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-3b8de0ba6d1df43261718ea05d3e9378092a0a27f8590252146bd130292a5c26 | A governor spends a million gold coins distributed equally among all his soldiers. The development council suggests decreasing the number of soldiers by 50 percent and raising the salaries of the remaining soldiers by 50 percent. How many coins would still be released for the development of this country if the governor... | [
"750000"
] | [
"A governor spends a million gold coins distributed equally among all his soldiers. The development council suggests decreasing the number of soldiers by 50 percent and raising the salaries of the remaining soldiers by 50 percent. How many coins would still be released for the development of this country if the gov... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a72c5512db0742ea9025ad6a78ab4918d83d10cfd2bde0ca96cba3c2026f4cbc | A grapefruit in the shape of a sphere has a diameter of 4 inches, with a peel that is 0.25 inches thick. Once the grapefruit is peeled, it has 12 congruent wedges. Find the volume of each wedge. | [
"\\(\\frac{343 \\pi}{576}\\)"
] | [
"A grapefruit in the shape of a sphere has a diameter of 4 inches, with a peel that is 0.25 inches thick. Once the grapefruit is peeled, it has 12 congruent wedges. Find the volume of each wedge."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-288c17b5c505b87e5f97e10642edacc3c099a42376f8b7365a93f3245c7ce2a4 | A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of the other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. If $np < k$, calculate the minimum number of vertices in $B$ that are connected to all vertices in $A$. | [
"k - np"
] | [
"A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of the other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. If $np < k$, calculate the minimum number of vertices in $B$ that are connected to all vertices in $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6364f52ef1bc2cf56322b8baba20cfcd1955c4b582ca3df39f9107a4f8e531e3 | A graph $G$ has 2019 vertices, each with a degree of at least 3. Let $C$ be the number of vertices in the cycle with the maximal length. Find the minimum value that $C$ can take. | [
"4"
] | [
"A graph $G$ has 2019 vertices, each with a degree of at least 3. Let $C$ be the number of vertices in the cycle with the maximal length. Find the minimum value that $C$ can take."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 7
} | 34 | 0 | 0 |
omr-ws-c188f2ae88c99a6d0bf3560fcd2897d77ab17925882b147990806590dc839302 | A graph $\Gamma$ has 100 vertices. If any 49 vertices are deleted along with all the edges connected to them, there always remains a cycle in the graph. What is the minimum number of edges this graph must have? | [
"150"
] | [
"A graph $\\Gamma$ has 100 vertices. If any 49 vertices are deleted along with all the edges connected to them, there always remains a cycle in the graph. What is the minimum number of edges this graph must have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b82198ab8438145cef2d936ab4f4a5fbcfa4dd9f62aae88023f0bdf9bc4f7a7 | A graph $\mathcal{G}$ has $2019$ vertices and every cycle in $\mathcal{G}$ has a length of exactly $3$. What is the maximum number of edges in $\mathcal{G}$? | [
"3027"
] | [
"A graph $\\mathcal{G}$ has $2019$ vertices and every cycle in $\\mathcal{G}$ has a length of exactly $3$. What is the maximum number of edges in $\\mathcal{G}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 43,
"genselect": 5
} | 48 | 0 | 0 |
omr-ws-cd1350679f1d02aa7180d215e5854e3ce64559271e4e93bd243b4ab7cf6a053d | A graph $\mathcal{G}$ has $2k$ vertices ($k\in\mathbb{Z}_{+}$) and every two vertices that aren't connected have a different degree. Determine the maximum number of colors needed to color the vertices of this graph such that every group of vertices with the same color form a clique. If the maximum number of colors need... | [
"13"
] | [
"A graph $\\mathcal{G}$ has $2k$ vertices ($k\\in\\mathbb{Z}_{+}$) and every two vertices that aren't connected have a different degree. Determine the maximum number of colors needed to color the vertices of this graph such that every group of vertices with the same color form a clique. If the maximum number of col... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-2739c2e00f67b4a399c0bda12c59a9661bb8b5e606e7ef7e506798c693814ec4 | A graph \( G \) is defined as "good" if for every set \( V \) of vertices with cardinality \( m \), and the set \( S \) of all neighbors of vertices in \( V \) with cardinality \( n \), we have \( n \geq m \). What is a necessary condition for a graph to be "good"? | [
"\\text{The graph has a perfect matching}"
] | [
"A graph \\( G \\) is defined as \"good\" if for every set \\( V \\) of vertices with cardinality \\( m \\), and the set \\( S \\) of all neighbors of vertices in \\( V \\) with cardinality \\( n \\), we have \\( n \\geq m \\). What is a necessary condition for a graph to be \"good\"?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-c3bcc7f1bdc46773de98220a28a0044e5477ce7643bb46dfb166ceed338c3984 | A graph contains $p$ vertices numbered from $1$ to $p$, and $q$ edges numbered from $p + 1$ to $p + q$. For each edge, the sum of the numbers of its ends and of the edge itself equals the same number $s$. Additionally, the numbers of edges starting in all vertices are equal. If $p = 10$ and $q = 15$, find the value of ... | [
"29"
] | [
"A graph contains $p$ vertices numbered from $1$ to $p$, and $q$ edges numbered from $p + 1$ to $p + q$. For each edge, the sum of the numbers of its ends and of the edge itself equals the same number $s$. Additionally, the numbers of edges starting in all vertices are equal. If $p = 10$ and $q = 15$, find the valu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c451d170ae22ab37a4fc4c5c3e310b297289a7330ff7ca9d9f533fa098444ef | A graph has $N$ vertices. An invisible hare sits in one of the vertices. Hunters can shoot at vertices simultaneously in each move. If the hare is in one of the target vertices, it is killed. Otherwise, the hare can stay in its vertex or jump to a neighboring vertex. Given that there exists an algorithm allowing the hu... | [
"1"
] | [
"A graph has $N$ vertices. An invisible hare sits in one of the vertices. Hunters can shoot at vertices simultaneously in each move. If the hare is in one of the target vertices, it is killed. Otherwise, the hare can stay in its vertex or jump to a neighboring vertex. Given that there exists an algorithm allowing t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f73c7a6ccc6f0eb168bc29ebd2c6c1d824c364621974db0f17340616aba367c6 | A graph has $n$ points and 100 edges. A move consists of picking a point, removing all its edges, and joining it to any points which it was not joined to immediately before the move. What is the smallest number of moves required to get a graph which has 2 points with no path between them? | [
"2"
] | [
"A graph has $n$ points and 100 edges. A move consists of picking a point, removing all its edges, and joining it to any points which it was not joined to immediately before the move. What is the smallest number of moves required to get a graph which has 2 points with no path between them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-2d8df64dbffede236bcfdaea1141c49578d75bb71701942074153458b9c1ec8b | A graph has $n$ vertices and $\frac{n^2 - 3n + 4}{2}$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $k$ of a circuit in such a graph. | [
"\\( n-1 \\)"
] | [
"A graph has $n$ vertices and $\\frac{n^2 - 3n + 4}{2}$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $k$ of a circuit in such a graph."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-36242f49bef91660aea1f92e4264386d72b90736a98585560f6813e659c75cdc | A graph has 12k points. Each point has 3k+6 edges. For any two points, the number of points joined to both is the same. Find k. | [
"3",
"\\( k = 3 \\)"
] | [
"A graph has 12k points. Each point has 3k+6 edges. For any two points, the number of points joined to both is the same. Find k."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 44,
"genselect": 8
