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-e44c285dc4a2a85f45b07ac07e8c9addf0e44b82079d8e3333b07516353a17bd | A triangle has vertices A(8, -3), B(12, 0), and C. Vertex C lies on a circle with center (4, 5) and radius 6, described by the equation \((x-4)^2 + (y-5)^2 = 36\). If the area of triangle ABC is the maximum possible under these conditions, find the area of triangle ABC. | [
"\\(\\frac{85}{2}\\)"
] | [
"A triangle has vertices A(8, -3), B(12, 0), and C. Vertex C lies on a circle with center (4, 5) and radius 6, described by the equation \\((x-4)^2 + (y-5)^2 = 36\\). If the area of triangle ABC is the maximum possible under these conditions, find the area of triangle ABC."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2d3131ca90c26b16a32b1ca5c668f2061d9712c38fd1a29a9888795f87bd462 | A triangle has vertices A, B, and C, with opposite sides of lengths \( a = 1 \), \( b = 3 \), and \( \angle C = 60^\circ \). Find \( \sin^2 B \). | [
"\\(\\frac{189}{196}\\)"
] | [
"A triangle has vertices A, B, and C, with opposite sides of lengths \\( a = 1 \\), \\( b = 3 \\), and \\( \\angle C = 60^\\circ \\). Find \\( \\sin^2 B \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1032d8c2fa0c303f8b7302dfce84872354fe7a7bd9f37e682b6d4f5614bf1239 | A triangle has vertices \( (0,0) \), \( (1,1) \), and \( (6m,0) \), and the line \( y = mx \) divides the triangle into two triangles of equal area. What is the sum of all possible values of \( m \)? | [
"\\(-\\frac{1}{6}\\)"
] | [
"A triangle has vertices \\( (0,0) \\), \\( (1,1) \\), and \\( (6m,0) \\), and the line \\( y = mx \\) divides the triangle into two triangles of equal area. What is the sum of all possible values of \\( m \\)?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
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"cot",
"tir"
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"cot": 16,
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} | 32 | 0 | 0 |
omr-ws-49a7c123e448132fd761579e07e372f9ef1a180bed01a8498ed57fd1f73caf07 | A triangle has vertices \(A, B\), and \(C\). Suppose point \(D\) is on the line segment \(\overline{AC}\) such that \(AB = AD\). If \(\angle ABC - \angle ACB = 30^\circ\), find \(\angle CBD\). | [
"15 degrees"
] | [
"A triangle has vertices \\(A, B\\), and \\(C\\). Suppose point \\(D\\) is on the line segment \\(\\overline{AC}\\) such that \\(AB = AD\\). If \\(\\angle ABC - \\angle ACB = 30^\\circ\\), find \\(\\angle CBD\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-96257624c9a0d569e4ec491f2b1327be839c6613f3d9ff9a89531b5a64660e06 | A triangle has vertices at $(-3,2),(6,-2),(3,5)$. How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth. | [
"25.5"
] | [
"A triangle has vertices at $(-3,2),(6,-2),(3,5)$. How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24d71be51bfc6b02fd28931ba4c5a4055966854b277d48156323f640bd345e78 | A triangle has vertices at $(0,0)$, $(4,4)$, and $(10,0)$. A square is drawn inside the triangle such that its base lies on the x-axis and its two other corners are on the sides of the triangle. What is the area of the square? | [
"\\(\\frac{25}{4}\\)"
] | [
"A triangle has vertices at $(0,0)$, $(4,4)$, and $(10,0)$. A square is drawn inside the triangle such that its base lies on the x-axis and its two other corners are on the sides of the triangle. What is the area of the square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-544c6328d1edb9676c7b8945d77dfb84c533627769529feda9ebbdde0b4eab12 | A triangle has vertices at \((0,0)\), \((1,2)\), and \((k,0)\). The line \(y = kx\) divides the triangle into two triangles of equal area. Find all possible values of \(k\). | [
"\\( k = 1 \\) or \\( k = -2 \\)"
] | [
"A triangle has vertices at \\((0,0)\\), \\((1,2)\\), and \\((k,0)\\). The line \\(y = kx\\) divides the triangle into two triangles of equal area. Find all possible values of \\(k\\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-ca07ebd7ac7820bad35bdb74700558c5d793298d0643409ce9c1f1e6ec8b5ff5 | A triangle has vertices at coordinates (2,2), (5,6) and (6,2). What is the number of units in the length of the longest side of the triangle? | [
"5"
] | [
"A triangle has vertices at coordinates (2,2), (5,6) and (6,2). What is the number of units in the length of the longest side of the triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e21521157582b4f4584cc91b85248cb763ead69247f14918af0d6878d084b653 | A triangle has vertices at the points \( A(3,4) \), \( B(5,1) \), and \( C(1,1) \). A circle is circumscribed around this triangle. If the circle has center \( (h,k) \) and radius \( r \), find \( h + k + r \). | [
"7"
] | [
"A triangle has vertices at the points \\( A(3,4) \\), \\( B(5,1) \\), and \\( C(1,1) \\). A circle is circumscribed around this triangle. If the circle has center \\( (h,k) \\) and radius \\( r \\), find \\( h + k + r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8ed818bb2b3f0e62d3d199a2a55782e6f2cb8b13d1f61e8b43ec1437a3a7981d | A triangle has vertices with the coordinates \(A(0,15)\), \(B(0,0)\), \(C(10,0)\). Find the sum of \(x\) and \(y\) such that the point \(D(x,y)\) on \(AC\) makes the area of \(\triangle ABD\) and \(\triangle BCD\) equal. | [
"\\dfrac{25}{2}"
] | [
"A triangle has vertices with the coordinates \\(A(0,15)\\), \\(B(0,0)\\), \\(C(10,0)\\). Find the sum of \\(x\\) and \\(y\\) such that the point \\(D(x,y)\\) on \\(AC\\) makes the area of \\(\\triangle ABD\\) and \\(\\triangle BCD\\) equal."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-082688ddf015f146e68a526efb0431e53e0b4d93556684a6905b535373f202a4 | A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth. | [
"17.5"
] | [
"A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b77f059152e8e57b6365982d62612701430380c52eb477a72b9e4f0e5eef842 | A triangle in the coordinate plane has vertices \( A(\log_2 1, \log_2 2) \), \( B(\log_2 3, \log_2 4) \), and \( C(\log_2 7, \log_2 8) \). What is the area of \(\triangle ABC\)? | [
"\\(\\log_2\\frac{3}{\\sqrt{7}}\\)"
] | [
"A triangle in the coordinate plane has vertices \\( A(\\log_2 1, \\log_2 2) \\), \\( B(\\log_2 3, \\log_2 4) \\), and \\( C(\\log_2 7, \\log_2 8) \\). What is the area of \\(\\triangle ABC\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-aedc02444d2768ca02477035767094a97b52e7df8f9f6ae4c31a051f4b79e027 | A triangle inscribed in a circle of radius 5 has two sides measuring 5 and 6 respectively. Find the measure of the third side of the triangle. | [
"\\( 3 \\sqrt{3} \\pm 4 \\)"
] | [
"A triangle inscribed in a circle of radius 5 has two sides measuring 5 and 6 respectively. Find the measure of the third side of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-01f9741cf45d2fd2cf25dcac9b1673032de857cdaa37844a1bdf55ae9d99e2a3 | A triangle inscribed in a circle of radius 5 has two sides of length 5 and 6. Find the two possible lengths of the third side of the triangle. | [
"\\(\\sqrt{43 + 24\\sqrt{3}}\\) and \\(\\sqrt{43 - 24\\sqrt{3}}\\)"
] | [
