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-de5869bbae4d0e20ac9f2c7f6252f76ece4f09a2613c47a755f7dde80b58aab2 | An entire arc of a circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. | [
"y = \\frac{b}{2}"
] | [
"An entire arc of a circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-36e773f497b64bc7f334143fe7194dafe20dc8bd62e4db8c16be46815a664864 | An entire arc of a circle is drawn through the vertices \( A \) and \( C \) of the rectangle \( ABCD \) lying inside the rectangle. Draw a line parallel to \( AB \) intersecting \( BC \) at point \( P \), \( AD \) at point \( Q \), and the arc \( AC \) at point \( R \) so that the sum of the areas of the figures \( AQR... | [
"\\frac{b}{2}"
] | [
"An entire arc of a circle is drawn through the vertices \\( A \\) and \\( C \\) of the rectangle \\( ABCD \\) lying inside the rectangle. Draw a line parallel to \\( AB \\) intersecting \\( BC \\) at point \\( P \\), \\( AD \\) at point \\( Q \\), and the arc \\( AC \\) at point \\( R \\) so that the sum of the ar... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-ab1cb5c3d3600a2880bc09b98f681334c5b73cb215963f6689909b545424e4c7 | An entity A disappears with probability 3/8, or produces two entities A with probability 5/8. What is the probability that A produces an infinite descent of entities? | [
"\\(\\frac{2}{5}\\)"
] | [
"An entity A disappears with probability 3/8, or produces two entities A with probability 5/8. What is the probability that A produces an infinite descent of entities?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-a6711ae0698cf62bf68a7e96fdd57808e2e657f8c4e1d24439abfab873dd6215 | An entity A has a probability of 3/8 to disappear and a probability of 5/8 to produce two new entities A. What is the probability that A produces an infinite descent of entities? Explanation: The problem statement is extracted directly from the forum post and rephrased for clarity. The context of the probability equati... | [
"\\(\\frac{2}{5}\\)"
] | [
"An entity A has a probability of 3/8 to disappear and a probability of 5/8 to produce two new entities A. What is the probability that A produces an infinite descent of entities?\n\nExplanation: The problem statement is extracted directly from the forum post and rephrased for clarity. The context of the probabilit... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-62b98b51e904410112cafb2ad470c5ea2487c6020f16a16cce9fe8e33e28d402 | An entomologist places a centipede with 100 feet on a scale, but notes that only 80 of its feet touch the scale; the rest are raised in the air. If the scale reads 0.16 grams, what is the true weight of the centipede in grams? | [
"0.16 grams"
] | [
"An entomologist places a centipede with 100 feet on a scale, but notes that only 80 of its feet touch the scale; the rest are raised in the air. If the scale reads 0.16 grams, what is the true weight of the centipede in grams?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45305257410a710b90438cc71110a2433fbf60962a45093f0c362161557b42c0 | An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more? | [
"\\(\\frac{1}{2}\\)"
] | [
"An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\\$20$ or more?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-018a7177b1d3458b2fccaac2948d363a4c58cdd18d764c7df9ea0f90bffe51e5 | An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20 or more? | [
"\\(\\frac{1}{2}\\)"
] | [
"An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20 or more?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e9849f2863611286b7b7614105e29fa4003384e9cd60238f3152e28cf6e2a513 | An envelope contains ten bills: $2$ ones, $2$ fives, $2$ tens, $2$ twenties, and $2$ thirties. Two bills are drawn at random without replacement. What is the probability that their sum is $50$ or more? | [
"\\(\\frac{32}{45}\\)"
] | [
"An envelope contains ten bills: $2$ ones, $2$ fives, $2$ tens, $2$ twenties, and $2$ thirties. Two bills are drawn at random without replacement. What is the probability that their sum is $50$ or more?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18b60ce4d1cdac29b5476d6eefebbb672acf48b5c0738ea3807ad164cf4a94ed | An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution, and exactly 27 of the new employees will be able to monitor both. (These 27 are included in the 85 and 73 mentioned abov... | [
"131"
] | [
"An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution, and exactly 27 of the new employees will be able to monitor both. (These 27 are included in the 85 and 73 mentioned... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-068a05f2de564d5496553e92547efdddc11996bf7e4020669dd9cc477963d385 | An equal number of juniors and seniors responded to the question "Do you like math," each responding with Yes or No. If 70% of those who said Yes were seniors and 80% of those who said No were juniors, then what percentage of the seniors polled said Yes? | [
"84%"
] | [
"An equal number of juniors and seniors responded to the question \"Do you like math,\" each responding with Yes or No. If 70% of those who said Yes were seniors and 80% of those who said No were juniors, then what percentage of the seniors polled said Yes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-589536152c94dbb740878069f9d52a21159a972f3160c35aa986f39afae0096a | An equal number of novels and textbooks are in hard covers; $\frac{2}{5}$ of the novels and $\frac{3}{4}$ of the textbooks are in hard covers. What fraction of the total number of books is in hard cover? | [
"\\(\\frac{12}{23}\\)"
] | [
"An equal number of novels and textbooks are in hard covers; $\\frac{2}{5}$ of the novels and $\\frac{3}{4}$ of the textbooks are in hard covers. What fraction of the total number of books is in hard cover?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-dee6d8dd8962a917ad073e8686e0320f3d24bce5d55f290ba55ec17281d744d6 | An equation \(x^4 - 16x^3 + 94x^2 + Ax + B = 0\) has two double roots. Find \(2(A + B)\). | [
"-30"
] | [
"An equation \\(x^4 - 16x^3 + 94x^2 + Ax + B = 0\\) has two double roots. Find \\(2(A + B)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-feccc0acd5cd49c5e9d4ba3cdd8691535cb6d346afbc27bb9924cccff240a9c0 | An equiangular $24$-gon has integer sides. If its $1$st side equals $10$, its $5$th side equals $13$, and its $13$th side equals $7$, what is the length of its $17$th side? (assume the sides are numbered consecutively) | [
"16"
] | [
"An equiangular $24$-gon has integer sides. If its $1$st side equals $10$, its $5$th side equals $13$, and its $13$th side equals $7$, what is the length of its $17$th side? (assume the sides are numbered consecutively)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-6cbc534741da0a41eee75b5a9d225431ec7e9c303da8c1ef7621725a5178d9a8 | An equiangular 12-gon has side lengths that alternate between 2 and $\sqrt{3}$. Find the area of the circumscribed circle of this 12-gon. | [
"\\(\\pi (9\\sqrt{2}+5\\sqrt{6}+7\\sqrt{3}+14)\\)"
] | [
"An equiangular 12-gon has side lengths that alternate between 2 and $\\sqrt{3}$. Find the area of the circumscribed circle of this 12-gon."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e10f3e628bbd0f5aa433c1a2d87242d1dd73abbba026140eef513517fbf8d8a | An equiangular 24-gon has integer sides. If its 1st side equals 10, its 5th side equals 13, and its 13th side equals 7, what is the length of its 17th side? (assume the sides are numbered consecutively) | [
"4"
] | [
"An equiangular 24-gon has integer sides. If its 1st side equals 10, its 5th side equals 13, and its 13th side equals 7, what is the length of its 17th side? (assume the sides are numbered consecutively)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-ad9686254d67981f73c660bee9cf3b03ad4fe5b5e8a70db021304296862e64e6 | An equiangular 24-gon has integer sides. If its 1st side equals 10, its 5th side equals 13, and its 13th side equals 7, what is the length of its 17th side? (assume the sides are numbered consecutively). | [
