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-ac740cfcf0e97c57a74074f16023370eae1abb7d5b915514d529ff31b7ef8992 | A container is shaped like a right circular cone with base diameter $18$ and height $12$. Four spheres, each with radius $3$, are placed inside the container such that the first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are each tangent to the cone and ta... | [
"704"
] | [
"A container is shaped like a right circular cone with base diameter $18$ and height $12$. Four spheres, each with radius $3$, are placed inside the container such that the first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are each tangent to the cone a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-8473d0424f26cc4ea61ec15cbe275718c66e375a3cc293d10f23f2ec27bba1e8 | A container is shaped like a square-based pyramid with a base side length of 23 centimeters and a height of 120 centimeters. The container is open at the base and stands with its vertex pointing down. After 5 centimeters of rain falls, find the depth in centimeters of the rainwater in the bottom of the container. | [
"60"
] | [
"A container is shaped like a square-based pyramid with a base side length of 23 centimeters and a height of 120 centimeters. The container is open at the base and stands with its vertex pointing down. After 5 centimeters of rain falls, find the depth in centimeters of the rainwater in the bottom of the container."... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-4158415e6dd829f896e1840e59147aad9ab3cc7b26681def67ebcc9c3fc7d2bf | A container originally full of mango juice has 8 litres of juice drawn out and replaced with water. This operation is repeated 3 more times. After these operations, the ratio of the quantity of mango juice left in the container to the total quantity is 16:81. How much mango juice did the container originally hold? | [
"24 litres"
] | [
"A container originally full of mango juice has 8 litres of juice drawn out and replaced with water. This operation is repeated 3 more times. After these operations, the ratio of the quantity of mango juice left in the container to the total quantity is 16:81. How much mango juice did the container originally hold?... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-cd69c12c53701bde4d1177e9cd3305a8a14b139d1c299fc72caa3b1469955e29 | A container with $3$ L of pure water repeatedly undergoes the following process: add $10$ grams of salt, mix thoroughly, pour out $1$ L, and add back another liter of pure water. After infinitely many repetitions, how many grams of salt are in the container if all of the water were to evaporate? | [
"20"
] | [
"A container with $3$ L of pure water repeatedly undergoes the following process: add $10$ grams of salt, mix thoroughly, pour out $1$ L, and add back another liter of pure water. After infinitely many repetitions, how many grams of salt are in the container if all of the water were to evaporate?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95dea135c2abb82adbb74dc0cadb6adedd5b8cc5e87900101de4094accee1513 | A container with height $14$ inches, length $20$ inches, and width $21$ inches is filled halfway with water. A cubical box with side length $10$ inches is placed in the water and floats such that its top appears $2$ inches above the top of the container. The water now reaches $\frac{p}{q}$ inches up the sides, where $p... | [
"1186"
] | [
"A container with height $14$ inches, length $20$ inches, and width $21$ inches is filled halfway with water. A cubical box with side length $10$ inches is placed in the water and floats such that its top appears $2$ inches above the top of the container. The water now reaches $\\frac{p}{q}$ inches up the sides, wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 8
} | 31 | 0 | 0 |
omr-ws-f45729f3ca32f3799b91b60d21f572ee8863288391f626dcae3699e05bb8981d | A continuous function \( f : \mathbb{R} \to \mathbb{R} \) satisfies the conditions \( f(1000) = 999 \) and \( f(x)f(f(x)) = 1 \) for all real \( x \). Determine \( f(500) \). | [
"\\(\\frac{1}{500}\\)"
] | [
"A continuous function \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfies the conditions \\( f(1000) = 999 \\) and \\( f(x)f(f(x)) = 1 \\) for all real \\( x \\). Determine \\( f(500) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-b1ddfd7f3593dafb4bf9225d3d69873a77c341bbe2d0c3cce6e18760c6ecec67 | A continuous function \( f(x) \) defined for \( x > 0 \) satisfies \[ 2f(x) + \frac{1}{x^2} f \left( \frac{1}{x} \right) = \frac{1}{x} + \frac{1}{x^2} \] for all positive \( x \). What is the value of \( \int_{\frac{1}{2}}^2 f(x) \, dx \)? | [
"\\(\\frac{1}{2} + \\frac{\\ln 4}{3}\\)"
] | [
"A continuous function \\( f(x) \\) defined for \\( x > 0 \\) satisfies\n\\[ 2f(x) + \\frac{1}{x^2} f \\left( \\frac{1}{x} \\right) = \\frac{1}{x} + \\frac{1}{x^2} \\]\nfor all positive \\( x \\). What is the value of \\( \\int_{\\frac{1}{2}}^2 f(x) \\, dx \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-ae2c39483caad89d7ede5a63a84b020d6b72588e5928bb3bb04ea65b67a11f52 | A continuous function \( f(x) \) on the domain \([0,1]\) satisfies \( f(x^2) = f(x)^2 \) and \( f(0) \neq 0 \). Determine the value of \( f\left(\frac{1}{4}\right) \). | [
"1"
] | [
"A continuous function \\( f(x) \\) on the domain \\([0,1]\\) satisfies \\( f(x^2) = f(x)^2 \\) and \\( f(0) \\neq 0 \\). Determine the value of \\( f\\left(\\frac{1}{4}\\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a49d0e34b03ddffffcbd416456a79d384133cc16ed4b3e89ece354bcd5bbccab | A continuous function \( f(x) \) satisfies \( f(0) = f(1) = 0 \) and \( |f'(x)| \le k \) for all \( x \). Calculate the maximum value of \( \int_0^1 f(x) \, dx \) in terms of \( k \). | [
"\\dfrac{k}{4}"
] | [
"A continuous function \\( f(x) \\) satisfies \\( f(0) = f(1) = 0 \\) and \\( |f'(x)| \\le k \\) for all \\( x \\). Calculate the maximum value of \\( \\int_0^1 f(x) \\, dx \\) in terms of \\( k \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7137a053dffad1f9fe2475d370b5ea4687dce35ca455061cf720faaa4276d67 | A continuous function \( f: \mathbb{R} \to \mathbb{R} \) satisfies \( f(x) f(f(x)) = 1 \) for every real \( x \) and \( f(2020) = 2019 \). What is the value of \( f(2018) \)? | [
"\\dfrac{1}{2018}"
] | [
"A continuous function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfies \\( f(x) f(f(x)) = 1 \\) for every real \\( x \\) and \\( f(2020) = 2019 \\). What is the value of \\( f(2018) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-5b9382ef1a18b3d4f6b913e921c44c89b89b70731a8199ee92262faf21032d9f | A contractor deployed some men to plant 1800 trees in a certain number of days. But in 1/3rd of the planned time, 120 fewer trees could be planted. So to fulfill the target for the rest of the days, 20 more trees were planted each day. This saved one day out of the initially planned number of days. How many trees did h... | [
"100"
] | [
"A contractor deployed some men to plant 1800 trees in a certain number of days. But in 1/3rd of the planned time, 120 fewer trees could be planted. So to fulfill the target for the rest of the days, 20 more trees were planted each day. This saved one day out of the initially planned number of days. How many trees ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e193c2d0d19dee89688188a546399f5df80a0f00eeae2d176bc8f69691a747eb | A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ work to... | [
"16"
] | [