"A triangle inscribed in a circle of radius 5 has two sides of length 5 and 6. Find the two possible lengths of the third side of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-86b99354484d1cd5895d7c5281fc7a8d41ebd2b8cf4217eb2d273bb57a165c82 | A triangle is bounded by the x- and y-axes and must pass through the point \( P(2,2) \). The vertices of the triangle are \( A(0, y) \) and \( B(x, 0) \). a) Find the area of the triangle as a function of \( x \). b) What is the domain of the function found in part (a)? | [
"\\(\\frac{x^2}{x - 2}\\) and \\(x > 0 \\text{ and } x \\neq 2\\)"
] | [
"A triangle is bounded by the x- and y-axes and must pass through the point \\( P(2,2) \\). The vertices of the triangle are \\( A(0, y) \\) and \\( B(x, 0) \\).\n\na) Find the area of the triangle as a function of \\( x \\).\n\nb) What is the domain of the function found in part (a)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-85b93e3c894755e0ac384328de6cd3145bafda147e4db960c68548e430b05cdc | A triangle is bounded by three squares with areas $74$, $116$, and $370$. What is the area of the triangle? | [
"11"
] | [
"A triangle is bounded by three squares with areas $74$, $116$, and $370$. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-89d001a3e9cee3d4a4634a484a933f8e63bc7a1876f1a93f9b16f5264724d59e | A triangle is called Heronian if its perimeter equals its area. Heronian triangle COP has sides 29, 6, and $x$. If the distance from its incenter to the circumcenter is expressed as $\frac{\sqrt{b}}{a}$ where $b$ is not divisible by the squares of any prime, find the remainder when $b$ is divided by $a$. | [
"1"
] | [
"A triangle is called Heronian if its perimeter equals its area. Heronian triangle COP has sides 29, 6, and $x$. If the distance from its incenter to the circumcenter is expressed as $\\frac{\\sqrt{b}}{a}$ where $b$ is not divisible by the squares of any prime, find the remainder when $b$ is divided by $a$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c41bdd35650337666943e8e9ad4e6f0c82f0d4ed37d870856468e7937bea35d | A triangle is called Heronian if its perimeter equals the area of the triangle. Heronian triangle $COP$ has sides $29$, $6$, and $x$. If the distance from its incenter to the circumcenter is expressed as $\frac{\sqrt{b}}{a}$ where $b$ is not divisible by squares of any prime, find the remainder when $b$ is divided by $... | [
"1"
] | [
"A triangle is called Heronian if its perimeter equals the area of the triangle. Heronian triangle $COP$ has sides $29$, $6$, and $x$. If the distance from its incenter to the circumcenter is expressed as $\\frac{\\sqrt{b}}{a}$ where $b$ is not divisible by squares of any prime, find the remainder when $b$ is divid... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5420432eeb225efd5dfb691a9453d9fabb51029d955955aecc70c90c4826221b | A triangle is composed of circular cells arranged in 5784 rows, with the first row having one cell, the second row having two cells, and so on. The cells are divided into pairs of adjacent cells such that each cell belongs to exactly one pair. A pair of adjacent cells is called diagonal if the two cells in it are not i... | [
"2892"
] | [
"A triangle is composed of circular cells arranged in 5784 rows, with the first row having one cell, the second row having two cells, and so on. The cells are divided into pairs of adjacent cells such that each cell belongs to exactly one pair. A pair of adjacent cells is called diagonal if the two cells in it are ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 24
} | 41 | 0 | 0 |
omr-ws-b9afa4556abaaa10594936c366ccd1a84454b03643c2ea6ea3a769b3bf68b74e | A triangle is constructed starting with the numbers from 1 to 2005 in the first row. Each number in the triangle, except those in the first row, is the sum of the two numbers directly above it. What number occupies the lowest vertex of the triangle? | [
"\\( 2^{2003} \\cdot 2006 \\)"
] | [
"A triangle is constructed starting with the numbers from 1 to 2005 in the first row. Each number in the triangle, except those in the first row, is the sum of the two numbers directly above it. What number occupies the lowest vertex of the triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 25
} | 72 | 0 | 0 |
omr-ws-5444c7a22dce5e01df63054376b8f6d6a35911e9e3a7ae9ab7843b8c0b536bed | A triangle is constructed with the first row containing the numbers 1, 2, 3, ..., 2000. Each subsequent number in the triangle is the sum of the two numbers immediately above it. Find the number at the bottom of the triangle. | [
"\\(2001 \\cdot 2^{1998}\\)"
] | [
"A triangle is constructed with the first row containing the numbers 1, 2, 3, ..., 2000. Each subsequent number in the triangle is the sum of the two numbers immediately above it. Find the number at the bottom of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-16d2d4526b2d38beee69058ba95bb82670c3ecce3f63a4a6302d341b7981e74a | A triangle is cut from the corner of a rectangle, resulting in a pentagon with sides of length 8, 10, 13, 15, and 20 units (though not necessarily in that order). What is the area of the pentagon? | [
"270"
] | [
"A triangle is cut from the corner of a rectangle, resulting in a pentagon with sides of length 8, 10, 13, 15, and 20 units (though not necessarily in that order). What is the area of the pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-f94f8c65731eb1d406c49c2fb89da12b391e12826e8b7c66b1738684bf0e608d | A triangle is cut into several (at least two) triangles, where one of them is isosceles (but not equilateral) and all others are equilateral. Determine the angles of the original triangle. | [
"\\(30^\\circ, 60^\\circ, 90^\\circ\\)"
] | [
"A triangle is cut into several (at least two) triangles, where one of them is isosceles (but not equilateral) and all others are equilateral. Determine the angles of the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-94fa89674dd38f0538ce2933ce0408c13e2cd7ffa9e0544e15a8afe9507e4bf1 | A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle. | [
"$30^\\circ$, $60^\\circ$, $90^\\circ$"
] | [
"A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-008528691e696f01e6503076dc391f7ff87c86ad1c5dae01ed3da9a9d0b784fb | A triangle is defined by the coordinates of its vertices $A (3,-2, 1)$, $B (3, 1, 5)$, $C (4, 0, 3)$. Calculate the lengths of the medians $AA_1$ and $BB_1$, the distance between the origin and the center of gravity of the triangle $ABC$, and the angles of the triangle. | [
"$|AA_1|=\\sqrt {\\frac{31}{2}},\\ |BB_1|=\\frac{\\sqrt {53}}{2},\\ |OG|=\\frac{\\sqrt {182}}{3},\\ \\arccos \\frac{14}{15}$"
] | [
"A triangle is defined by the coordinates of its vertices $A (3,-2, 1)$, $B (3, 1, 5)$, $C (4, 0, 3)$. Calculate the lengths of the medians $AA_1$ and $BB_1$, the distance between the origin and the center of gravity of the triangle $ABC$, and the angles of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d90d03a5228874651a48277311839d3385ae1cb95a5826fa0d4ae7841df8f9e | A triangle is dissected into $k$ congruent triangles. Let $f(k)$ be the smallest positive integer such that a triangle can be dissected into $f(k)$ congruent triangles where $f(k)$ is a perfect square and $f(k) \geq k$. Find $f(18)$. | [
"25"
] | [