"10"
] | [
"An equiangular 24-gon has integer sides. If its 1st side equals 10, its 5th side equals 13, and its 13th side equals 7, what is the length of its 17th side? (assume the sides are numbered consecutively)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f78418256f7ad7b42b4252bb66e977f7b0b8e7589f3d6f7f7962070b3e032353 | An equiangular hexagon has consecutive side lengths of $3, 4, 6$, and $6$. The area of this hexagon can be written as $\frac{a\sqrt{b}}{c}$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Find $a + b + c$. | [
"130"
] | [
"An equiangular hexagon has consecutive side lengths of $3, 4, 6$, and $6$. The area of this hexagon can be written as $\\frac{a\\sqrt{b}}{c}$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Find $a + b + c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-2356751f572a35cfbd9bb5b4fbaf092f8ea42eb28d37c3634d611914d515f109 | An equiangular hexagon has four consecutive sides of lengths $5$, $3$, $6$, and $7$ in this order. Find the lengths of the other two sides. | [
"\\(a + b = 1\\)"
] | [
"An equiangular hexagon has four consecutive sides of lengths $5$, $3$, $6$, and $7$ in this order. Find the lengths of the other two sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-071aca6d6471435158c1e397842a06bc83997fddc8a6f7b131756dfab40f4cc1 | An equiangular hexagon has four consecutive sides with lengths $5, 3, 6,$ and $7$. Determine the lengths of the other two sides. | [
"1 and 8"
] | [
"An equiangular hexagon has four consecutive sides with lengths $5, 3, 6,$ and $7$. Determine the lengths of the other two sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 15
} | 18 | 0 | 0 |
omr-ws-3e8b69e30db629cbfe6b27e0d2692cdb9e2c93f75e221f5e16b8eefbdadee3a4 | An equiangular hexagon with side lengths \( s_1, s_1, s_1, s_1, s_2, s_2 \) is inscribed in a circle. If the circle's area is \( \frac{25\pi}{2} \), what is the minimum value of \( 8s_1^2 + 2s_2^2 \)? | [
"\\(\\frac{135}{4}\\)"
] | [
"An equiangular hexagon with side lengths \\( s_1, s_1, s_1, s_1, s_2, s_2 \\) is inscribed in a circle. If the circle's area is \\( \\frac{25\\pi}{2} \\), what is the minimum value of \\( 8s_1^2 + 2s_2^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dccdcd69eb9242986592859016d71c6694da79aef77ba3d0cd1aba9b0d319fe | An equiangular octagon \( ABCDEFGH \) has sides of lengths \( 2, 2\sqrt{2}, 4, 4\sqrt{2}, 6, 7, 7, 8 \). Given that \( AB = 8 \), find the length of \( EF \). | [
"6"
] | [
"An equiangular octagon \\( ABCDEFGH \\) has sides of lengths \\( 2, 2\\sqrt{2}, 4, 4\\sqrt{2}, 6, 7, 7, 8 \\). Given that \\( AB = 8 \\), find the length of \\( EF \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 23
} | 23 | 0 | 0 |
omr-ws-68c0940b98701db7a40361192e9489324fff1d85b4cb014d8eb47f63fc21173a | An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \in \overline{AB}$, $Q \in \overline{AC}$, and $N, P \in \overline{BC}$. Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}$. Denote by $\ell$ the angle bisector of $\angle MSQ$. If $O$ is the circumcenter of ... | [
"2"
] | [
"An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \\in \\overline{AB}$, $Q \\in \\overline{AC}$, and $N, P \\in \\overline{BC}$. Let $S$ be the intersection of $\\overleftrightarrow{MN}$ and $\\overleftrightarrow{PQ}$. Denote by $\\ell$ the angle bisector of $\\angle MSQ$. If $O$ is the c... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-313b9c79150cccad2a28aba1fc2b13e891218babbcc5810cf187075b5a42d0d9 | An equilateral pentagon in three-dimensional space has four right angles. What can be its fifth angle? | [
"\\(180^\\circ\\)"
] | [
"An equilateral pentagon in three-dimensional space has four right angles. What can be its fifth angle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-c102efe52caba20c085df6133f9893984e20208732b6c02c7a2b286e5346117e | An equilateral trapezoid is circumscribed about a circle. The lateral side of the trapezoid is \( l \), and one of its bases is equal to \( a \). Find the area of the trapezoid. | [
"l \\sqrt{a(2l - a)}"
] | [
"An equilateral trapezoid is circumscribed about a circle. The lateral side of the trapezoid is \\( l \\), and one of its bases is equal to \\( a \\). Find the area of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fae0a8417d0d126ae7eb6e2be1fea317c501c0a4bd9a5930a0887d54bdbc68c0 | An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of side length $1$. A path is a continuous path starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side they share, without passing through any sm... | [
"\\( n^2 \\)"
] | [
"An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of side length $1$. A path is a continuous path starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side they share, without passing through a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 8
} | 26 | 0 | 0 |
omr-ws-45d8c43df04c0e28071ce76911bb6eabc9a71f4e5feebe2b92edd2ecc8e53f1d | An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of the small triangles that can be chosen so that no two of them lie on a line parallel to any of the sides of triangle $ABC$? | [
"10"
] | [
"An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of the small triangles that can be chosen so that no two of them lie on a line parallel to any of the sides of triangle $ABC$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-b6fe0d8bbf3e50a581495c5b581221277ac26709975ed67d9c7442a30d648b8b | An equilateral triangle $ABC$ is inscribed in a circle of radius $R$. The side $BC$ is divided into three equal parts and a straight line is drawn through the point of division which is closest to $C$. The straight line passes through the vertex $A$ and cuts the circle at a point $D$. Find the perimeter of the triangle... | [
"$\\frac{R\\sqrt 3\\left(\\sqrt 7+5\\right)}{\\sqrt 7}$"
] | [
"An equilateral triangle $ABC$ is inscribed in a circle of radius $R$. The side $BC$ is divided into three equal parts and a straight line is drawn through the point of division which is closest to $C$. The straight line passes through the vertex $A$ and cuts the circle at a point $D$. Find the perimeter of the tri... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c48a035a9d312fbd927fa6172ba31ec64a7af052a1dfadfc3dd14a8f1c87fb08 | An equilateral triangle $ABC$ is inscribed in a circle with radius 6. Find the area of the region enclosed by $AB, AC,$ and the minor arc $BC$. | [
"\\( 6(3\\sqrt{3} + 2\\pi) \\)"
] | [
"An equilateral triangle $ABC$ is inscribed in a circle with radius 6. Find the area of the region enclosed by $AB, AC,$ and the minor arc $BC$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-78da4297f1a354453bae363bee7f4350540601e90734744c04bb445df7ea5ee4 | An equilateral triangle $ABC$ with side $a$ is given in a plane $P$. A line segment $A S = a$ is laid off on a perpendicular to plane $P$ at point $A$. Find the tangent of the acute angle between $AB$ and $SC$ and the shortest distance between them. | [
"$\\tan\\varphi =\\sqrt 7$, $a\\sqrt {3/7}$"
] | [
"An equilateral triangle $ABC$ with side $a$ is given in a plane $P$. A line segment $A S = a$ is laid off on a perpendicular to plane $P$ at point $A$. Find the tangent of the acute angle between $AB$ and $SC$ and the shortest distance between them."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cbe6d3728c1af95372d14b8d2c3c683c08797762105c4b6e59e3c223e0d84f0 | An equilateral triangle $ABC$ with sides of integer length $N$ is divided into smaller equilateral triangular cells of side length $1$. A continuous route starts inside the cell containing vertex $A$ and moves from one cell to another through shared edges, without visiting any cell more than once. Let $f(N)$ be the gre... | [