} | 52 | 0 | 0 |
omr-ws-14103be988b802e4f34a59860d1749375a0b5d0d230af2ec779746a9cb306146 | A graph has 1991 points and every point has a degree of at least 1593. Find the minimum number of edges in a complete subgraph (clique) that must exist within this graph. | [
"15"
] | [
"A graph has 1991 points and every point has a degree of at least 1593. Find the minimum number of edges in a complete subgraph (clique) that must exist within this graph."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56afdf89b42e6b8b052f3f0312c682fe7454c6c4eaacdbe6321b71a780198172 | A graph has 2002 points. Given any three distinct points \( A \), \( B \), and \( C \), there is a path from \( A \) to \( B \) that does not pass through \( C \). A move consists of taking any cycle, removing its edges, and adding a new point \( X \) connected to each point of the cycle. After a series of such moves, ... | [
"1001"
] | [
"A graph has 2002 points. Given any three distinct points \\( A \\), \\( B \\), and \\( C \\), there is a path from \\( A \\) to \\( B \\) that does not pass through \\( C \\). A move consists of taking any cycle, removing its edges, and adding a new point \\( X \\) connected to each point of the cycle. After a ser... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-dbb2e3c12b85edc3a9d2c251469564b10186a1c9403cb69d7eb12cf28487a3da | A graph has 30 vertices, 105 edges, and 4822 unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph. | [
"5"
] | [
"A graph has 30 vertices, 105 edges, and 4822 unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 15
} | 30 | 0 | 0 |
omr-ws-76f4a2a284626adea9358df6ddbbde93be3799bf0e61e88acd7e53e147263bff | A graph has \( n \) points and 100 edges. A move consists of picking a point, removing all its edges, and joining it to any points which it was not joined to immediately before the move. What is the smallest number of moves required to get a graph which has two points with no path between them? | [
"2"
] | [
"A graph has \\( n \\) points and 100 edges. A move consists of picking a point, removing all its edges, and joining it to any points which it was not joined to immediately before the move. What is the smallest number of moves required to get a graph which has two points with no path between them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4e7e312d9e346b3cf7bec276fbe3e7f5902a602944d5f81e0fa253fa92c774ac | A graph has \( n \) vertices and \( \frac{1}{2}(n^2 - 3n + 4) \) edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length \( k \) of a circuit in such a graph. | [
"\\( n - 1 \\)"
] | [
"A graph has \\( n \\) vertices and \\( \\frac{1}{2}(n^2 - 3n + 4) \\) edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length \\( k \\) of a circuit in such a graph."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-8f9616a0720a109c26a975ab34c5251083cd8e773ea00213555a25392c45b65b | A graph has \( n \) vertices and \( \frac{n^2 - 3n + 4}{2} \) edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length \( k \) of a circuit in such a graph. | [
"n-1"
] | [
"A graph has \\( n \\) vertices and \\( \\frac{n^2 - 3n + 4}{2} \\) edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length \\( k \\) of a circuit in such a graph."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eeb488541f2b6afa2cba31d3ad9a158edad3a6088191c0d7bb1d8687dbfc1496 | A graph is defined parametrically by the equations \( X = \frac{(\sin t)^2}{\cos t} \) and \( Y = \cos t \). Describe the general shape of this graph. | [
"\\text{a hyperbola}"
] | [
"A graph is defined parametrically by the equations \\( X = \\frac{(\\sin t)^2}{\\cos t} \\) and \\( Y = \\cos t \\). Describe the general shape of this graph."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-4fb7caf519b03414088483adbe97d015ca074654f71a72b331dbdd5462952bb1 | A graph of the function \(y = x^2\) is drawn on the Cartesian plane. The chord \(AB\) of this parabola is parallel to the abscissa, for each point \(C\) lying on the parabola and different from points \(A\) and \(B\), we find point \(C_1\), which lies on the circle circumscribed around the triangle \(ABC\), with the ch... | [
"y = a^2 + 1"
] | [
"A graph of the function \\(y = x^2\\) is drawn on the Cartesian plane. The chord \\(AB\\) of this parabola is parallel to the abscissa, for each point \\(C\\) lying on the parabola and different from points \\(A\\) and \\(B\\), we find point \\(C_1\\), which lies on the circle circumscribed around the triangle \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1cf36133dbe53537fe0d06ff095053756b220ccbd0b40e1aa0f937d7e9979f0c | A graph with $n$ vertices has each vertex with degree $k$. What values can $n$ take? | [
"\\( n \\geq k + 1 \\) and \\( nk \\) must be even."
] | [
"A graph with $n$ vertices has each vertex with degree $k$. What values can $n$ take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 8,
"tir": 10
} | 40 | 0 | 0 |
omr-ws-d7f94fa461bf697ab629055373faee5993e89479b7e1ff2e693a8b8e415bf2b4 | A grasshopper can jump along a checkered strip for 8, 9, or 10 cells in any direction. A natural number \( n \) is called jumpable if the grasshopper can start from some cell of a strip of length \( n \) and visit every cell exactly once. Find at least one non-jumpable number \( n > 50 \). | [
"62 or 63"
] | [
"A grasshopper can jump along a checkered strip for 8, 9, or 10 cells in any direction. A natural number \\( n \\) is called jumpable if the grasshopper can start from some cell of a strip of length \\( n \\) and visit every cell exactly once. Find at least one non-jumpable number \\( n > 50 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 5,
"tir": 10
} | 53 | 0 | 0 |
omr-ws-50ec3e2330fcf3c2516cfb3f9dc003cec6eb26ec5b75eef700960dd01ce18c61 | A grasshopper hops from an integer point to another integer point in the plane, where every even jump has a length $\sqrt{98}$ and every odd jump has a length $\sqrt{149}$. What is the least number of jumps the grasshopper has to make in order to return to its starting point after an odd number of jumps? | [
"7"
] | [
"A grasshopper hops from an integer point to another integer point in the plane, where every even jump has a length $\\sqrt{98}$ and every odd jump has a length $\\sqrt{149}$. What is the least number of jumps the grasshopper has to make in order to return to its starting point after an odd number of jumps?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-403d7f66a6c00c92362cf01bfccb59501995056c53b6685f4d8e0b563f5149a9 | A grasshopper is in the left above corner of a $10 \times 10$ square. At each step, it can jump to a square below or to the right. Additionally, it can fly from a cell in the bottom row to a cell in the top row, and from a cell in the rightmost column to a cell in the leftmost column. If the grasshopper visits each cel... | [
"9"
] | [
"A grasshopper is in the left above corner of a $10 \\times 10$ square. At each step, it can jump to a square below or to the right. Additionally, it can fly from a cell in the bottom row to a cell in the top row, and from a cell in the rightmost column to a cell in the leftmost column. If the grasshopper visits ea... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fa6cf206924484aff87417a3c4022200ba06c61f4ae155a048c759aff6cfa00 | A grasshopper is initially at position 0 on the number line. In the $k$-th step, it jumps $k$ units. If $k$ is even, it jumps to the left; if $k$ is odd, it jumps to the right. Let $f(n)$ be the smallest positive integer such that the grasshopper visits the integer $n$ for the first time after exactly $f(n)$ steps. Fin... | [
"4045"
] | [
"A grasshopper is initially at position 0 on the number line. In the $k$-th step, it jumps $k$ units. If $k$ is even, it jumps to the left; if $k$ is odd, it jumps to the right. Let $f(n)$ be the smallest positive integer such that the grasshopper visits the integer $n$ for the first time after exactly $f(n)$ steps... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-debaa9abb6c6a302ed32a297aca6cca456e3d14e8148770a6fd06c58ad4a72e6 | A grasshopper is jumping clockwise on a circumference with a constant jump length \( L \). The circumference has a pool of width \( W \). If the grasshopper starts at point \( 0 \) and never lands on a point it has already visited, determine the minimum number of jumps \( N \) the grasshopper must make to guarantee lan... | [