"A triangle is dissected into $k$ congruent triangles. Let $f(k)$ be the smallest positive integer such that a triangle can be dissected into $f(k)$ congruent triangles where $f(k)$ is a perfect square and $f(k) \\geq k$. Find $f(18)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0104b3fa6e3e0eba9e287e2acd36fff4c99ef8e0852bac8e2185dd92c0894c2f | A triangle is divided into 9 strips by segments parallel to one of the sides and equidistant. The strips are alternately gray and white, with the strip adjacent to the side being gray. If the sum of the areas of the gray bands is equal to 145, what is the sum of the areas of the white bands? | [
"116"
] | [
"A triangle is divided into 9 strips by segments parallel to one of the sides and equidistant. The strips are alternately gray and white, with the strip adjacent to the side being gray. If the sum of the areas of the gray bands is equal to 145, what is the sum of the areas of the white bands?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-39244c4a1b633093d9304ef77528be44decbdad66370b14d4f794b6a96a3850b | A triangle is divided into \( n \) smaller triangles without overlapping. Let \( h_1, h_2, \ldots, h_n \) be the lengths of the smallest altitudes of these smaller triangles and \( h \) be the length of the smallest altitude of the original triangle. If the area of the original triangle is 1, find the minimum possible ... | [
"2"
] | [
"A triangle is divided into \\( n \\) smaller triangles without overlapping. Let \\( h_1, h_2, \\ldots, h_n \\) be the lengths of the smallest altitudes of these smaller triangles and \\( h \\) be the length of the smallest altitude of the original triangle. If the area of the original triangle is 1, find the minim... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-026896106421945f8454b6c64c512ef55e232703227fc2250dd8ac73357bb931 | A triangle is divided into \( n \) strips, with \( n \) being an odd number greater than or equal to 3. The strips are alternately gray and white, with the strip adjacent to the side being gray. What is the ratio of the sum of the areas of the gray bands to the sum of the areas of the white bands, in terms of \( n \)? | [
"\\(\\frac{n + 1}{n - 1}\\)"
] | [
"A triangle is divided into \\( n \\) strips, with \\( n \\) being an odd number greater than or equal to 3. The strips are alternately gray and white, with the strip adjacent to the side being gray. What is the ratio of the sum of the areas of the gray bands to the sum of the areas of the white bands, in terms of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-19ec3d906b6351575028e5e8dc45a8c080a18193a0bac289a063fa3677c4d74e | A triangle is divided into nine smaller triangles, with counters numbered zero at each of the ten vertices. A movement consists of choosing one of the nine triangles and applying the same operation (adding or subtracting a unit) to the three counters of that triangle. Determine all integer numbers \( n \) such that it ... | [
"n \\equiv 0 \\pmod{3}"
] | [
"A triangle is divided into nine smaller triangles, with counters numbered zero at each of the ten vertices. A movement consists of choosing one of the nine triangles and applying the same operation (adding or subtracting a unit) to the three counters of that triangle. Determine all integer numbers \\( n \\) such t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-3d835595c8b6a4c75b1a29b27d46a3f73f06d58237fb3b12839dd567abc6a139 | A triangle is drawn on the coordinate plane with vertices at points with integer coordinates. It is known that inside this triangle there is exactly one point with integer coordinates. What is the largest number of points with integer coordinates (excluding vertices) that can lie on the sides of this triangle? | [
"3"
] | [
"A triangle is drawn on the coordinate plane with vertices at points with integer coordinates. It is known that inside this triangle there is exactly one point with integer coordinates. What is the largest number of points with integer coordinates (excluding vertices) that can lie on the sides of this triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68821d77f3818263b85980c6215482e15f10d920003f5c2fde62ce22fd2411f3 | A triangle is formed between the points $A(0,0)$, $B(3,0)$, and $C(3,4)$. Points $A$ and $C$ are the foci of an ellipse, and point $B$ lies on the ellipse. What is the area of this ellipse? | [
"\\(\\frac{7\\pi\\sqrt{6}}{2}\\)"
] | [
"A triangle is formed between the points $A(0,0)$, $B(3,0)$, and $C(3,4)$. Points $A$ and $C$ are the foci of an ellipse, and point $B$ lies on the ellipse. What is the area of this ellipse?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-356d699e06513551b2ba8f929cdb9f7ad769fe7e1e097c2e01746a38ad8cb0fd | A triangle is formed between the points $A(0,0)$, $B(3,0)$, and $C(3,4)$. Points $A$ and $C$ are the foci of an ellipse, and point $B$ lies on the ellipse. What is the size of the smallest rectangle that can contain the ellipse? | [
"\\( 14\\sqrt{6} \\)"
] | [
"A triangle is formed between the points $A(0,0)$, $B(3,0)$, and $C(3,4)$. Points $A$ and $C$ are the foci of an ellipse, and point $B$ lies on the ellipse. What is the size of the smallest rectangle that can contain the ellipse?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-696db24679368351cf77de4602d399505c7e1a4de0875ea89d2b6242bad88dd1 | A triangle is formed using 3 of the vertices of a regular decagon. How many such triangles can be drawn having no edge in common with the decagon? | [
"150"
] | [
"A triangle is formed using 3 of the vertices of a regular decagon. How many such triangles can be drawn having no edge in common with the decagon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5eda46c2911f75c00f995f1f5685846f576ea9cace185f4919acbe3849c138fb | A triangle is formed with edges along the line \( y = \frac{2}{3}x + 5 \), the \( x \)-axis, and the line \( x = k \). If the area of the triangle is less than 20, find the sum of all possible integral values of \( k \). | [
"-120"
] | [
"A triangle is formed with edges along the line \\( y = \\frac{2}{3}x + 5 \\), the \\( x \\)-axis, and the line \\( x = k \\). If the area of the triangle is less than 20, find the sum of all possible integral values of \\( k \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-ba0ac862a4891bcce963f919fc9c31f3b3f75f4bb36e34d34bd2b6eb99f41cc2 | A triangle is formed with one vertex at the vertex of the parabola $y=x^2-1$ and the other two vertices at the intersections of the line $y=r$ and the parabola. If the area of the triangle is between $8$ and $64$ inclusive, find all possible values of $r$. Express your answer in interval notation. | [
"[3,15]"
] | [
"A triangle is formed with one vertex at the vertex of the parabola $y=x^2-1$ and the other two vertices at the intersections of the line $y=r$ and the parabola. If the area of the triangle is between $8$ and $64$ inclusive, find all possible values of $r$. Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43a31a186a3efdfaad518984682f79b21148434ef33c33d069adc8f1b3a57684 | A triangle is given with $\angle C=90^{\circ}$. Angle bisectors of $\angle BAC$ and $\angle ABC$ intersect sides $\overline{BC}$ and $\overline{CA}$, respectively, at points $P,Q$. Let $M,N$ be the feet of perpendiculars from $P,Q$ onto $\overline{AB}$. Determine $\angle MCN$. | [