"A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ wo... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-0e8145310560f9912324da9e76c04bf971c16679453bc7094f083f1314b46cec | A control board controls 100 lightbulbs arranged in a 10 by 10 grid. Flipping a switch changes the state of the corresponding lightbulb and its 8 adjacent lightbulbs (if they exist). If 10 lightbulbs are initially lit, compute the minimum number of switches needed to turn all of them off. | [
"2"
] | [
"A control board controls 100 lightbulbs arranged in a 10 by 10 grid. Flipping a switch changes the state of the corresponding lightbulb and its 8 adjacent lightbulbs (if they exist). If 10 lightbulbs are initially lit, compute the minimum number of switches needed to turn all of them off."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-904f010fd88fdf8a471ad217dbe1d56dc53224855131fa20b55da3f46a30bc1b | A convergent, positive geometric series is given. When the first 3 terms are removed, the sum of the series decreases by 2. When the 4th through 6th terms are removed, the sum of the series is reduced by 10%. Compute the sum of the original series. | [
"\\(10 - 2\\sqrt{15}\\)"
] | [
"A convergent, positive geometric series is given. When the first 3 terms are removed, the sum of the series decreases by 2. When the 4th through 6th terms are removed, the sum of the series is reduced by 10%. Compute the sum of the original series."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 8
} | 11 | 0 | 0 |
omr-ws-8b844294cdc498e182874f53bc4e21206db7aadfbe3651742ab54aa8effd8dbf | A convex $N$-gon is divided by diagonals into triangles such that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles sharing a common side are painted in different colors. For each $N$, find the maximal difference between the numbers of black and white ... | [
"1"
] | [
"A convex $N$-gon is divided by diagonals into triangles such that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles sharing a common side are painted in different colors. For each $N$, find the maximal difference between the numbers of black and w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-e8a82ef9317828d6370470357197c667b6bbfb3bba0c8302cf2b52283b086733 | A convex $N$-gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle, creating two new segments outside the polygon for each side. Let $S_R$ be the sum of the lengths of all the segments painted in red and $S_B$ be the sum of the lengths of all th... | [
"0"
] | [
"A convex $N$-gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle, creating two new segments outside the polygon for each side. Let $S_R$ be the sum of the lengths of all the segments painted in red and $S_B$ be the sum of the lengths of a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfd7df53531c72ea470ee6ed14621b2dd9536e6734ad3ad55c10c69d52624754 | A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals. For every vertex, the number of sides issuing from it is even, except for the vertices $A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. If $k > 0$, calculate the value of $n \mod 3$ given that $i_1 = 2... | [
"0"
] | [
"A convex $n$-gon $A_1A_2\\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals. For every vertex, the number of sides issuing from it is even, except for the vertices $A_{i_1},A_{i_2},\\dots,A_{i_k}$, where $1\\leq i_1<\\dots<i_k\\leq n$. If $k > 0$, calculate the value of $n \\mod 3$ given th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-e07743581bc93cc2993916b8a066d15d840129e85a9b5f966f1e0e16ddbff581 | A convex $n$-gon is split into three convex polygons. One of them has $n$ sides, the second one has more than $n$ sides, and the third one has less than $n$ sides. Find all possible values of $n$. | [
"5"
] | [
"A convex $n$-gon is split into three convex polygons. One of them has $n$ sides, the second one has more than $n$ sides, and the third one has less than $n$ sides. Find all possible values of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-3d3e6a186a7188c6e1869f95c7280def57658a3a64b9f607522611e8e3044c7c | A convex 12-gon is divided into 20 triangles. Let $S$ be the set of sides of the polygon that are also sides of one of the triangles. Calculate the minimum number of elements in $S$. | [
"2"
] | [
"A convex 12-gon is divided into 20 triangles. Let $S$ be the set of sides of the polygon that are also sides of one of the triangles. Calculate the minimum number of elements in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-d707bf176f29ca19dcb4997f10c85b82eac8be548a94bf81d0c615b5eb88cc60 | A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw? | [
"4016"
] | [
"A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-ef6ad4452d6db370a29e294a83351d56045d86579a6a882cb183e4fd327416ea | A convex 2018-gon is drawn on the board. Parker keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Parker can draw? Note that two diagonals sharing a vertex are not considered to be intersecti... | [
"4030"
] | [
"A convex 2018-gon is drawn on the board. Parker keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Parker can draw? Note that two diagonals sharing a vertex are not considered to be inter... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 3,
"tir": 2
} | 28 | 0 | 0 |
omr-ws-6f50958c4af9d4b423a9ef49bcda6143e68a369629aecb37645285c3c89cf1ab | A convex \( n \)-gon is divided by diagonals that do not intersect in the interior points into triangles. Determine the maximum number of vertices \( k \) that can be chosen such that no two chosen vertices are connected by either a side or a diagonal of the partitions. Calculate \( k \) for \( n = 15 \). | [
"7"
] | [
"A convex \\( n \\)-gon is divided by diagonals that do not intersect in the interior points into triangles. Determine the maximum number of vertices \\( k \\) that can be chosen such that no two chosen vertices are connected by either a side or a diagonal of the partitions. Calculate \\( k \\) for \\( n = 15 \\)."... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2c77b6e6444cb141ba0e29650b95edc3f0d29c2036423ee2f8a7041eb16a4f5 | A convex \(n\)-gon is split into three convex polygons. One of them has \(n\) sides, the second one has more than \(n\) sides, the third one has less than \(n\) sides. Find all possible values of \(n\). | [
"4"
] | [
"A convex \\(n\\)-gon is split into three convex polygons. One of them has \\(n\\) sides, the second one has more than \\(n\\) sides, the third one has less than \\(n\\) sides. Find all possible values of \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-e3e96af32890385d52c22ab01d65a7029be61af662b138efca2b043cc86ad429 | A convex cyclic quadrilateral has side lengths $1, \sqrt{3}, \sqrt{7}, \sqrt{21}$, in that order around the polygon. What is the distance between the two midpoints of the diagonals? | [
"\\(\\frac{3}{2}\\)"
] | [
"A convex cyclic quadrilateral has side lengths $1, \\sqrt{3}, \\sqrt{7}, \\sqrt{21}$, in that order around the polygon. What is the distance between the two midpoints of the diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-a88c64630d036ac5f18f7f7230fa25699cae05ff0b9281079fe3f4b572a7b085 | A convex cyclic quadrilateral has sides of length $1, \sqrt{7}, \sqrt{3}, \sqrt{21}$ in that order. What is the distance between the midpoints of the two diagonals? | [
"1"
] | [