"b"
] | [
"A grasshopper is jumping clockwise on a circumference with a constant jump length \\( L \\). The circumference has a pool of width \\( W \\). If the grasshopper starts at point \\( 0 \\) and never lands on a point it has already visited, determine the minimum number of jumps \\( N \\) the grasshopper must make to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-162c3c16ad338e76638e96bf3a927d6ce4a8696a92a871815d64b4002ad18d43 | A grasshopper is sitting at a point \( M \) with non-integer coordinates outside the square defined by \( 0 \le x \le 1 \) and \( 0 \le y \le 1 \). The grasshopper jumps to a new point which is symmetrical to \( M \) with respect to the leftmost vertex of the square from its point of view. Let \( C \) be the center of ... | [
"d + 100"
] | [
"A grasshopper is sitting at a point \\( M \\) with non-integer coordinates outside the square defined by \\( 0 \\le x \\le 1 \\) and \\( 0 \\le y \\le 1 \\). The grasshopper jumps to a new point which is symmetrical to \\( M \\) with respect to the leftmost vertex of the square from its point of view. Let \\( C \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-446bcd5d103909bec3cc67ce2b47a795dcd4b3803db9535d4911a5f4cce7fbe7 | A grasshopper is sitting at an integer point $(x_0, y_0)$ in the Euclidean plane and jumps each second with a constant jump vector $(a, b)$. A hunter can choose one integer point $(x_n, y_n)$ each second to check. Suppose the hunter follows a strategy to check points in the order $(0,0)$, $(1,0)$, $(0,1)$, $(-1,0)$, $(... | [
"25"
] | [
"A grasshopper is sitting at an integer point $(x_0, y_0)$ in the Euclidean plane and jumps each second with a constant jump vector $(a, b)$. A hunter can choose one integer point $(x_n, y_n)$ each second to check. Suppose the hunter follows a strategy to check points in the order $(0,0)$, $(1,0)$, $(0,1)$, $(-1,0)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c78a6c82417c446097ad11a156691aa01d731a6492431b9d5e140b928389523c | A grasshopper is sitting at an integer point $(x_0, y_0)$ in the Euclidean plane and jumps to another integer point each second with a constant jump vector $(a, b)$. A hunter can choose one integer point each second to check if the grasshopper is there. The hunter devises a strategy to check points $(x, y)$ in a specif... | [
"4"
] | [
"A grasshopper is sitting at an integer point $(x_0, y_0)$ in the Euclidean plane and jumps to another integer point each second with a constant jump vector $(a, b)$. A hunter can choose one integer point each second to check if the grasshopper is there. The hunter devises a strategy to check points $(x, y)$ in a s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5775bac101455c9ae80300fe887055a9a2825c4f11735ecf6909ce98f1953f2d | A grasshopper is sitting at the edge of a circle with a radius of 3 inches. He can hop exactly 4 inches in any direction, as long as he stays within the circle. Determine which points inside the circle the grasshopper can reach if he can make as many jumps as he likes. | [
"\\text{All points at least 1 inch from the center within the circle}"
] | [
"A grasshopper is sitting at the edge of a circle with a radius of 3 inches. He can hop exactly 4 inches in any direction, as long as he stays within the circle. Determine which points inside the circle the grasshopper can reach if he can make as many jumps as he likes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-6a225c45a460840d553e9cb2a29467e49884519af8277a1951f0a4c2bfbd6ed6 | A grasshopper is situated at the origin $(0, 0)$ on the Cartesian plane. If the grasshopper is at a point $(m, n)$, it can jump to either $(m + 6, n + 9)$ or $(m + 9, n + 6)$. After a finite sequence of jumps, the grasshopper lands on a point $(m, n)$ such that the sum of the roots of the polynomial $18x^2 + mx + n$ is... | [
"12"
] | [
"A grasshopper is situated at the origin $(0, 0)$ on the Cartesian plane. If the grasshopper is at a point $(m, n)$, it can jump to either $(m + 6, n + 9)$ or $(m + 9, n + 6)$. After a finite sequence of jumps, the grasshopper lands on a point $(m, n)$ such that the sum of the roots of the polynomial $18x^2 + mx + ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2167b6f21586095bc4638ae7050b5f134e747b4cc1f00fa55353b1763957432 | A grasshopper is situated at the origin $(0,0)$ on the Cartesian plane. If the grasshopper is at a point $(m,n)$, it can jump to either $(m + 6, n + 9)$ or $(m + 9, n + 6)$. After a finite sequence of jumps, the grasshopper lands on a point $(m_k, n_k)$ such that the sum of the roots of the polynomial $18x^2 + m_kx + n... | [
"12"
] | [
"A grasshopper is situated at the origin $(0,0)$ on the Cartesian plane. If the grasshopper is at a point $(m,n)$, it can jump to either $(m + 6, n + 9)$ or $(m + 9, n + 6)$. After a finite sequence of jumps, the grasshopper lands on a point $(m_k, n_k)$ such that the sum of the roots of the polynomial $18x^2 + m_k... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-928301942eb3c06830dd540ac3ef5a08eb25d353dabce8cb3c246bc0e401e569 | A grasshopper is situated at the origin $(0,0)$ on the Cartesian plane. It can jump to $(m+6, n+9)$ or $(m+9, n+6)$ from any point $(m,n)$. After a finite sequence of jumps, the grasshopper lands on a point $(m_k, n_k)$ such that the sum of the roots of the polynomial $18x^2 + m_kx + n_k$ is $-2$. How many different pa... | [
"12"
] | [
"A grasshopper is situated at the origin $(0,0)$ on the Cartesian plane. It can jump to $(m+6, n+9)$ or $(m+9, n+6)$ from any point $(m,n)$. After a finite sequence of jumps, the grasshopper lands on a point $(m_k, n_k)$ such that the sum of the roots of the polynomial $18x^2 + m_kx + n_k$ is $-2$. How many differe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b04191431029a0c10239511c39daa362942307ad9218c4fb4f6b58ee7714b05d | A grasshopper is situated at the origin (0, 0) on the Cartesian plane. If the grasshopper is at a given point (m, n), it can jump to either (m + 6, n + 9) or (m + 9, n + 6). After a finite sequence of jumps, the grasshopper lands on some point (mk, nk) such that the sum of the roots of the polynomial \(18x^2 + mkx + nk... | [
"12"
] | [
"A grasshopper is situated at the origin (0, 0) on the Cartesian plane. If the grasshopper is at a given point (m, n), it can jump to either (m + 6, n + 9) or (m + 9, n + 6). After a finite sequence of jumps, the grasshopper lands on some point (mk, nk) such that the sum of the roots of the polynomial \\(18x^2 + mk... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b36150aa7644e47bf84430ad165ee9da46ae426dc3c78dad61b71a7225e905f | A grasshopper is situated at the origin (0, 0) on the Cartesian plane. If the grasshopper is at a point (m, n), it can jump to either (m+6, n+9) or (m+9, n+6). After a finite sequence of jumps, the grasshopper lands on a point (m, n) such that the sum of the roots of the polynomial $18x^2 + mx + n$ is −2. How many diff... | [
"12"
] | [
"A grasshopper is situated at the origin (0, 0) on the Cartesian plane. If the grasshopper is at a point (m, n), it can jump to either (m+6, n+9) or (m+9, n+6). After a finite sequence of jumps, the grasshopper lands on a point (m, n) such that the sum of the roots of the polynomial $18x^2 + mx + n$ is −2. How many... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7fd989adad52e5e35c09817882354a96d41296bbb8c38820610fbbf382bc046d | A grasshopper is situated at the origin of the Cartesian plane. If the grasshopper is at a given point \((m, n)\), it jumps to the point \((m + 6, n + 9)\) or \((m + 9, n + 6)\). After a finite sequence of jumps, the grasshopper lands on some point \((m_k, n_k)\) such that the sum of the roots of the polynomial \(18x^2... | [
"12"
] | [