"45^\\circ"
] | [
"A triangle is given with $\\angle C=90^{\\circ}$. Angle bisectors of $\\angle BAC$ and $\\angle ABC$ intersect sides $\\overline{BC}$ and $\\overline{CA}$, respectively, at points $P,Q$. Let $M,N$ be the feet of perpendiculars from $P,Q$ onto $\\overline{AB}$. Determine $\\angle MCN$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-ce338d193f1f57929fff5aada81291d175031aeb7ce34662dbc26d8ca130575f | A triangle is inscribed in a circle of diameter $2$ so that one side of the triangle is a diameter. If one of the remaining sides of the triangle subtends an arc with an angular measure of $30^\circ$, find the area of the circle this triangle circumscribes. | [
"$\\frac{6-3\\sqrt{3}}{2}\\pi$"
] | [
"A triangle is inscribed in a circle of diameter $2$ so that one side of the triangle is a diameter. If one of the remaining sides of the triangle subtends an arc with an angular measure of $30^\\circ$, find the area of the circle this triangle circumscribes."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c073728cecc4b2578ecdf56424cdf2d31c5f1adbfc0d64c236b7176d58fb7a2 | A triangle is inscribed in a circle with a line from a point on the circumference extending to a vertex of the triangle opposite to that point. The point from which the line extends is the origin of two other lines that intersect the triangle's sides, dividing one side into two parts and the adjacent side into two part... | [
"2\\sqrt{2}"
] | [
"A triangle is inscribed in a circle with a line from a point on the circumference extending to a vertex of the triangle opposite to that point. The point from which the line extends is the origin of two other lines that intersect the triangle's sides, dividing one side into two parts and the adjacent side into two... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-812dcd081a649471b3a99db2395df48e5ede107062cb58ec72018de60da0cf12 | A triangle is inscribed in a circle with a radius of $5$. The measures of two of the triangle's sides are $5$ and $6$. What are the possible measures of the third side? | [
"\\(3\\sqrt{3} + 4, 3\\sqrt{3} - 4\\)"
] | [
"A triangle is inscribed in a circle with a radius of $5$. The measures of two of the triangle's sides are $5$ and $6$. What are the possible measures of the third side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 23
} | 24 | 0 | 0 |
omr-ws-5a657c6f65eb312c4d22c74dbc26edc86923691f4f0a8c1e9f2b785f6a804ca8 | A triangle is inscribed in a circle with a radius of 5. Two sides of the triangle measure 5 and 6 respectively. Find the measure of the third side of the triangle. | [
"\\(\\sqrt{61 - 30\\sqrt{3}}\\)"
] | [
"A triangle is inscribed in a circle with a radius of 5. Two sides of the triangle measure 5 and 6 respectively. Find the measure of the third side of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3d80d6afcb7da585f2ef18908f2b6a62290820c15ea254dc384350a62f6e55f | A triangle is inscribed in a circle, and another triangle is circumscribed around the same circle such that it is similar to the inscribed triangle. Find the ratio of the perimeter of the inscribed triangle to the perimeter of the circumscribed triangle. | [
"\\(\\frac{r}{R}\\)"
] | [
"A triangle is inscribed in a circle, and another triangle is circumscribed around the same circle such that it is similar to the inscribed triangle. Find the ratio of the perimeter of the inscribed triangle to the perimeter of the circumscribed triangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb376c6dc3b19e0d20ff4afe6c5ab4d209d568bc3ca1003b10d2197c0c8eb91a | A triangle is inscribed in a circle, dividing the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle? | [
"$\\frac{27+9\\sqrt{3}}{\\pi^2}$",
"\\(\\frac{27 + 9\\sqrt{3}}{\\pi^2}\\)"
] | [
"A triangle is inscribed in a circle, dividing the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 61
} | 92 | 0 | 0 |
omr-ws-16f97cc33b7f813bb59eaa9ea4a10fa81e5c9a3c17ad6f69b8e3524c6919eb14 | A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle? | [
"\\(\\frac{9}{\\pi^2} (\\sqrt{3} + 3)\\)"
] | [
"A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$, $4$, and $5$. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e918a63813e46dfc7373b04e07d9a0f10e35eb920c787998e9a61e897fd90683 | A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle? | [
"\\(\\frac{9}{\\pi^2} (\\sqrt{3} + 3)\\)"
] | [
"A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f50b6a95e056517fc79e4e2250740bdeae51f14dec17053118c9a37a1a25f1c2 | A triangle is inscribed in a semicircle with two side lengths measuring 7 cm and 24 cm. The smallest possible difference, in square centimeters, of the area inside the semicircle and outside the triangle can be written as $a\pi -\dfrac{b}{c}\sqrt{d},$ where $\dfrac{b}{c}$ is reduced and $d$ is not divisible by the squa... | [
"608"
] | [
"A triangle is inscribed in a semicircle with two side lengths measuring 7 cm and 24 cm. The smallest possible difference, in square centimeters, of the area inside the semicircle and outside the triangle can be written as $a\\pi -\\dfrac{b}{c}\\sqrt{d},$ where $\\dfrac{b}{c}$ is reduced and $d$ is not divisible by... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 19
} | 32 | 0 | 0 |
omr-ws-78aea3e9cafe2c775712f4b2880b259cbb57d2299546d3aa93be60119de51c54 | A triangle is inscribed in an ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, and its centroid is at the center of the ellipse. Find the locus of the circumcenter of this triangle. | [
"\\(\\frac{x^2}{\\frac{c^2}{a^2}} + \\frac{y^2}{\\frac{c^2}{b^2}} = 1\\)"
] | [
"A triangle is inscribed in an ellipse given by the equation $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$, and its centroid is at the center of the ellipse. Find the locus of the circumcenter of this triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-88c4fa58c5caf62927b948da94f8bf1159c4bd5b31ebc4a30ec94770d0d2e6da | A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sticks t... | [
"6"
] | [
"A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sti... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4215a4110342128199125be21b3640b3548f489ff07632f8e8d51ae48c5a3a48 | A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7. What is the area of the shaded quadrilateral? | [
"18"
] | [
"A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7. What is the area of the shaded quadrilateral?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 4,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-1c5a34608af5dd6241454dc5996447a8759ee88b6ae8cdab35f282465261b166 | A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7. What is the area of the shaded quadrilateral? Note: The problem statement is extracted directly from the forum post and is clearly stated. The ... | [
"18"
] | [
