | 32 | 0 | 0 |
omr-ws-a361b526946f7b5c4c43b17a13e00f3dce9d2036acffd7463490a98eca0aeefc | An ant moves along the segments of a cubic grid from $A_0$ to $K_2$. How many shortest ways are there for the ant if it only moves along the edges on the cube's surface? | [
"54"
] | [
"An ant moves along the segments of a cubic grid from $A_0$ to $K_2$. How many shortest ways are there for the ant if it only moves along the edges on the cube's surface?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 9
} | 11 | 0 | 0 |
omr-ws-0cfa73267bd21dc07d0319a114d313946b772f67aa8e24b6cb79f6c333fa85f1 | An ant moves on a regular tetrahedron \(OABC\) with edge length 2. It starts from the midpoint \(M\) of \(OA\) and wants to reach the midpoint \(N\) of \(BC\), the opposite edge. The shortest distance \(MN\) on the surface of the tetrahedron is ... | [
"2"
] | [
"An ant moves on a regular tetrahedron \\(OABC\\) with edge length 2. It starts from the midpoint \\(M\\) of \\(OA\\) and wants to reach the midpoint \\(N\\) of \\(BC\\), the opposite edge. The shortest distance \\(MN\\) on the surface of the tetrahedron is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a5dc5cf1f2c4fd569e2475b383427539bf2d6d1c7a44ea8708584188344d501 | An ant moves on the coordinate plane starting from point $(0, -1)$ along a straight line until it reaches the $x$-axis at point $(x, 0)$ where $x$ is a real number. After turning $90^\circ$ to the left, it moves along a straight line until it reaches the $y$-axis. Then it turns left again and moves along a straight lin... | [
"42"
] | [
"An ant moves on the coordinate plane starting from point $(0, -1)$ along a straight line until it reaches the $x$-axis at point $(x, 0)$ where $x$ is a real number. After turning $90^\\circ$ to the left, it moves along a straight line until it reaches the $y$-axis. Then it turns left again and moves along a straig... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f5b855246a8b351cf97f31488c5ef949b14c82da89ccdb27e11d6bdf9442812 | An ant of negligible dimensions starts at the origin of a Cartesian coordinate system. It walks 1 unit right and then \(\frac{1}{2}\) unit up, then \(\frac{1}{4}\) unit left, and \(\frac{1}{8}\) unit down, and so on. In each move, it always turns counter-clockwise at a 90-degree angle and goes half the distance it went... | [
"\\(\\left(\\frac{4}{5}, \\frac{2}{5}\\right)\\)"
] | [
"An ant of negligible dimensions starts at the origin of a Cartesian coordinate system. It walks 1 unit right and then \\(\\frac{1}{2}\\) unit up, then \\(\\frac{1}{4}\\) unit left, and \\(\\frac{1}{8}\\) unit down, and so on. In each move, it always turns counter-clockwise at a 90-degree angle and goes half the di... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-14ef9af00f57a7c89ebfa74a26788a5988c1ad8a2d09ca67a91e64857450826e | An ant on a hexagon has a $\frac{1}{2}$ chance of moving to the vertex to its left and a $\frac{1}{2}$ chance of moving to the vertex to its right. What is the probability that the ant will move from a vertex to the opposite vertex? | [
"1"
] | [
"An ant on a hexagon has a $\\frac{1}{2}$ chance of moving to the vertex to its left and a $\\frac{1}{2}$ chance of moving to the vertex to its right. What is the probability that the ant will move from a vertex to the opposite vertex?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b6f9c30dba1c95900872111ea46fabdb1df6fc166b222ba3e0fc035333691fe | An ant on an edge of a cube with side length 8 must travel on the surface and return to the starting point, visiting the interior points of each of the six faces exactly once. Find the length of the shortest path the ant can take. | [
"\\(8\\sqrt{13}\\)"
] | [
"An ant on an edge of a cube with side length 8 must travel on the surface and return to the starting point, visiting the interior points of each of the six faces exactly once. Find the length of the shortest path the ant can take."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-22a3c9049ced2f73421d7c66194367e195aab6cdc262602d20ecc79b230bb194 | An ant on the coordinate plane starts at the origin and takes 8 steps up, down, left, or right. What is the probability that the ant reaches the same point at least twice? | [
"\\dfrac{14905}{16384}"
] | [
"An ant on the coordinate plane starts at the origin and takes 8 steps up, down, left, or right. What is the probability that the ant reaches the same point at least twice?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-560776b73ab41384cb7454dd2d31252a841bf24b53e5d5a6080eb1bdb9dbc878 | An ant rests on the centroid of one of the faces of a regular tetrahedron with side length 3 cm. What is the shortest distance the ant may travel to reach the centroid of an adjacent triangular face? | [
"\\(\\sqrt{3}\\)"