"A grasshopper is situated at the origin of the Cartesian plane. If the grasshopper is at a given point \\((m, n)\\), it jumps to the point \\((m + 6, n + 9)\\) or \\((m + 9, n + 6)\\). After a finite sequence of jumps, the grasshopper lands on some point \\((m_k, n_k)\\) such that the sum of the roots of the polyn... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f5907941a3e45d36c2cdafd42b76f16dd90a29b236b03b11bc4ad86f8c37c875 | A grasshopper is situated at the origin of the complex plane. It can jump to either \((m+6, n+9)\) or \((m+9, n+6)\) from any point \((m, n)\). After a finite sequence of jumps, the grasshopper lands at some point \((m_k, n_k)\) such that the sum of the roots of the polynomial \(18x^2 + m_k x + n_k\) is \(-2\). How man... | [
"12"
] | [
"A grasshopper is situated at the origin of the complex plane. It can jump to either \\((m+6, n+9)\\) or \\((m+9, n+6)\\) from any point \\((m, n)\\). After a finite sequence of jumps, the grasshopper lands at some point \\((m_k, n_k)\\) such that the sum of the roots of the polynomial \\(18x^2 + m_k x + n_k\\) is ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d98956d25a74ca52aa32f9f3c81bbd6d481cf370c09f3a8f10440aca70de330a | A grasshopper jumps either 364 or 715 units on the real number line. If it starts from the point 0, what is the smallest distance that the grasshopper can be away from the point 2010? | [
"5"
] | [
"A grasshopper jumps either 364 or 715 units on the real number line. If it starts from the point 0, what is the smallest distance that the grasshopper can be away from the point 2010?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-a538a649817880ee61741cb3e9675d25ac751e3ed224af78b06ff2ef87adaef8 | A grasshopper jumps either \( 364 \) or \( 715 \) units on the real number line. If it starts from the point \( 0 \), the distance of the points, which the grasshopper landed on, to the point \( 2010 \) can be at least? | [
"5"
] | [
"A grasshopper jumps either \\( 364 \\) or \\( 715 \\) units on the real number line. If it starts from the point \\( 0 \\), the distance of the points, which the grasshopper landed on, to the point \\( 2010 \\) can be at least?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-b23daaa9a9ab357001002bbcf857c97044dd5027bb454efa221e78c2e38ef006 | A grasshopper jumps on the plane from an integer point to another integer point according to the following rules: His first jump is of length $\sqrt{98}$, his second jump is of length $\sqrt{149}$, his next jump is of length $\sqrt{98}$, and so on, alternatively. What is the least possible odd number of moves in which ... | [
"7"
] | [
"A grasshopper jumps on the plane from an integer point to another integer point according to the following rules: His first jump is of length $\\sqrt{98}$, his second jump is of length $\\sqrt{149}$, his next jump is of length $\\sqrt{98}$, and so on, alternatively. What is the least possible odd number of moves i... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-66d9a17562def7dd9c6ba170c64d7e3e6af8192fe107bf23d75ec39b4b2a06a4 | A grasshopper lies on a line at position $0$. It can jump one unit left ($-1$) or one unit right ($+1$) with equal probability. What is the probability that the grasshopper lands on position $-1$? | [
"1"
] | [
"A grasshopper lies on a line at position $0$. It can jump one unit left ($-1$) or one unit right ($+1$) with equal probability. What is the probability that the grasshopper lands on position $-1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f195ac9d82c1a3313d607524c06c13757936f5ef4030988166e03c895340ab51 | A grasshopper makes three jumps on the plane with lengths of 1 m, 2 m, and 4 m, respectively. Determine the figure formed by all points on the plane from which the grasshopper can reach a given point after making these three jumps. | [
"A circle with radius 7 m."
] | [
"A grasshopper makes three jumps on the plane with lengths of 1 m, 2 m, and 4 m, respectively. Determine the figure formed by all points on the plane from which the grasshopper can reach a given point after making these three jumps."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-c162ef630d0cb8d628499d266dc830066744221ab21623f9b9ab261cef418ddb | A grasshopper moves along a straight line and can make jumps of 11 cm or 17 cm in either direction. After a number of these jumps, she finds herself exactly 9 centimeters from her starting point. What is the minimum number of jumps she made? | [
"11"
] | [
"A grasshopper moves along a straight line and can make jumps of 11 cm or 17 cm in either direction. After a number of these jumps, she finds herself exactly 9 centimeters from her starting point. What is the minimum number of jumps she made?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3c74c0602997ad1ad5025019498b828dbb5be67b33efd05f8735aa63e1beb28 | A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac{1}{2}.$ $(a)$ Find all the positive integral points $(m... | [
"(m, n) \\text{ with } \\gcd(m, n) = 1"
] | [
"A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\\triangle AOB$ is equal to $\\frac{1}{2}.$\n$(a)$ Find all the positive integral po... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-e9a1d914f120c1170a76e19500709ddf4f25724efb182f0c481da782e43565a7 | A grasshopper rests on the point $(1,1)$ on the plane. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac{1}{2}.$ Find all the positive integral points $(m,n)$ which can be covered by the grasshopper aft... | [
"\\(\\gcd(m,n) = 1\\)"
] | [
"A grasshopper rests on the point $(1,1)$ on the plane. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\\triangle AOB$ is equal to $\\frac{1}{2}.$ Find all the positive integral points $(m,n)$ which can be covered by the grasshop... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-6eeb25c7379dc1317589add579ae5f6ae50c8dd161826d3eb1a16c72c8917cdd | A grasshopper sits in the center of a $100 \times 100$ square. Every second, it jumps by one unit in one of the four directions (North, South, East, West) randomly. What is the expected number of seconds for the grasshopper to leave the square? (Being on the edge does not count as "leaving"). | [
"2500"
] | [
"A grasshopper sits in the center of a $100 \\times 100$ square. Every second, it jumps by one unit in one of the four directions (North, South, East, West) randomly. What is the expected number of seconds for the grasshopper to leave the square? (Being on the edge does not count as \"leaving\")."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-bec0e4b96bbe36b05a1f0487d5f80eb1caf17f469f4f2086d4575256d5b6c407 | A grasshopper sits on a circle of length $1$. Every second, it jumps an irrational distance counterclockwise and marks its landing point with a number corresponding to the jump number. After $n$ jumps, it stops. Let $a$ and $b$ be the jump numbers of the marked points closest to the final position from both sides. Calc... | [
"197"
] | [
"A grasshopper sits on a circle of length $1$. Every second, it jumps an irrational distance counterclockwise and marks its landing point with a number corresponding to the jump number. After $n$ jumps, it stops. Let $a$ and $b$ be the jump numbers of the marked points closest to the final position from both sides.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-5621b3f4208d47f35973444cc16525737260a62cdc665e77ee1dc4f0de5753dd | A grasshopper starts at (0,0). Each minute it jumps one unit in a random direction in the plane. Find the expected number of minutes before the grasshopper is at least 100 units away from the starting point. | [
"10000"
] | [
"A grasshopper starts at (0,0). Each minute it jumps one unit in a random direction in the plane. Find the expected number of minutes before the grasshopper is at least 100 units away from the starting point."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-06bec547433432371d5276b47bf3fecc4134d109af19cea264b9b7018c9d2c1c | A grasshopper starts at \( x = 0 \) on an integer number line. During each jump, the grasshopper can jump either to the right or the left, and the length of its \( n \)-th jump is exactly \( n^2 \) units long. Let \( S \) be the set of all integers that the grasshopper can reach after exactly 5 jumps. Find the sum of a... | [
"0"
] | [