"A convex cyclic quadrilateral has sides of length $1, \\sqrt{7}, \\sqrt{3}, \\sqrt{21}$ in that order. What is the distance between the midpoints of the two diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-9906807107061b0ed4228eb2490ad5379ccdffdcff4a8af45a7b2dd9b558f23a | A convex equilateral pentagon \(ABCDE\) is inscribed in a circle. Each side of the pentagon is extended to intersect the circle, creating additional segments. The segments extending from \(A\) to the circle beyond \(B\), \(B\) to the circle beyond \(C\), \(C\) to the circle beyond \(D\), \(D\) to the circle beyond \(E\... | [
"10r \\sin \\frac{2\\pi}{5}"
] | [
"A convex equilateral pentagon \\(ABCDE\\) is inscribed in a circle. Each side of the pentagon is extended to intersect the circle, creating additional segments. The segments extending from \\(A\\) to the circle beyond \\(B\\), \\(B\\) to the circle beyond \\(C\\), \\(C\\) to the circle beyond \\(D\\), \\(D\\) to t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e7ffcbc79bf580b3d6db5ca84b2c2557c04c5de5beda98ded437ccb4d2a89eb | A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. If the lengths of the sides of the hexagon are $AB = a, BC = b, CD = c, DE = d, EF = e,$ and $FA = f$, find the value of $\frac{a + b + c + d + ... | [
"1"
] | [
"A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. If the lengths of the sides of the hexagon are $AB = a, BC = b, CD = c, DE = d, EF = e,$ and $FA = f$, find the value of $\\frac{a + b + c ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e5a78092d5c639e5656cad320b9e5cf0af4af5483c19d86be7c5b7bea971a1a1 | A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB = CD = EF$ and diagonals $AD$, $BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. If $\frac{AC}{CE} = \sqrt{2}$, find the value of $\frac{CP}{PE}$. | [
"\\sqrt{2}"
] | [
"A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB = CD = EF$ and diagonals $AD$, $BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. If $\\frac{AC}{CE} = \\sqrt{2}$, find the value of $\\frac{CP}{PE}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-d39d885dcb5e7e167f9e555e9c4ff8adf1393634efa01287020d238fb4e8f4d9 | A convex hexagon $AC_1BA_1CB_1$ satisfies the conditions $AB_1=AC_1$, $BC_1=BA_1$, $CA_1=CB_1$, and $\angle A+\angle B+\angle C=\angle A_1+\angle B_1+\angle C_1$. If the area of the hexagon is $120$, find the area of $\triangle ABC$. | [
"60"
] | [
"A convex hexagon $AC_1BA_1CB_1$ satisfies the conditions $AB_1=AC_1$, $BC_1=BA_1$, $CA_1=CB_1$, and $\\angle A+\\angle B+\\angle C=\\angle A_1+\\angle B_1+\\angle C_1$. If the area of the hexagon is $120$, find the area of $\\triangle ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-658f724eba85aee4c5c0a8b30271e729fe4866686b988e303bd7997f7d26141a | A convex hexagon $A_1B_1A_2B_2A_3B_3$ is inscribed in a circle $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, and $A_3B_1$ are concurrent at point $X$. For each $i=1,2,3$, let $\omega_i$ be a circle tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain ... | [
"10"
] | [
"A convex hexagon $A_1B_1A_2B_2A_3B_3$ is inscribed in a circle $\\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, and $A_3B_1$ are concurrent at point $X$. For each $i=1,2,3$, let $\\omega_i$ be a circle tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\\Omega$ that does not ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-a6751f9a16fc8d3206a0690203fcd9e943d988016dbaf8ede81a4d2e90703dbd | A convex hexagon $A_1B_1A_2B_2A_3B_3$ is inscribed in a circle $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, and $A_3B_1$ are concurrent at point $X$. For each $i=1,2,3$, let $\omega_i$ be a circle tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain ... | [
"\\dfrac{R}{2}"
] | [
"A convex hexagon $A_1B_1A_2B_2A_3B_3$ is inscribed in a circle $\\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, and $A_3B_1$ are concurrent at point $X$. For each $i=1,2,3$, let $\\omega_i$ be a circle tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\\Omega$ that does not ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d3377549c34cfa68bc165094f442ea3691d888148099d6fb295866702a4dfb90 | A convex hexagon \(ABCDEF\) is inscribed in a circle such that \(AB = CD = EF\) and diagonals \(AD\), \(BE\), and \(CF\) are concurrent. Let \(P\) be the intersection of \(AD\) and \(CE\). If \(AC = 6\) and \(CE = 8\), calculate the value of \(\frac{CP}{PE}\). | [
"\\dfrac{9}{16}"
] | [
"A convex hexagon \\(ABCDEF\\) is inscribed in a circle such that \\(AB = CD = EF\\) and diagonals \\(AD\\), \\(BE\\), and \\(CF\\) are concurrent. Let \\(P\\) be the intersection of \\(AD\\) and \\(CE\\). If \\(AC = 6\\) and \\(CE = 8\\), calculate the value of \\(\\frac{CP}{PE}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-462d3d5e2501dd780f402f7d3f3028e4b0094d0b690c7ffe1b573ae062bc2e2e | A convex hexagon \(ABCDEF\) with an interior point \(P\) is given. Assume that \(BCEF\) is a square and both \(ABP\) and \(PCD\) are right isosceles triangles with right angles at \(B\) and \(C\), respectively. Lines \(AF\) and \(DE\) intersect at \(G\). Calculate the length of \(GP\) if \(BC = 2\). | [
"2"
] | [
"A convex hexagon \\(ABCDEF\\) with an interior point \\(P\\) is given. Assume that \\(BCEF\\) is a square and both \\(ABP\\) and \\(PCD\\) are right isosceles triangles with right angles at \\(B\\) and \\(C\\), respectively. Lines \\(AF\\) and \\(DE\\) intersect at \\(G\\). Calculate the length of \\(GP\\) if \\(B... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-dfb56ee997469a4c8fdf0236a5c4553781b52b84c082502537ecb5532dafea44 | A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r$. | [
"2"
] | [
"A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1b504a183e5321549c6c56fbde31b300670fcaa4ff1f93d913b5672a3cbeb4cb | A convex hexagon is inscribed in a circle with successive sides of lengths 2, 2, 7, 7, 11, and 11. Compute the diameter of the circle. | [
"14"
] | [
"A convex hexagon is inscribed in a circle with successive sides of lengths 2, 2, 7, 7, 11, and 11. Compute the diameter of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-d87fbc0f51de6401a8e46409bc280ac100c2a6d78aca9888c981f0c79a91fedc | A convex hexagon is inscribed in a circle with successive sides of lengths 2, 2, 7, 7, 11, and 11. Compute the diameter of the circumscribed circle. | [
"14"
] | [
"A convex hexagon is inscribed in a circle with successive sides of lengths 2, 2, 7, 7, 11, and 11. Compute the diameter of the circumscribed circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 10
} | 74 | 0 | 0 |
omr-ws-74922b67cf681148a6d085e2f6382feb71dea0c3509bab3b453cbfcff6909dbe | A convex hexagon is symmetrical about the point $O$. The point $O$ was reflected with respect to the midpoints of the small diagonals of the hexagon, and none of the images fell on the sides of the hexagon. How many of these images could fall inside the hexagon? | [
"3"
] | [
"A convex hexagon is symmetrical about the point $O$. The point $O$ was reflected with respect to the midpoints of the small diagonals of the hexagon, and none of the images fell on the sides of the hexagon. How many of these images could fall inside the hexagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc018edaaeff9425d5315bff89350bd3ae7b71375f7b30eb0f6b03735a575cfc | A convex hexagon with side lengths $2, 2, 7, 7, 11, 11$ (in that order) is inscribed in a circle. What is the diameter of the circle? | [
"14"
] | [