] | [
"An ant rests on the centroid of one of the faces of a regular tetrahedron with side length 3 cm. What is the shortest distance the ant may travel to reach the centroid of an adjacent triangular face?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 20
} | 28 | 0 | 0 |
omr-ws-d37c1cfbd44ff6087ec8a3a72ff6a946bbc96415d94e5804ca03b6fa8b3216c6 | An ant sits at point ($1$ mm, $2$ mm, $0$ mm) inside a rectangular matchbox measuring $10$ mm $\times$ $20$ mm $\times$ $30$ mm. A sugar crumb is located at point ($9$ mm, $3$ mm, $30$ mm). What is the shortest distance the ant must crawl over the surface of the box to reach the sugar crumb? | [
"\\(\\sqrt{785}\\)"
] | [
"An ant sits at point ($1$ mm, $2$ mm, $0$ mm) inside a rectangular matchbox measuring $10$ mm $\\times$ $20$ mm $\\times$ $30$ mm. A sugar crumb is located at point ($9$ mm, $3$ mm, $30$ mm). What is the shortest distance the ant must crawl over the surface of the box to reach the sugar crumb?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-6ed3044a9e12ead8ab31cb59630d3e89e6296c49cbdddedb053a596e31aecc17 | An ant sits at vertex $A$ of a unit square $ABCD$. It needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that the ant must take to get into the ... | [
"2"
] | [
"An ant sits at vertex $A$ of a unit square $ABCD$. It needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that the ant must take to get into... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-cc131d1f233f74deb0936572cedfc927c0b9d179d01b30baa6a33acbf5f128b5 | An ant stands at the origin in a 4-dimensional space. Every second he may move to any of the 80 points which are adjacent to him. (Thus he may move 1, \( \sqrt{2} \), \( \sqrt{3} \), or even 2 units in one second.) The ant may not visit a previously-visited gridpoint, but he may cross his path. After 10 seconds, how ma... | [
"80 \\times 79^{9}"
] | [
"An ant stands at the origin in a 4-dimensional space. Every second he may move to any of the 80 points which are adjacent to him. (Thus he may move 1, \\( \\sqrt{2} \\), \\( \\sqrt{3} \\), or even 2 units in one second.) The ant may not visit a previously-visited gridpoint, but he may cross his path. After 10 seco... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-dc83783bdbf07dda645587138f96211a81c6ed0a17309ea5fd069bdaf4065ecc | An ant stands at the origin in a 4-dimensional space. Every second, it may move to any of the 80 adjacent points. The ant may not visit a previously-visited gridpoint but may cross its path. After 10 seconds, how many different paths could the ant have taken? | [
"80 \\times 79^{9}"
] | [
"An ant stands at the origin in a 4-dimensional space. Every second, it may move to any of the 80 adjacent points. The ant may not visit a previously-visited gridpoint but may cross its path. After 10 seconds, how many different paths could the ant have taken?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-d2a65a34870c6dec5613c7bb2f3e3acaba390a8c1b4efd8592d383ab1aeea6e4 | An ant starts at $(0, 0)$ and wants to walk to $(A, B)$ where $A \ge B > 0$ and $2 \mid (A - B)$. Each step, the ant can change his position by $(1, -1)$ or $(1, 1)$ but he can never move below $y = 0$ or above $y = B$. How many different walks can the ant take? | [
"\\dbinom{A}{\\frac{A + B}{2}} - \\dbinom{A}{\\frac{A + B}{2} - (B + 1)}"
] | [
"An ant starts at $(0, 0)$ and wants to walk to $(A, B)$ where $A \\ge B > 0$ and $2 \\mid (A - B)$. Each step, the ant can change his position by $(1, -1)$ or $(1, 1)$ but he can never move below $y = 0$ or above $y = B$. How many different walks can the ant take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-93c03cc79cf52e909d79e524cdad7110a59baeeaeb7383d9e3749808242e6db3 | An ant starts at $(0, 0)$ on the coordinate plane and takes a path moving only one unit upwards or one unit to the right each step to $(3, 3)$. Once reaching $(3, 3)$, it takes a path moving only one unit downwards or one unit to the left each step back to $(0, 0)$, such that this second path does not intersect the fir... | [
"20"
] | [
"An ant starts at $(0, 0)$ on the coordinate plane and takes a path moving only one unit upwards or one unit to the right each step to $(3, 3)$. Once reaching $(3, 3)$, it takes a path moving only one unit downwards or one unit to the left each step back to $(0, 0)$, such that this second path does not intersect th... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-69385c431ee2dcffca9c207b36989c34526e16d2b909316bfd4fb46e0e24d26d | An ant starts at $(0,0)$ on a 2D-plane and moves to adjacent lattice points. If the ant moves from $(a,b)$ to $(a,b+1)$ or $(a+1,b)$ only, what is the probability of reaching $(m,n)$ after $m+n$ steps, where $m,n \ge 0$? | [