"A grasshopper starts at \\( x = 0 \\) on an integer number line. During each jump, the grasshopper can jump either to the right or the left, and the length of its \\( n \\)-th jump is exactly \\( n^2 \\) units long. Let \\( S \\) be the set of all integers that the grasshopper can reach after exactly 5 jumps. Find... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-222280bb27faa357f9b0171b7cb393230a0eb81e4929faa4ae38d25959a17792 | A grasshopper starts at a point on a circle of radius $1$ and jumps to points on a straight line not passing through the circle's center, and then back to the circle, with each jump having a length of $1$. Let $N$ be the maximum number of distinct points the grasshopper can visit on the circle before it starts repeatin... | [
"3"
] | [
"A grasshopper starts at a point on a circle of radius $1$ and jumps to points on a straight line not passing through the circle's center, and then back to the circle, with each jump having a length of $1$. Let $N$ be the maximum number of distinct points the grasshopper can visit on the circle before it starts rep... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0c7b676d721eafb340ee2db3856f55425e8f602a2d0cda97846345a2419aa26 | A grasshopper starts at point \( O \) and makes \( n \) jumps to the right with lengths \( a_1, a_2, \ldots, a_n \) respectively, where \( a_1, a_2, \ldots, a_k, \ldots, a_n \) is a sequence of distinct positive real numbers such that \( a_1 < a_2 < \cdots < a_k \) and \( a_k > a_{k+1} > \cdots > a_n \). After reaching... | [
"32"
] | [
"A grasshopper starts at point \\( O \\) and makes \\( n \\) jumps to the right with lengths \\( a_1, a_2, \\ldots, a_n \\) respectively, where \\( a_1, a_2, \\ldots, a_k, \\ldots, a_n \\) is a sequence of distinct positive real numbers such that \\( a_1 < a_2 < \\cdots < a_k \\) and \\( a_k > a_{k+1} > \\cdots > a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-07340f2f1a313e1feba469db991b7e78e389ac978fb36955829ca00d64cac23d | A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$? | [
"578"
] | [
"A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-684dede7deeae2452ef3570d5e2e3ef1d250ffab7a71e8723a7b8eb9e0ead4e2 | A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers. What is the smallest number of hops needed for the grasshopper to reach the point (2021, 2021)? | [
"578"
] | [
"A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers. What is the smallest number of hops needed for the grasshopper to reach the point (2021, 2021)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-fb307656d98cf0307aef372fbf7d399b174405ef32cfd93c02028bbc974e6894 | A grasshopper starts at the point $(0, 0)$ on a grid and can jump to one of the following points: $(a + 1, b)$, $(a + 2, b)$, $(a + 1, b + 1)$, $(a, b + 2)$, or $(a, b + 1)$. In how many ways can the grasshopper reach the line $x + y = 2014$? | [
"\\dfrac{3^{2015} + 1}{4}"
] | [
"A grasshopper starts at the point $(0, 0)$ on a grid and can jump to one of the following points: $(a + 1, b)$, $(a + 2, b)$, $(a + 1, b + 1)$, $(a, b + 2)$, or $(a, b + 1)$. In how many ways can the grasshopper reach the line $x + y = 2014$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dbf37fb34ff3578b904c345cc6d8e3ffd84ee899389c89d5a77c589c3c2be8dc | A grasshopper wants to climb a staircase with many steps, but can only take two types of jumps: three steps up or four steps down. Starting at the ground, what is the least number of jumps the grasshopper has to take to rest on the $22 \text{nd}$ step? | [
"12"
] | [
"A grasshopper wants to climb a staircase with many steps, but can only take two types of jumps: three steps up or four steps down. Starting at the ground, what is the least number of jumps the grasshopper has to take to rest on the $22 \\text{nd}$ step?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-c41749bb0c0f6f6fa446d92d6af5192986ecf89ceecd2e83c4cdb6c7ba59e352 | A gray unit square is divided in half by a diagonal, and the incircles of the resulting right triangles are drawn in white. Then, the incircle of the square is drawn on top in gray, forming two congruent white crescent moon shapes. Find the area of one of these crescent moons. | [
"no_valid_answer_found"
] | [
"A gray unit square is divided in half by a diagonal, and the incircles of the resulting right triangles are drawn in white. Then, the incircle of the square is drawn on top in gray, forming two congruent white crescent moon shapes. Find the area of one of these crescent moons."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2be951b668b91fe302ebb390b334bf25dc7d80a8cc5748137951167f8e253119 | A greyhound and a hare are on a real number line. The hare starts at position 0 and the greyhound starts at position $d$. The hare can run at a maximum speed of $v_h$ and the greyhound can run at a maximum speed of $v_g$, with $v_h < v_g$. The hare can change direction at any time, but the greyhound must always move to... | [
"\\dfrac{d}{v_g - v_h}"
] | [
"A greyhound and a hare are on a real number line. The hare starts at position 0 and the greyhound starts at position $d$. The hare can run at a maximum speed of $v_h$ and the greyhound can run at a maximum speed of $v_g$, with $v_h < v_g$. The hare can change direction at any time, but the greyhound must always mo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cce8b394df6e84fba519b53f980ee3a30994ef83a32ab7215a250fbcc0b893ec | A grid consists of $2n$ vertical and $2n$ horizontal lines, each group disposed at equal distances. The lines are all painted in red and black, such that exactly $n$ vertical and $n$ horizontal lines are red. Find the smallest $n$ such that for any painting satisfying the above condition, there is a square formed by th... | [
"3"
] | [
"A grid consists of $2n$ vertical and $2n$ horizontal lines, each group disposed at equal distances. The lines are all painted in red and black, such that exactly $n$ vertical and $n$ horizontal lines are red. Find the smallest $n$ such that for any painting satisfying the above condition, there is a square formed ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbc443935e693633455e3cd999d8ef92f6f6a1a13885790d50d39b2028e89da5 | A grid has 12 lines in one direction and 9 lines in the other direction, with parallel lines 11 inches apart. What is the minimum number of 12-inch-by-12-inch floor tiles needed to cover all of the line intersections on the grid? The tiles do not have to touch each other and must remain intact. | [
"30"
] | [
"A grid has 12 lines in one direction and 9 lines in the other direction, with parallel lines 11 inches apart. What is the minimum number of 12-inch-by-12-inch floor tiles needed to cover all of the line intersections on the grid? The tiles do not have to touch each other and must remain intact."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-472e37ab0f9f379620f2516a04fea702aeac69f25250158bf63bbe09ec3051eb | A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain? | [
"10"
] | [
"A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22f892c04eb99f4f3a1fea5fd7f5abe939e5b4fc72e6fa9c8a0e818cd5139382 | A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack? | [
"100"
] | [
"A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-2a2f0e691efbf6e0e32199bcf8f6c7bc7b0ed337d6e604695f96597ef1617649 | A group consists of $2005$ members. Each member holds exactly one secret. Each member can send a letter to any other member to convey all the secrets he holds. The number of letters that need to be sent for all the group members to know the whole secret is ... | [
"4008"
] | [
"A group consists of $2005$ members. Each member holds exactly one secret. Each member can send a letter to any other member to convey all the secrets he holds. The number of letters that need to be sent for all the group members to know the whole secret is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4c9b905c8760a545b8542670ab6192efbff9e2d4a7117cdac72869ff977e2f5 | A group hug of \( n \) people is a circular formation where each person puts their arms around the shoulders of the two people adjacent to them, facing inward. If there are 314 people in a room, in how many ways can they group up into some number of group hugs? Group hugs of 2 people (i.e., a normal hug) or even 1 pers... | [
"314!"