"A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7. What is the area of the shaded quadrilateral?\n\nNote: The problem statement is extracted directly from the forum post and is clearly stat... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5fd48a9473ab0465c138eb058cceac7ee0f4351f403639c9792264af8a88851e | A triangle is picked randomly and uniformly from the set of triangles with real side lengths. Let $\alpha$, $\beta$, and $\gamma$ be its angle measures. Let $p$ be the probability that $\cos(\alpha)\sin(\beta)\sin(\gamma) + \sin(\alpha)\cos(\beta)\sin(\gamma) + \sin(\alpha)\sin(\beta)\cos(\gamma) \leq 1$. Then $p$ can ... | [
"7"
] | [
"A triangle is picked randomly and uniformly from the set of triangles with real side lengths. Let $\\alpha$, $\\beta$, and $\\gamma$ be its angle measures. Let $p$ be the probability that $\\cos(\\alpha)\\sin(\\beta)\\sin(\\gamma) + \\sin(\\alpha)\\cos(\\beta)\\sin(\\gamma) + \\sin(\\alpha)\\sin(\\beta)\\cos(\\gam... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ee3d2c5297fec0d495e8fd7c57c973160d9d4ede8338f549802879629f9f7e78 | A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. What is the number in the lowermost row? | [
"\\(2001 \\cdot 2^{1998}\\)"
] | [
"A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. What is the number in the lowermost row?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-dc6abffc963dcaf8e322d953b80178d9d3bd731677c122bfc2deba191f31f7fb | A triangle was projected onto straight lines parallel to its sides, and three segments of length \(a\), \(b\), and \(c\) were obtained. It is known that the lengths of the sides of a triangle can be uniquely restored from the numbers \(a\), \(b\), and \(c\). What can be its largest angle? | [
"120^\\circ"
] | [
"A triangle was projected onto straight lines parallel to its sides, and three segments of length \\(a\\), \\(b\\), and \\(c\\) were obtained. It is known that the lengths of the sides of a triangle can be uniquely restored from the numbers \\(a\\), \\(b\\), and \\(c\\). What can be its largest angle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-ca3d46fd164621e6cea0d93fb8c4b119ac3fd1a538e1e7c1ffa14c3d9d5b5447 | A triangle with an area of 1 is folded from its top vertex such that the fold is parallel to the base. Find the minimum area of the resulting figure. | [
"\\(\\frac{2}{3}\\)"
] | [
"A triangle with an area of 1 is folded from its top vertex such that the fold is parallel to the base. Find the minimum area of the resulting figure."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-1359c9026c7276e06b6fba5cfde7691e7eb7f32c1ee05e769989d848fa0a203e | A triangle with angles \( A, B, C \) satisfies \( \frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \frac{12}{7} \) and \( \sin A \sin B \sin C = \frac{12}{25} \). Given that \( \sin C \) takes on three possible values \( s_1, s_2, \) and \( s_3 \), find \( 100 s_1 s_2 s_3 \). | [
"48"
] | [
"A triangle with angles \\( A, B, C \\) satisfies \\( \\frac{\\sin A + \\sin B + \\sin C}{\\cos A + \\cos B + \\cos C} = \\frac{12}{7} \\) and \\( \\sin A \\sin B \\sin C = \\frac{12}{25} \\). Given that \\( \\sin C \\) takes on three possible values \\( s_1, s_2, \\) and \\( s_3 \\), find \\( 100 s_1 s_2 s_3 \\)."... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-c994b3afec2ca2d7be0c93b1d6e5c253fd0936cc09b15afbf7f4ce7d9e7de984 | A triangle with area $\sqrt 3$ has side lengths $a, b, c.$ Determine the maximum value of \[ | a^2 + b^2 + c^2 - 2(ab + bc + ca)|. \] | [
"12"
] | [
"A triangle with area $\\sqrt 3$ has side lengths $a, b, c.$ Determine the maximum value of \\[ | a^2 + b^2 + c^2 - 2(ab + bc + ca)|. \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-53ce7ca05c7e228a84518efd98d37f5f0fb318652f7785a3c3f1cbf25e64552b | A triangle with area \( T \) is divided into six regions by lines drawn through a point inside the triangle parallel to the sides. The three triangular regions have areas \( T_1 \), \( T_2 \), and \( T_3 \). If \( T = 36 \), find the value of \( \sqrt{T_1} + \sqrt{T_2} + \sqrt{T_3} \). | [
"6"
] | [
"A triangle with area \\( T \\) is divided into six regions by lines drawn through a point inside the triangle parallel to the sides. The three triangular regions have areas \\( T_1 \\), \\( T_2 \\), and \\( T_3 \\). If \\( T = 36 \\), find the value of \\( \\sqrt{T_1} + \\sqrt{T_2} + \\sqrt{T_3} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d587b37d2c8ce5a20a110725249c6b7ca69659b0bb46418746366388df4e2a21 | A triangle with each side \( n \) cm long has \( n-1 \) points marked on each side to divide it into \( n \) equal parts of 1 cm each. From each of these points, two lines are drawn parallel to the other sides to meet the other sides, creating a mesh of triangles. How many triangles are there in this diagram? | [
"\\( n^2 \\)"
] | [
"A triangle with each side \\( n \\) cm long has \\( n-1 \\) points marked on each side to divide it into \\( n \\) equal parts of 1 cm each. From each of these points, two lines are drawn parallel to the other sides to meet the other sides, creating a mesh of triangles. How many triangles are there in this diagram... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-67ef9548a0a24789e714d059d1c71bc5ce551b08df3c91a871154f3eb52bb0d7 | A triangle with perimeter $2p$ has sides $a$, $b$, and $c$. If possible, a new triangle with sides $p-a$, $p-b$, and $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely? | [
"\\(a = b = c\\)"
] | [
"A triangle with perimeter $2p$ has sides $a$, $b$, and $c$. If possible, a new triangle with sides $p-a$, $p-b$, and $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9cde72579ab5a61ed85d962f85ad3fdffa3872e4000c3ad8a6859fe81f18acb3 | A triangle with perimeter 7 has integer side lengths. What is the maximum possible area of such a triangle? | [
"\\(\\frac{3\\sqrt{7}}{4}\\)"
] | [
"A triangle with perimeter 7 has integer side lengths. What is the maximum possible area of such a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-9eaf3abee8da54b2512c50e46788779a50ac57e10224a509c4d79e138e7e3c76 | A triangle with side lengths $x$, $x-1$, and $x+1$ is inscribed in a circle of radius 2. Find the inradius of the triangle. | [
"\\(\\frac{1}{2}\\)"
] | [
"A triangle with side lengths $x$, $x-1$, and $x+1$ is inscribed in a circle of radius 2. Find the inradius of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-4fc0f441abcf2f1f3169c4d63644039eadc31e8928a995789a1b63aa53126a36 | A triangle with side lengths 16, 30, and 34 is inscribed inside Circle A and circumscribed about Circle B. What is the ratio of the area of Circle B to the area of Circle A? | [
"\\(\\frac{36}{289}\\)"
] | [
"A triangle with side lengths 16, 30, and 34 is inscribed inside Circle A and circumscribed about Circle B. What is the ratio of the area of Circle B to the area of Circle A?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-855f25ce53104425ca0969e18f66f5b712bafcb8b4b3171562ec4cb36ec5b1b4 | A triangle with side lengths \(a\), \(b\), and \(c\) is tossed at random onto a horizontal surface ruled with parallel lines spaced \(d\) apart, where \(d\) is greater than any side of the triangle. What is the probability that the triangle intersects one of the lines? | [
"\\dfrac{a + b + c}{\\pi d}"
] | [
