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-6265dc9a6fdedef1c8134bd590b65eaa413b8400fa97a124a73df9279b656286 | An equilateral triangle has a side length of 8. If one corner of the triangle is removed by slicing off an equilateral triangle of side length 2, find the perimeter of the remaining quadrilateral. | [
"22"
] | [
"An equilateral triangle has a side length of 8. If one corner of the triangle is removed by slicing off an equilateral triangle of side length 2, find the perimeter of the remaining quadrilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a3d2ba03e35aaae011fbe579d461e60912b8b3991b2d918fb465403227fb38fa | An equilateral triangle has a side length of \(26\sqrt{3}\) cm. A point \(H\) is placed inside the triangle such that its perpendicular distances to two sides of the triangle are 9 cm and 14 cm, respectively. Segments are drawn from point \(H\) to the three vertices of the triangle, forming three smaller triangles. Wha... | [
"\\(208\\sqrt{3}\\)"
] | [
"An equilateral triangle has a side length of \\(26\\sqrt{3}\\) cm. A point \\(H\\) is placed inside the triangle such that its perpendicular distances to two sides of the triangle are 9 cm and 14 cm, respectively. Segments are drawn from point \\(H\\) to the three vertices of the triangle, forming three smaller tr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-7416242a0675bfc9f28b2cb655d7a02aed966a8cb808ab45b9af5521758c0664 | An equilateral triangle has a vertex at point \( A = (2,1) \) and its opposite side is contained in the line with equation \( x - 3y + 4 = 0 \). Determine the Cartesian equations of the lines containing the other two sides of the triangle. | [
"(6 - 5\\sqrt{3})x - 3y = 9 - 10\\sqrt{3}"
] | [
"An equilateral triangle has a vertex at point \\( A = (2,1) \\) and its opposite side is contained in the line with equation \\( x - 3y + 4 = 0 \\). Determine the Cartesian equations of the lines containing the other two sides of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a959600cce4b024ef8d41ae64742813a8ba82058e0c2a139d48534c5a9dfae19 | An equilateral triangle has an area of $64\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased? | [
"28\\sqrt{3}"
] | [
"An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f499e1a6da545d4df411a2ef96dc944034bd30fa66358dffafcccd25198ddeb5 | An equilateral triangle has an area of \(4\sqrt{3}\). A square is inscribed in this triangle. What is the length of the side of the square? | [
"\\(4 (2 \\sqrt{3} - 3)\\)"
] | [
"An equilateral triangle has an area of \\(4\\sqrt{3}\\). A square is inscribed in this triangle. What is the length of the side of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-dd42a60d181700c17beb1d90de25ce645f2f5716bb3c075981267685175c1317 | An equilateral triangle has marked points at vertices, a center, and points dividing sides into three equal parts (two on each side). How many exist triangles with vertices at marked points such that none of their sides is not parallel to any side of the original triangle? | [
"2"
] | [
"An equilateral triangle has marked points at vertices, a center, and points dividing sides into three equal parts (two on each side). How many exist triangles with vertices at marked points such that none of their sides is not parallel to any side of the original triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bfd2749358fa7cc266b01f73ef4d5fca3e7125259c835dcf74fbdb33e935e2f5 | An equilateral triangle has nails driven into points \( K \), \( L \), and \( M \) on its sides such that the triangle cannot move. Points \( K \) and \( L \) divide their respective sides in the ratios \( 2:1 \) and \( 3:2 \). Determine the ratio in which point \( M \) divides its side of the triangle. | [
"\\( 5:3 \\)"
] | [
"An equilateral triangle has nails driven into points \\( K \\), \\( L \\), and \\( M \\) on its sides such that the triangle cannot move. Points \\( K \\) and \\( L \\) divide their respective sides in the ratios \\( 2:1 \\) and \\( 3:2 \\). Determine the ratio in which point \\( M \\) divides its side of the tria... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5527dfac50480b92ad02b59f9985cb8cb4e169c53e71172c94cfefa4240e064 | An equilateral triangle has one angle trisected, one angle quintisected, and one angle heptasected. Compute the number of intersections. | [
"59"
] | [
"An equilateral triangle has one angle trisected, one angle quintisected, and one angle heptasected. Compute the number of intersections."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34013f59ba3d67e2dfa4d703873450f4791a4155347b61cf6daf79b5d5a2187c | An equilateral triangle has side length $2$. Consider the set of all squares whose vertices lie on this triangle. Find the area of the region that belongs to at least $2$ of these squares. | [
"1"
] | [
"An equilateral triangle has side length $2$. Consider the set of all squares whose vertices lie on this triangle. Find the area of the region that belongs to at least $2$ of these squares."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-ca18a02eae96c25b9624e87b2c2383bb9134b4ffb716310a4c6a7b2f6263e2c7 | An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction. | [
"\\frac{1}{3}"
] | [
"An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-845b9f482cfcfdf7b3d1d22c7f44e44f788f716538c5a601cbe36a1be60572a3 | An equilateral triangle has sides of length $4$ cm. At each vertex, a circle with radius $2$ cm is drawn. The total area of the shaded regions of the three circles is $a \cdot \pi \text{cm}^2$. Determine $a$. | [
"2"
] | [
"An equilateral triangle has sides of length $4$ cm. At each vertex, a circle with radius $2$ cm is drawn. The total area of the shaded regions of the three circles is $a \\cdot \\pi \\text{cm}^2$. Determine $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df200632a6a9df54b1d9e63502f0aa6bd5fcf77d2c57c1b7e5c56c5448bde8a5 | An equilateral triangle has sides of length 2 units. A second equilateral triangle is formed having sides that are $150\%$ of the length of the sides of the first triangle. A third equilateral triangle is formed having sides that are $150\%$ of the length of the sides of the second triangle. The process is continued un... | [
"237.5\\%"
] | [
"An equilateral triangle has sides of length 2 units. A second equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the first triangle. A third equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the second triangle. The process is conti... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c3486d6910fbd4334a811839dd2667096f853cc10521fcc4e8a68b6b8a5f68f | An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form. [asy] draw((-1,0)--(11,0),Arrows); draw((0,-1)--(0,12),Arrows); for(int i=0;i<11;++i) {draw((i,-0.1)--(i,0.1));} for(int j=0;j<11;++j) {d... | [
"5+4\\sqrt{3}"
] | [
"An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form. [asy]\ndraw((-1,0)--(11,0),Arrows);\ndraw((0,-1)--(0,12),Arrows);\nfor(int i=0;i<11;++i)\n{draw((i,-0.1)--(i,0.1));}\nfor(int j=0;j<... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c17cea9d063e19c0efa147af175f1df7c8b4cd03dfa82092dbfe8355f3b309ee | An equilateral triangle has vertices at \((0,0)\), \((a,5)\), and \((b,13)\). Find the area of the triangle. | [
"\\(25\\sqrt{3}\\)"
] | [
"An equilateral triangle has vertices at \\((0,0)\\), \\((a,5)\\), and \\((b,13)\\). Find the area of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45a5069b4b92ee484247a51e9b9ec6e5b07c6ab626f011a0a2815b9e830e35b1 | An equilateral triangle in the Cartesian plane has one of its sides lying on the line \( y = 2x - 4 \). If one of its vertices lies on the origin, what is the perimeter of the triangle? | [
"\\(\\frac{8\\sqrt{15}}{5}\\)"
] | [
