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-b46d150f0f779828fd7d6929f62f935789146412e166d73a9c56a5c43cf68e7f | Two regular hexagons share a side. If the perimeter of the quadrilateral ABCD is $56 + 28\sqrt{3}$, find the area of triangle ABE. | [
"\\(49\\sqrt{3}\\)"
] | [
"Two regular hexagons share a side. If the perimeter of the quadrilateral ABCD is $56 + 28\\sqrt{3}$, find the area of triangle ABE."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9759a19396488651a63e826a243505a57cbe04df5f0db1ecaef7b844b58936ec | Two regular octagons and one square completely cover the part of a plane around a point without any overlapping. Find all other possible combinations of these three regular polygons (two are congruent and one different). | [
"\\((5, 10)\\) and \\((12, 3)\\)"
] | [
"Two regular octagons and one square completely cover the part of a plane around a point without any overlapping. Find all other possible combinations of these three regular polygons (two are congruent and one different)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 22
} | 24 | 0 | 0 |
omr-ws-6e476e2f001049798ff7d76a93a07586da950e86a2b167f94143190c598e1b8a | Two regular octagons have sides of length 21 cm and 28 cm. Find: a) The scale factor. b) The ratio of the perimeters. c) The ratio of the areas of the octagons. | [
"\\dfrac{16}{9}"
] | [
"Two regular octagons have sides of length 21 cm and 28 cm. Find:\na) The scale factor.\nb) The ratio of the perimeters.\nc) The ratio of the areas of the octagons."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-e488fb56061c612a8cc36b30ad623cdda1e8c34b253878a1e9db7db0c0047790 | Two regular pentagons $ABCDE$ and $AEKPL$ are placed in space such that $\angle DAK = 60^\circ$. Let $P_1$ and $P_2$ be the planes $ACK$ and $BAL$, respectively. Calculate the value of $\cos \theta$, where $\theta$ is the angle between $P_1$ and $P_2$. | [
"\\dfrac{\\sqrt{5}-1}{4}"
] | [
"Two regular pentagons $ABCDE$ and $AEKPL$ are placed in space such that $\\angle DAK = 60^\\circ$. Let $P_1$ and $P_2$ be the planes $ACK$ and $BAL$, respectively. Calculate the value of $\\cos \\theta$, where $\\theta$ is the angle between $P_1$ and $P_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39780e973ef81ada5d958d34a2617c2da6ceca09a619a1a156de1d98b25e16a5 | Two regular pentagons $OKLMN$ and $OPRST$ are given, with point $P$ lying on the segment $ON$, and points $R, S, T$ outside the pentagon $OKLMN$. Find the angle between lines $KP$ and $MS$. | [
"36^\\circ"
] | [
"Two regular pentagons $OKLMN$ and $OPRST$ are given, with point $P$ lying on the segment $ON$, and points $R, S, T$ outside the pentagon $OKLMN$. Find the angle between lines $KP$ and $MS$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-6a71f5d6d205ca9311bada86292c5bdee002ed6d25b73c900d73ca9118559285 | Two regular pentagons ABCDE and AEKPL are situated in space such that the angle \( \angle DAK = 60^\circ \). Let \( \theta \) be the angle between the planes \( (ACK) \) and \( (BAL) \). Calculate \( \cos(\theta) \). | [
"\\dfrac{1}{2}"
] | [
"Two regular pentagons ABCDE and AEKPL are situated in space such that the angle \\( \\angle DAK = 60^\\circ \\). Let \\( \\theta \\) be the angle between the planes \\( (ACK) \\) and \\( (BAL) \\). Calculate \\( \\cos(\\theta) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-b618bb34cb53735f785cb1cfeaddc7c25251d0e81cb63dc07e2998e0fc81d4c4 | Two regular polygons have the same perimeter. If the first has 38 sides and a side length twice as long as the second, how many sides does the second have? | [
"76"
] | [
"Two regular polygons have the same perimeter. If the first has 38 sides and a side length twice as long as the second, how many sides does the second have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62e04e6196ca6dbd736d7816c772cacf8d4323ee72d9f118129d39a28fc0fe74 | Two regular polygons of side $1$ are called [i]friendly[/i] if: - They have a side in common that leaves them in opposite half-planes. - The two sides, one from each polygon, that meet at a vertex of the common side, are sides of an equilateral triangle. Find all pairs of friendly polygons. | [
"$(12,12), (10, 15), (9, 18), (8, 24), (7, 42)$"
] | [