"\\(\\binom{m+n}{m} \\left(\\frac{1}{2}\\right)^{m+n}\\)"
] | [
"An ant starts at $(0,0)$ on a 2D-plane and moves to adjacent lattice points. If the ant moves from $(a,b)$ to $(a,b+1)$ or $(a+1,b)$ only, what is the probability of reaching $(m,n)$ after $m+n$ steps, where $m,n \\ge 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-5e57630152138c1307d9eaa12e2e78fa4c9f29ee5904798ff4107426c761521b | An ant starts at $(0,0)$ on a 2D-plane and moves to any adjacent lattice points. What is the probability of reaching $(m,n)$ after exactly $k$ steps, where $m,n \in \mathbb{Z}$? | [
"0"
] | [
"An ant starts at $(0,0)$ on a 2D-plane and moves to any adjacent lattice points. What is the probability of reaching $(m,n)$ after exactly $k$ steps, where $m,n \\in \\mathbb{Z}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4748611f3b9d91c75c7668bd79ced0e12170d43caba49ed31ccd4240a4da9f61 | An ant starts at $(1,1)$ in the Cartesian plane. Every second, the ant moves from its position $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. However, it cannot ever move to a point whose $x-$ and $y-$ coordinates are both divisible by $3$. Let $N$ be the number of paths that the ant can take from $(1,1)$ to $(11,11)$. Find... | [
"14494"
] | [
"An ant starts at $(1,1)$ in the Cartesian plane. Every second, the ant moves from its position $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. However, it cannot ever move to a point whose $x-$ and $y-$ coordinates are both divisible by $3$. Let $N$ be the number of paths that the ant can take from $(1,1)$ to $(11,11)$.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-5f76fb929b24312bced60b6643e2372c0ea445ff1ea6709f0b4ca1f41c6c35f5 | An ant starts at (0,0) and makes 16 moves, each moving one unit along lattice points. How many paths are there such that the ant returns to (0,0) after 16 moves? | [
"\\(\\sum_{k=0}^8 \\frac{16!}{(k!)^2((8-k)!)^2}\\)"
] | [
"An ant starts at (0,0) and makes 16 moves, each moving one unit along lattice points. How many paths are there such that the ant returns to (0,0) after 16 moves?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b1e21811ad806af06b35fb6c66c81b49f7a27e474678c7bf3849f6448ef75b8 | An ant starts at 0 on a number line confined to the interval [0,10]. If the ant takes 20 unit steps either to the right or left and cannot go outside the interval, how many ways can it get to 10? | [
"6954"
] | [
"An ant starts at 0 on a number line confined to the interval [0,10]. If the ant takes 20 unit steps either to the right or left and cannot go outside the interval, how many ways can it get to 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-20dd67bdaff4e4f4abafb8e7a7bcd95a46b31991c99acf6fffe8c468da8ee24b | An ant starts at 1 on the number line and has a 50% chance of moving left and a 50% chance of moving right by 1 unit every turn. Once the ant reaches 0, it stops. Find the expected number of turns the ant will take before it stops. | [
"\\(\\infty\\)"
] | [
"An ant starts at 1 on the number line and has a 50% chance of moving left and a 50% chance of moving right by 1 unit every turn. Once the ant reaches 0, it stops. Find the expected number of turns the ant will take before it stops."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e1a62c47c917329cdaffae752c9ecc4b356cfeb4357f83ca710746730eb1b3f7 | An ant starts at a junction $A$ on a lattice frame of a $2 \times 2 \times 2$ cube formed with $54$ metal shafts of unit length. The ant moves along the shafts by turning to a perpendicular shaft at each junction and never revisits the same junction. What is the maximum possible length of the ant’s path before it retur... | [
"26"
] | [
"An ant starts at a junction $A$ on a lattice frame of a $2 \\times 2 \\times 2$ cube formed with $54$ metal shafts of unit length. The ant moves along the shafts by turning to a perpendicular shaft at each junction and never revisits the same junction. What is the maximum possible length of the ant’s path before i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-ee3d8a3dc7802eb4277e86b365fa810ce21d4a5271f12c0ba450869f6356a74a | An ant starts at point $(0,0)$ on the $Oxy$ coordinate plane. In each second, it moves 1 unit of distance either upward, downward, to the left, or to the right with equal probability. Given positive integers $a$ and $s$, what is the probability that at some point during the $s$ seconds, the ant is outside the square de... | [
"1 - \\sum_{k=0}^{s} \\binom{s}{k} \\left(\\frac{1}{2}\\right)^s \\left( \\sum_{m=-a}^{a} \\binom{k}{\\frac{k+m}{2}} \\left(\\frac{1}{2}\\right)^k \\right) \\left( \\sum_{n=-a}^{a} \\binom{s-k}{\\frac{s-k+n}{2}} \\left(\\frac{1}{2}\\right)^{s-k} \\right)"
] | [