"A convex hexagon with side lengths $2, 2, 7, 7, 11, 11$ (in that order) is inscribed in a circle. What is the diameter of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-31575c1b05cc388bd77803e22fbc47ecb773c629788e474e05c3ceea94cf163b | A convex n-polygon is divided into triangles by some of its diagonals. Each triangle is colored either black or white such that any two triangles sharing an edge have different colors. Find the largest possible difference between the number of black and white triangles. | [
"1"
] | [
"A convex n-polygon is divided into triangles by some of its diagonals. Each triangle is colored either black or white such that any two triangles sharing an edge have different colors. Find the largest possible difference between the number of black and white triangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-823f5b73ee4f52674c3cfc3894f057cd9aabca4b2bea2d4efa6c12fe11c71f9e | A convex n-sided polygon \( A_1A_2\cdots A_n \) satisfies that \( A_iA_{i+1} \parallel A_{i-1}A_{i+2} \) for all \( 1 \le i \le n \), where \( A_{n+1} = A_1 \) and \( A_{n+2} = A_2 \). It is given that all triangles \( [A_iA_{i+1}A_{i+2}] \) have the same area \( S \). For a fixed \( S \), find the smallest value of th... | [
"2S"
] | [
"A convex n-sided polygon \\( A_1A_2\\cdots A_n \\) satisfies that \\( A_iA_{i+1} \\parallel A_{i-1}A_{i+2} \\) for all \\( 1 \\le i \\le n \\), where \\( A_{n+1} = A_1 \\) and \\( A_{n+2} = A_2 \\). It is given that all triangles \\( [A_iA_{i+1}A_{i+2}] \\) have the same area \\( S \\). For a fixed \\( S \\), find... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-66a5100c8917dbfe228e48b53d797de5bd75164699ad46d39092a25a7271b966 | A convex octagon is inscribed in a circle and has four consecutive sides each of length 3, and four other sides each of length 2. Find the area of the octagon. | [
"\\( 9 + 12\\sqrt{2} \\)"
] | [
"A convex octagon is inscribed in a circle and has four consecutive sides each of length 3, and four other sides each of length 2. Find the area of the octagon."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-497c7a1851cca8d69fea816943305e629bae96a1da05c5e5f1fd9a0b669d0fb9 | A convex octagon is inscribed in a circle of radius \( R \). The octagon has four consecutive sides of length 3 units and four consecutive sides of length 2 units. Find the area of the octagon. | [
"13 + 12\\sqrt{2}"
] | [
"A convex octagon is inscribed in a circle of radius \\( R \\). The octagon has four consecutive sides of length 3 units and four consecutive sides of length 2 units. Find the area of the octagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e5d6a91af1f2b22cd3bfcb812d586ec33229d2abf2e824276c209fa93838233d | A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\sqrt{17}$, $\sqrt{37}$, $\sqrt{52}$. A second triangular face $F_0$ has side lengths $\sqrt{13}$, $\sqrt{2... | [
"\\dfrac{77}{2}"
] | [
"A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\\sqrt{17}$, $\\sqrt{37}$, $\\sqrt{52}$. A second triangular face $F_0$ has side lengths $\\sqrt{13}$,... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-1f5315b087507a7ed592d01dcd3a2a75d4a588d61491e8236876ea62141d1543 | A convex pentagon $ABCDE$ is inscribed in a circle. The distances from point $A$ to the lines $BC$, $CD$, and $DE$ are $a$, $b$, and $c$, respectively. Compute the distance from point $A$ to the line $BE$. | [
"\\(\\frac{a \\cdot c}{b}\\)"
] | [
"A convex pentagon $ABCDE$ is inscribed in a circle. The distances from point $A$ to the lines $BC$, $CD$, and $DE$ are $a$, $b$, and $c$, respectively. Compute the distance from point $A$ to the line $BE$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-d9b4051202d7cab7d661dea6eac5d6dcf31b1f1015a28d0c569e6fb472ce01e1 | A convex pentagon $ABCDE$ is inscribed in a unit circle with $AE$ as its diameter. Given $AB = a$, $BC = b$, $CD = c$, $DE = d$, and $ab = cd = \frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d$. | [
"\\( 4(4 - a^2 - b^2 - c^2 - d^2) \\)"
] | [
"A convex pentagon $ABCDE$ is inscribed in a unit circle with $AE$ as its diameter. Given $AB = a$, $BC = b$, $CD = c$, $DE = d$, and $ab = cd = \\frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-126af345b354aef72185382108962c714febf7d9e85ab05fe664cc92e768edf2 | A convex pentagon \( P \) is divided by all its diagonals into ten triangles and one smaller pentagon \( P' \). Let \( N \) be the sum of the areas of the five triangles adjacent to the sides of \( P \) decreased by the area of \( P' \). The same operations are performed with the pentagon \( P' \), let \( N' \) be the ... | [
"28"
] | [
"A convex pentagon \\( P \\) is divided by all its diagonals into ten triangles and one smaller pentagon \\( P' \\). Let \\( N \\) be the sum of the areas of the five triangles adjacent to the sides of \\( P \\) decreased by the area of \\( P' \\). The same operations are performed with the pentagon \\( P' \\), let... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-ce2e3d845d2ec1b005a858007bbca48078f5ea1df781d516c18e86997ee1d6ca | A convex pentagon \(ABCDE\) has an inscribed circle that is tangent to \(AB\), \(BC\), \(CD\), \(DE\), and \(EA\) at points \(P\), \(Q\), \(R\), \(S\), and \(T\) respectively. Let \(AP = a\), \(BQ = b\), \(CR = c\), \(DS = d\), \(ET = e\), and the radius of the incircle be \(r\). Given \(a = 3\), \(b = 4\), \(c = 5\), ... | [
"45"
] | [
"A convex pentagon \\(ABCDE\\) has an inscribed circle that is tangent to \\(AB\\), \\(BC\\), \\(CD\\), \\(DE\\), and \\(EA\\) at points \\(P\\), \\(Q\\), \\(R\\), \\(S\\), and \\(T\\) respectively. Let \\(AP = a\\), \\(BQ = b\\), \\(CR = c\\), \\(DS = d\\), \\(ET = e\\), and the radius of the incircle be \\(r\\). ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-218a080c52bbbfd6ce58370ae41bc30d8f19b9c4760ea149ab7f5c052a2e43ac | A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle? | [
"179"
] | [
"A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f7f9a58dd29d6eff0071ccba384559dede072744b82bd9b722b585db13c604c | A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon. | [
"\\( 2 + \\frac{\\sqrt{5} + 1}{2} \\)"
] | [
"A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 6
} | 27 | 0 | 0 |
omr-ws-866c6e8e3e5e0e10f611b68141df02d3711a05d659ad5e4190b6c02385108470 | A convex pentagon is inscribed in a circle with sides of lengths 16, 16, 16, 16, and 6. The area of the pentagon is given by \( k \sqrt{7} \). Determine the value of \( k \). | [
"123"
] | [
"A convex pentagon is inscribed in a circle with sides of lengths 16, 16, 16, 16, and 6. The area of the pentagon is given by \\( k \\sqrt{7} \\). Determine the value of \\( k \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-1cf4bbb6617e90d49eefe8d92a3e5d23894eb245fc992c203fc79926e9f405cc | A convex pentagon is inscribed in a circle with sides of lengths 16, 16, 16, 16, and 6. The area of the pentagon is given by \( k\sqrt{7} \). Determine the value of \( k \). | [
"123"
] | [
"A convex pentagon is inscribed in a circle with sides of lengths 16, 16, 16, 16, and 6. The area of the pentagon is given by \\( k\\sqrt{7} \\). Determine the value of \\( k \\)."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 7,
"tir": 15
} | 68 | 0 | 0 |
omr-ws-5dfdf7b706b24f4ee3138b7b1ed074219889431407476d89d92282e3364b1660 | A convex polygon \( M \) has been decomposed into red equilateral triangles (not necessarily equal) and blue quadrilaterals with all angles strictly between \( 80^\circ \) and \( 100^\circ \). If \( M \) has all angles strictly greater than \( 60^\circ \), let \( R \) be the number of completely red sides of \( M \). F... | [
"0"
] | [