] | [
"A group hug of \\( n \\) people is a circular formation where each person puts their arms around the shoulders of the two people adjacent to them, facing inward. If there are 314 people in a room, in how many ways can they group up into some number of group hugs? Group hugs of 2 people (i.e., a normal hug) or even... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-174c6ebc129f101290a9da1f5a8681af20d58b8167bb88be3797c95655665e6a | A group is generated by elements \( x \) and \( y \) such that \( x^2 = y^3 \) and \( x^4 = y^6 = e \). Find the number of elements in this group. | [
"12"
] | [
"A group is generated by elements \\( x \\) and \\( y \\) such that \\( x^2 = y^3 \\) and \\( x^4 = y^6 = e \\). Find the number of elements in this group."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-c56bb3156513ce11decb363aadb0d1b3757f86942f70ae06941e123cf2169c02 | A group is generated by elements \( x \) and \( y \) such that \( x^{-1}y^2x = y^3 \) and \( y^{-1}x^2y = x^3 \). Find the number of elements in this group. | [
"1"
] | [
"A group is generated by elements \\( x \\) and \\( y \\) such that \\( x^{-1}y^2x = y^3 \\) and \\( y^{-1}x^2y = x^3 \\). Find the number of elements in this group."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0de5c09da5b00f3e21386257b26c02953fe526304f553ebf22e0a55f5feed520 | A group of $200$ high school students visited the University of New Mexico for Lobo Day. The students could participate in at most two of the following three workshops: 1) Algorithms, 2) Bioinformatics, and 3) Coding. Suppose $100$ students participated in Algorithms, $90$ participated in Bioinformatics, and $80$ parti... | [
"14"
] | [
"A group of $200$ high school students visited the University of New Mexico for Lobo Day. The students could participate in at most two of the following three workshops: 1) Algorithms, 2) Bioinformatics, and 3) Coding. Suppose $100$ students participated in Algorithms, $90$ participated in Bioinformatics, and $80$ ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-323b30e17f92c63cf3ac272a712007cad44dda34b7c5d5272f30acc135167b80 | A group of $2022$ people stand in a circle, numbered from $1$ through $2022$ counterclockwise, each starting with $1$ life. They count positive integers in ascending order clockwise. If someone counts a number divisible by $5$ or containing $5$, they gain a life. If someone counts a number divisible by $7$ or containin... | [
"0"
] | [
"A group of $2022$ people stand in a circle, numbered from $1$ through $2022$ counterclockwise, each starting with $1$ life. They count positive integers in ascending order clockwise. If someone counts a number divisible by $5$ or containing $5$, they gain a life. If someone counts a number divisible by $7$ or cont... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d9134338405e6897ac6db979c21ebce6a99bdaf61de7c9fec21b45f37dbdfa9b | A group of $4n$ persons are participating in a carrom tournament lasting $k$ days. There are $n$ boards in total, and $4$ players play at each board. Each day, every person plays a game, and no two persons play together more than once. Determine the maximum value of $k$ as a function of $n$. | [
"\\left\\lfloor \\dfrac{4n - 1}{3} \\right\\rfloor"
] | [
"A group of $4n$ persons are participating in a carrom tournament lasting $k$ days. There are $n$ boards in total, and $4$ players play at each board. Each day, every person plays a game, and no two persons play together more than once. Determine the maximum value of $k$ as a function of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c49f9e432fc7246c812a657f27c9cde4e91624c564845ad690fdd93a0cc9837e | A group of $N$ people play a game $N$ times. The loser of each game must give to each of the non-losers the amount of money that each non-loser currently has. Initially, each player has a different number of dollars, and at the end of $N$ games, each player has $64$ dollars. Sally lost only the third and fourth games, ... | [
"73"
] | [
"A group of $N$ people play a game $N$ times. The loser of each game must give to each of the non-losers the amount of money that each non-loser currently has. Initially, each player has a different number of dollars, and at the end of $N$ games, each player has $64$ dollars. Sally lost only the third and fourth ga... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-f36b901303390ad4bb4e04ab9da6c872d4d4838a515a13dd0c0b50a21a325ad6 | A group of $n$ children form circles by holding hands, with a child in the center of each circle. Let $h(n)$ be the number of ways this can be done. Set $h(0)=1$. Find a simple expression for the generating function $F(x) = \sum_{n \geq 0} h(n)x^n/n!$. A circle may consist of just one child holding his or her own hands... | [
"\\( (1-x)^{-x} \\)"
] | [
"A group of $n$ children form circles by holding hands, with a child in the center of each circle. Let $h(n)$ be the number of ways this can be done. Set $h(0)=1$. Find a simple expression for the generating function $F(x) = \\sum_{n \\geq 0} h(n)x^n/n!$. A circle may consist of just one child holding his or her ow... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 8,
"tir": 4
} | 46 | 0 | 0 |
omr-ws-83507c4e2a15be0f208d631489cfd1e80529fb2c3de347d53032aa4f359e3e58 | A group of $n$ friends play "Secret Santa" by randomly assigning each person the name of another member of the group to buy a gift for. The deal is considered a "success" if nobody is assigned their own name. For $n = 10$, calculate the exact probability of a successful deal and express it as a fraction in lowest terms... | [
"61281"
] | [
"A group of $n$ friends play \"Secret Santa\" by randomly assigning each person the name of another member of the group to buy a gift for. The deal is considered a \"success\" if nobody is assigned their own name. For $n = 10$, calculate the exact probability of a successful deal and express it as a fraction in low... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04f492d0d19dd0703a743ab2c033be9acc98867c11bad1f53e2ee1a17097dd3f | A group of $n$ people is divided over two buses such that there are as many seats in total as people. The probability that two randomly chosen friends are seated on the same bus is $\frac{1}{2}$. Find the sum of all possible values of $n$ that satisfy this condition. | [
"13"
] | [
"A group of $n$ people is divided over two buses such that there are as many seats in total as people. The probability that two randomly chosen friends are seated on the same bus is $\\frac{1}{2}$. Find the sum of all possible values of $n$ that satisfy this condition."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bcd2aaa72425506b37dce3d1fcb42dc1220be9bc9872e35d1df5426917d690fc | A group of $n$ sages are arranged in a row, each wearing a black or white hat. Each sage can see the hats of those in front of them but not their own. They take turns, starting from the last sage, to call out a color (white or black) and a natural number. Exactly $k$ of the sages are insane and will call out a color an... | [
"n - k - 1"
] | [
"A group of $n$ sages are arranged in a row, each wearing a black or white hat. Each sage can see the hats of those in front of them but not their own. They take turns, starting from the last sage, to call out a color (white or black) and a natural number. Exactly $k$ of the sages are insane and will call out a col... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 6,
"tir": 1
} | 30 | 0 | 0 |
omr-ws-42bcae5935c1f7986ed13bbf4e2f5b67681c11eae242b816f6820b5b3d0692b0 | A group of $n$ students went on a summer trip. The following facts were discovered: 1. If student $A$ was friends with student $B$, then they did not have any mutual friends among the other students in the group. 2. If student $A$ was not friends with student $B$, then they had exactly 2 mutual friends among the other ... | [