"An ant starts at point $(0,0)$ on the $Oxy$ coordinate plane. In each second, it moves 1 unit of distance either upward, downward, to the left, or to the right with equal probability. Given positive integers $a$ and $s$, what is the probability that at some point during the $s$ seconds, the ant is outside the squa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-0bd450e458713fc373f0af2b42794d29c1c5ec1dadf920536f2c040118bdf9b6 | An ant starts at point $A$ and follows the route: $1 \text{ cm}$ north, $2 \text{ cm}$ east, $3 \text{ cm}$ south, $4 \text{ cm}$ west, $5 \text{ cm}$ north, $6 \text{ cm}$ east, and so on, finally $41 \text{ cm}$ north. Calculate the straight-line distance between the starting point $A$ and the ending point $B$. | [
"29"
] | [
"An ant starts at point $A$ and follows the route: $1 \\text{ cm}$ north, $2 \\text{ cm}$ east, $3 \\text{ cm}$ south, $4 \\text{ cm}$ west, $5 \\text{ cm}$ north, $6 \\text{ cm}$ east, and so on, finally $41 \\text{ cm}$ north. Calculate the straight-line distance between the starting point $A$ and the ending poin... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-bff4fb14364fb71686ff73b5024f584b20cd516b260cbfa8f9f5cab0e31b12c8 | An ant starts at point $A$ and follows the route: 1 cm north, 2 cm east, 3 cm south, 4 cm west, 5 cm north, 6 cm east, and so on, until it moves 41 cm north and ends at point $B$. Calculate the straight-line distance between points $A$ and $B$. | [
"29"
] | [
"An ant starts at point $A$ and follows the route: 1 cm north, 2 cm east, 3 cm south, 4 cm west, 5 cm north, 6 cm east, and so on, until it moves 41 cm north and ends at point $B$. Calculate the straight-line distance between points $A$ and $B$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-4e70ba66b2bd0eecc241f755ae583efd44f14a7d7758e5a71bf30932a7b132de | An ant starts at point \( A = (0,0) \) in square \( ABCD \) with vertices \( A = (0,0) \), \( B = (20,0) \), \( C = (20,20) \), and \( D = (0,20) \). The ant faces point \( \left(20, \frac{20}{3}\right) \) and moves in a straight line. Upon hitting a side of the square, it turns 90 degrees counterclockwise and continue... | [
"250"
] | [
"An ant starts at point \\( A = (0,0) \\) in square \\( ABCD \\) with vertices \\( A = (0,0) \\), \\( B = (20,0) \\), \\( C = (20,20) \\), and \\( D = (0,20) \\). The ant faces point \\( \\left(20, \\frac{20}{3}\\right) \\) and moves in a straight line. Upon hitting a side of the square, it turns 90 degrees counter... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-6a5a03ec945401b099fa2095fff393e81bf6111b677f54d218303527ec7112d9 | An ant starts at point \( A = (0,0,0) \) in a \( 3 \times 3 \times 3 \) cube \( ABCD-EFGH \) and travels to point \( G \), then returns to \( A \). The ant can only turn by \( 90 \) degrees at each lattice point and cannot retrace its path or enter the interior of the cube. Additionally, while traveling to \( G \), the... | [
"384"
] | [
"An ant starts at point \\( A = (0,0,0) \\) in a \\( 3 \\times 3 \\times 3 \\) cube \\( ABCD-EFGH \\) and travels to point \\( G \\), then returns to \\( A \\). The ant can only turn by \\( 90 \\) degrees at each lattice point and cannot retrace its path or enter the interior of the cube. Additionally, while travel... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-007ee24469871f725862aaf4a9a872e317d15b2eb099b0510f96049d3b43b62b | An ant starts at position (0,0) and walks north, east, south, or west, one unit at a time. How many different paths of length 8 units finish at (0,0)? | [
"4900"
] | [
"An ant starts at position (0,0) and walks north, east, south, or west, one unit at a time. How many different paths of length 8 units finish at (0,0)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-7107a0a2643dbb0316e387d2412c9c0c558d4eb2aebb456b71ce4fce51b01b7a | An ant starts at the bottom left corner (A) of a $40$ by $40$ grid and wants to reach the upper right corner (B) by traveling only up and right. However, the $20$ by $20$ grid in the bottom right corner is demolished. If $n$ is the number of ways the ant can still travel from A to B, find the remainder when $n$ is divi... | [
"500"
] | [
"An ant starts at the bottom left corner (A) of a $40$ by $40$ grid and wants to reach the upper right corner (B) by traveling only up and right. However, the $20$ by $20$ grid in the bottom right corner is demolished. If $n$ is the number of ways the ant can still travel from A to B, find the remainder when $n$ is... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ab0cd580bd56ab654b186d90ecc4e1beced862f4bc1c5e8a1db166c26374686 | An ant starts at the center of a shape consisting of 7 points connected by segments of equal length. In each movement, the ant moves randomly to a neighboring point. Calculate the probability that the ant returns to the center after 3 movements. | [