"A convex polygon \\( M \\) has been decomposed into red equilateral triangles (not necessarily equal) and blue quadrilaterals with all angles strictly between \\( 80^\\circ \\) and \\( 100^\\circ \\). If \\( M \\) has all angles strictly greater than \\( 60^\\circ \\), let \\( R \\) be the number of completely red... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-c4dd7077ca1d940b44ec4f6f4798779f4f17787956f910826dc1af72318d1424 | A convex polygon \( P \) has been decomposed into red equilateral triangles (not necessarily equal) and blue quadrilaterals with all angles strictly between \( 80^\circ \) and \( 100^\circ \). If all angles of \( P \) are strictly greater than \( 60^\circ \), let \( N \) be the number of completely red sides of \( P \)... | [
"0"
] | [
"A convex polygon \\( P \\) has been decomposed into red equilateral triangles (not necessarily equal) and blue quadrilaterals with all angles strictly between \\( 80^\\circ \\) and \\( 100^\\circ \\). If all angles of \\( P \\) are strictly greater than \\( 60^\\circ \\), let \\( N \\) be the number of completely ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8e47156fa20f387d089c9cdf406a5d3cfc968cf324d7dec9848928fc9a01deb6 | A convex polygon \(A_1, A_2, \ldots, A_n\) is inscribed in a circle with center \(O\) and radius \(R\) so that \(O\) lies inside the polygon. Let the inradii of the triangles \(A_1A_2A_3, A_1A_3A_4, \ldots, A_1A_{n-1}A_n\) be denoted by \(r_1, r_2, \ldots, r_{n-2}\). Calculate the maximum possible value of \(r_1 + r_2 ... | [
"no_valid_answer_found"
] | [
"A convex polygon \\(A_1, A_2, \\ldots, A_n\\) is inscribed in a circle with center \\(O\\) and radius \\(R\\) so that \\(O\\) lies inside the polygon. Let the inradii of the triangles \\(A_1A_2A_3, A_1A_3A_4, \\ldots, A_1A_{n-1}A_n\\) be denoted by \\(r_1, r_2, \\ldots, r_{n-2}\\). Calculate the maximum possible v... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f03e37be59036df5a5c363962d5186d1691998e706e9b9372ac0e8759321de39 | A convex polygon contains N non-overlapping circles. The polygon is cut into N smaller polygons, each containing exactly one of the circles. Let P(N) be the minimum number of cuts required to achieve this for any configuration of the circles. Find P(5). | [
"4"
] | [
"A convex polygon contains N non-overlapping circles. The polygon is cut into N smaller polygons, each containing exactly one of the circles. Let P(N) be the minimum number of cuts required to achieve this for any configuration of the circles. Find P(5)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59c2d47d722e8afe5c1d76d41e193bee1a7ea44b44b3adc61156078da57a2c4f | A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? | [
"5"
] | [
"A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-26007a29a9395766336e2fe2ace3c95a1ead05b670acad661fa56a935602b98e | A convex polygon has integer interior angle measures which form an arithmetic sequence with a common difference of 3. What is the largest possible measure of an angle in this polygon? | [
"180"
] | [
"A convex polygon has integer interior angle measures which form an arithmetic sequence with a common difference of 3. What is the largest possible measure of an angle in this polygon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3588ec0476889bd1f75bbbaa67233377b5568ff8dbf3c99148781aab88c1a2fe | A convex polygon has its interior angles in arithmetic progression, with the least angle being \(120^\circ\) and the common difference being \(5^\circ\). Find the number of sides of the polygon. | [
"\\( n \\in \\{9, 16\\} \\)"
] | [
"A convex polygon has its interior angles in arithmetic progression, with the least angle being \\(120^\\circ\\) and the common difference being \\(5^\\circ\\). Find the number of sides of the polygon."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5761ffa6010d2041c97095a216c860bb328c89db140bbf0116fea107bd5374dc | A convex polygon has the property that the number of sides is equal to the number of diagonals. If this polygon has $n$ sides, find the value of $n^2 - 3n + 2$. | [
"12"
] | [
"A convex polygon has the property that the number of sides is equal to the number of diagonals. If this polygon has $n$ sides, find the value of $n^2 - 3n + 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3119c1067fa67efbeba5c995436f7b4b653a6e5dc8b55c69a360b06a71a66eb9 | A convex polygon is circumscribed around a circle. The points of contact of its sides with the circle form a polygon with the same set of angles (though the order of angles may differ). If the original polygon has 10 sides, calculate the sum of all possible values of the ratio of the perimeter of the original polygon t... | [
"2"
] | [
"A convex polygon is circumscribed around a circle. The points of contact of its sides with the circle form a polygon with the same set of angles (though the order of angles may differ). If the original polygon has 10 sides, calculate the sum of all possible values of the ratio of the perimeter of the original poly... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-799482924ca26f9a773a4b18663b35a283316edded065aac8a90fabe469b5851 | A convex polygon is composed of $n > 1$ equal (possibly overlapping) equilateral triangles such that the sides of all triangles are pairwise non-parallel. What is the smallest possible value of $n$? | [
"4"
] | [
"A convex polygon is composed of $n > 1$ equal (possibly overlapping) equilateral triangles such that the sides of all triangles are pairwise non-parallel. What is the smallest possible value of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
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"cot": 4
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omr-ws-c25fa55cd747f946ab9c44398822e063b2e751df6b0efdb3eaf7fcb60ecda28a | A convex polygon is given in the plane. A vertex \( A \) is called "good" if the point symmetrical to \( A \) with respect to the midpoint of the diagonal connecting two vertices adjacent to \( A \) belongs to the polygon (perhaps its boundary). What is the smallest number of good vertices a convex polygon can have? | [
"2"
] | [
"A convex polygon is given in the plane. A vertex \\( A \\) is called \"good\" if the point symmetrical to \\( A \\) with respect to the midpoint of the diagonal connecting two vertices adjacent to \\( A \\) belongs to the polygon (perhaps its boundary). What is the smallest number of good vertices a convex polygon... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
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"cot": 1
} | 1 | 0 | 0 |
omr-ws-f6dd62b49ed53c01864ea4710b0161d1c1197f8607f9aff4f86fe1bafc6c2338 | A convex polygon with 12 sides is inscribed in a circle. The polygon has 6 sides of length $\sqrt{2}$ and 6 sides of length $\sqrt{24}$. What is the radius of the circle? | [
"$\\sqrt{38}$"
] | [
"A convex polygon with 12 sides is inscribed in a circle. The polygon has 6 sides of length $\\sqrt{2}$ and 6 sides of length $\\sqrt{24}$. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
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"cot",
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omr-ws-cfcb6bf29ea9b0872e920f3f18241534e39536a1dabd93c85c90d4c2a6f5708a | A convex polygon with \( n \) sides has all angles equal to \( 150 \) degrees, with the possible exception of one angle. List all the possible values of \( n \). | [
"\\( n = 8, 9, 10, 11, 12 \\)"
] | [
"A convex polygon with \\( n \\) sides has all angles equal to \\( 150 \\) degrees, with the possible exception of one angle. List all the possible values of \\( n \\)."