"An equilateral triangle in the Cartesian plane has one of its sides lying on the line \\( y = 2x - 4 \\). If one of its vertices lies on the origin, what is the perimeter of the triangle?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-db554dc3cf6030ae82677106c42a404e8d4b03f8cb0f6524ecee5d6f8bafdf49 | An equilateral triangle is decomposed into smaller equilateral subtriangles. If the total number of subtriangles is 61, find the sum of all possible values for the side length ratio between the largest subtriangle and the original triangle. | [
"\\dfrac{9}{8}"
] | [
"An equilateral triangle is decomposed into smaller equilateral subtriangles. If the total number of subtriangles is 61, find the sum of all possible values for the side length ratio between the largest subtriangle and the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5ee9159f780c7188bc1c5df79bf3fbaaf8f4af4eeb907e105dc6f00778dd1a11 | An equilateral triangle is divided into $9$ smaller triangles, each initially labeled with the number $0$. Marco plays a game where at each move, he selects two adjacent triangles and either adds or subtracts $1$ from the numbers in both triangles. After several moves, the numbers on the $9$ triangles are $n, n+1, \ldo... | [
"2"
] | [
"An equilateral triangle is divided into $9$ smaller triangles, each initially labeled with the number $0$. Marco plays a game where at each move, he selects two adjacent triangles and either adds or subtracts $1$ from the numbers in both triangles. After several moves, the numbers on the $9$ triangles are $n, n+1,... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-07040c9be959c548e9b6216af5f5ec4c8bda30180456df01e360d3178261d83f | An equilateral triangle is divided into $9000000$ congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is colored in one of three colors. Let $N$ be the number of monochromatic triangles (triangles with all vertices of the same color) with sides parallel to the lines of the... | [
"1000000"
] | [
"An equilateral triangle is divided into $9000000$ congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is colored in one of three colors. Let $N$ be the number of monochromatic triangles (triangles with all vertices of the same color) with sides parallel to the lines o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-439c70655caaa6a73b033f19e10b27a3cc1e44fc7048360f4dfcf93c39708438 | An equilateral triangle is divided into $n^2$ congruent equilateral triangles. A spider starts at one vertex and a fly starts at another. They alternate moving to a neighboring vertex. If $n=7$, and the spider always moves optimally to catch the fly, calculate the maximum number of moves the spider will need to guarant... | [
"13"
] | [
"An equilateral triangle is divided into $n^2$ congruent equilateral triangles. A spider starts at one vertex and a fly starts at another. They alternate moving to a neighboring vertex. If $n=7$, and the spider always moves optimally to catch the fly, calculate the maximum number of moves the spider will need to gu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-c512ac95afb97d35ca205a86a9bd57509d24e71a2a23922366fab5bed2305606 | An equilateral triangle is divided into $n^2$ congruent small equilateral triangles. What is the maximum number of vertices we can choose (among the vertices of the small triangles) so that no line passing through any two of the chosen points is parallel to a side of the equilateral triangle? | [
"n"
] | [
"An equilateral triangle is divided into $n^2$ congruent small equilateral triangles. What is the maximum number of vertices we can choose (among the vertices of the small triangles) so that no line passing through any two of the chosen points is parallel to a side of the equilateral triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-8ec95c977f90214ff16267baf00d42ec6e88f6927919a6183e758a3a05588385 | An equilateral triangle is divided into $n^2$ smaller equilateral triangles by lines parallel to its sides. A "line of triangles" is defined as all triangles between two adjacent parallel lines forming the grid, including corner triangles. The goal is to paint all triangles black by selecting a line of triangles that c... | [
"Smallest: \\( n \\), Largest: \\( 3n-2 \\)"
] | [
"An equilateral triangle is divided into $n^2$ smaller equilateral triangles by lines parallel to its sides. A \"line of triangles\" is defined as all triangles between two adjacent parallel lines forming the grid, including corner triangles. The goal is to paint all triangles black by selecting a line of triangles... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f9def9c63d07327b34b835a7dbac2fb8b7f11b6445b47431354f1da662b2f6a | An equilateral triangle is divided into 25 equal equilateral triangles labelled by 1 through 25. What is the minimum possible difference between the labels of two triangles that share a common side? | [
"5"
] | [
"An equilateral triangle is divided into 25 equal equilateral triangles labelled by 1 through 25. What is the minimum possible difference between the labels of two triangles that share a common side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-8a9d1d703905b5957b63f817926315e3210b6fdb7d217df725341c346e27a9fc | An equilateral triangle is divided into 36 smaller equilateral triangles by dividing each side into 6 parts and joining the corresponding points on adjacent sides of the triangle. There is a drunk ant at each of the 28 vertices, and they all start to walk along the edges (in random directions) with the same speed. Sinc... | [
"7"
] | [
"An equilateral triangle is divided into 36 smaller equilateral triangles by dividing each side into 6 parts and joining the corresponding points on adjacent sides of the triangle. There is a drunk ant at each of the 28 vertices, and they all start to walk along the edges (in random directions) with the same speed.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87157574dcdb272e357c53adae41621783f3694f6304af7fb577f9263a0024c3 | An equilateral triangle is divided into 9 pentagons. If \( x \) is the greatest internal angle of the pentagons, find the smallest value of \( x \). | [
"150"
] | [
"An equilateral triangle is divided into 9 pentagons. If \\( x \\) is the greatest internal angle of the pentagons, find the smallest value of \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-f1b5dc9d832aabe6a69cdf3d5612bbbedca3fdbfecd53e06961ec71344360883 | An equilateral triangle is divided into \( n^2 \) congruent equilateral triangles. Let \( V \) be the set of all vertices and \( E \) be the set of all edges of these triangles. Find all \( n \) for which we can paint all edges black or white such that for every vertex, the number of incident edges of the black color i... | [
"n \\equiv 0 \\text{ or } 3 \\pmod{4}"
] | [
"An equilateral triangle is divided into \\( n^2 \\) congruent equilateral triangles. Let \\( V \\) be the set of all vertices and \\( E \\) be the set of all edges of these triangles. Find all \\( n \\) for which we can paint all edges black or white such that for every vertex, the number of incident edges of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-32965c5208fa07fe8fb1ac6939e5505425bbf6d66efa0ae6b6b8ed0b94047057 | An equilateral triangle is divided into nine rooms labeled $P$ and $Q$ among others. A ball starts from room $P$ and moves to an adjacent room every second with equal probability. Determine the probability that the ball will be in room $Q$ after $n$ seconds. | [
"\\(\\frac{1-2^{-n}}{3}\\)"
] | [