"Two regular polygons of side $1$ are called [i]friendly[/i] if:\n- They have a side in common that leaves them in opposite half-planes.\n- The two sides, one from each polygon, that meet at a vertex of the common side, are sides of an equilateral triangle.\nFind all pairs of friendly polygons."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-e9135795a998b1b839bb84226305b9e75ce3b5be8cd6434ec84a0bc807706e22 | Two regular polygons, each with \( n \) sides, are circumscribed to and inscribed in a given circle. If an angular point of the circumscribed polygon is joined to each of the angular points of the inscribed polygon, calculate the value of \( \frac{S}{A} \), where \( S \) is the sum of the squares of the straight lines ... | [
"\\dfrac{2}{\\sin \\dfrac{2\\pi}{n}}"
] | [
"Two regular polygons, each with \\( n \\) sides, are circumscribed to and inscribed in a given circle. If an angular point of the circumscribed polygon is joined to each of the angular points of the inscribed polygon, calculate the value of \\( \\frac{S}{A} \\), where \\( S \\) is the sum of the squares of the str... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-3b8a29a4ee1c67a7d124cf813d71742bf9e27cd8ca9279cafdb8659240988cdd | Two regular triangular pyramids have a common side face and no other common points. Spheres of radius $r$ are inscribed in each pyramid. A third ball of radius $R$ externally touches both pyramids and the inscribed balls. If $R:r = 2:1$, find the flat angle at the top of the pyramids. | [
"\\(\\pi - \\arccos\\left(\\frac{1}{4}\\right)\\)"
] | [
"Two regular triangular pyramids have a common side face and no other common points. Spheres of radius $r$ are inscribed in each pyramid. A third ball of radius $R$ externally touches both pyramids and the inscribed balls. If $R:r = 2:1$, find the flat angle at the top of the pyramids."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-e84878e4f091a494c7c1d8b98c90710db801fc950aa14e198d1fc08c033bb1f4 | Two right circular cones have parallel bases, and the apex of each is at the center of the base of the other. The cones intersect in a circle \( C \). If the areas of the bases are 400 and 900, compute the area of circle \( C \). | [
"144"
] | [
"Two right circular cones have parallel bases, and the apex of each is at the center of the base of the other. The cones intersect in a circle \\( C \\). If the areas of the bases are 400 and 900, compute the area of circle \\( C \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-380c468fc9ee54107f750c0e14b3aad63d440da1a2ccabbf8a5f8e94373a7119 | Two right circular cones have vertices facing down. The first cone has a radius of $1$ and a height of $7;$ the second cone has a radius of $4$ and a height of $4$. Emily fills the first cone up with water until the very top. If she then pours the water from the first cone to the second until both have the same height,... | [
"\\(\\frac{7\\sqrt[3]{20}}{10}\\)"
] | [
"Two right circular cones have vertices facing down. The first cone has a radius of $1$ and a height of $7;$ the second cone has a radius of $4$ and a height of $4$. Emily fills the first cone up with water until the very top. If she then pours the water from the first cone to the second until both have the same he... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-3f11144ebe8e32e781af1a4b40cd6910ec52a7b357e74abf511f33bc36441157 | Two right circular cones with vertices facing down contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liqu... | [
"4"
] | [
"Two right circular cones with vertices facing down contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \\text{ cm}$ and $6 \\text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \\text{ cm}$, which sinks to the bottom and is completely submerged without spilling ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-09a5e94fe72d89e452b79e9f14b6fba784f37a07114d6b81ab948130bdb385a4 | Two right circular cones with vertices facing down contain the same amount of liquid. The radii of the tops of the liquid surfaces are 3 cm and 6 cm. Into each cone is dropped a spherical marble of radius 1 cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the r... | [
"4"
] | [
"Two right circular cones with vertices facing down contain the same amount of liquid. The radii of the tops of the liquid surfaces are 3 cm and 6 cm. Into each cone is dropped a spherical marble of radius 1 cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 79
} | 79 | 0 | 0 |
omr-ws-04fce37f88222d73d8ee9a7c224e130046cc96283023875f270b1973ce8a2f8d | Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders? | [
"The first height is 21% more than the second."