] | [
"aops_c4_high_school_math"
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"cot",
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"cot": 2,
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omr-ws-1c9b36098a82e82343f12e09f2145f7f7acb5925ec2e66468b79ec51eb2ca8b6 | A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have? | [
"241"
] | [
"A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have?\n"
] | [
"MATH_training_set"
] | [
"cot"
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"cot": 16
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omr-ws-cefb1bc43c86d8e9feed560b40ce8397e04c53f77334596b18730276211c4933 | A convex polyhedron $P$ has all faces as congruent isosceles (but not equilateral) triangles. Each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. Find the sum of the numbers of faces, edges, and vertices of $P$. | [
"182"
] | [
"A convex polyhedron $P$ has all faces as congruent isosceles (but not equilateral) triangles. Each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. Find the sum of the numbers of faces, edges, and vertices of $P$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c14adcb868bbd18a298b7eeee578ea35b8e8cfd46fbeff53fb8449cb60da555f | A convex polyhedron $P$ has two square faces and eight equilateral triangular faces such that each square shares an edge with four triangles and each triangle shares an edge with exactly one square. If each edge of the polyhedron has length 1, what is the distance between the two square faces? | [
"\\(\\frac{\\sqrt[4]{8}}{2}\\)"
] | [
"A convex polyhedron $P$ has two square faces and eight equilateral triangular faces such that each square shares an edge with four triangles and each triangle shares an edge with exactly one square. If each edge of the polyhedron has length 1, what is the distance between the two square faces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
"cot": 11,
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} | 21 | 0 | 0 |
omr-ws-d8391719a4642890449653a8b94c17c143172225bdf71172abe1cae468b9ff34 | A convex polyhedron $Q$ has vertices $V_1, V_2, \ldots, V_n$, and 100 edges. The polyhedron is cut by planes $P_1, P_2, \ldots, P_n$ such that plane $P_k$ cuts only those edges that meet at vertex $V_k$. No two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges doe... | [
"300"
] | [
"A convex polyhedron $Q$ has vertices $V_1, V_2, \\ldots, V_n$, and 100 edges. The polyhedron is cut by planes $P_1, P_2, \\ldots, P_n$ such that plane $P_k$ cuts only those edges that meet at vertex $V_k$. No two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many ed... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d946991e304501393b311df868342f5678a038f4ef214d36e4061e86563ceb90 | A convex polyhedron \( P \) has 26 vertices, 60 edges, 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. How many space diagonals does \( P \) have? | [
"241"
] | [
"A convex polyhedron \\( P \\) has 26 vertices, 60 edges, 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. How many space diagonals does \\( P \\) have?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
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"cot": 16,
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} | 27 | 0 | 0 |
omr-ws-321a74723c7f1514e6c830a8fb1ce1180488ffa324b46a642a559aa63af6b2f0 | A convex polyhedron \( P \) has 26 vertices, 60 edges, and 36 faces, with 24 triangular faces and 12 quadrilateral faces. How many space diagonals does \( P \) have? | [
"241"
] | [
"A convex polyhedron \\( P \\) has 26 vertices, 60 edges, and 36 faces, with 24 triangular faces and 12 quadrilateral faces. How many space diagonals does \\( P \\) have?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
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"cot",
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"cot": 48,
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} | 93 | 0 | 0 |
omr-ws-369d1ab23e8cf718605d44633aaa8c74ff1b8b54b44566f5ad6c3ab5b1503d40 | A convex polyhedron \( P \) has 30 vertices, 100 edges, and 42 faces, with 32 quadrilateral faces and 10 pentagonal faces. How many space diagonals does \( P \) have? | [
"293"
] | [
"A convex polyhedron \\( P \\) has 30 vertices, 100 edges, and 42 faces, with 32 quadrilateral faces and 10 pentagonal faces. How many space diagonals does \\( P \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02132b43ed3f6dfe62a0718d6d71260ef90c604447f66f8e9c865ee1f14fd67d | A convex polyhedron \( Q \) has vertices \( V_1, V_2, \ldots, V_n \), and 100 edges. The polyhedron is cut by planes \( P_1, P_2, \ldots, P_n \) such that plane \( P_k \) cuts only those edges that meet at vertex \( V_k \). No two planes intersect inside or on \( Q \). The cuts produce \( n \) pyramids and a new polyhe... | [
"300"
] | [
"A convex polyhedron \\( Q \\) has vertices \\( V_1, V_2, \\ldots, V_n \\), and 100 edges. The polyhedron is cut by planes \\( P_1, P_2, \\ldots, P_n \\) such that plane \\( P_k \\) cuts only those edges that meet at vertex \\( V_k \\). No two planes intersect inside or on \\( Q \\). The cuts produce \\( n \\) pyra... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 46,
"genselect": 7
} | 53 | 0 | 0 |
omr-ws-b01dbbe512aa866c4ec39d397ebcde9ff57389548e2d053cec8fe89eda44169f | A convex polyhedron \( Q \) is obtained from a convex polyhedron \( P \) with 36 edges by cutting off a pyramid at each vertex of \( P \) using a flat plane that does not intersect inside \( P \). Determine the number of edges of \( Q \). | [
"108"
] | [
"A convex polyhedron \\( Q \\) is obtained from a convex polyhedron \\( P \\) with 36 edges by cutting off a pyramid at each vertex of \\( P \\) using a flat plane that does not intersect inside \\( P \\). Determine the number of edges of \\( Q \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-680aaece3aa7069f60023eb716e8fcdb39f2e0f916b31974dcf5ce056895acd8 | A convex polyhedron \( S \) with \( n \) vertices has the property that for any three of its edges, there are two that lie in the same plane. If \( S \) is a pyramid, determine the maximum possible value of \( n \). | [
"5"
] | [
"A convex polyhedron \\( S \\) with \\( n \\) vertices has the property that for any three of its edges, there are two that lie in the same plane. If \\( S \\) is a pyramid, determine the maximum possible value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c547516a6be8e8f3483a08caff01c6a213e70a9cf0ad278c714befa915d14a82 | A convex polyhedron has $2n$ triangular faces ($n \ge 3$). What is the largest number of vertices at which exactly 3 edges converge? | [
"8 - 2n"
] | [
"A convex polyhedron has $2n$ triangular faces ($n \\ge 3$). What is the largest number of vertices at which exactly 3 edges converge?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-6bbbb329fb2de653307c27a7b94cc99e7a4330d23c3b9b8b9d0c89d41cbb3eb5 | A convex polyhedron has $9$ vertices, one of which is $A$. Parallel translations that send $A$ into each of the other vertices form $8$ equal polyhedra. Let $N$ be the number of pairs of these $8$ polyhedra that have an inner point in intersection. Calculate the minimum possible value of $N$. | [
"12"
] | [