"An equilateral triangle is divided into nine rooms labeled $P$ and $Q$ among others. A ball starts from room $P$ and moves to an adjacent room every second with equal probability. Determine the probability that the ball will be in room $Q$ after $n$ seconds."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7ae416091bd105ad714cdd995078bc02b296f22264b9c872ad896d401877ef2 | An equilateral triangle is divided into smaller equilateral triangles by lines parallel to its sides. One of the smaller triangles (whose vertices do not belong to the boundary of the large triangle) is colored black. We can change the color of a triangle from black to white or vice versa by "cutting" it with a line pa... | [
"3"
] | [
"An equilateral triangle is divided into smaller equilateral triangles by lines parallel to its sides. One of the smaller triangles (whose vertices do not belong to the boundary of the large triangle) is colored black. We can change the color of a triangle from black to white or vice versa by \"cutting\" it with a ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8dd011450cacbc21d10d966b4c1869e9517b4a73b51bedde4ef015828dc6cece | An equilateral triangle is divided into two triangles such that every side in each of the two new triangles is a whole number of centimetres long. What is the minimum length of a side of the initial triangle? | [
"8"
] | [
"An equilateral triangle is divided into two triangles such that every side in each of the two new triangles is a whole number of centimetres long. What is the minimum length of a side of the initial triangle?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 61,
"genselect": 7,
"tir": 17
} | 85 | 0 | 0 |
omr-ws-06bebb7ebb27e7cacc3202dd73e5bd6238770bc299f97e991e3eb4f893c37ffa | An equilateral triangle is filled to the maximum with \( n \) rows of congruent circles. What is the limit as \( n \) approaches \( \infty \) of the ratio of the area of the circles to the area of the triangle? | [
"\\dfrac{\\pi}{2\\sqrt{3}}"
] | [
"An equilateral triangle is filled to the maximum with \\( n \\) rows of congruent circles. What is the limit as \\( n \\) approaches \\( \\infty \\) of the ratio of the area of the circles to the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64e0727dad25fa97aa05ab6307538fb5b8bf98d68902b34d3c2b65818d4fc66b | An equilateral triangle is formed in the Cartesian plane with vertices at the origin $(0,0)$, $(a,11)$, and $(b,7)$. Find the value of $a+b$. | [
"\\(4\\sqrt{3}\\) or \\(-4\\sqrt{3}\\)"
] | [
"An equilateral triangle is formed in the Cartesian plane with vertices at the origin $(0,0)$, $(a,11)$, and $(b,7)$. Find the value of $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-064b9c6443a8b5a5d2c8f088fdee17b5e20732d0381ad29171b367ea8224b5a3 | An equilateral triangle is inscribed in a circle of area $1$ m$^2$. A second circle is inscribed in the triangle. Find the radius of the second circle. | [
"\\(\\frac{1}{2}\\)"
] | [
"An equilateral triangle is inscribed in a circle of area $1$ m$^2$. A second circle is inscribed in the triangle. Find the radius of the second circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cdcc1471c98c2d58fdab6c878b01deb1d0b3fd165e7a72663a470f999f88ac4 | An equilateral triangle is inscribed in a circle of radius $r$. Another smaller equilateral triangle is inscribed in the region bounded by the base of the larger equilateral triangle and the lower part of the circle in such a way that the heights of each triangle are on the same line. What is the ratio of the area of t... | [
"16"
] | [
"An equilateral triangle is inscribed in a circle of radius $r$. Another smaller equilateral triangle is inscribed in the region bounded by the base of the larger equilateral triangle and the lower part of the circle in such a way that the heights of each triangle are on the same line. What is the ratio of the area... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1972f262441580189a45264df5f54ebf689974f2cde6e747ed4dcfaac00361b | An equilateral triangle is inscribed in a circle of radius 6. A second equilateral triangle is circumscribed about the circle. If the sides of the triangles are parallel, what is the shortest distance from a point on one triangle to a point on the other? | [
"3"
] | [
"An equilateral triangle is inscribed in a circle of radius 6. A second equilateral triangle is circumscribed about the circle. If the sides of the triangles are parallel, what is the shortest distance from a point on one triangle to a point on the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-2239a21435e7da6eae387f872107040f1b9fd453924cde583fcb658f5707fe06 | An equilateral triangle is inscribed in a circle of radius \( r \). Express the circumference \( C \) of the circle as a function of the length \( x \) of a side of the triangle. HINT GIVEN: First show that \( r^2 = \frac{x^2}{3} \). | [
"C(x) = \\dfrac{2\\sqrt{3}}{3} \\pi x"
] | [
"An equilateral triangle is inscribed in a circle of radius \\( r \\). Express the circumference \\( C \\) of the circle as a function of the length \\( x \\) of a side of the triangle. \n\nHINT GIVEN: First show that \\( r^2 = \\frac{x^2}{3} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5071022336422d508760c3ec07804f435738176667ff8da31ad8eb67fcb68d25 | An equilateral triangle is inscribed in a circle which is circumscribed by a square. This square is inscribed in a circle which is circumscribed by a regular hexagon. If the area of the equilateral triangle is $1$, what is the area of the hexagon? | [
"\\dfrac{16}{3}"
] | [
"An equilateral triangle is inscribed in a circle which is circumscribed by a square. This square is inscribed in a circle which is circumscribed by a regular hexagon. If the area of the equilateral triangle is $1$, what is the area of the hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af9c1f1bbd3caf0a367980a36072dffcde59cb71b70b125ddb879be3f7e33c97 | An equilateral triangle is inscribed in a circle. A chord parallel to one side of the triangle intersects the other two sides of the triangle, and the resulting chord is $\frac{5}{7}$ inside the triangle. All sides of the triangle and the segments inside and outside the triangle formed by the chord are integers. How fa... | [
"\\dfrac{13\\sqrt{3}}{6}"
] | [
"An equilateral triangle is inscribed in a circle. A chord parallel to one side of the triangle intersects the other two sides of the triangle, and the resulting chord is $\\frac{5}{7}$ inside the triangle. All sides of the triangle and the segments inside and outside the triangle formed by the chord are integers. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-1907493a0595c76d4c0786fb6753ccb3c2043cb63a4012562e400a30af51f385 | An equilateral triangle is inscribed in a circle. A chord parallel to one side of the triangle intersects the other two sides of the triangle, and the resulting chord is $\frac{5}{7}$ inside the triangle. All sides of the triangle and the segments inside and outside the triangle formed by the chord are integers. What i... | [
"31"
] | [
"An equilateral triangle is inscribed in a circle. A chord parallel to one side of the triangle intersects the other two sides of the triangle, and the resulting chord is $\\frac{5}{7}$ inside the triangle. All sides of the triangle and the segments inside and outside the triangle formed by the chord are integers. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 6,
"tir": 1
} | 45 | 0 | 0 |
omr-ws-b53c280c91690cf9c03b16084893f5fd0798b1b25c1fc37ec3385390cfc37a59 | An equilateral triangle is inscribed in a circle. A line segment parallel to one side of the triangle divides the triangle into two parts of equal area. An inscribed circle is drawn inside the smaller triangle. What is the ratio of the area of this inscribed circle to the area of the original circle? | [
"\\dfrac{1}{8}"
] | [