"\\(\\frac{2}{9}\\)"
] | [
"An ant starts at the center of a shape consisting of 7 points connected by segments of equal length. In each movement, the ant moves randomly to a neighboring point. Calculate the probability that the ant returns to the center after 3 movements."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 24
} | 70 | 0 | 0 |
omr-ws-26257659a0cd486e6db7d13519d86025f8579757baceeb19d4b3978822481d43 | An ant starts at the origin of a coordinate plane and takes four steps, each step being either up, down, left, or right. What is the probability that the ant ends up at the origin after four steps? | [
"\\dfrac{9}{64}"
] | [
"An ant starts at the origin of a coordinate plane and takes four steps, each step being either up, down, left, or right. What is the probability that the ant ends up at the origin after four steps?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7409a76a15e3c8aaff9964f52632a6200038e33928380c6b30e2ba86a4f95288 | An ant starts at the origin of a number line with integers $-50, -49, -48, \ldots, 44, 45$. The ant moves to an adjacent number every second. What is the expected number of seconds until the ant reaches $-50$ or $45$? | [
"2250"
] | [
"An ant starts at the origin of a number line with integers $-50, -49, -48, \\ldots, 44, 45$. The ant moves to an adjacent number every second. What is the expected number of seconds until the ant reaches $-50$ or $45$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78db2f3a21cc1a735dc4d8a03f34d101f7b43ee62aacd54d09e5761b6cdd8bc7 | An ant starts at the origin of the Cartesian plane and moves one unit at a time either left, right, up, or down. Compute the number of ways for the ant to reach (4,7) in exactly 11 moves. | [
"330"
] | [
"An ant starts at the origin of the Cartesian plane and moves one unit at a time either left, right, up, or down. Compute the number of ways for the ant to reach (4,7) in exactly 11 moves."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-b310eab95977455d2390cc38ef93f0432b7d77b84710541dae19ef40389d6d23 | An ant starts at the point $(0, 0)$ and travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. What is the probability that the ant reaches $(20, 23)$ without passing through $(20, 20)$? | [
"\\(\\frac{1510}{1763}\\)"
] | [
"An ant starts at the point $(0, 0)$ and travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. What is the probability that the ant reaches $(20, 23)$ without passing through $(20, 20)$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-ce1f1ebe736f8ba8d2dbeab6d9565323761adbe73cb44e23cc8a7b2f108a244f | An ant starts at the point $(0,0)$ on a Cartesian coordinate system and can move to either $(x,y+1)$ or $(x+1,y)$ from any point $(x,y)$. A sequence of points $x_i = (a_i, b_i)$ becomes toxic after the $i$-th move and remains toxic forever. Determine the maximum number of moves the ant can make without stepping on a to... | [
"4"
] | [
"An ant starts at the point $(0,0)$ on a Cartesian coordinate system and can move to either $(x,y+1)$ or $(x+1,y)$ from any point $(x,y)$. A sequence of points $x_i = (a_i, b_i)$ becomes toxic after the $i$-th move and remains toxic forever. Determine the maximum number of moves the ant can make without stepping on... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-d2c72be8cafddf8687005762bdebfa860fffd9a4b487a06efed1d932534c88e0 | An ant starts at the point $(0,0)$. Every second, it randomly either walks $1$ unit in the positive $x$ direction or $1$ unit in the positive $y$ direction. After $2n$ seconds, what is the expected number of times the ant touches the line $y=x$, including $(0,0)$? | [
"\\dfrac{(2n + 1) \\binom{2n}{n}}{4^n}"
] | [
"An ant starts at the point $(0,0)$. Every second, it randomly either walks $1$ unit in the positive $x$ direction or $1$ unit in the positive $y$ direction. After $2n$ seconds, what is the expected number of times the ant touches the line $y=x$, including $(0,0)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f594f3372004991900988b5499b9fc1a9617d8995920944c60bd8dd87a86a45 | An ant starts at the point $(1,2)$ and moves one unit per second in one of the four coordinate directions. Whenever the ant reaches $(x,\pm 3)$, it is moved to $(-x,\mp 2)$, and whenever it reaches $(\pm 2,y)$, it is moved to $(\mp 1,y)$. If $p$ and $q$ are relatively prime positive integers such that $\tfrac pq$ is th... | [
"95"
] | [