"100"
] | [
"An equilateral triangle $ABC$ with sides of integer length $N$ is divided into smaller equilateral triangular cells of side length $1$. A continuous route starts inside the cell containing vertex $A$ and moves from one cell to another through shared edges, without visiting any cell more than once. Let $f(N)$ be th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a38ced053464059ec5d4a7152f0ebdb38d64e84cbcf1086c1e49c66c5dbd3f14 | An equilateral triangle $ABC$ with sides of integer length $N$ is divided into smaller equilateral triangular cells of side length $1$. A continuous route starts inside the cell containing vertex $A$ and moves from one cell to another through shared edges, without visiting any cell more than once. Let $f(N)$ be the lar... | [
"100"
] | [
"An equilateral triangle $ABC$ with sides of integer length $N$ is divided into smaller equilateral triangular cells of side length $1$. A continuous route starts inside the cell containing vertex $A$ and moves from one cell to another through shared edges, without visiting any cell more than once. Let $f(N)$ be th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-81ee81f01fe7824a871986c21fcb722f2d5bb29a116ec3fb9036f03a789b617b | An equilateral triangle $DCE$ is placed outside a square $ABCD$. The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ is denoted as $S$. Calculate the length of $CM$ if $AB = 2$. | [
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"An equilateral triangle $DCE$ is placed outside a square $ABCD$. The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ is denoted as $S$. Calculate the length of $CM$ if $AB = 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-887593cbf3261182a50687e20081719fce692111e0853ad1876a70c8a41ffbdb | An equilateral triangle $T$ has a side length of $10\sqrt{3}$ and a height of $15$. A circle $E$ is inscribed such that it touches the base of the triangle and the altitude from the opposite vertex. If the radius of the circle $E$ is $x$, find the value of $x$. | [
"\\dfrac{15 - 5\\sqrt{3}}{2}"
] | [
"An equilateral triangle $T$ has a side length of $10\\sqrt{3}$ and a height of $15$. A circle $E$ is inscribed such that it touches the base of the triangle and the altitude from the opposite vertex. If the radius of the circle $E$ is $x$, find the value of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-89bdd2dab73d154ae248f408a47840931ae1f9a378eb86028d29dcfc811e47ca | An equilateral triangle $T$ with side length $111$ is partitioned into smaller equilateral triangles with side length $1$ using lines parallel to the sides of $T$. Every point except the center of $T$ is marked. A set of marked points is called $\textit{linear}$ if the points lie on a line parallel to a side of $T$. In... | [
"3^{111}"
] | [
"An equilateral triangle $T$ with side length $111$ is partitioned into smaller equilateral triangles with side length $1$ using lines parallel to the sides of $T$. Every point except the center of $T$ is marked. A set of marked points is called $\\textit{linear}$ if the points lie on a line parallel to a side of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-a2b8d77dcc330a430be4e95cd7a303ff1a14f02fef7286c6cedcd7495e5d22b7 | An equilateral triangle $T$ with side length $2022$ is divided into smaller equilateral triangles with side length $1$. Ivan colors some of these smaller triangles black. Peter selects a set $S$ of these smaller triangles, and Ivan informs Peter of the parity of the number of black triangles in $S$. Peter must then det... | [
"12120"
] | [
"An equilateral triangle $T$ with side length $2022$ is divided into smaller equilateral triangles with side length $1$. Ivan colors some of these smaller triangles black. Peter selects a set $S$ of these smaller triangles, and Ivan informs Peter of the parity of the number of black triangles in $S$. Peter must the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ca200183762eb01c0ef6664125488456b337a245fc04988b0418857a1a3cb9d | An equilateral triangle $T$ with side length $\ell$ is divided into 23 smaller equilateral triangles. Among these, 16 have side length 1, 6 have side length 2, and one has side length $a$ where $a \neq 1$ and $a \neq 2$. Determine all possible values of $\ell$ and $a$. | [
"\\(\\ell = 11, a = 9\\) or \\(\\ell = 7, a = 3\\)"
] | [
"An equilateral triangle $T$ with side length $\\ell$ is divided into 23 smaller equilateral triangles. Among these, 16 have side length 1, 6 have side length 2, and one has side length $a$ where $a \\neq 1$ and $a \\neq 2$. Determine all possible values of $\\ell$ and $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 7
} | 8 | 0 | 0 |
omr-ws-7ab4748365941f95e20ee370fc2b0b197e78d3d768df805fd03514b096541892 | An equilateral triangle $\Delta$ is partitioned into $n$ smaller equilateral triangles with side lengths $a_1, a_2, \dots, a_n$. Let $S$ be the set of all possible values of $\sum_{1 \le i < j \le n} \epsilon_i \epsilon_j a_i a_j$ where $\epsilon_i \in \{-1, 1\}$ for all $i$. Find the sum of all elements in $S$. | [
"0"
] | [
"An equilateral triangle $\\Delta$ is partitioned into $n$ smaller equilateral triangles with side lengths $a_1, a_2, \\dots, a_n$. Let $S$ be the set of all possible values of $\\sum_{1 \\le i < j \\le n} \\epsilon_i \\epsilon_j a_i a_j$ where $\\epsilon_i \\in \\{-1, 1\\}$ for all $i$. Find the sum of all element... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b1f1dcf75f481922e76b50814a27a82e89ebdbdf79f2439af16fc8325816ede | An equilateral triangle $\Delta$ of side length $L > 0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\Delta$, such that each unit equilateral triangle has sides parallel to $\Delta$, but with opposite orientation. Find the largest possible valu... | [
"15"
] | [
"An equilateral triangle $\\Delta$ of side length $L > 0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\\Delta$, such that each unit equilateral triangle has sides parallel to $\\Delta$, but with opposite orientation. Find the largest possi... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-344c4e0bff1b2d835ace50f2c15cbf310852c7a0b2a8d5faf142fb57ddecfdd1 | An equilateral triangle $\mathcal{T}$ with side length 111 is divided into smaller equilateral triangles with side length 1. The vertices of these small triangles, except the center of $\mathcal{T}$, are marked. A set of marked points is called [i]linear[/i] if: - The marked points lie on a line $\ell$ parallel to one ... | [
"3^{110}"
] | [
"An equilateral triangle $\\mathcal{T}$ with side length 111 is divided into smaller equilateral triangles with side length 1. The vertices of these small triangles, except the center of $\\mathcal{T}$, are marked. A set of marked points is called [i]linear[/i] if:\n- The marked points lie on a line $\\ell$ paralle... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-9e4a6c1bc17ce12ebf15614092f2cfb5efc8d0b02182352409c5f284d8141cde | An equilateral triangle $\triangle ABC$ is divided into 4 triangles of equal area: three congruent triangles $\triangle ABX$, $\triangle BCY$, $\triangle CAZ$, and a smaller equilateral triangle $\triangle XYZ$. If the side length of $\triangle ABC$ is $12$, find the radius of the circle that passes through points $X$,... | [
"\\sqrt{21}"
] | [
"An equilateral triangle $\\triangle ABC$ is divided into 4 triangles of equal area: three congruent triangles $\\triangle ABX$, $\\triangle BCY$, $\\triangle CAZ$, and a smaller equilateral triangle $\\triangle XYZ$. If the side length of $\\triangle ABC$ is $12$, find the radius of the circle that passes through ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-fae21fa7217e198ff00f00cb420e6e68b9b9faba271246c37c53c84c7d0e9da4 | An equilateral triangle $\triangle PQR$ is rotated $30^{\circ}$ counterclockwise about its center to form its image $\triangle XYZ$. What is the ratio of the overlap of these two triangles and the area of one of the triangles? | [