"An equilateral triangle is inscribed in a circle. A line segment parallel to one side of the triangle divides the triangle into two parts of equal area. An inscribed circle is drawn inside the smaller triangle. What is the ratio of the area of this inscribed circle to the area of the original circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9187230f8660a08f5fdb90ec9fcf15576dd6efb8b9271876d5d8346cc679d97b | An equilateral triangle is inscribed in a circle. A line segment parallel to one side of the triangle divides the triangle into two regions of equal area. An inscribed circle is drawn inside the smaller of these two regions. What is the ratio of the area of this inscribed circle to the area of the original circle? | [
"\\( \\frac{1}{8} \\)"
] | [
"An equilateral triangle is inscribed in a circle. A line segment parallel to one side of the triangle divides the triangle into two regions of equal area. An inscribed circle is drawn inside the smaller of these two regions. What is the ratio of the area of this inscribed circle to the area of the original circle?... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-a8bfb0c673e2894df5e073dc3255e1116edc48a485f7ea0c5289996bf47c6fa9 | An equilateral triangle is inscribed in a circle. Points D and E are midpoints of sides AB and BC, respectively, and F is the point where DE meets the circle. Find the ratio DE/EF. | [
"\\(\\frac{\\sqrt{5}+1}{2}\\)"
] | [
"An equilateral triangle is inscribed in a circle. Points D and E are midpoints of sides AB and BC, respectively, and F is the point where DE meets the circle. Find the ratio DE/EF."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-03811f4ed4ba19028ca02d688fd93e0e024ccbc330f30c1def18b44ef42d3f13 | An equilateral triangle is inscribed in a circle. Points D and E are the midpoints of sides AB and BC, respectively, and F is the point where DE meets the circle. Find the ratio \( \frac{DE}{EF} \). | [
"\\(\\frac{1+\\sqrt{5}}{2}\\)"
] | [
"An equilateral triangle is inscribed in a circle. Points D and E are the midpoints of sides AB and BC, respectively, and F is the point where DE meets the circle. Find the ratio \\( \\frac{DE}{EF} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-209f0f210e2c99cddbee1d440c4ebb15d4632a796a8851aac133cb6df009fc4b | An equilateral triangle is inscribed in a circle. What is the probability that a randomly drawn chord in the circle is longer than the side length of the triangle? | [
"$\\frac13$"
] | [
"An equilateral triangle is inscribed in a circle. What is the probability that a randomly drawn chord in the circle is longer than the side length of the triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ea4d73bc8f30a48d9040f42e2717f9f1b9803a3b39e28c5b5a7872c0e2900f33 | An equilateral triangle is inscribed in a quarter circle with radius 1 such that two of its vertices are on the two radii and one vertex is on the arc. What is the side length of the equilateral triangle? | [
"\\(\\sqrt{3} - 1\\)"
] | [
"An equilateral triangle is inscribed in a quarter circle with radius 1 such that two of its vertices are on the two radii and one vertex is on the arc. What is the side length of the equilateral triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-5b6d6373b138e16ac61215d763b88b7d85022b485cf0f45e2bfff94e853f3b52 | An equilateral triangle is inscribed in a regular hexagon of area $96$. Find the area of this triangle. | [
"39"
] | [
"An equilateral triangle is inscribed in a regular hexagon of area $96$. Find the area of this triangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-822fd1169b12c1ae408e6b88081d26dd6e29f052dce0bb799e6c1bac25d19af0 | An equilateral triangle is inscribed in the ellipse \(x^2 + 4y^2 = 4\), with one vertex at \((0,1)\) and the corresponding altitude along the y-axis. Find its side length. | [
"\\(\\frac{16\\sqrt{3}}{13}\\)"
] | [
"An equilateral triangle is inscribed in the ellipse \\(x^2 + 4y^2 = 4\\), with one vertex at \\((0,1)\\) and the corresponding altitude along the y-axis. Find its side length."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 55
} | 86 | 0 | 0 |
omr-ws-b42fcf1851a0a1c03bac1c78f1185617a7ad76298c553c26f62bd87f27a01c10 | An equilateral triangle is inscribed in the ellipse given by $\frac{x^2}{4} + y^2 = 1$, with one vertex at $(0, 1)$. Find the height of the triangle. | [
"\\dfrac{24}{13}"
] | [
"An equilateral triangle is inscribed in the ellipse given by $\\frac{x^2}{4} + y^2 = 1$, with one vertex at $(0, 1)$. Find the height of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-636251f7c679bb7192912bb37db14fb761ee55afdeec889bc0383d8f84abc9f3 | An equilateral triangle is inscribed in the ellipse given by the equation $x^2 + 4y^2 = 4$. One vertex of the triangle is at $(0,1)$, and one altitude is contained in the $y$-axis. The length of each side of the triangle is $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"937"
] | [
"An equilateral triangle is inscribed in the ellipse given by the equation $x^2 + 4y^2 = 4$. One vertex of the triangle is at $(0,1)$, and one altitude is contained in the $y$-axis. The length of each side of the triangle is $\\sqrt{\\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-48d93838064bf5a0f31dcfc3d99414b38013c9ca155d64099e7479c5fa6341f0 | An equilateral triangle is inscribed in the ellipse given by the equation \( x^2 + 9y^2 = 9 \). One vertex of the triangle is at (0,1), and one altitude is contained in the \( y \)-axis. The area of the equilateral triangle is in the form \( \frac{m \sqrt{n}}{o} \), where \( n \) is square-free and \( m \) and \( o \) ... | [
"442"
] | [
"An equilateral triangle is inscribed in the ellipse given by the equation \\( x^2 + 9y^2 = 9 \\). One vertex of the triangle is at (0,1), and one altitude is contained in the \\( y \\)-axis. The area of the equilateral triangle is in the form \\( \\frac{m \\sqrt{n}}{o} \\), where \\( n \\) is square-free and \\( m... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-3043cb2714e17f89dc88d1525f88736bb27c53592f4e0b7194edd4037d5b86c7 | An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"937"
] | [
"An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17c6ce020146ab362fa4d0006e95fe2bb0946c3831327785d05df432d0724e56 | An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Contents 1 Problem 2 Solut... | [
"937"
] | [
"An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. \nContents 1 ... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3a6dce19f1687fc3d3b5c52b183e86165b28a31f1cde43747439f46f681fa066 | An equilateral triangle is inscribed on three concentric circles with each vertex being on a different circle. The radii of the circles are 3, 4, and 5 respectively. Find the length of the side of the triangle. | [
"\\( \\sqrt{37} \\)"
] | [
"An equilateral triangle is inscribed on three concentric circles with each vertex being on a different circle. The radii of the circles are 3, 4, and 5 respectively. Find the length of the side of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da79d1563d6271e23b67f3aaf571df74153bfd0bf61e353ffbe0596128c5d11f | An equilateral triangle is on top of a square, sharing a side of length $10$. Both shapes are inscribed in a circle. What is the radius of the circle? | [
"10"
] | [
"An equilateral triangle is on top of a square, sharing a side of length $10$. Both shapes are inscribed in a circle. What is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-7b3df9f12095f62ad9fd6865a0a5f5a37c8287e6cd1145167fe60c03123e6245 | An equilateral triangle is painted on the floor. From a certain point, the distances to the three corners of the triangle are 3 meters, 4 meters, and 5 meters, respectively. What is the side length of the equilateral triangle? | [
"\\(6.7664\\) meters"
] | [