"7, 11"
] | [
"A group of $n$ students went on a summer trip. The following facts were discovered:\n1. If student $A$ was friends with student $B$, then they did not have any mutual friends among the other students in the group.\n2. If student $A$ was not friends with student $B$, then they had exactly 2 mutual friends among the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6a2ac09eae30559b04018990f3e413f2f68775a760a596cf8d628afac755a9c | A group of 10 friends attend an amusement park. Each has visited three different attractions. Leaving the park and talking to each other, they found that any pair of friends visited at least one attraction in common. Determine the minimum number of friends who could have visited the most visited attraction. | [
"4"
] | [
"A group of 10 friends attend an amusement park. Each has visited three different attractions. Leaving the park and talking to each other, they found that any pair of friends visited at least one attraction in common. Determine the minimum number of friends who could have visited the most visited attraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6abb704b77e977ce8eb8c87f531f8ef76ef78874f574413a715c1e33730ec576 | A group of 10 people went to a bookstore. Each person bought exactly 3 books, and for every two persons, there is at least one book that both of them bought. What is the least number of people that could have bought the book purchased by the greatest number of people? | [
"5"
] | [
"A group of 10 people went to a bookstore. Each person bought exactly 3 books, and for every two persons, there is at least one book that both of them bought. What is the least number of people that could have bought the book purchased by the greatest number of people?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-ca78e9369d09b22ddd3ce88a27261fd23c114cfeb54dce6a4d5331408f685521 | A group of 100 contestants numbered 1 through 100 play the following game: The judge writes the numbers 1 through 100 on 100 cards, turns them over, and places them on a table in an arbitrary order. For any \( j \), contestant number \( j \) enters the room and flips 50 cards. The contestant gains one point if the numb... | [
"Yes"
] | [
"A group of 100 contestants numbered 1 through 100 play the following game: The judge writes the numbers 1 through 100 on 100 cards, turns them over, and places them on a table in an arbitrary order. For any \\( j \\), contestant number \\( j \\) enters the room and flips 50 cards. The contestant gains one point if... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-36776e504ff77258d407c34b06436e26fa930f3245985385e85da9a2b80607d7 | A group of 100 kids each takes a deck of 101 cards numbered from 0 to 100. Each kid shuffles the deck, takes the cards one by one, and writes down the average of the numbers on the cards taken up to that moment (excluding the last card). Each kid writes down 100 numbers. Calculate the minimum number of distinct values ... | [
"101"
] | [
"A group of 100 kids each takes a deck of 101 cards numbered from 0 to 100. Each kid shuffles the deck, takes the cards one by one, and writes down the average of the numbers on the cards taken up to that moment (excluding the last card). Each kid writes down 100 numbers. Calculate the minimum number of distinct va... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-cbb76d27e903f23c2a39af39ef55dd5fb7a573900f5517682de8090b37277f1b | A group of 100 pirates has found a box with 100 gold coins. The pirates are ordered by strength from the strongest to the weakest. The strongest pirate suggests a way to divide the treasure, and the proposal is accepted if at least half of the pirates vote for it. If the proposal is not accepted, the proposing pirate i... | [
"1"
] | [
"A group of 100 pirates has found a box with 100 gold coins. The pirates are ordered by strength from the strongest to the weakest. The strongest pirate suggests a way to divide the treasure, and the proposal is accepted if at least half of the pirates vote for it. If the proposal is not accepted, the proposing pir... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-62f15a913d41d10b6be5677944615e4d84407a1134eb3207ba422ee0a4c4594f | A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and for every pair of students \(A\) and \(B\), student \(A\) speaks some language that student \(B\) does not speak, and student \(B\) speaks some language that student \(A\) does not s... | [
"9"
] | [
"A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and for every pair of students \\(A\\) and \\(B\\), student \\(A\\) speaks some language that student \\(B\\) does not speak, and student \\(B\\) speaks some language that student \... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 13
} | 61 | 0 | 0 |
omr-ws-e99c72c2aaea2e2c530d71ba174e8908f657601b27bb68a1c62a4932ec59234b | A group of 100 students numbered 1 through 100 are playing a game. The judge writes the numbers 1 through 100 on 100 cards, places them on the table in an arbitrary order, and turns them over. Each student flips 50 of the cards. If among the cards flipped by student \( j \) there is card \( j \), he gains one point. Th... | [
"Yes"
] | [
"A group of 100 students numbered 1 through 100 are playing a game. The judge writes the numbers 1 through 100 on 100 cards, places them on the table in an arbitrary order, and turns them over. Each student flips 50 of the cards. If among the cards flipped by student \\( j \\) there is card \\( j \\), he gains one ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0787d60ebc046c73496b4253d800eb551c95e5c21382980d64a323a1a4268a8 | A group of 100 students took examinations in English, Science, and Mathematics. Among them, 92 passed in English, 75 in Science, and 63 in Mathematics. At most 65 passed in both English and Science, at most 54 in both English and Mathematics, and at most 48 in both Science and Mathematics. Find the largest possible num... | [
"37"
] | [
"A group of 100 students took examinations in English, Science, and Mathematics. Among them, 92 passed in English, 75 in Science, and 63 in Mathematics. At most 65 passed in both English and Science, at most 54 in both English and Mathematics, and at most 48 in both Science and Mathematics. Find the largest possibl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-bbdc3183e04183867704e93a8b492ea961818af9e217847b551f1c3476ec1a04 | A group of 11 scientists are working on a secret project, the materials of which are kept in a safe. In order to permit the opening of the safe only when a majority of the scientists is present, each scientist is given the keys to a certain number of locks. Determine the number of locks required and the number of keys ... | [
"462 locks, 252 keys per scientist."
] | [
"A group of 11 scientists are working on a secret project, the materials of which are kept in a safe. In order to permit the opening of the safe only when a majority of the scientists is present, each scientist is given the keys to a certain number of locks. Determine the number of locks required and the number of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 30
} | 35 | 0 | 0 |
omr-ws-64d75da95498a8a740ada2d13f09c4219c4f40098b678360ca9ce5accc306155 | A group of 11 scientists want to ensure that a safe can only be opened when a majority of the group is present. The safe is equipped with several locks, and each scientist is given keys to certain locks. Determine the minimum number of locks required and the number of keys each scientist must have to satisfy this condi... | [
"462 locks, 210 keys per scientist."