"\\dfrac{1}{2}"
] | [
"An equilateral triangle $\\triangle PQR$ is rotated $30^{\\circ}$ counterclockwise about its center to form its image $\\triangle XYZ$. What is the ratio of the overlap of these two triangles and the area of one of the triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-760d7d6f8de03a10ec97e19240223e835d6bcd270914e7ec69f4c4b4f7a46306 | An equilateral triangle ABC has been folded such that vertex A now rests at A' on BC. If BA' = 1 and A'C = 2, find the length of the crease. | [
"\\(\\frac{7\\sqrt{21}}{20}\\)"
] | [
"An equilateral triangle ABC has been folded such that vertex A now rests at A' on BC. If BA' = 1 and A'C = 2, find the length of the crease."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3e0148f1059ee35294dcc196b9412a5dcd59071e6d91d794e14656b9617d3a8e | An equilateral triangle ABC has side length 1. Point D lies on the exterior of triangle ABC such that triangle BCD is an isosceles right triangle. Compute the minimum value of AD. | [
"\\( \\frac{\\sqrt{6} - \\sqrt{2}}{2} \\)"
] | [
"An equilateral triangle ABC has side length 1. Point D lies on the exterior of triangle ABC such that triangle BCD is an isosceles right triangle. Compute the minimum value of AD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 17
} | 30 | 0 | 0 |
omr-ws-250d4a84f66fa1945d0a655371640421bb289d9d12082afa6b25e22662097cfe | An equilateral triangle ABC has side length 2. A square PQRS is such that P lies on AB, Q lies on BC, and R and S lie on AC. The points P, Q, R, and S move so that P, Q, and R always remain on the sides of the triangle and S moves from AC to AB through the interior of the triangle. If the points P, Q, R, and S always f... | [
"4 - 2\\sqrt{3}"
] | [
"An equilateral triangle ABC has side length 2. A square PQRS is such that P lies on AB, Q lies on BC, and R and S lie on AC. The points P, Q, R, and S move so that P, Q, and R always remain on the sides of the triangle and S moves from AC to AB through the interior of the triangle. If the points P, Q, R, and S alw... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-e883104a20376d18b28945f478a9c11277817a320a160e161e6be50bcc32942d | An equilateral triangle ABC has side length 6. A circle is tangent to BC at D and intersects AC at E, where E is one of the trisection points of side AC. Find the length of CD. | [
"2"
] | [
"An equilateral triangle ABC has side length 6. A circle is tangent to BC at D and intersects AC at E, where E is one of the trisection points of side AC. Find the length of CD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-25d97b18ca591a61ea95e6a99ad8159eb60bb1ee81ea39adbf10fc63da9eaec4 | An equilateral triangle ABC is inscribed in a circle. The point D is on the minor arc AC of the circle. The line CD meets the line BA at E and the line AD meets the line BC at F. The length of CF is $21\sqrt{3}$ and the length of AE is $84\sqrt{3}$. Find the radius of the circle. | [
"28"
] | [
"An equilateral triangle ABC is inscribed in a circle. The point D is on the minor arc AC of the circle. The line CD meets the line BA at E and the line AD meets the line BC at F. The length of CF is $21\\sqrt{3}$ and the length of AE is $84\\sqrt{3}$. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-25662fac62c574281d8f765a885a5e5223e7f20be3e5a5b44fa4ad60a264e18d | An equilateral triangle ABC is inscribed in a triangle KLM such that the vertices A, B, and C belong to the sides KL, LM, and MK, respectively. It is also given that KA = LB = MC. If the side length of triangle KLM is denoted as $s$, find the ratio of the side length of triangle ABC to $s$. | [
"\\dfrac{1}{2}"
] | [
"An equilateral triangle ABC is inscribed in a triangle KLM such that the vertices A, B, and C belong to the sides KL, LM, and MK, respectively. It is also given that KA = LB = MC. If the side length of triangle KLM is denoted as $s$, find the ratio of the side length of triangle ABC to $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5993667809ac279c1822edd1b3b47412c2eb52eceabb2325b38312b326185ab2 | An equilateral triangle ABC with side length x has the same area as a regular hexagon DEFGHI with side length y. Find x/y. Express as a simplified fraction in the form a/b. | [
"\\(\\sqrt{6}\\)"
] | [
"An equilateral triangle ABC with side length x has the same area as a regular hexagon DEFGHI with side length y. Find x/y. Express as a simplified fraction in the form a/b."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-97e3d261d667f96e8a7f7562a1728c62b1f53a0d601d211bcc528f1d1391d9ca | An equilateral triangle ABC with sides of integer length $N$ is divided into smaller equilateral triangular cells of side length 1. A continuous route starts inside the cell containing vertex A and moves from one cell to another by crossing through shared edges, without visiting any cell more than once. Find the greate... | [
"91"
] | [
"An equilateral triangle ABC with sides of integer length $N$ is divided into smaller equilateral triangular cells of side length 1. A continuous route starts inside the cell containing vertex A and moves from one cell to another by crossing through shared edges, without visiting any cell more than once. Find the g... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-fcf7fcd7e32c1bb908f3d5d1d41f258b37fddd19b5fef6fe248ea743d5d88d8f | An equilateral triangle ABP is drawn inside a square ABCD. Point E is the midpoint of CD, and a circle is drawn with diameter EP. Find the ratio of the area inside the square but outside the circle and triangle to the total area of the square. | [
"\\dfrac{16 - 4\\sqrt{3} - 7\\pi + 4\\sqrt{3}\\pi}{16}"
] | [
"An equilateral triangle ABP is drawn inside a square ABCD. Point E is the midpoint of CD, and a circle is drawn with diameter EP. Find the ratio of the area inside the square but outside the circle and triangle to the total area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e1e69b10aca4ba55d38c68d61e268407f7eb52f7ad0c9cc7ce495d4cd8a6887 | An equilateral triangle \( ABC \) is partitioned into \( n^2 \) smaller equilateral triangles. Let \( P(n) \) be the total number of vertices in this partition, including the vertices of the original triangle. Find \( P(5) \). | [
"21"
] | [
"An equilateral triangle \\( ABC \\) is partitioned into \\( n^2 \\) smaller equilateral triangles. Let \\( P(n) \\) be the total number of vertices in this partition, including the vertices of the original triangle. Find \\( P(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-a339970c685cc9adcd59f24c4a48dc5c7279182088537d29de5202a18aa628b5 | An equilateral triangle \( ABC \) of side length 1 has center \( O \). A point \( P \) is selected on side \( AB \) such that \( \frac{BP}{AP} = 3 \). Let \( Q \) be a point on \( AC \) such that \( OP = OQ \). The maximum possible area of quadrilateral \( APOQ \) can be expressed as \( \frac{\sqrt{m}}{n} \) for positi... | [
"15"
] | [
"An equilateral triangle \\( ABC \\) of side length 1 has center \\( O \\). A point \\( P \\) is selected on side \\( AB \\) such that \\( \\frac{BP}{AP} = 3 \\). Let \\( Q \\) be a point on \\( AC \\) such that \\( OP = OQ \\). The maximum possible area of quadrilateral \\( APOQ \\) can be expressed as \\( \\frac{... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c43030d8bcdd84b008cc71a13f2c5d423832971c9409111e3f18b8c6b5b60410 | An equilateral triangle \( ABC \) of side length 1 is given, and \( S \) is the set of all its interior points (including the boundary points). For any point \( M \) in \( S \), let \( a(M) \), \( b(M) \), and \( c(M) \) denote the distances from \( M \) to \( BC \), \( CA \), and \( AB \) respectively. Define the func... | [