"An equilateral triangle is painted on the floor. From a certain point, the distances to the three corners of the triangle are 3 meters, 4 meters, and 5 meters, respectively. What is the side length of the equilateral triangle?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-fe03391ac23f8ced6f747d59e113e8481d6ac3984e61187cf1f6d101a3ba3557 | An equilateral triangle is partitioned into $n$ smaller equilateral triangular pieces. If all pieces have distinct sizes, what is the minimum possible value of $n$? | [
"4"
] | [
"An equilateral triangle is partitioned into $n$ smaller equilateral triangular pieces. If all pieces have distinct sizes, what is the minimum possible value of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-79374e3b960df4ec5398d84412fe5dce09f34e7bcccdcfdb963114e70c3f12b5 | An equilateral triangle is partitioned into five isosceles triangles. If the side length of the original equilateral triangle is $s$, and the base of one of the isosceles triangles formed is $b$, find the sum of all possible values of $\frac{b}{s}$. | [
"1"
] | [
"An equilateral triangle is partitioned into five isosceles triangles. If the side length of the original equilateral triangle is $s$, and the base of one of the isosceles triangles formed is $b$, find the sum of all possible values of $\\frac{b}{s}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-28bbfb43ebc874183e8b28399c7e6f90b3e3eccbc2e6e148be47db90be9ccc69 | An equilateral triangle is rotated about its center by \(3^\circ\), then \(9^\circ\), \(27^\circ\), and so on (on the \(n\)-th step by \(3^n\) degrees). How many different positions will the triangle take during this activity (including the initial position)? | [
"4"
] | [
"An equilateral triangle is rotated about its center by \\(3^\\circ\\), then \\(9^\\circ\\), \\(27^\\circ\\), and so on (on the \\(n\\)-th step by \\(3^n\\) degrees). How many different positions will the triangle take during this activity (including the initial position)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3c714f536fe620e2c237ae7892036098ae8e0050a06d15f07fc1a8db6048a47 | An equilateral triangle is split into 3 sections with the same area by cuts parallel to one of the triangle’s sides. What is the ratio between the heights of the upper triangle \(ABC\) and the lower trapezoid \(DEFG\)? | [
"\\(\\frac{\\sqrt{3}}{3}\\)"
] | [
"An equilateral triangle is split into 3 sections with the same area by cuts parallel to one of the triangle’s sides. What is the ratio between the heights of the upper triangle \\(ABC\\) and the lower trapezoid \\(DEFG\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae9d770f4008be15ad90f391c0e87a089f0384e237b4643678ff87c2a1a0f2f7 | An equilateral triangle is surrounded by 3 congruent rectangles of width 1, all inscribed in a circle of radius 3. Find the length of the side of the equilateral triangle. | [
"\\(\\frac{-\\sqrt{3} + 3\\sqrt{11}}{2}\\)"
] | [
"An equilateral triangle is surrounded by 3 congruent rectangles of width 1, all inscribed in a circle of radius 3. Find the length of the side of the equilateral triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 13
} | 33 | 0 | 0 |
omr-ws-e36b35bc2f1d6e9a758dc01fb3827559b359b5ccadc25f5fb499ff03ca347e84 | An equilateral triangle of side $N$ is divided into $N^2$ small equilateral triangles of side $1$ by parallels to its sides. At each vertex of a small triangle, there is an ant. At the same instant, all the ants begin to walk at the same speed along the lines of the triangulation. When they reach another vertex, they t... | [
"Yes"
] | [
"An equilateral triangle of side $N$ is divided into $N^2$ small equilateral triangles of side $1$ by parallels to its sides. At each vertex of a small triangle, there is an ant. At the same instant, all the ants begin to walk at the same speed along the lines of the triangulation. When they reach another vertex, t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf1e6eb169c89281d3c4184b7859de55cb9c5ff301f73637dd9332250c66ddfd | An equilateral triangle of side $N$ is divided into $N^2$ small equilateral triangles of side $1$ by parallels to its sides. At each vertex of a small triangle, there is an ant. At the same instant, all the ants begin to walk at the same speed along the lines of the triangulation. When they reach another vertex, they t... | [
"\\text{No}"
] | [
"An equilateral triangle of side $N$ is divided into $N^2$ small equilateral triangles of side $1$ by parallels to its sides. At each vertex of a small triangle, there is an ant. At the same instant, all the ants begin to walk at the same speed along the lines of the triangulation. When they reach another vertex, t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-6aaa4511d11e45624ce32b1d5aa091f7191b812013622fb591a4a4e797912134 | An equilateral triangle of side 2a is partitioned symmetrically into a quadrilateral, an isosceles triangle, and two other congruent triangles. If the inradii of the quadrilateral and the isosceles triangle are equal, find this radius. | [
"\\( (\\sqrt{3} - \\sqrt{2})a \\)"
] | [
"An equilateral triangle of side 2a is partitioned symmetrically into a quadrilateral, an isosceles triangle, and two other congruent triangles. If the inradii of the quadrilateral and the isosceles triangle are equal, find this radius."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6efac3f1a043edf4836c3a584a763f2d486ab5c093931544719d989b0287fb87 | An equilateral triangle of side \( n \) has been divided into little equilateral triangles of side 1 in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the \(\frac{(n+1)(n+2)}{2}\) vertices. What is the minimum number of times the path can cha... | [
"\\( n \\)"
] | [
"An equilateral triangle of side \\( n \\) has been divided into little equilateral triangles of side 1 in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the \\(\\frac{(n+1)(n+2)}{2}\\) vertices. What is the minimum number of times the pa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-774e38476ba4686f7ab4b2ea662a3befecf61ee6f1ec69ca12a14c7a87b22100 | An equilateral triangle of side length $3n$ is divided into $9n^2$ unit equilateral triangles, each colored red, yellow, or blue such that each color appears $3n^2$ times. A "standard trapezoid" is formed by three unit equilateral triangles. If a "standard trapezoid" contains all three colors, it is called a "colorful ... | [
"\\( 18n^2 - 9n \\)"
] | [
"An equilateral triangle of side length $3n$ is divided into $9n^2$ unit equilateral triangles, each colored red, yellow, or blue such that each color appears $3n^2$ times. A \"standard trapezoid\" is formed by three unit equilateral triangles. If a \"standard trapezoid\" contains all three colors, it is called a \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-29068914e519d01758caf70965c4bfc041cc49cfc01a7b405fef7b6b34017a6d | An equilateral triangle of side length $8$ is tiled by $64$ equilateral triangles of unit side length to form a triangular grid. Initially, each triangular cell is either living or dead. The grid evolves over time under the following rule: every minute, if a dead cell is edge-adjacent to at least two living cells, then... | [
"8"
] | [
"An equilateral triangle of side length $8$ is tiled by $64$ equilateral triangles of unit side length to form a triangular grid. Initially, each triangular cell is either living or dead. The grid evolves over time under the following rule: every minute, if a dead cell is edge-adjacent to at least two living cells,... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-f00d34f7caf4a37167f02d9ea14a07c44396998ed3e98cf3cf7a5cca85fb0c7f | An equilateral triangle of side length $n$ is divided into $n^2$ smaller triangles of side length 1. Determine the maximum number of vertices of the smaller triangles that can be colored such that no two colored vertices lie along a boundary of the division. | [