"An ant starts at the point $(1,2)$ and moves one unit per second in one of the four coordinate directions. Whenever the ant reaches $(x,\\pm 3)$, it is moved to $(-x,\\mp 2)$, and whenever it reaches $(\\pm 2,y)$, it is moved to $(\\mp 1,y)$. If $p$ and $q$ are relatively prime positive integers such that $\\tfrac... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-0ec078a7fd93d48bd6060936ac54e1f86ad49bf60bec9d68e832c3c3ff59fefe | An ant starts at the point \((0,0)\) in the coordinate plane. It can make moves from lattice point \((x_1,y_1)\) to lattice point \((x_2,y_2)\) whenever \(x_2 \geq x_1\), \(y_2 \geq y_1\), and \((x_1,y_1) \neq (x_2,y_2)\). For all nonnegative integers \(m, n\), define \(a_{m,n}\) to be the number of possible sequences ... | [
"\\(\\frac{81}{62}\\)"
] | [
"An ant starts at the point \\((0,0)\\) in the coordinate plane. It can make moves from lattice point \\((x_1,y_1)\\) to lattice point \\((x_2,y_2)\\) whenever \\(x_2 \\geq x_1\\), \\(y_2 \\geq y_1\\), and \\((x_1,y_1) \\neq (x_2,y_2)\\). For all nonnegative integers \\(m, n\\), define \\(a_{m,n}\\) to be the numbe... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-659cd097c84abaa782aaae3bb4195a9f9c4b282f4f96210dcc48ee7860529164 | An ant starts at the top of a circular rubber band with a circumference of 1 meter. Each minute, the ant crawls 1 centimeter relative to the rubber band, and then the rubber band uniformly elongates by 1 meter. Let $d_n$ be the distance the ant has traveled relative to the original circumference after $n$ minutes. Calc... | [
"5.187"
] | [
"An ant starts at the top of a circular rubber band with a circumference of 1 meter. Each minute, the ant crawls 1 centimeter relative to the rubber band, and then the rubber band uniformly elongates by 1 meter. Let $d_n$ be the distance the ant has traveled relative to the original circumference after $n$ minutes.... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-c48e875a1003c1da1859aad1ce12c35dc0aa88432cd254504a8e7d2a2bf2d140 | An ant starts at the top of a cube and moves to one of the three adjacent faces. In subsequent moves, the ant does not return to the previous face but moves to one of the two adjacent faces. The probability that the ant is on the north face after 8 moves is $\frac{m}{n}$, where $m$ and $n$ are coprime. Find $m + n$. | [
"4"
] | [
"An ant starts at the top of a cube and moves to one of the three adjacent faces. In subsequent moves, the ant does not return to the previous face but moves to one of the two adjacent faces. The probability that the ant is on the north face after 8 moves is $\\frac{m}{n}$, where $m$ and $n$ are coprime. Find $m + ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-e2adfbc540a7ebb1abe161eb23e28de775aef2ada80849c8ee7d4c9af1ea800b | An ant starts at the top vertex of a tetrahedron. On each move, he crawls to an adjacent vertex. The number of ways such that after \(x\) turns, the ant is at the top vertex, can be expressed as \(\frac{3^a + 3(-1)^x}{3^c + 3^d}\). Find \(a + c + d\) in terms of \(x\). | [
"x + 1"
] | [
"An ant starts at the top vertex of a tetrahedron. On each move, he crawls to an adjacent vertex. The number of ways such that after \\(x\\) turns, the ant is at the top vertex, can be expressed as \\(\\frac{3^a + 3(-1)^x}{3^c + 3^d}\\). Find \\(a + c + d\\) in terms of \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b64c624f95c94fe5896a6d52c8507653fab751cab35dad892ea5a80512b5888d | An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days? | [
"\\(\\frac{2}{9}\\)"
] | [
"An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-38de2c5767a9985ea8a6ecd807902ddaf5f568d3b03f39020b9a2ed23650bae0 | An ant starts at vertex A of a graph with vertices A, B, C, and D, where each edge is 10 km long. The ant wants to walk along the edges and return to vertex A after traveling exactly $10n$ km. How many ways can the ant do this? | [
"\\( \\frac{1}{4}(3^n) + \\frac{3}{4}(-1)^n \\)"
] | [
"An ant starts at vertex A of a graph with vertices A, B, C, and D, where each edge is 10 km long. The ant wants to walk along the edges and return to vertex A after traveling exactly $10n$ km. How many ways can the ant do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-e831a21b2fc66e1789df6858c137d72e1265552e2f09b84ba4c0ca99f5c2238a | An ant starts from a corner of a sugar cube with a side length of 2 cm and travels on the shortest path on the side surface of the cube to the opposite corner. Calculate the length of the path traveled by the ant. | [
"\\(2\\sqrt{5}\\)"
] | [
"An ant starts from a corner of a sugar cube with a side length of 2 cm and travels on the shortest path on the side surface of the cube to the opposite corner. Calculate the length of the path traveled by the ant."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-73c3cc80be3925f62647799c9ed02785cddd1488bfa048f1344e1890e7f1a2ac | An ant starts from a juncture $A$ in a $2 \times 2 \times 2$ cubic structure formed by 27 points $(a, b, c)$ where $a, b, c \in \{0, 1, 2\}$ and moves along rods of length 1. At each juncture, the ant turns $90^\circ$ and changes rods. If the ant returns to $A$ without visiting any other juncture more than once, what i... | [