"\\(\\text{Minimum value: } -\\frac{\\sqrt{3}}{2}, \\text{Maximum value: } \\frac{\\sqrt{3}}{2}\\)"
] | [
"An equilateral triangle \\( ABC \\) of side length 1 is given, and \\( S \\) is the set of all its interior points (including the boundary points). For any point \\( M \\) in \\( S \\), let \\( a(M) \\), \\( b(M) \\), and \\( c(M) \\) denote the distances from \\( M \\) to \\( BC \\), \\( CA \\), and \\( AB \\) re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d860b322a318561d1acd6ce7bf4c430447f9bc789b784e8031a315dafd1f4931 | An equilateral triangle \( BDE \) is constructed on the diagonal \( BD \) of the square \( ABCD \), and the point \( C \) is located inside the triangle \( BDE \). Let \( M \) be the midpoint of \( BE \). Find the angle between the lines \( MC \) and \( DE \). | [
"75^\\circ"
] | [
"An equilateral triangle \\( BDE \\) is constructed on the diagonal \\( BD \\) of the square \\( ABCD \\), and the point \\( C \\) is located inside the triangle \\( BDE \\). Let \\( M \\) be the midpoint of \\( BE \\). Find the angle between the lines \\( MC \\) and \\( DE \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-75e46fe91f6e80550659663f70138171b7037adb61aa16a8404de5d1e2df02d3 | An equilateral triangle \( T \) and a circle \( C \) are on the same plane. Each side length of \( T \) is \( 6\sqrt{3} \) and the radius of \( C \) is \( 2 \). The distance between the centers of \( T \) and \( C \) is \( 15 \). For every two points \( X \) on \( T \) and \( Y \) on \( C \), let \( M(X, Y) \) be the m... | [
"\\(\\pi + 15\\sqrt{3}\\)"
] | [
"An equilateral triangle \\( T \\) and a circle \\( C \\) are on the same plane. Each side length of \\( T \\) is \\( 6\\sqrt{3} \\) and the radius of \\( C \\) is \\( 2 \\). The distance between the centers of \\( T \\) and \\( C \\) is \\( 15 \\). For every two points \\( X \\) on \\( T \\) and \\( Y \\) on \\( C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19c56308fea8c41df6da005bcafda07dccec3c0bb0e11f2796590f094d7b2700 | An equilateral triangle \( T \) is inscribed in circle \( A \), which has radius 10. Circle \( B \) with radius 3 is internally tangent to circle \( A \) at one vertex of \( T \). Circles \( C \) and \( D \), both with radius 2, are internally tangent to circle \( A \) at the other two vertices of \( T \). Circles \( B... | [
"32"
] | [
"An equilateral triangle \\( T \\) is inscribed in circle \\( A \\), which has radius 10. Circle \\( B \\) with radius 3 is internally tangent to circle \\( A \\) at one vertex of \\( T \\). Circles \\( C \\) and \\( D \\), both with radius 2, are internally tangent to circle \\( A \\) at the other two vertices of ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 7
} | 60 | 0 | 0 |
omr-ws-8dcc0e5fda59f9d0cddad65d419ba9e8798b40bcf769307d9ad16251ace72a38 | An equilateral triangle \( T \) with side length \( 2022 \) is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures consisting of \( 4 \) equilateral unit triangles, which can be rotated by any angle \( k \cdot 60^\circ \) for \( k \in \{... | [
"1011"
] | [
"An equilateral triangle \\( T \\) with side length \\( 2022 \\) is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures consisting of \\( 4 \\) equilateral unit triangles, which can be rotated by any angle \\( k \\cdot 60^\\circ \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-daafc68ca36267817dbc7ae1d179e48d1a5039312796dff54bebb2184fc0eb22 | An equilateral triangle \( \triangle ABC \) is inscribed inside an isosceles triangle \( \triangle AXY \) such that \( AX = AY = \sqrt{28} \), \( XY = 2 \), and \( XB = YC = 4 \). Find the length of \( AB \). | [
"2\\sqrt{7} - 4"
] | [
"An equilateral triangle \\( \\triangle ABC \\) is inscribed inside an isosceles triangle \\( \\triangle AXY \\) such that \\( AX = AY = \\sqrt{28} \\), \\( XY = 2 \\), and \\( XB = YC = 4 \\). Find the length of \\( AB \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-c6a5eeb89317bcee80d6b0886b8e70c3c6ba9feee1f0f228a153d42658f6c7f0 | An equilateral triangle \( \triangle EQL \) is inscribed in a circle with radius 1. Find the area of the segment of the circle bounded by segment \( EQ \) and the minor arc \( EQ \). | [
"\\(\\frac{\\pi}{3} - \\frac{\\sqrt{3}}{4}\\)"
] | [
"An equilateral triangle \\( \\triangle EQL \\) is inscribed in a circle with radius 1. Find the area of the segment of the circle bounded by segment \\( EQ \\) and the minor arc \\( EQ \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-54eced49a2298f01902ee8668a6b2dad552f7da70b60a4f1273ede5fc646ca36 | An equilateral triangle \( \vartriangle ABC \) has a center \( O \) and side length \( 12\sqrt{3} \). The triangles \( \vartriangle AOB \), \( \vartriangle BOC \), and \( \vartriangle COA \) are colored red, blue, and green, respectively. A circle with radius 1 is randomly placed such that it is completely contained wi... | [
"276"
] | [
"An equilateral triangle \\( \\vartriangle ABC \\) has a center \\( O \\) and side length \\( 12\\sqrt{3} \\). The triangles \\( \\vartriangle AOB \\), \\( \\vartriangle BOC \\), and \\( \\vartriangle COA \\) are colored red, blue, and green, respectively. A circle with radius 1 is randomly placed such that it is c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-d903372e699ef7a1962b51e1b9b8c1e6fa94222182f950519b6923079a3f2b5a | An equilateral triangle \(ABC\) has a circumcircle \(\Gamma\) with center \(O\) and circumradius 10. Another circle \(\Gamma_1\) is drawn inside \(\Gamma\) such that it is tangential to radii \(OC\) and \(OB\) and circle \(\Gamma\). The radius of \(\Gamma_1\) can be expressed in the form \(a\sqrt{b} - c\), where \(a\),... | [
"53"
] | [
"An equilateral triangle \\(ABC\\) has a circumcircle \\(\\Gamma\\) with center \\(O\\) and circumradius 10. Another circle \\(\\Gamma_1\\) is drawn inside \\(\\Gamma\\) such that it is tangential to radii \\(OC\\) and \\(OB\\) and circle \\(\\Gamma\\). The radius of \\(\\Gamma_1\\) can be expressed in the form \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-a8eed47838773733fc68b710e2481966d2f58dedc977a126e557d1b5a1e4bb92 | An equilateral triangle \(ABC\) has a side length of 7. A ball starts at vertex \(A\), rolls through the interior of the triangle, bounces off side \(BC\), and settles at point \(P\). Given that \(|BP| = 3\) and \(|CP| = 5\), find the total distance that the ball travels. | [
"\\(2\\sqrt{10}\\)"
] | [
"An equilateral triangle \\(ABC\\) has a side length of 7. A ball starts at vertex \\(A\\), rolls through the interior of the triangle, bounces off side \\(BC\\), and settles at point \\(P\\). Given that \\(|BP| = 3\\) and \\(|CP| = 5\\), find the total distance that the ball travels."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33624147a16af504608eb798bd02e47735d5a337c36de006fbd54d24985f4c07 | An equilateral triangle \(ABC\) has a side length of 8. An interior point \(P\) inside the triangle is such that \(PA = 4\) and \(PB = 6\). Find the length of \(PC\). | [
"\\(\\sqrt{58 - 18\\sqrt{5}}\\)"
] | [
"An equilateral triangle \\(ABC\\) has a side length of 8. An interior point \\(P\\) inside the triangle is such that \\(PA = 4\\) and \\(PB = 6\\). Find the length of \\(PC\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1c34e9f689b7fc94ab0ea59f29ac98d2f3ba0863583225c17e09422f345da8be | An equilateral triangle \(ABC\) has circumcircle \(\Omega\) and incircle \(\omega\). Points \(P\) and \(Q\) are chosen on \(AC\) and \(AB\) respectively such that \(PQ\) is tangent to \(\omega\). Let \(X\) be the common point of circles \(\Omega_B = \text{C}(P, PB)\), \(\Omega_C = \text{C}(Q, QC)\), and \(\Omega\). If ... | [
"\\dfrac{2\\sqrt{3}}{3} s"
] | [