"\\(\\left\\lfloor \\frac{(n+1)(n+2)}{4} \\right\\rfloor\\)"
] | [
"An equilateral triangle of side length $n$ is divided into $n^2$ smaller triangles of side length 1. Determine the maximum number of vertices of the smaller triangles that can be colored such that no two colored vertices lie along a boundary of the division."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 3
} | 12 | 0 | 0 |
omr-ws-227f3c1a63ec27453ca948c5f7fd542aad601932dea54adeff3a2b2a2cd18592 | An equilateral triangle of side length $n$ is divided into $n^2$ unit equilateral triangles. Find the number $f(n)$ of parallelograms bounded by sides in the so-obtained grid. | [
"\\( 3 \\binom{n+2}{4} \\)"
] | [
"An equilateral triangle of side length $n$ is divided into $n^2$ unit equilateral triangles. Find the number $f(n)$ of parallelograms bounded by sides in the so-obtained grid."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 10
} | 28 | 0 | 0 |
omr-ws-fb6590ebb73751fe1da9ea7f5f54f9de4215fc51768e633e20eadca5600ec988 | An equilateral triangle of side length $n$ is divided into equilateral triangles of side length $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle. | [
"\\( n(n+1) \\)"
] | [
"An equilateral triangle of side length $n$ is divided into equilateral triangles of side length $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-2aa6a31ad1ed693eca83cc601d9a29a5d93f268cd2ab6bdf5ca3ff0e62647d9c | An equilateral triangle of side length $n$ is divided into equilateral triangles of unit side. How many "castles" can be placed on the lattice points of the triangular lattice so that no two of them attack each other? ("Castles" can move parallel to the sides of the triangle, in six directions altogether.) | [
"\\( n \\)"
] | [
"An equilateral triangle of side length $n$ is divided into equilateral triangles of unit side. How many \"castles\" can be placed on the lattice points of the triangular lattice so that no two of them attack each other? (\"Castles\" can move parallel to the sides of the triangle, in six directions altogether.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-0d94e1d48fd7b533037f6129148b03456c0b6690892daa526af17395431b7c30 | An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect at the centroid of the triangle. What is the maximum possible area of one of these pieces? | [
"\\(\\frac{\\sqrt{3}}{3}\\)"
] | [
"An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect at the centroid of the triangle. What is the maximum possible area of one of these pieces?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-d552b022b255cb256a1915ef4c7f4a24138cca8aaacc64ee9cea8c47af613a6f | An equilateral triangle of side length 3 cm is formed using 18 bars, divided into 9 smaller equilateral triangles, each 1 cm on each side. Find the largest possible sum of the numbers written on the 9 bars that form the border of the large triangle. | [
"126"
] | [
"An equilateral triangle of side length 3 cm is formed using 18 bars, divided into 9 smaller equilateral triangles, each 1 cm on each side. Find the largest possible sum of the numbers written on the 9 bars that form the border of the large triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-e193ca7b31b12f1c71b115654bcc455d743fd75ae94479311cb7a2bcc9bd1365 | An equilateral triangle of side length \( n \) is divided into unit triangles. Let \( f(n) \) be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or r... | [
"\\( 2004! \\)"
] | [
"An equilateral triangle of side length \\( n \\) is divided into unit triangles. Let \\( f(n) \\) be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d769892a9c53cadf3c8363ab927379b1eeacbaa9b2d33906ad106199f3d2359e | An equilateral triangle sits atop a square of area 64. A circle passes through the vertices not shared by the square and the triangle. What is the area of this circle? | [
"\\( 64\\pi \\)"
] | [
"An equilateral triangle sits atop a square of area 64. A circle passes through the vertices not shared by the square and the triangle. What is the area of this circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-adf2d50816843d6d7716107893c99364dba46b830a0a22cf91bb606a5ea821ec | An equilateral triangle with a side length of 14 centimeters is revolved about an altitude to form a cone. What is the volume of the cone in cubic centimeters, expressed to the nearest whole number? | [
"622"
] | [
"An equilateral triangle with a side length of 14 centimeters is revolved about an altitude to form a cone. What is the volume of the cone in cubic centimeters, expressed to the nearest whole number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-0bb25d282df0dd2039a3597dd31d9b1bc99c156cbb8bfe23162378b63f489061 | An equilateral triangle with a side length of 17 is divided into smaller equilateral triangles by lines parallel to its sides. Two players take turns drawing convex polygons along the grid lines, ensuring that the constructed polylines do not share common points. The player who cannot make a move loses. Determine who h... | [
"\\text{First player}"
] | [
"An equilateral triangle with a side length of 17 is divided into smaller equilateral triangles by lines parallel to its sides. Two players take turns drawing convex polygons along the grid lines, ensuring that the constructed polylines do not share common points. The player who cannot make a move loses. Determine ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e1cbd5b2014c30e6a61114a29b719a4e637f28c9cf9492852bff0f27cd1544b | An equilateral triangle with circumference \( o \) is tilted sideways twice such that each tip travels in a circular arc. Calculate how far each tip has traveled, expressed in terms of \( o \). | [
"\\(\\frac{4\\pi o}{9}\\)"
] | [
"An equilateral triangle with circumference \\( o \\) is tilted sideways twice such that each tip travels in a circular arc. Calculate how far each tip has traveled, expressed in terms of \\( o \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-2f61bbe2c336f5c97ae14cb5f80e0613fef25f99967554ad8275a5d106ba6179 | An equilateral triangle with integer side length \( n \geq 3 \) is divided into \( n^2 \) equilateral triangles with side length 1. The vertices of these triangles form a triangular lattice with \(\frac{(n+1)(n+2)}{2}\) lattice points. Let \( H(n) \) be the number of all regular hexagons that can be found by connecting... | [
"98769120"
] | [
"An equilateral triangle with integer side length \\( n \\geq 3 \\) is divided into \\( n^2 \\) equilateral triangles with side length 1. The vertices of these triangles form a triangular lattice with \\(\\frac{(n+1)(n+2)}{2}\\) lattice points. Let \\( H(n) \\) be the number of all regular hexagons that can be foun... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-281068569edb136f8de1a4ac3d67f1089b2399275bc66a65ede44aa532bc892a | An equilateral triangle with side $20$ is cut into $400$ equilateral triangles with side $1$ by three sets of lines parallel to the sides of the triangle. What is the largest number of such triangles that can be crossed by one straight line at the internal points? | [
"39"
] | [
"An equilateral triangle with side $20$ is cut into $400$ equilateral triangles with side $1$ by three sets of lines parallel to the sides of the triangle. What is the largest number of such triangles that can be crossed by one straight line at the internal points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bf5fab216b5f475b8cfaf0435496c4c565790664e5d60245397dfdb95cf865d3 | An equilateral triangle with side \(n\) is cut into \(n^2\) triangles with side 1. How many parallelograms exist with sides parallel to the sides of the large triangle? | [