"26"
] | [
"An ant starts from a juncture $A$ in a $2 \\times 2 \\times 2$ cubic structure formed by 27 points $(a, b, c)$ where $a, b, c \\in \\{0, 1, 2\\}$ and moves along rods of length 1. At each juncture, the ant turns $90^\\circ$ and changes rods. If the ant returns to $A$ without visiting any other juncture more than o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6477b7ddc8809b1f313b35434238111ba2bb04f409a78f4315b9aaee22b2f11 | An ant starts from one corner of a square with side length 1 meter and walks along the sides to reach the diagonally opposite corner, traveling a total distance of 2 meters. A step-like structure is created, and the ant walks the same distance of 2 meters along this structure. As the number of steps increases to infini... | [
"2"
] | [
"An ant starts from one corner of a square with side length 1 meter and walks along the sides to reach the diagonally opposite corner, traveling a total distance of 2 meters. A step-like structure is created, and the ant walks the same distance of 2 meters along this structure. As the number of steps increases to i... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72e16581915d8945d5ee6927c01bcd2dd16bd5f06a4466d9d87dfb70ca887a05 | An ant starts from the anthill and travels in a straight line a section of \(d\) cm, then turns \(90^\circ\) and travels in a straight line another section of \(d/2\) cm, then turns \(90^\circ\) again and travels a section of \(d/(2^2)\) cm, and so on. The direction in which it rotates is decided at each vertex. What i... | [
"\\dfrac{d\\sqrt{5}}{3}"
] | [
"An ant starts from the anthill and travels in a straight line a section of \\(d\\) cm, then turns \\(90^\\circ\\) and travels in a straight line another section of \\(d/2\\) cm, then turns \\(90^\\circ\\) again and travels a section of \\(d/(2^2)\\) cm, and so on. The direction in which it rotates is decided at ea... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-64fe7286b2ecdd0b846ea9f50aaa2c2ea28799d9d7023aa2b39d5a8f254cd093 | An ant starts from the bottom edge of a can and makes 2 complete revolutions to the top, ending up directly above where it started. If the can is 18 cm tall and has a radius of 3 cm, how far did the ant travel along the can? | [
"\\( 6\\sqrt{4\\pi^2 + 9} \\)"
] | [
"An ant starts from the bottom edge of a can and makes 2 complete revolutions to the top, ending up directly above where it started. If the can is 18 cm tall and has a radius of 3 cm, how far did the ant travel along the can?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c26342c60a10643f13cd1d02b97bf3fc4cac87faca9d1a1112ae0a0c9569db83 | An ant takes 7 random steps on a square grid (i.e., it moves a unit distance up, down, left, or right with equal probability at each step independently). What is the probability it ends up exactly a knight's hop away from where it started? | [
"\\(735 / 2048\\)"
] | [
"An ant takes 7 random steps on a square grid (i.e., it moves a unit distance up, down, left, or right with equal probability at each step independently). What is the probability it ends up exactly a knight's hop away from where it started?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-99412c77ac63415fbd890eb06400a333558f798c604421fa8c27ae9d9c88e661 | An ant walks 4 feet east, makes a 160-degree turn to the right, walks another 4 feet east, and continues this pattern. Find the distance covered in feet for the ant to return to the anthill. | [
"36"
] | [
"An ant walks 4 feet east, makes a 160-degree turn to the right, walks another 4 feet east, and continues this pattern. Find the distance covered in feet for the ant to return to the anthill."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-a1e51f24f196887d81d7183bb8796ce89078b538230bfce58e0f0d28c4fd87f3 | An ant walks a distance \( A = 10^9 \) millimeters. A bear walks \( B = 10^6 \) feet. A chicken walks \( C = 10^8 \) inches. What is the correct ordering of \( A, B, C \)? (Note: There are 25.4 millimeters in an inch, and there are 12 inches in a foot.) | [
"B < A < C"
] | [
"An ant walks a distance \\( A = 10^9 \\) millimeters. A bear walks \\( B = 10^6 \\) feet. A chicken walks \\( C = 10^8 \\) inches. What is the correct ordering of \\( A, B, C \\)? (Note: There are 25.4 millimeters in an inch, and there are 12 inches in a foot.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e13a927efe4768b17bbc52dbfc8e246d3dd776286a876943d70b401e22b7251e | An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming its march, it rotates $60^\circ$ alternating the direction (if the last time it turned $60^\circ$ to its right, the next one does it $60^\circ$ to its left, and vice vers... | [