"An equilateral triangle \\(ABC\\) has circumcircle \\(\\Omega\\) and incircle \\(\\omega\\). Points \\(P\\) and \\(Q\\) are chosen on \\(AC\\) and \\(AB\\) respectively such that \\(PQ\\) is tangent to \\(\\omega\\). Let \\(X\\) be the common point of circles \\(\\Omega_B = \\text{C}(P, PB)\\), \\(\\Omega_C = \\te... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4503fe600988ee78c47fe27d7f933db72258926ddbdb434b3290e627a5af86b5 | An equilateral triangle \(ABC\) has its rightmost vertex \(B\) 2 units up from the x-axis. A point \(P\) is inside triangle \(ABC\) such that \(PA = 5\), \(PB = 12\), and \(PC = 13\). What are the coordinates of this equilateral triangle after it is rotated 60 degrees to the right around its leftmost vertex \(A\) at co... | [
"A(0,0), \\left(\\frac{\\sqrt{165}}{2} + \\sqrt{3}, 1 - \\frac{\\sqrt{495}}{2}\\right), (\\sqrt{165}, 2)"
] | [
"An equilateral triangle \\(ABC\\) has its rightmost vertex \\(B\\) 2 units up from the x-axis. A point \\(P\\) is inside triangle \\(ABC\\) such that \\(PA = 5\\), \\(PB = 12\\), and \\(PC = 13\\). What are the coordinates of this equilateral triangle after it is rotated 60 degrees to the right around its leftmost... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-7e51188b56cf1974ee668b6f50072fd786683e927b9489c00c894abeec714eab | An equilateral triangle \(ABC\) has side length \(AB = 1\) and \(K\) as its circumcircle. A square has one side parallel to \(BC\), with two vertices on \(AB\) and \(AC\), and the remaining two vertices on the circumcircle \(K\). What is the side length of the square? | [
"\\( \\frac{\\sqrt{3}}{3} \\)"
] | [
"An equilateral triangle \\(ABC\\) has side length \\(AB = 1\\) and \\(K\\) as its circumcircle. A square has one side parallel to \\(BC\\), with two vertices on \\(AB\\) and \\(AC\\), and the remaining two vertices on the circumcircle \\(K\\). What is the side length of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-ffa392a548b579ff20c9fa9b69c4e9f7825bff92f3aeb70198b20a0fd59e2a42 | An equilateral triangle \(ABC\) has side length \(a\). A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side \(a\) is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above ... | [
"\\dfrac{3\\sqrt{3}}{2}a^2"
] | [
"An equilateral triangle \\(ABC\\) has side length \\(a\\). A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side \\(a\\) is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-a73bbb6f8e81f63202d4ccabca7776f3f1fc2f6ae69ecc64f7f473744d7e6503 | An equilateral triangle \(ABC\) intersects a circle such that all sides of the triangle go through the circle. Points \(F\), \(G\), \(H\), \(J\), \(D\), and \(E\) all lie on both the circle and the triangle. \(G\) and \(F\) are on \(AC\) (with \(G\) closer to \(A\)) such that \(AG = 2\), \(GF = 16\), and \(FC = 3\). \(... | [
"\\( 2\\sqrt{7} \\)"
] | [
"An equilateral triangle \\(ABC\\) intersects a circle such that all sides of the triangle go through the circle. Points \\(F\\), \\(G\\), \\(H\\), \\(J\\), \\(D\\), and \\(E\\) all lie on both the circle and the triangle. \\(G\\) and \\(F\\) are on \\(AC\\) (with \\(G\\) closer to \\(A\\)) such that \\(AG = 2\\), ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-33d3f013e534e1c93adf117363538d546a1c89d35d6b5e7dcf2b876a7fea0b7c | An equilateral triangle \(ABC\) is circumscribed in a circle. Point \(D\) lies on the minor arc \(AC\). Given \(AD = 12\), \(CD = 6\), and \(AC\) and \(BD\) intersect at \(E\), find the length of \(BE\). | [
"14"
] | [
"An equilateral triangle \\(ABC\\) is circumscribed in a circle. Point \\(D\\) lies on the minor arc \\(AC\\). Given \\(AD = 12\\), \\(CD = 6\\), and \\(AC\\) and \\(BD\\) intersect at \\(E\\), find the length of \\(BE\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 8
} | 20 | 0 | 0 |
omr-ws-bc6eb2b6c6ab6ba7dc10d280f5849f22b4ab7e5f358a6d7306f2099d166d7bc8 | An equilateral triangle \(ABC\) is given. Point \(D\) is such that \(\angle BDC = 90^\circ\) and \(D\) and \(A\) lie in different half-planes with respect to line \(BC\). Point \(M\) is the midpoint of side \(AB\). Find the angle \(\angle BDM\). | [
"\\(30^\\circ\\)"
] | [
"An equilateral triangle \\(ABC\\) is given. Point \\(D\\) is such that \\(\\angle BDC = 90^\\circ\\) and \\(D\\) and \\(A\\) lie in different half-planes with respect to line \\(BC\\). Point \\(M\\) is the midpoint of side \\(AB\\). Find the angle \\(\\angle BDM\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 19
} | 37 | 0 | 0 |
omr-ws-661f4462f7c27094231afa94d83f55e5ff596b3e9c96d122ad5cfd403be2fc1a | An equilateral triangle \(ABC\) is given. Points \(D\), \(E\), and \(F\) are chosen such that \(D\) is on the extension of side \(AB\) beyond \(A\), \(E\) is on the extension of side \(BC\) beyond \(C\), and \(F\) is on the extension of side \(AC\) beyond \(C\). It is given that \(CF = AD\) and \(AC + EF = DE\). Find t... | [
"\\(120^\\circ\\)"
] | [
"An equilateral triangle \\(ABC\\) is given. Points \\(D\\), \\(E\\), and \\(F\\) are chosen such that \\(D\\) is on the extension of side \\(AB\\) beyond \\(A\\), \\(E\\) is on the extension of side \\(BC\\) beyond \\(C\\), and \\(F\\) is on the extension of side \\(AC\\) beyond \\(C\\). It is given that \\(CF = A... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d130bfc3607957b7c9d27829cf722a7d90b171005fc0210c3da1342b72853696 | An equilateral triangle \(ABC\) is inscribed in a circle \(O\) with radius 1. Let \(DA\), \(BE\), and \(CF\) be diameters of the circle. The intersection of \(CF\) and \(AB\) is \(G\). The line \(DG\) extended intersects the circle at point \(H\). Find the measure of angle \(HOF\). | [
"\\(60 - 2\\sin^{-1}\\left(\\frac{\\sqrt{21}}{14}\\right)\\)"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a circle \\(O\\) with radius 1. Let \\(DA\\), \\(BE\\), and \\(CF\\) be diameters of the circle. The intersection of \\(CF\\) and \\(AB\\) is \\(G\\). The line \\(DG\\) extended intersects the circle at point \\(H\\). Find the measure of angle \\(HOF\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdf41320eae9f21eaf4a18acc0ce46ebd918b7419dc27055f359f63ab9e0603e | An equilateral triangle \(ABC\) is inscribed in a circle \(\Omega\) and has an incircle \(\omega\). Points \(P\) and \(Q\) are chosen on sides \(AC\) and \(AB\), respectively, such that the segment \(PQ\) is tangent to \(\omega\). Circle \(\Omega_B\) has center \(P\) and passes through \(B\), and circle \(\Omega_C\) ha... | [
"\\dfrac{1}{3}"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a circle \\(\\Omega\\) and has an incircle \\(\\omega\\). Points \\(P\\) and \\(Q\\) are chosen on sides \\(AC\\) and \\(AB\\), respectively, such that the segment \\(PQ\\) is tangent to \\(\\omega\\). Circle \\(\\Omega_B\\) has center \\(P\\) and passes through \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-037c76741df04119a73d65453323c451966ff919b0526f481980d09499190f32 | An equilateral triangle \(ABC\) is inscribed in a circle \(\Omega\) and has an incircle \(\omega\). Points \(P\) and \(Q\) are on segments \(AC\) and \(AB\), respectively, such that \(PQ\) is tangent to \(\omega\). The circle \(\Omega_B\) has center \(P\) and radius \(PB\), and the circle \(\Omega_C\) has center \(Q\) ... | [
"\\dfrac{2\\sqrt{21}}{3}R"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a circle \\(\\Omega\\) and has an incircle \\(\\omega\\). Points \\(P\\) and \\(Q\\) are on segments \\(AC\\) and \\(AB\\), respectively, such that \\(PQ\\) is tangent to \\(\\omega\\). The circle \\(\\Omega_B\\) has center \\(P\\) and radius \\(PB\\), and the circ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ddb9ff2c85856187a22025a88c7106d2c436e9ca934a49e78ef8d51d9e6a3f28 | An equilateral triangle \(ABC\) is inscribed in a circle \(k\). Points \(M\) and \(N\) are chosen on sides \(AC\) and \(AB\) respectively such that \(AM = 2MC\) and \(AN = NB\). The ray \(MN\) meets the circle \(k\) at point \(P\). Calculate the value of \(\frac{PA}{PB} - \frac{PB}{PA}\). | [