"\\(\\left(\\frac{(n+1)n}{2}\\right)^3\\)"
] | [
"An equilateral triangle with side \\(n\\) is cut into \\(n^2\\) triangles with side 1. How many parallelograms exist with sides parallel to the sides of the large triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06d4bafbf5db5bdc982e371355de9db532262fdc558efea1c238a232a8577de4 | An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells. Initially, all cells contain $0$. A move consists of selecting two adjacent cells and either increasing or decreasing the numbers in both cells by $1$ simultaneously. Determine all positive integers $n$ such that after performi... | [
"\\( n = 2 \\)"
] | [
"An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells. Initially, all cells contain $0$. A move consists of selecting two adjacent cells and either increasing or decreasing the numbers in both cells by $1$ simultaneously. Determine all positive integers $n$ such that after per... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-e5ef1f2c15127a3a13a776326ec5e618c4f216bd51fcf30e06242dba0f317663 | An equilateral triangle with side length $a$ is given. Points $P$, $M$, and $N$ are on sides $AB$, $BC$, and $AC$, respectively, such that $MP \perp AB$, $NM \perp BC$, and $PN \perp AC$. Find the length of the segment $MP$. | [
"\\(\\frac{a \\sqrt{3}}{3}\\)"
] | [
"An equilateral triangle with side length $a$ is given. Points $P$, $M$, and $N$ are on sides $AB$, $BC$, and $AC$, respectively, such that $MP \\perp AB$, $NM \\perp BC$, and $PN \\perp AC$. Find the length of the segment $MP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 18
} | 65 | 0 | 0 |
omr-ws-ab7896ea0f2555afb29b808a2ca6ed17dde8af4579d50d4b2a6489f83cee4a03 | An equilateral triangle with side length $n$ is divided into $n^2$ smaller equilateral triangles with side length $1$. Initially, exactly one of these small triangles is colored red, and it does not share any point with the boundary of the large triangle. Each move consists of selecting a triangle and changing its colo... | [
"No"
] | [
"An equilateral triangle with side length $n$ is divided into $n^2$ smaller equilateral triangles with side length $1$. Initially, exactly one of these small triangles is colored red, and it does not share any point with the boundary of the large triangle. Each move consists of selecting a triangle and changing its... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-5b458bc8ee3c7ed91578368f236f029960f5adb5865e5ba2c9b63c3df249cb4c | An equilateral triangle with side length $n$ is divided into smaller equilateral triangles with side length 1, forming a triangular lattice. Each segment of length 1 is colored either red, blue, or green. A coloring is called "good" if every small equilateral triangle has sides of three different colors. How many good ... | [
"\\( 6 \\cdot 2^{\\frac{(n+1)(n+2)}{2} - 3} \\)"
] | [
"An equilateral triangle with side length $n$ is divided into smaller equilateral triangles with side length 1, forming a triangular lattice. Each segment of length 1 is colored either red, blue, or green. A coloring is called \"good\" if every small equilateral triangle has sides of three different colors. How man... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-239af3df5ef3633063a210587d5d48a51de15d92911b0325b14314b6876b3afa | An equilateral triangle with side length 10 is divided into 100 smaller equilateral triangles, each with side length 1, by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles that can be formed using the intersection points of these parallel lines as vertices, with sid... | [
"227"
] | [
"An equilateral triangle with side length 10 is divided into 100 smaller equilateral triangles, each with side length 1, by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles that can be formed using the intersection points of these parallel lines as vertices, wit... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62c355c92143374f4593a44b361af781ce000fd4f95d2221f0743ce4acb45e73 | An equilateral triangle with side length 100 is divided into many small equilateral triangles with side length 1 by parallel lines in three directions. If 77 vertices of the resulting triangular lattice are painted blue, find the minimum number of pairs of blue vertices that lie on a line parallel to one of the sides o... | [
"0"
] | [
"An equilateral triangle with side length 100 is divided into many small equilateral triangles with side length 1 by parallel lines in three directions. If 77 vertices of the resulting triangular lattice are painted blue, find the minimum number of pairs of blue vertices that lie on a line parallel to one of the si... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-689e88beff4bf74152059ec86b4995d8986e7dc9ce288fe5e04e9fe900c0f544 | An equilateral triangle with side length 101 is divided into smaller equilateral triangles with side length 1. All sides of these smaller triangles are colored red, including the border of the large triangle. An equilateral triangle is called a "mirror" triangle if its sides are parallel to the sides of the original tr... | [
"101"
] | [
"An equilateral triangle with side length 101 is divided into smaller equilateral triangles with side length 1. All sides of these smaller triangles are colored red, including the border of the large triangle. An equilateral triangle is called a \"mirror\" triangle if its sides are parallel to the sides of the orig... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1b5bf3a717571117ad359aeb3767c2a6406ee1da1f4066122e9f3f6d505ece0 | An equilateral triangle with side length 12 is inscribed in a circle. The diameter AB of the circle is parallel to one side of the triangle and intersects the other two sides at points C and D, with point C closer to A. Find the length of AC. | [
"\\(4\\sqrt{3} - 4\\)"
] | [
"An equilateral triangle with side length 12 is inscribed in a circle. The diameter AB of the circle is parallel to one side of the triangle and intersects the other two sides at points C and D, with point C closer to A. Find the length of AC."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-829073fc7e95b4d53ca19d7c74aa8db4ac99a33f5b0baff954dd7f4e0cf5df21 | An equilateral triangle with side length 2 has centroid \( G \). Let \( G_1 \) be the reflection of \( G \) with respect to the line \( AC \). Let \( G_2 \) be the reflection of \( G_1 \) with respect to the line \( BC \), and let \( G_3 \) be the reflection of \( G_2 \) with respect to the line \( AB \). Find the dist... | [
"\\(\\frac{2\\sqrt{21}}{3}\\)"
] | [
"An equilateral triangle with side length 2 has centroid \\( G \\). Let \\( G_1 \\) be the reflection of \\( G \\) with respect to the line \\( AC \\). Let \\( G_2 \\) be the reflection of \\( G_1 \\) with respect to the line \\( BC \\), and let \\( G_3 \\) be the reflection of \\( G_2 \\) with respect to the line ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2665a9f344e06790475800acc5a84aae13b6ed9c0a695ac9a898124a650b6336 | An equilateral triangle with side length 20 is divided into 400 smaller equilateral triangles with side length 1 by a series of parallel lines. What is the maximum number of these small triangles that can be crossed internally by a single line? | [
"39"
] | [
"An equilateral triangle with side length 20 is divided into 400 smaller equilateral triangles with side length 1 by a series of parallel lines. What is the maximum number of these small triangles that can be crossed internally by a single line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0571640292c9bc63e8717b8f21176178bab0fb986cbed3318599c3879d5d6ee7 | An equilateral triangle with side length 21 is partitioned into \(21^2\) unit equilateral triangles. The number of parallelograms formed by these unit triangles is \(21k\). The value of \(k\) is \(\ldots\). | [