] | [
"A group of 11 scientists want to ensure that a safe can only be opened when a majority of the group is present. The safe is equipped with several locks, and each scientist is given keys to certain locks. Determine the minimum number of locks required and the number of keys each scientist must have to satisfy this ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-477a90602942e454e7a43029e944717fc1bdd736de2833061df5fa249b69b025 | A group of 12 friends goes to a cinema complex that is showing 6 different movies. If the group splits up into subgroups based on movie preferences, how many subgroup combinations are possible? | [
"\\(\\binom{17}{5}\\)"
] | [
"A group of 12 friends goes to a cinema complex that is showing 6 different movies. If the group splits up into subgroups based on movie preferences, how many subgroup combinations are possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-656aabcd52a507840b9e3b7507fd227bba97892b51622e9e1778cd8b03a64331 | A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^\text{th}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to re... | [
"1925"
] | [
"A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^\\text{th}$ pirate to take a share takes $\\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirat... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-8af69cbe26f5877ae92572f09250977f4ff00656227e60d2411090b3583e64cf | A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows: the $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to ... | [
"1925"
] | [
"A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows: the $k^{\\text{th}}$ pirate to take a share takes $\\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pir... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-a82cc032f509974bc59618fb9e7a20335217920394b8315bbaf9ea286d2e15e7 | A group of 12 pirates agree to divide a treasure chest of gold coins among themselves such that the kth pirate to take a share takes k/12 of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole... | [
"1925"
] | [
"A group of 12 pirates agree to divide a treasure chest of gold coins among themselves such that the kth pirate to take a share takes k/12 of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-5f2687b30b1b48044e84d55a20e6c6aa088e8a31f4fc7e3c2981b0c158b032df | A group of 12 pirates of different ages share 2022 coins such that each pirate (except the youngest) has one more coin than the next youngest. Each day, a pirate with at least eleven coins gives one coin to each of the other pirates. Find the largest number of coins that a pirate can have. | [
"174"
] | [
"A group of 12 pirates of different ages share 2022 coins such that each pirate (except the youngest) has one more coin than the next youngest. Each day, a pirate with at least eleven coins gives one coin to each of the other pirates. Find the largest number of coins that a pirate can have."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0841227779ba2685d0cba49786dbc115845906d10b57ca77b02acac78525858 | A group of 15 executives are to share 5 secretaries. Each executive is assigned exactly 1 secretary and no secretary is assigned to more than 4 executives. Let $x$ be the number of secretaries assigned to 3 or more executives. Find the minimum possible value of $x$. | [
"3"
] | [
"A group of 15 executives are to share 5 secretaries. Each executive is assigned exactly 1 secretary and no secretary is assigned to more than 4 executives. Let $x$ be the number of secretaries assigned to 3 or more executives. Find the minimum possible value of $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6be8ddf8632ab133cccd005b8bde5d40dc9598564ee0c49594299e9e5cb6bcd | A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$ | [
"5"
] | [
"A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$"... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 6
} | 54 | 0 | 0 |
omr-ws-34d1e73abf97aa4d7ae3e3403db48603d851b5c2ebd58e9d74b1f6f0514e23b8 | A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by 3. What is $r$? | [
"5"
] | [
"A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by 3. What is $r$?"
... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 1
} | 56 | 0 | 0 |
omr-ws-6d5f00057a24cc3e085fe4216cb40186c558beac9996e8bac200eadb3a6385d8 | A group of 17 persons, labeled $A, B, \dots, Q$, are suspects in a case. Each person gave four names of people with whom they were not in the same group. The groups consist of one person who committed the crime alone and one group of three people. Given the following information, determine who committed the crime and h... | [
"B"
] | [
"A group of 17 persons, labeled $A, B, \\dots, Q$, are suspects in a case. Each person gave four names of people with whom they were not in the same group. The groups consist of one person who committed the crime alone and one group of three people. Given the following information, determine who committed the crime... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-5965fa4086fa7f8376082e2951efe3ddfd4b6fdfdc31ffe4f9f9150ed02d407a | A group of 2004 pirates has found a box with 100 gold coins. The pirates are ordered by strength from the strongest to the weakest. The strongest pirate suggests a way to divide the treasure, and the proposal is accepted if at least half of the pirates vote for it. If the proposal is not accepted, the proposing pirate ... | [
"1803"
] | [
"A group of 2004 pirates has found a box with 100 gold coins. The pirates are ordered by strength from the strongest to the weakest. The strongest pirate suggests a way to divide the treasure, and the proposal is accepted if at least half of the pirates vote for it. If the proposal is not accepted, the proposing pi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-a4f6cab3ffa60583c9294f73a0865764d719c5587e7760ef52b78598c672360e | A group of 2017 students sit in a circle and consecutively count numbers 1, 2, and 3, repeating the sequence. Starting from some student A with number 1 and counting clockwise, students who count numbers 2 and 3 must leave the circle until only one student remains. Determine which student is the last one remaining in t... | [
"1"
] | [
"A group of 2017 students sit in a circle and consecutively count numbers 1, 2, and 3, repeating the sequence. Starting from some student A with number 1 and counting clockwise, students who count numbers 2 and 3 must leave the circle until only one student remains. Determine which student is the last one remaining... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5dce87415343eff0147dd87092325f6eb607cfba2f1adf98d48155059cae8713 | A group of 25 friends were discussing a large positive integer. ``It can be divided by 1,'' said the first friend. ``It can be divided by 2,'' said the second friend. ``And by 3,'' said the third friend. ``And by 4,'' added the fourth friend. This continued until everyone had made such a comment. If exactly two friends... | [
"787386600"
] | [
"A group of 25 friends were discussing a large positive integer. ``It can be divided by 1,'' said the first friend. ``It can be divided by 2,'' said the second friend. ``And by 3,'' said the third friend. ``And by 4,'' added the fourth friend. This continued until everyone had made such a comment. If exactly two fr... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3927f256dea22d340833ea4b911aaa87a7a6f053004d6931c8866c07efd5b68a | A group of 25 people stand in a ring on a field. All the people have different heights. Each person is asked whether they are taller than their neighbors, shorter than their neighbors, or in-between. 5 people said "taller". How many said "in-between"? | [
"15"
] | [
"A group of 25 people stand in a ring on a field. All the people have different heights. Each person is asked whether they are taller than their neighbors, shorter than their neighbors, or in-between. 5 people said \"taller\". How many said \"in-between\"?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1fab08f17feeda7930fdb0635d824b2e7a2051317ce09242cd3a457dffc1a579 | A group of 25 students wants to travel 105 miles to Disneyland. They have a bus that can carry up to 10 people and travels at 60 miles per hour, and the students can run at 9 miles per hour. Each student can board the bus only once. What is the least number of minutes it will take for all students to reach Disneyland? ... | [
"375"
] | [
"A group of 25 students wants to travel 105 miles to Disneyland. They have a bus that can carry up to 10 people and travels at 60 miles per hour, and the students can run at 9 miles per hour. Each student can board the bus only once. What is the least number of minutes it will take for all students to reach Disneyl... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-23428f95c7c4cf948763a87012710cd41fd5d2cd1eca2831641b7d71300405ad | A group of 3 girls and 3 boys are going on a trip and will travel to the airport in 2 taxis. One girl refuses to travel with a certain boy because he was rude. In how many ways can the groups be allocated to the taxis? | [
"24"
] | [
"A group of 3 girls and 3 boys are going on a trip and will travel to the airport in 2 taxis. One girl refuses to travel with a certain boy because he was rude. In how many ways can the groups be allocated to the taxis?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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