"\\dfrac{1}{2}"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a circle \\(k\\). Points \\(M\\) and \\(N\\) are chosen on sides \\(AC\\) and \\(AB\\) respectively such that \\(AM = 2MC\\) and \\(AN = NB\\). The ray \\(MN\\) meets the circle \\(k\\) at point \\(P\\). Calculate the value of \\(\\frac{PA}{PB} - \\frac{PB}{PA}\\).... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-781ce88e3d9b9e95634511777543636f36dcbc9c0a820cfca8e1a89f7ef7f6b9 | An equilateral triangle \(ABC\) is inscribed in a circle. A second circle is tangent internally to the circumcircle at \(T\) and tangent to sides \(AB\) and \(AC\) at points \(P\) and \(Q\). Determine the length of the segment \(PQ\) if side \(BC\) has length \(12\). | [
"8"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a circle. A second circle is tangent internally to the circumcircle at \\(T\\) and tangent to sides \\(AB\\) and \\(AC\\) at points \\(P\\) and \\(Q\\). Determine the length of the segment \\(PQ\\) if side \\(BC\\) has length \\(12\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-bc7cb4220aabd13e08d8c76a39e4931756b76bfd1ed615de764d766e38840164 | An equilateral triangle \(ABC\) is inscribed in a circle. Let \(BD\) be a diameter of this circle. Points \(M\) and \(N\) are taken on sides \(AB\) and \(CB\) respectively, such that \(MN = AM + CN\). Calculate the measure of \(\angle MDN\) in degrees. | [
"60"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a circle. Let \\(BD\\) be a diameter of this circle. Points \\(M\\) and \\(N\\) are taken on sides \\(AB\\) and \\(CB\\) respectively, such that \\(MN = AM + CN\\). Calculate the measure of \\(\\angle MDN\\) in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ba01b6b2e8bbec68ae1b85b539bbc9c93343e62e03130753bacde560461c2720 | An equilateral triangle \(ABC\) is inscribed in a square with side length 1, with each vertex of the triangle on a different side of the square. Determine the greatest and smallest possible values of the side length of \(\Delta ABC\). | [
"\\(\\sqrt{6} - \\sqrt{2}\\) (maximum) and \\(1\\) (minimum)"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a square with side length 1, with each vertex of the triangle on a different side of the square. Determine the greatest and smallest possible values of the side length of \\(\\Delta ABC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-2032b5bcb285dfccab7d31710d783bafecacf401d0e1f178a82c2f6a9bd9ae7a | An equilateral triangle \(ABC\) is inscribed in a triangle \(KLM\) such that the vertices \(A\), \(B\), and \(C\) belong to the sides \(KL\), \(LM\), and \(MK\) respectively. It is also given that \(KA = LB = MC\). If \(KA = x\), find the value of \(x\) when the perimeter of triangle \(KLM\) is 36. | [
"4"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a triangle \\(KLM\\) such that the vertices \\(A\\), \\(B\\), and \\(C\\) belong to the sides \\(KL\\), \\(LM\\), and \\(MK\\) respectively. It is also given that \\(KA = LB = MC\\). If \\(KA = x\\), find the value of \\(x\\) when the perimeter of triangle \\(KLM\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-70eb5b88ae72ba70d1920b69104846bf396969996e70bdc4440173703888ad85 | An equilateral triangle \(ABC\) is inscribed in a unit circle. What is the radius of the largest circle that can fit between the line \(AB\) and the minor arc \(AB\)? | [
"\\(\\frac{1}{4}\\)"
] | [
"An equilateral triangle \\(ABC\\) is inscribed in a unit circle. What is the radius of the largest circle that can fit between the line \\(AB\\) and the minor arc \\(AB\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 14
} | 34 | 0 | 0 |
omr-ws-d0f9b51241b09b7847e8c4261f6e84a903720309b90d801068c9640b0fe3f630 | An equilateral triangle \(ABC\) with side length 1 has a square drawn such that one vertex of the square is at point \(A\) and the opposite vertex of the square is at the midpoint of side \(BC\). Find the area of the region enclosed within the intersection of the interiors of the triangle and the square. | [
"\\(\\frac{3\\sqrt{3}-3}{8}\\)"
] | [
"An equilateral triangle \\(ABC\\) with side length 1 has a square drawn such that one vertex of the square is at point \\(A\\) and the opposite vertex of the square is at the midpoint of side \\(BC\\). Find the area of the region enclosed within the intersection of the interiors of the triangle and the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-984154ac40b54948f060926d27951d28ad1da14d2aa5fc49d7c6920fabda448b | An equilateral triangle \(ABC\) with side length 1 is bent such that vertex \(A\) reaches a point on segment \(BC\). Find the maximum value of the area of the quadrilateral \(BCED\). | [
"\\(\\frac{3\\sqrt{3}}{16}\\)"
] | [
"An equilateral triangle \\(ABC\\) with side length 1 is bent such that vertex \\(A\\) reaches a point on segment \\(BC\\). Find the maximum value of the area of the quadrilateral \\(BCED\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 12
} | 17 | 0 | 0 |
omr-ws-7f1112491d8da810304651ef4c1c75b1b0a88ce87a147f530829fcd9ce90486c | An equilateral triangle \(ABC\) with side length 12 is folded such that vertex \(A\) touches a point on side \(\overline{BC}\) that is 9 units from point \(B\). The length of the line segment along which the triangle is folded can be written as \(\frac{m\sqrt{p}}{n}\), where \(m\), \(n\), and \(p\) are positive integer... | [
"113"
] | [
"An equilateral triangle \\(ABC\\) with side length 12 is folded such that vertex \\(A\\) touches a point on side \\(\\overline{BC}\\) that is 9 units from point \\(B\\). The length of the line segment along which the triangle is folded can be written as \\(\\frac{m\\sqrt{p}}{n}\\), where \\(m\\), \\(n\\), and \\(p... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 2
} | 57 | 0 | 0 |
omr-ws-1eb018a102283d097341ac260cc3155dbbe516a505f1f348464b3fc81f66115e | An equilateral triangle \(ABC\) with side length 2 has three squares \(ABDE\), \(BCFG\), and \(CAHI\) constructed externally on its sides. What is the area of the smallest equilateral triangle that can cover all these squares and the original triangle? | [
"\\(7\\sqrt{3} + 12\\)"
] | [
"An equilateral triangle \\(ABC\\) with side length 2 has three squares \\(ABDE\\), \\(BCFG\\), and \\(CAHI\\) constructed externally on its sides. What is the area of the smallest equilateral triangle that can cover all these squares and the original triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-259bfee3e767f2bce1b275696745be24d3c2653bd12121da13a96c890800eaf3 | An equilateral triangle \(ABC\) with side length 2 has three squares \(ABDE\), \(BCFG\), and \(CAHI\) constructed externally on its sides. What is the area of the smallest triangle that contains these three squares? | [
"\\(12 + 7\\sqrt{3}\\)"
] | [
"An equilateral triangle \\(ABC\\) with side length 2 has three squares \\(ABDE\\), \\(BCFG\\), and \\(CAHI\\) constructed externally on its sides. What is the area of the smallest triangle that contains these three squares?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-543cad224a25f347d33b731dde224cd70d84be8d5e53f7d4ecbbcbda8bf40cfc | An equilateral triangle \(ABC\) with sides \(AB = BC = 2\sqrt{2}\) is inscribed in a semicircle such that \(AC\) is the diameter of the semicircle. What is the area of the two segments bounded by the chords \(AB\) and \(BC\) and their respective arcs? | [
"\\(\\frac{2\\pi}{3} - \\sqrt{3}\\)"
] | [