"6300"
] | [
"An equilateral triangle with side length 21 is partitioned into \\(21^2\\) unit equilateral triangles. The number of parallelograms formed by these unit triangles is \\(21k\\). The value of \\(k\\) is \\(\\ldots\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-6ca54c65f6c01c45c17da6cd5d957f1a31d94016ebd8390f451b9611f0c3b578 | An equilateral triangle with side length 3 is divided into 9 smaller equilateral triangles with side length 1. Each vertex of the smaller triangles is numbered with a different number from 1 to 10. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Calculate the minimum possib... | [
"54"
] | [
"An equilateral triangle with side length 3 is divided into 9 smaller equilateral triangles with side length 1. Each vertex of the smaller triangles is numbered with a different number from 1 to 10. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Calculate the minimum p... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-dbe8c0b1650c404681d5bb7de537675f574a37d4730b3d0c494127b09863e437 | An equilateral triangle with side length 3 is to be dissected into $n$ smaller equilateral triangles, where each smaller triangle is considered "big" if all its sides are greater than 1. Find the smallest possible value of $n$ such that $n > 100$. | [
"121"
] | [
"An equilateral triangle with side length 3 is to be dissected into $n$ smaller equilateral triangles, where each smaller triangle is considered \"big\" if all its sides are greater than 1. Find the smallest possible value of $n$ such that $n > 100$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-09d4c3c6eb54fec91f5cba1e6e340355b4787426b5b54eb917cf67b837848df3 | An equilateral triangle with side length 6 is circumscribed about a circle, which is circumscribed about a square. Determine the area of the square. | [
"6"
] | [
"An equilateral triangle with side length 6 is circumscribed about a circle, which is circumscribed about a square. Determine the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a35f135c470251b38b60a03af86ba247d2b3b2689c6834ac3ef9e297fbb57781 | An equilateral triangle with side length 6 is inscribed inside a circle. What is the diameter of the largest circle that can fit inside the circle but outside the triangle? | [
"\\(6 - 3\\sqrt{3}\\)"
] | [
"An equilateral triangle with side length 6 is inscribed inside a circle. What is the diameter of the largest circle that can fit inside the circle but outside the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61e916b2d45ffc6768b0f3fa479ca30ab5d512ce33c25780dc3402ec0cade1d2 | An equilateral triangle with side length 8 is divided into smaller equilateral triangles with side length 1. What is the largest number of these smaller triangles that can be colored such that no two colored triangles share a common vertex? | [
"32"
] | [
"An equilateral triangle with side length 8 is divided into smaller equilateral triangles with side length 1. What is the largest number of these smaller triangles that can be colored such that no two colored triangles share a common vertex?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0507466b74e2cc418718c2850e3a8217b1857322181affdca342dcc73f17a8db | An equilateral triangle with side length 9 is divided into 81 congruent smaller equilateral triangles by segments parallel to its sides. What is the maximum number of non-overlapping parallelograms with sides of lengths 1 and 2 that can be formed within this division? | [
"36"
] | [
"An equilateral triangle with side length 9 is divided into 81 congruent smaller equilateral triangles by segments parallel to its sides. What is the maximum number of non-overlapping parallelograms with sides of lengths 1 and 2 that can be formed within this division?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c448f84183e39ea1e227e30857a48689ee677efb967165b425b990f14c44a26f | An equilateral triangle with side length \( n \) is built using \( n^2 \) plates, each of which is an equilateral triangle with side length 1. Each plate has one side black and the other side white. A move consists of selecting a plate \( P \) that shares a side with at least two other plates, all of which have the sam... | [
"\\text{Yes}"
] | [
"An equilateral triangle with side length \\( n \\) is built using \\( n^2 \\) plates, each of which is an equilateral triangle with side length 1. Each plate has one side black and the other side white. A move consists of selecting a plate \\( P \\) that shares a side with at least two other plates, all of which h... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-36055686b7af21fd0d7ca69ae1e4d3d4dc3e7a0ec45ceacb75eaa1b7fffd8aa2 | An equilateral triangle with side length \( n \) is divided into smaller equilateral triangles of side length 1 by drawing lines through the points of division parallel to the sides of the triangle. Nils has rhombic tiles, each of side length 1 and angles \( 60^\circ \) and \( 120^\circ \), and wants to tile the triang... | [
"\\(\\frac{n^2 - n}{2}\\)"
] | [
"An equilateral triangle with side length \\( n \\) is divided into smaller equilateral triangles of side length 1 by drawing lines through the points of division parallel to the sides of the triangle. Nils has rhombic tiles, each of side length 1 and angles \\( 60^\\circ \\) and \\( 120^\\circ \\), and wants to ti... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-afd3d004d65f146bd74b21d5abd8d631783577286dffb56efa59eddb877e3128 | An equilateral triangle with sides equal to $n$ is divided into regular triangles with sides $1$. Semyon drew a broken line, the links of which run along the segments of the partition. This broken line passed through all the vertices of the partition exactly once. What is the smallest number of links could such a broke... | [
"\\dfrac{n(n + 3)}{2}"
] | [
"An equilateral triangle with sides equal to $n$ is divided into regular triangles with sides $1$. Semyon drew a broken line, the links of which run along the segments of the partition. This broken line passed through all the vertices of the partition exactly once. What is the smallest number of links could such a ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-2102a4afafc38bae670bc3728e40617b1097bfd2cd13daf991ccf433a8c1c796 | An equilateral triangle with sides of $10.4$ cm is to be cut from a square sheet of paper with sides of $10$ cm. If it were possible, calculate the area of the largest equilateral triangle that could be cut from the square, rounded to the nearest hundredth. | [
"46.41"
] | [
"An equilateral triangle with sides of $10.4$ cm is to be cut from a square sheet of paper with sides of $10$ cm. If it were possible, calculate the area of the largest equilateral triangle that could be cut from the square, rounded to the nearest hundredth."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba788a0d1b49f0c12ce111e2b420d772e957cafd43571fa6319eb84d781d20e5 | An equilateral triangle with sides of length $n=12$ is divided into $n^2$ triangular cells. Some cells are initially infected. A cell that is not yet infected becomes infected if it shares adjacent sides with at least two already infected cells. Determine the least number of initially infected cells required to eventua... | [
"45"
] | [
"An equilateral triangle with sides of length $n=12$ is divided into $n^2$ triangular cells. Some cells are initially infected. A cell that is not yet infected becomes infected if it shares adjacent sides with at least two already infected cells. Determine the least number of initially infected cells required to ev... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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