"6"
] | [
"An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming its march, it rotates $60^\\circ$ alternating the direction (if the last time it turned $60^\\circ$ to its right, the next one does it $60^\\circ$ to its left, and v... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-ac5e9fbc0c078f661df95c0ab654cf7c60fabad5837c47cc9f425d969eb7b315 | An ant walks around a square of insect repellent that is 0.5 meters on each side, always remaining exactly 20 cm away from it. What is the total area enclosed by the ant's path in square centimeters? | [
"\\(2500 + 400\\pi\\)"
] | [
"An ant walks around a square of insect repellent that is 0.5 meters on each side, always remaining exactly 20 cm away from it. What is the total area enclosed by the ant's path in square centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ea4d82662d9d3aac1786a90eaf26cc4764aa6ad3cc1eb1c6399ae1dcc09ac93 | An ant walks from the bottom left corner of a \( 10 \times 10 \) square grid to the diagonally opposite corner, always walking along grid lines and taking as short a route as possible. Let \( N(k) \) be the number of different paths that the ant could follow if it makes exactly \( k \) turns. Find \( N(6) - N(5) \). | [
"3456"
] | [
"An ant walks from the bottom left corner of a \\( 10 \\times 10 \\) square grid to the diagonally opposite corner, always walking along grid lines and taking as short a route as possible. Let \\( N(k) \\) be the number of different paths that the ant could follow if it makes exactly \\( k \\) turns. Find \\( N(6) ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-9368e10645ef74961d4ca93a9e3858e6a37080dfa24f1d9e76f0a42538e4a0f6 | An ant walks on an infinite lattice, moving only upwards or to the right. There is an infinite sequence of points \( A_i \) such that immediately after the ant takes the \( i \)-th step, the point \( A_i \) becomes poisoned. Let \( P(n) \) be the number of distinct infinite paths the ant can take starting from the orig... | [
"16"
] | [
"An ant walks on an infinite lattice, moving only upwards or to the right. There is an infinite sequence of points \\( A_i \\) such that immediately after the ant takes the \\( i \\)-th step, the point \\( A_i \\) becomes poisoned. Let \\( P(n) \\) be the number of distinct infinite paths the ant can take starting ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-28eb10893be9a905129bb3e67d3f7df3aa23db2203a8c354433c3e6540df580a | An ant walks on the plane as follows: initially, it walks 1 cm in any direction. After each step, it changes the direction of the path by 60º to the left or right and walks 1 cm in that direction. If the ant returns to its starting point after 2009 steps, calculate the sum of all possible values of the number of steps ... | [
"671676"
] | [
"An ant walks on the plane as follows: initially, it walks 1 cm in any direction. After each step, it changes the direction of the path by 60º to the left or right and walks 1 cm in that direction. If the ant returns to its starting point after 2009 steps, calculate the sum of all possible values of the number of s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-a9000aa21b4b376b7a2896572b9070b515b964fb58e26242aeec9a5635f8bbfd | An ant wanders over the coordinate plane. Whenever it finds itself stationary at a point \((x, y)\), it walks to the point \((-3x - y, 7x + ky)\), where \(k\) is an integer. Determine the value of \(k\) such that the ant returns to its original point \((x, y)\) after a finite number of stops. | [
"\\( k = 2 \\)"
] | [
"An ant wanders over the coordinate plane. Whenever it finds itself stationary at a point \\((x, y)\\), it walks to the point \\((-3x - y, 7x + ky)\\), where \\(k\\) is an integer. Determine the value of \\(k\\) such that the ant returns to its original point \\((x, y)\\) after a finite number of stops."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9f29d4eb0fc1b1a8b31689899ff3721231357707b7c80f9636b850b0e2b805fa | An antelope is a chess piece which moves similarly to the knight: two cells $(x_1,y_1)$ and $(x_2,y_2)$ are joined by an antelope move if and only if $\{|x_1-x_2|,|y_1-y_2|\}=\{3,4\}$. The numbers from $1$ to $10^{12}$ are placed in the cells of a $10^6 \times 10^6$ grid. Let $D$ be the set of all absolute differences ... | [
"0"
] | [
"An antelope is a chess piece which moves similarly to the knight: two cells $(x_1,y_1)$ and $(x_2,y_2)$ are joined by an antelope move if and only if $\\{|x_1-x_2|,|y_1-y_2|\\}=\\{3,4\\}$. The numbers from $1$ to $10^{12}$ are placed in the cells of a $10^6 \\times 10^6$ grid. Let $D$ be the set of all absolute di... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
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