"A triangle with side lengths \\(a\\), \\(b\\), and \\(c\\) is tossed at random onto a horizontal surface ruled with parallel lines spaced \\(d\\) apart, where \\(d\\) is greater than any side of the triangle. What is the probability that the triangle intersects one of the lines?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-7ddf50d277e69c357e6c8f58804f6c538bb0aa1f675a6e1ba5eb8e59aafe7017 | A triangle with side lengths in the ratio $3:4:5$ is inscribed in a circle of radius $3$. What is the area of the triangle? | [
"8.64"
] | [
"A triangle with side lengths in the ratio $3:4:5$ is inscribed in a circle of radius $3$. What is the area of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e691d94bf4fafa08cf522dc71c74246ac263db492f4c1c303c558ae97e2d44b6 | A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth. | [
"8.64"
] | [
"A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8c9e94b8ff8b8eaee32213a0ea51823e3f2e66baafdf7c7ca5139ae63e2c03b | A triangle with sides $13$, $14$, and $15$ sits around the top half of a sphere of radius $5$ (touching the sphere at three points). How far is the plane of the triangle from the center of the sphere? | [
"3"
] | [
"A triangle with sides $13$, $14$, and $15$ sits around the top half of a sphere of radius $5$ (touching the sphere at three points). How far is the plane of the triangle from the center of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
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omr-ws-b0efaaec65910bbe66dc358f53d2c40cc5113949162b08676de7a39299a89fbc | A triangle with sides $2, 3$, and $3$ was cut into four triangles similar to it. What can be the similarity ratios of these smaller triangles to the original triangle? | [
"\\(\\frac{1}{2}\\)"
] | [
"A triangle with sides $2, 3$, and $3$ was cut into four triangles similar to it. What can be the similarity ratios of these smaller triangles to the original triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
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omr-ws-f52959578b019a79ff9f1c8a2fc6aac1cd7d3301a2f705e1a7bec84690f0cea1 | A triangle with sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of length $\frac{3}{2}$. Find the maximum value of the coincident area. | [
"$\\frac{9}{28}$"
] | [
"A triangle with sides $\\frac{3}{2}, \\frac{\\sqrt{5}}{2}, \\sqrt{2}$ is folded along a variable line perpendicular to the side of length $\\frac{3}{2}$. Find the maximum value of the coincident area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
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omr-ws-a3655a8ec4e418900bdf8b005529892aa58d1f77112c9ef11b773bf7660605ce | A triangle with sides 13, 14, and 15 sits around the top half of a sphere of radius 5. How far is the plane of the triangle from the center of the sphere? | [
"3"
] | [
"A triangle with sides 13, 14, and 15 sits around the top half of a sphere of radius 5. How far is the plane of the triangle from the center of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1794766cce2f6478b0d41638af00259ec7437c654c330438f33cf560c2f27541 | A triangle with sides 7, 9, and \( x \) has an area of \( 6\sqrt{5} \). What is \( x \)? | [
"\\( x = 4, 2\\sqrt{61} \\)"
] | [
"A triangle with sides 7, 9, and \\( x \\) has an area of \\( 6\\sqrt{5} \\). What is \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
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omr-ws-f81a7c7bd933be5ba3557814288e1ec43af8cf815330600217148c8db8397b36 | A triangle with sides 9, 28, and 29 has one of its sides increased by \( m \), a second by \( 2m \), and a third by \( 3m \). If \( m \) is an integer such that \( m > 0 \), find the perimeter of the newly formed triangle if the newly formed triangle is a right triangle. | [
"84"
] | [
"A triangle with sides 9, 28, and 29 has one of its sides increased by \\( m \\), a second by \\( 2m \\), and a third by \\( 3m \\). If \\( m \\) is an integer such that \\( m > 0 \\), find the perimeter of the newly formed triangle if the newly formed triangle is a right triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
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} | 32 | 0 | 0 |
omr-ws-1e339d5be3b51b047e7811959776bff9bb6d068db08469978cb60c80f268987d | A triangle with sides \(\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}\) is folded along a variable line perpendicular to the side of \(\frac{3}{2}\). Find the maximum value of the coincident area. | [
"\\dfrac{9}{28}"
] | [
"A triangle with sides \\(\\frac{3}{2}, \\frac{\\sqrt{5}}{2}, \\sqrt{2}\\) is folded along a variable line perpendicular to the side of \\(\\frac{3}{2}\\). Find the maximum value of the coincident area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-4f375c39a59609e50fb3fdbd39dacabdda699d1b6471de6ab9a720e115a7d976 | A triangle with sides \(a < b < c\) is called log-right if \(\log a^2 + \log b^2 = \log c^2\). Compute the largest possible value of \(a\) in a right triangle that is also log-right. | [
"\\(\\sqrt{2}\\)"
] | [
"A triangle with sides \\(a < b < c\\) is called log-right if \\(\\log a^2 + \\log b^2 = \\log c^2\\). Compute the largest possible value of \\(a\\) in a right triangle that is also log-right."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 42 | 0 | 0 |
omr-ws-22117a12f3a73ee6b03af92aaa40783f77239ffa12b554ae90fa49a680d65d00 | A triangle with sides \(a, b, c\) is called a good triangle if \(a^2, b^2, c^2\) can form a triangle. Determine how many of the following triangles are good: (i) \(40^\circ, 60^\circ, 80^\circ\) (ii) \(10^\circ, 10^\circ, 160^\circ\) (iii) \(110^\circ, 35^\circ, 35^\circ\) (iv) \(50^\circ, 30^\circ, 100^\circ\) (v) \(9... | [
"2"
] | [
"A triangle with sides \\(a, b, c\\) is called a good triangle if \\(a^2, b^2, c^2\\) can form a triangle. Determine how many of the following triangles are good:\n\n(i) \\(40^\\circ, 60^\\circ, 80^\\circ\\)\n\n(ii) \\(10^\\circ, 10^\\circ, 160^\\circ\\)\n\n(iii) \\(110^\\circ, 35^\\circ, 35^\\circ\\)\n\n(iv) \\(50... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
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} | 16 | 0 | 0 |
omr-ws-c0c22aa1e8f4a24c9631e87323371c23d18610bcd166325283ef9afc615b9f1d | A triangle with sides \(a, b, c\) is called a good triangle if \(a^2, b^2, c^2\) can form a triangle. How many of the following triangles are good? (i) \(40^\circ, 60^\circ, 80^\circ\) (ii) \(10^\circ, 10^\circ, 160^\circ\) (iii) \(110^\circ, 35^\circ, 35^\circ\) (iv) \(50^\circ, 30^\circ, 100^\circ\) (v) \(90^\circ, 4... | [
"2"
] | [
"A triangle with sides \\(a, b, c\\) is called a good triangle if \\(a^2, b^2, c^2\\) can form a triangle. How many of the following triangles are good?\n(i) \\(40^\\circ, 60^\\circ, 80^\\circ\\)\n(ii) \\(10^\\circ, 10^\\circ, 160^\\circ\\)\n(iii) \\(110^\\circ, 35^\\circ, 35^\\circ\\)\n(iv) \\(50^\\circ, 30^\\circ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d869401d375679d7fe40648071f506bbbe738e3d3c6dbfa0e731ba462327aed9 | A triangle with sides \(a, b, c\) is folded along a line \(\ell\) so that a vertex \(C\) is on side \(c\). Find the segments on which point \(C\) divides \(c\), given that the angles adjacent to \(\ell\) are equal. | [
"\\(\\frac{ac}{a+b}\\) and \\(\\frac{bc}{a+b}\\)"
] | [