"A convex polyhedron has $9$ vertices, one of which is $A$. Parallel translations that send $A$ into each of the other vertices form $8$ equal polyhedra. Let $N$ be the number of pairs of these $8$ polyhedra that have an inner point in intersection. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-51318ed069767a35632dafcbe86dc65b59eb916f194bde6b9b134863dfd9b6bd | A convex polyhedron has 12 squares, 8 hexagons, and 6 octagons for faces. At every vertex, exactly one square, one hexagon, and one octagon concur. Consider the line segments that join two vertices of the polyhedron and determine how many of them are neither edges nor contained in some face of the polyhedron. | [
"840"
] | [
"A convex polyhedron has 12 squares, 8 hexagons, and 6 octagons for faces. At every vertex, exactly one square, one hexagon, and one octagon concur. Consider the line segments that join two vertices of the polyhedron and determine how many of them are neither edges nor contained in some face of the polyhedron."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-266104ed4b2a71c087ae3359aebc15981918f730e9450e140ab3fb7e0c20d74d | A convex polyhedron has 12 squares, 8 regular hexagons, and 6 regular octagons as its faces. At each vertex, one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face? | [
"840"
] | [
"A convex polyhedron has 12 squares, 8 regular hexagons, and 6 regular octagons as its faces. At each vertex, one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-310e27947f7a121be73ed830a53dcee58b6732e6546af319410ecd1d1f6c233d | A convex polyhedron has 26 faces, specifically 6 regular octagons, 8 regular hexagons, and 12 squares. At every vertex, an octagon, a hexagon, and a square meet. How many segments joining vertices of the polyhedron are in the interior of the polyhedron and not on the boundary? | [
"840"
] | [
"A convex polyhedron has 26 faces, specifically 6 regular octagons, 8 regular hexagons, and 12 squares. At every vertex, an octagon, a hexagon, and a square meet. How many segments joining vertices of the polyhedron are in the interior of the polyhedron and not on the boundary?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-c6adb06f6d76af38ea7f2c9a4535dd250c9cf92baad8325356c0dfb619e1c7ff | A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. Given that the polyhedron satisfies Euler's formula $V - E + F = 2$, determine the value of $100P + 10T + V$. | [
"250"
] | [
"A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. Given that the polyhedron satisfies Euler's formula $V - E + F = 2$, determine the value of $100P + 10T + V$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-6e5c48dc30005010ff0634fee149c20d764332880806f08135fda6e8f98bc04f | A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$? | [
"250"
] | [
"A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-b9865aeee72c7a3d8859a2bd2edc8dd06c7b98ff84744f7ca13a5b5ccaa0470c | A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face? | [
"840"
] | [
"A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9efadf68de8c271192566891e4697007f90826956d14e6cd0a91e30db8228462 | A convex polyhedron has only one face with 100 vertices. From each vertex, there are at least 4 edges to other vertices. What is the minimum number of triangular faces this polyhedron can have? | [
"196"
] | [
"A convex polyhedron has only one face with 100 vertices. From each vertex, there are at least 4 edges to other vertices. What is the minimum number of triangular faces this polyhedron can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-7b5a22fac085aa6fff544decbb95a78692e80d6c34fd9f0253d81dee5b6e706a | A convex polyhedron has three triangular faces. The sum of the measures of all its angles, except one, is 2014 degrees. How many vertices from which three edges depart does it have? Note: The angles added are those on the faces polygonal. One degree is worth \( \frac{n}{180} \) radians. | [
"6"
] | [
"A convex polyhedron has three triangular faces. The sum of the measures of all its angles, except one, is 2014 degrees. How many vertices from which three edges depart does it have? Note: The angles added are those on the faces polygonal. One degree is worth \\( \\frac{n}{180} \\) radians."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-5094375e834b1465f4c403e364007678bffabd30ef988fcd1c8af4824c06dabe | A convex polyhedron has two triangular and three quadrilateral faces. Choose one of the triangular faces and label its vertices $A$, $B$, and $C$. Connect each vertex of this triangular face with the intersection point of the diagonals in the quadrilateral face opposite to it. If the intersection point of these three l... | [
"\\dfrac{1}{9}"
] | [
"A convex polyhedron has two triangular and three quadrilateral faces. Choose one of the triangular faces and label its vertices $A$, $B$, and $C$. Connect each vertex of this triangular face with the intersection point of the diagonals in the quadrilateral face opposite to it. If the intersection point of these th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b22016ff02f05bc03d75e8f8bb702bac4f16f725adf4f15b6d4e4b5298faef33 | A convex polyhedron with 100 edges has all its vertices cut off by planes that do not intersect inside or on the surface of the polyhedron. Find the number of edges of the new polyhedron. | [
"300"
] | [
"A convex polyhedron with 100 edges has all its vertices cut off by planes that do not intersect inside or on the surface of the polyhedron. Find the number of edges of the new polyhedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75fc95bf00bee335a6d498ecadaeca0e6df294f45ac14835d5b059482cfbea66 | A convex polyhedron with 100 edges has all its vertices cut off by planes that do not intersect inside or on the surface of the polyhedron. Find the number of vertices of the new polyhedron. | [
"200"
] | [
"A convex polyhedron with 100 edges has all its vertices cut off by planes that do not intersect inside or on the surface of the polyhedron. Find the number of vertices of the new polyhedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96ff7e35a984ef02db6e87d749321700866e5c7b601c192614376fc5972cb8a5 | A convex quadrilateral $ABCD$ has $AD=3$. The diagonals $AC$ and $BD$ intersect at point $E$. The areas of triangles $ABE$ and $DCE$ are both $1$. Given that the area of quadrilateral $ABCD$ does not exceed $4$, find the length of side $BC$. | [
"3"
] | [
"A convex quadrilateral $ABCD$ has $AD=3$. The diagonals $AC$ and $BD$ intersect at point $E$. The areas of triangles $ABE$ and $DCE$ are both $1$. Given that the area of quadrilateral $ABCD$ does not exceed $4$, find the length of side $BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1224514f221abb330900cbc57f03646de2558bfac7adc61df092fae79eca4143 | A convex quadrilateral $ABCD$ has $\angle BAC = \angle ADC$. Let $M$ be the midpoint of the diagonal $AC$. The side $AD$ contains a point $E$ such that $ABME$ is a parallelogram. Let $N$ be the midpoint of the line segment $AE$. If the circumcircle of triangle $DMN$ intersects $AC$ again at point $P$, calculate the rat... | [
"\\dfrac{1}{2}"
] | [