"An equilateral triangle \\(ABC\\) with sides \\(AB = BC = 2\\sqrt{2}\\) is inscribed in a semicircle such that \\(AC\\) is the diameter of the semicircle. What is the area of the two segments bounded by the chords \\(AB\\) and \\(BC\\) and their respective arcs?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8bae7019e77d76768031a9e0a4dc5a63e77a9ea125547e4867c267c47b07b116 | An equilateral triangle and a point inside it are given on the plane. This point is reflected several times with respect to the lines containing the sides of the triangle, and at the end the point is again inside the triangle. If the point starts at $(0,0)$ and the vertices of the equilateral triangle are at $(0, \sqrt... | [
"0"
] | [
"An equilateral triangle and a point inside it are given on the plane. This point is reflected several times with respect to the lines containing the sides of the triangle, and at the end the point is again inside the triangle. If the point starts at $(0,0)$ and the vertices of the equilateral triangle are at $(0, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-061787f702fe722411eadc73f765bf37304d9e65de868c36e72ff77e574adda7 | An equilateral triangle and a regular 7-sided polygon are inscribed in the same circle of circumference 84 cm, dividing the circle into ten arcs. What is the maximum possible length, in cm, of the shortest arc? | [
"6"
] | [
"An equilateral triangle and a regular 7-sided polygon are inscribed in the same circle of circumference 84 cm, dividing the circle into ten arcs. What is the maximum possible length, in cm, of the shortest arc?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-097ab54ff601a70544b1ad071dd5d0932110b9fa8573664f9870fdeb1bf99446 | An equilateral triangle and a regular hexagon each have an area of 2023. In simplest radical form, what is the ratio of the perimeter of the hexagon to the perimeter of the triangle? | [
"\\dfrac{\\sqrt{6}}{3}"
] | [
"An equilateral triangle and a regular hexagon each have an area of 2023. In simplest radical form, what is the ratio of the perimeter of the hexagon to the perimeter of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1cded6829ffa25afa57ddf4bb03b2012e824aa5bdbf42a815102699a4517d1a | An equilateral triangle and a regular hexagon have equal areas. The ratio of the perimeter of the triangle to the area of the hexagon can be written as $\frac{\sqrt{a}}{b}$, where $a$ is not divisible by the square of any number. Find $a+b$. | [
"8"
] | [
"An equilateral triangle and a regular hexagon have equal areas. The ratio of the perimeter of the triangle to the area of the hexagon can be written as $\\frac{\\sqrt{a}}{b}$, where $a$ is not divisible by the square of any number. Find $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1549052a247e0c3e21b1bff3fbe19ef1d34b9b38a547f0071ed69df8586687c | An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 2, find the area of the hexagon. | [
"3"
] | [
"An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 2, find the area of the hexagon."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ca8d170f5a4d24f3457a7d1b1bc621aeed7e81021641165d0b5ed0f9ed2e607 | An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon? | [
"6"
] | [
"An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6b3b27a18f3ad4dbbffa9fa6e5b08e8f5d32f8c1172f998d5ffe9bc517e3145 | An equilateral triangle and a square both have perimeters of 48 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction. | [
"\\frac43"
] | [
"An equilateral triangle and a square both have perimeters of 48 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d26caf7062ac6ca67dd34313b4d92e7c88815f1a3d69ae6a16c38b0f7e806099 | An equilateral triangle and a square have equal areas. Each shape has a circle inscribed within it. What is the ratio of the area of the circle inscribed in the triangle to the area of the circle inscribed in the square? | [
"\\(\\frac{4\\sqrt{3}}{9}\\)"
] | [
"An equilateral triangle and a square have equal areas. Each shape has a circle inscribed within it. What is the ratio of the area of the circle inscribed in the triangle to the area of the circle inscribed in the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-9c37ba7591c89316d5ad889be8830c8530c6a0d9c5a4ab79054fbeb062ad970e | An equilateral triangle and a square have the same perimeter of 12 inches. What is the ratio of the side length of the triangle to the side length of the square? Express your answer as a common fraction. | [
"\\frac{4}{3}"
] | [
"An equilateral triangle and a square have the same perimeter of 12 inches. What is the ratio of the side length of the triangle to the side length of the square? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-486775e80b96d3798a90ab0270d2ec1b42c1e6840539a9f1fd04588220d9ad40 | An equilateral triangle consists of three congruent semicircles, with the center of each semicircle lying on each side of the triangle at a constant distance from the vertices. Determine the ratio of the area of the triangle to the sum of the areas of the semicircles. | [
"\\dfrac{2\\sqrt{3}}{3\\pi}"
] | [
"An equilateral triangle consists of three congruent semicircles, with the center of each semicircle lying on each side of the triangle at a constant distance from the vertices. Determine the ratio of the area of the triangle to the sum of the areas of the semicircles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c84bcaf8756cbf866e859112b236952b06fedb62032e001dbaf3680b47fbd5d | An equilateral triangle flag is suspended by two of its corners from the tops of two vertical poles of lengths 8 and 11 inches. The third corner of the flag just touches the ground. What is the area of the flag? | [
"\\dfrac{97\\sqrt{3}}{3}"
] | [
"An equilateral triangle flag is suspended by two of its corners from the tops of two vertical poles of lengths 8 and 11 inches. The third corner of the flag just touches the ground. What is the area of the flag?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e0adc550f166eaa0aba756a3515056234517ace2b4c46207478aa7483d0445b | An equilateral triangle has 6, 8, and 10 points on each of its sides, respectively (excluding the vertices). All lines are drawn between points that are not on the same side, and no three lines intersect at a single point. How many regions are created within the triangle? | [
"5229"
] | [
"An equilateral triangle has 6, 8, and 10 points on each of its sides, respectively (excluding the vertices). All lines are drawn between points that are not on the same side, and no three lines intersect at a single point. How many regions are created within the triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-dc24bf8997f5f2f500b6fb35ca52a4de4e3e80253c2e87eb8b61a3dd829700fd | An equilateral triangle has a point inside it such that the distances from this point to the vertices are 5, 12, and 13. Find the side length of the equilateral triangle. | [
"\\( \\sqrt{169 + 60\\sqrt{3}} \\)",
"\\(\\sqrt{169 + 60 \\sqrt{3}}\\)"
] | [
"An equilateral triangle has a point inside it such that the distances from this point to the vertices are 5, 12, and 13. Find the side length of the equilateral triangle."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 89,
"tir": 55
} | 144 | 0 | 0 |
omr-ws-b3a80877d0372d333970a8b6d320b19dc893271c6e6ab39401f37b7f4b014157 | An equilateral triangle has a side length of 3 inches. Squares are drawn on each of the sides of the equilateral triangle, such that no point in the equilateral triangle is also in the square. The vertices of the squares that are not also vertices of the triangle are the vertices of a hexagon that contains the triangle... | [
"\\( 27 + 9\\sqrt{3} \\)"
] | [
"An equilateral triangle has a side length of 3 inches. Squares are drawn on each of the sides of the equilateral triangle, such that no point in the equilateral triangle is also in the square. The vertices of the squares that are not also vertices of the triangle are the vertices of a hexagon that contains the tri... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 18
} | 27 | 0 | 0 |
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