"A triangle with sides \\(a, b, c\\) is folded along a line \\(\\ell\\) so that a vertex \\(C\\) is on side \\(c\\). Find the segments on which point \\(C\\) divides \\(c\\), given that the angles adjacent to \\(\\ell\\) are equal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 11
} | 13 | 0 | 0 |
omr-ws-00b208637a47657e2e280b57fdacdfe05a1731411148bd1c6a614b81534ad64f | A triangle with sides of 36, 40, and 52 has one of its sides increased by \( x \), a second by \( 2x \), and a third by \( 3x \), where \( x \) is a positive integer. If the newly formed triangle is a right triangle, find the sum of all possible distinct perimeters of the newly formed triangle. | [
"448"
] | [
"A triangle with sides of 36, 40, and 52 has one of its sides increased by \\( x \\), a second by \\( 2x \\), and a third by \\( 3x \\), where \\( x \\) is a positive integer. If the newly formed triangle is a right triangle, find the sum of all possible distinct perimeters of the newly formed triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-d7a58fe43856c4192af3c99ae24069060eac1810b660a01179ec068e22e50323 | A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? | [
"\\frac{\\sqrt{65}}{2}"
] | [
"A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca170cc0a44a743c9c976ccde95cc92a3bf003303ceea5e9512a1a7801ef903b | A triangle with sides of 9, 28, and 29 has one of its sides increased by \( m \), a second by \( 2m \), and a third by \( 3m \). If \( m \) is an integer and \( m > 0 \), find the perimeter of the newly formed triangle if the newly formed triangle is a right triangle. | [
"84"
] | [
"A triangle with sides of 9, 28, and 29 has one of its sides increased by \\( m \\), a second by \\( 2m \\), and a third by \\( 3m \\). If \\( m \\) is an integer and \\( m > 0 \\), find the perimeter of the newly formed triangle if the newly formed triangle is a right triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-6acad7851cf55bb083ef4a4d31ce5a1aa7c3deda5f52855a5f2f5fb574c4a4ef | A triangle with sides of length $a, b, c$ has the same area as the triangles with side lengths (i) $7, b, c$, (ii) $a, 8, c$, and (iii) $a, b, 10$. However, $a \neq 7$, $b \neq 8$, and $c \neq 10$. Find the values of $a, b, c$. | [
"\\( a = \\sqrt{31}, b = \\sqrt{26}, c = \\sqrt{14} \\)"
] | [
"A triangle with sides of length $a, b, c$ has the same area as the triangles with side lengths (i) $7, b, c$, (ii) $a, 8, c$, and (iii) $a, b, 10$. However, $a \\neq 7$, $b \\neq 8$, and $c \\neq 10$. Find the values of $a, b, c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 19
} | 22 | 0 | 0 |
omr-ws-de5772aeaef48976f2615a43429b5b0191272af02dca66b72622fb96ebd584ba | A triangle with sides of lengths 6, 10, and 14 is inscribed in a circle. Find the area of the largest triangle that can be inscribed in the circle. | [
"\\( 49 \\sqrt{3} \\)"
] | [
"A triangle with sides of lengths 6, 10, and 14 is inscribed in a circle. Find the area of the largest triangle that can be inscribed in the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-903d22505da6e04f82ccb7ed50ea61d76b9cad04583e079d49ce51bf77c5a09d | A triangle with sides of lengths 7, 8, and 9 inches is inscribed in a circle. Find the radius of the circle. | [
"\\dfrac{21\\sqrt{5}}{10}"
] | [
"A triangle with sides of lengths 7, 8, and 9 inches is inscribed in a circle. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98364cfe7ae7722f77e01f51465774cdc1402e06fe0679ff436af2dd8d4e46ae | A triangle with vertices \( A(0, 0) \), \( B(3, 4) \), and \( C(2, c) \) has an area of 5. Find all possible values of \( c \). | [
"\\( c = 6 \\) or \\( c = -\\frac{2}{3} \\)"
] | [
"A triangle with vertices \\( A(0, 0) \\), \\( B(3, 4) \\), and \\( C(2, c) \\) has an area of 5. Find all possible values of \\( c \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 25
} | 27 | 0 | 0 |
omr-ws-a684c82ca3e47771bae3ea178180e13019a4631059fd616874af12d34db074bc | A triangle with vertices \((6,5)\), \((8,-3)\), and \((9,1)\) is reflected about the line \(x=8\) to create a second triangle. What is the area of the union of the two triangles? | [
"\\(\\frac{32}{3}\\)"
] | [
"A triangle with vertices \\((6,5)\\), \\((8,-3)\\), and \\((9,1)\\) is reflected about the line \\(x=8\\) to create a second triangle. What is the area of the union of the two triangles?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 8,
"tir": 15
} | 52 | 0 | 0 |
omr-ws-5328bd9ff25a99d8d81bcdf372a9b1de1eb365e36a1e97e9a6e33f581d26ee14 | A triangle with vertices at $3 + i$, $i(1 + \sqrt{3})$, and $i(1 - \sqrt{3})$ is rotated about its circumcenter through an angle $\pi/3$ in the counterclockwise direction. Find the new positions of its vertices. | [
"2 + (1 - \\sqrt{3})i"
] | [
"A triangle with vertices at $3 + i$, $i(1 + \\sqrt{3})$, and $i(1 - \\sqrt{3})$ is rotated about its circumcenter through an angle $\\pi/3$ in the counterclockwise direction. Find the new positions of its vertices."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cedee5b5018366990df4b0e9fac2f4320218432379beaabaa741dd647f15c3b | A triangle, a line, and three rectangles are located on a plane such that one of the sides of each rectangle is parallel to the line and the rectangles completely cover all sides of the triangle. If the area of the triangle is $T$, and the total area of the three rectangles is $R$, find the minimum possible value of $\... | [
"2"
] | [
"A triangle, a line, and three rectangles are located on a plane such that one of the sides of each rectangle is parallel to the line and the rectangles completely cover all sides of the triangle. If the area of the triangle is $T$, and the total area of the three rectangles is $R$, find the minimum possible value ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f2213ffe266a7ee5ec318ef4556b12a7fc4d0820d0a443eb47c13ad95043ab3 | A triangle, one of whose angles is $40^\circ$, was cut along its angle bisectors into six triangles, among which there are right triangles. What can be the measures of the other two angles of the original triangle? | [
"\\(70^\\circ, 70^\\circ\\) or \\(40^\\circ, 100^\\circ\\)"
] | [
"A triangle, one of whose angles is $40^\\circ$, was cut along its angle bisectors into six triangles, among which there are right triangles. What can be the measures of the other two angles of the original triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-219d7348478177773e0e3d28c496efbae4b3d1bb959adadefb33131c9735cee2 | A triangular array of numbers has a first row consisting of the odd integers $1, 3, 5, \ldots, 99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it... | [
"17"
] | [
"A triangular array of numbers has a first row consisting of the odd integers $1, 3, 5, \\ldots, 99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally ab... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 16,
"tir": 13
} | 59 | 0 | 0 |
omr-ws-93d19b2e78cab5761d582cbf63a7977d70082f3b971b3c2de63cf7fb9e0691f1 | A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in ... | [
"17"
] | [
"A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above ... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d939e8c19c6dd63b860c4e68d619c05656a307ca3e3b6fd06f81269a6511fb4c | A triangular colony area is divided into four fields. The distances $AF$, $FD$, $BF$, and $FE$ have the lengths $5$, $2$, $4$, and $2$ respectively (in $10$s of meters). Which lot has the largest area? | [