"A convex quadrilateral $ABCD$ has $\\angle BAC = \\angle ADC$. Let $M$ be the midpoint of the diagonal $AC$. The side $AD$ contains a point $E$ such that $ABME$ is a parallelogram. Let $N$ be the midpoint of the line segment $AE$. If the circumcircle of triangle $DMN$ intersects $AC$ again at point $P$, calculate ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-e75bbb4f67856c261e0ee12f20a82852bcc4cf3bd26f5d8a5f33038466824892 | A convex quadrilateral $ABCD$ has an incircle. In each corner, a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Let $r_A$, $r_B$, $r_C$, and $r_D$ be the radii of the circles inscribed at vertices $A$, $B$, $C$, and $D$ respectively. If the sum of the radii of the ci... | [
"1"
] | [
"A convex quadrilateral $ABCD$ has an incircle. In each corner, a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Let $r_A$, $r_B$, $r_C$, and $r_D$ be the radii of the circles inscribed at vertices $A$, $B$, $C$, and $D$ respectively. If the sum of the radii of t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4cff0cb3cb3813d434cd9310c27857c555fc742dbdaaaca90f1e8e877b3802fe | A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a$ and $I_c$ be the incenters of triangles $DAB$ and $BCD$, respectively. Suppose that the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $X$. Calculate the value of $\angle XI_aI_c + \angle XI_cI_a$ in degrees. | [
"90"
] | [
"A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a$ and $I_c$ be the incenters of triangles $DAB$ and $BCD$, respectively. Suppose that the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $X$. Calculate the value of $\\angle XI_aI_c + \\angle XI_cI_a$ in degrees... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2540247df039196392d949bd1b2543c042b5bfdb17060cd8a2962d5c1d4e8ad1 | A convex quadrilateral $ABCD$ has no parallel sides. The angles formed by the diagonal $AC$ and the four sides are $55^\circ, 55^\circ, 19^\circ,$ and $16^\circ$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$. | [
"$87^\\circ$"
] | [
"A convex quadrilateral $ABCD$ has no parallel sides. The angles formed by the diagonal $AC$ and the four sides are $55^\\circ, 55^\\circ, 19^\\circ,$ and $16^\\circ$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-8b5c0b00dfb6a9abc0d1fa826a77dc3d291e65dfcca7a18c4d2a1ec95145a0e2 | A convex quadrilateral $ABCD$ is called [i]rude[/i] if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of $ABCD$ such that the sum of the diagonals of $PQRS$ is larger than the sum of the diagonals of $ABCD$. Given that $ABCD$ is not rude, but every quadrilateral $A'BCD$ ... | [
"150^{\\circ}"
] | [
"A convex quadrilateral $ABCD$ is called [i]rude[/i] if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of $ABCD$ such that the sum of the diagonals of $PQRS$ is larger than the sum of the diagonals of $ABCD$. Given that $ABCD$ is not rude, but every quadrilateral $A'... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39fbb17906312fbf4de3054cff10eb085fcc90259e7f1c3a9aadd5e807672ca7 | A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\frac{1}{AC^2}+\frac{1}{BD^2}$? | [
"\\dfrac{1}{4r^2}"
] | [
"A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\\frac{1}{AC^2}+\\frac{1}{BD^2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-5817ccd3f28b6c23e6dc790c981f98c2e9d6831df06a6c2ab4bc1440eb85296c | A convex quadrilateral $ABCD$ is defined on the coordinate plane $xOy$, with all vertices lying on the graph of the function $y = \frac{1}{x}$. The abscissas of points $A$ and $D$ are negative, the abscissas of points $B$ and $C$ are positive, and $x_B < x_C$. The segment $AC$ passes through the origin. If the coordina... | [
"no_valid_answer_found"
] | [
"A convex quadrilateral $ABCD$ is defined on the coordinate plane $xOy$, with all vertices lying on the graph of the function $y = \\frac{1}{x}$. The abscissas of points $A$ and $D$ are negative, the abscissas of points $B$ and $C$ are positive, and $x_B < x_C$. The segment $AC$ passes through the origin. If the co... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad06e87f98b248f4fb742ae3ca266ae016fa14ce9386a798e5747ad8ed15f1e8 | A convex quadrilateral $ABCD$ is divided by its diagonals into four triangles. If the incircles of each of the four triangles have equal radii, find the ratio of the perimeter of the quadrilateral to the length of one of its sides. | [
"4"
] | [
"A convex quadrilateral $ABCD$ is divided by its diagonals into four triangles. If the incircles of each of the four triangles have equal radii, find the ratio of the perimeter of the quadrilateral to the length of one of its sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e933db798475b506f52a05b9fdc0757bbc0b0de2c54d21363ebb102230179dd | A convex quadrilateral $ABCD$ is given on the plane. Denote by $K, L, M, N$ the midpoints of its sides $AB$, $BC$, $CD$, $DA$, respectively. Calculate the maximum possible value of $\frac{KL + KM + KN}{AC + BD}$, where $AC$ and $BD$ are the diagonals of the quadrilateral $ABCD$. | [
"1"
] | [
"A convex quadrilateral $ABCD$ is given on the plane. Denote by $K, L, M, N$ the midpoints of its sides $AB$, $BC$, $CD$, $DA$, respectively. Calculate the maximum possible value of $\\frac{KL + KM + KN}{AC + BD}$, where $AC$ and $BD$ are the diagonals of the quadrilateral $ABCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0c430d7e07bed1304c061d9c29f7d757d3dcc966c1c0838ba2bf6e7c52372ef | A convex quadrilateral $ABCD$ is given. Let $n$ be the number of pairs of parallel lines dividing the broken lines $ABC$ and $ADC$ into the same ratio (counting from $A$). Find the minimum possible value of $n$. | [
"1"
] | [
"A convex quadrilateral $ABCD$ is given. Let $n$ be the number of pairs of parallel lines dividing the broken lines $ABC$ and $ADC$ into the same ratio (counting from $A$). Find the minimum possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-010d12d267bcdacdcd1818e92f203162ccbd714f33eccb1c2b1aab5ef8108a12 | A convex quadrilateral $ABCD$ is inscribed in a circle $k$. The rays $DA$ and $CB$ meet at a point $E$ such that $CD^2 = AD \cdot ED$. The perpendicular to $ED$ at $A$ intersects $k$ again at point $F$. If the circumcenter of $\triangle ABE$ lies on $ED$, find the ratio $\frac{AD}{CF}$. | [
"1"
] | [
"A convex quadrilateral $ABCD$ is inscribed in a circle $k$. The rays $DA$ and $CB$ meet at a point $E$ such that $CD^2 = AD \\cdot ED$. The perpendicular to $ED$ at $A$ intersects $k$ again at point $F$. If the circumcenter of $\\triangle ABE$ lies on $ED$, find the ratio $\\frac{AD}{CF}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6f0816395323394d5aaed5f41bcd54ee7c8ef5daace976ad805f858b6fec0d62 | A convex quadrilateral $ABCD$ is inscribed in a circle and circumscribes another circle. The lengths of its sides $AB$, $BC$, $CD$, and $DA$ are integers whose product is a perfect square. If the side lengths are $a$, $b$, $c$, and $d$ respectively, find the smallest possible value of $a + b + c + d$ given that $a < b ... | [
"18"
] | [
"A convex quadrilateral $ABCD$ is inscribed in a circle and circumscribes another circle. The lengths of its sides $AB$, $BC$, $CD$, and $DA$ are integers whose product is a perfect square. If the side lengths are $a$, $b$, $c$, and $d$ respectively, find the smallest possible value of $a + b + c + d$ given that $a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
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