"$DFEC$"
] | [
"A triangular colony area is divided into four fields. The distances $AF$, $FD$, $BF$, and $FE$ have the lengths $5$, $2$, $4$, and $2$ respectively (in $10$s of meters). Which lot has the largest area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8481ba1e3a297c5f5a3aa16a6f63d1ed820ad5b9d0c711e4d3816ba41e3f1c96 | A triangular home base section has a perimeter of $P$ meters and an area of $A$ square meters. The home park is a field consisting of points that are 5 meters away from the nearest edge of the house. Determine the combined area of the park and the house. | [
"\\( A + 5P + 25\\pi \\)"
] | [
"A triangular home base section has a perimeter of $P$ meters and an area of $A$ square meters. The home park is a field consisting of points that are 5 meters away from the nearest edge of the house. Determine the combined area of the park and the house."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 16
} | 63 | 0 | 0 |
omr-ws-364459c4b2a338420985f7e83c66d3203e7f415918ba8349df1ab78953ef4c17 | A triangular parcel of ground has sides of lengths 725 ft, 650 ft, and 575 ft. Find the measure of the largest angle. | [
"72.8°"
] | [
"A triangular parcel of ground has sides of lengths 725 ft, 650 ft, and 575 ft. Find the measure of the largest angle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7b5b2fa8520297f86a5e1e6044be3e9985d5cd9365f124c9500d6e56bd52849 | A triangular pencil has each side of the triangle measuring 7 mm and a total length of 15 cm. One end is cut straight, revealing a triangular cross-section. The other end is sharpened to form a cone with a vertex angle of 20 degrees. Calculate the volume of the pencil. | [
"2.99"
] | [
"A triangular pencil has each side of the triangle measuring 7 mm and a total length of 15 cm. One end is cut straight, revealing a triangular cross-section. The other end is sharpened to form a cone with a vertex angle of 20 degrees. Calculate the volume of the pencil."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-9826a88b006218f52b5d885a5166caace15ce5559c4b94f411e1376ea9dabd86 | A triangular piece of paper ABC is folded along the line ED, which is parallel to its side AB. The area of the folded part that projects below the base AB (i.e., the area of triangle FHG) is 25% of the area of the whole triangle ABC. What is the ratio of the overlapping area (trapezoid FGDE) to the area of the whole tr... | [
"\\( \\frac{5}{32} \\)"
] | [
"A triangular piece of paper ABC is folded along the line ED, which is parallel to its side AB. The area of the folded part that projects below the base AB (i.e., the area of triangle FHG) is 25% of the area of the whole triangle ABC. What is the ratio of the overlapping area (trapezoid FGDE) to the area of the who... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4c201b40cb5fe438130f88424467afd09c6300d613dcff10edc7d87e5fa44bb | A triangular piece of paper with an area of 1 is folded along a line parallel to one of its sides and pressed flat. What is the minimum possible area of the resulting figure? | [
"\\dfrac{2}{3}"
] | [
"A triangular piece of paper with an area of 1 is folded along a line parallel to one of its sides and pressed flat. What is the minimum possible area of the resulting figure?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 5
} | 31 | 0 | 0 |
omr-ws-b90a8c9a4aff708fda080311a1d132c180e3078d41da3f254da889004bcc4b26 | A triangular pyramid (tetrahedron) is inscribed in a sphere with radius 3 units. If all sides of the pyramid are equilateral triangles, find the surface area of the pyramid. | [
"\\( 96 \\sqrt{3} \\)"
] | [
"A triangular pyramid (tetrahedron) is inscribed in a sphere with radius 3 units. If all sides of the pyramid are equilateral triangles, find the surface area of the pyramid."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d312f76d76fe16c48d699d7eb65f7e83fb0705c1f0320328046208e78a63cf7b | A triangular pyramid \( O.ABC \) with base \( ABC \) has the property that the lengths of the altitudes from \( A \), \( B \), and \( C \) are not less than \( \frac{OB + OC}{2} \), \( \frac{OC + OA}{2} \), and \( \frac{OA + OB}{2} \), respectively. Given that the area of \( ABC \) is \( S \), calculate the volume of t... | [
"\\dfrac{\\sqrt{2} \\, S^{3/2}}{3^{7/4}}"
] | [
"A triangular pyramid \\( O.ABC \\) with base \\( ABC \\) has the property that the lengths of the altitudes from \\( A \\), \\( B \\), and \\( C \\) are not less than \\( \\frac{OB + OC}{2} \\), \\( \\frac{OC + OA}{2} \\), and \\( \\frac{OA + OB}{2} \\), respectively. Given that the area of \\( ABC \\) is \\( S \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-4069d8fb9400ec449fd9de5bb0fb948901191f32beac35f09a535aa588970c59 | A triangular pyramid has all plane angles at one vertex as right angles. There exists a point such that its distance from this vertex is $3$, and its distances from the other three vertices are $\sqrt{5}$, $\sqrt{6}$, and $\sqrt{7}$, respectively. Find the radius of the sphere circumscribed around this pyramid. | [
"\\(\\frac{3}{2}\\)"
] | [
"A triangular pyramid has all plane angles at one vertex as right angles. There exists a point such that its distance from this vertex is $3$, and its distances from the other three vertices are $\\sqrt{5}$, $\\sqrt{6}$, and $\\sqrt{7}$, respectively. Find the radius of the sphere circumscribed around this pyramid.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 14
} | 60 | 0 | 0 |
omr-ws-2d0aace5f63a3ab3dc31fc4733028eb280d17fb6f767044d80c02d241024c4fa | A triangular pyramid is cut by a plane into two polyhedrons. Find the ratio of volumes of these polyhedrons if it is known that the cutting plane divides three concurrent lateral edges of the pyramid so that the ratios of the segments of these edges adjacent to the common vertex to the remaining parts of the edges are ... | [
"\\(\\frac{2}{25}\\)"
] | [
"A triangular pyramid is cut by a plane into two polyhedrons. Find the ratio of volumes of these polyhedrons if it is known that the cutting plane divides three concurrent lateral edges of the pyramid so that the ratios of the segments of these edges adjacent to the common vertex to the remaining parts of the edges... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40e059bc17eba4793a9152036426b319bfd921cf4a27a595da97064ccdb04aad | A triangular pyramid is defined by the coordinates of its vertices: $A (3, 0, 1)$, $B ( - 1 , 4, 1)$, $C (5, 2, 3)$, $D (0, - 5 , 4)$. Calculate the length of the vector $\overrightarrow{AG}$, if $G$ is the point of intersection of the medians of the face $BCD$. | [
"$|AG|=\\frac{\\sqrt {51}}{3}$"
] | [
"A triangular pyramid is defined by the coordinates of its vertices: $A (3, 0, 1)$, $B ( - 1 , 4, 1)$, $C (5, 2, 3)$, $D (0, - 5 , 4)$. Calculate the length of the vector $\\overrightarrow{AG}$, if $G$ is the point of intersection of the medians of the face $BCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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