] | [
"Two right circular cylinders have the same volume. The radius of the second cylinder is $10\\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 29
} | 47 | 0 | 0 |
omr-ws-d750c72fb71942ff4f173663bc395ba0251dafb3b9d86d38db8e92257a2055b6 | Two right triangles ABC and A'B'C' are supported by a line XY. The legs of the triangles are AB = 10 cm, AC = 12 cm, A'B' = 16 cm, and A'C' = 8 cm. A line parallel to XY intersects sides AC, BC, A'C', and B'C' at points D, E, D', and E' respectively, such that DE = D'E'. Calculate the lengths of AD and DE. Answers: AD ... | [
"AD = 5.14 cm and DE = 5.71 cm."
] | [
"Two right triangles ABC and A'B'C' are supported by a line XY. The legs of the triangles are AB = 10 cm, AC = 12 cm, A'B' = 16 cm, and A'C' = 8 cm. A line parallel to XY intersects sides AC, BC, A'C', and B'C' at points D, E, D', and E' respectively, such that DE = D'E'. Calculate the lengths of AD and DE.\n\nAnsw... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 20
} | 21 | 0 | 0 |
omr-ws-892e42ea78b5ca0c8e25d8bdf8bd7705de7b7c1e2cb1e387e858c4d31a7dc38c | Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Let $\theta$ be the angle formed by the hypotenuses of the two triangles and $\phi$ be the angle formed by one of the legs of one triangle and one of the legs of the other triangle. If $\theta = 120^\circ... | [
"60^\\circ"
] | [
"Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Let $\\theta$ be the angle formed by the hypotenuses of the two triangles and $\\phi$ be the angle formed by one of the legs of one triangle and one of the legs of the other triangle. If $\\theta = 1... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-5a2ced06924f4cf6f9742322cf26e622b9a7d1f5e0028119722db678094c32cd | Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Let the angle formed by one of the legs of one of the triangles and one of the legs of the other triangle be $\theta$, and the angle formed by the hypotenuses be $\phi$. If $\phi = 60^\circ$, find the val... | [
"30"
] | [
"Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Let the angle formed by one of the legs of one of the triangles and one of the legs of the other triangle be $\\theta$, and the angle formed by the hypotenuses be $\\phi$. If $\\phi = 60^\\circ$, fin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-4f26b5943d0f042c3ec3a2ed5620dfe4dfb6cb6a41a22a12f10867a2285b51c8 | Two right triangles have sides of the form \(\sqrt{x}\) where \(x\) is an integer. The hypotenuse of one triangle of length 2 is joined together with the other to produce a quadrilateral. Find the diagonals of the quadrilateral. | [
"\\sqrt{3}"
] | [
"Two right triangles have sides of the form \\(\\sqrt{x}\\) where \\(x\\) is an integer. The hypotenuse of one triangle of length 2 is joined together with the other to produce a quadrilateral. Find the diagonals of the quadrilateral."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-5809ecb578558034a7911e2dc19d1ad56b5ec589d815145eb8a6feb12d8ee636 | Two right triangles share a side as follows: [asy] pair pA, pB, pC, pD, pE; pA = (0, 0); pB = pA + 6 * dir(0); pC = pA + 10 * dir(90); pD = pB + 6 * dir(90); pE = (6 * pA + 10 * pD) / 16; draw(pA--pB--pC--pA); draw(pA--pB--pD--pA); label("$A$", pA, SW); label("$B$", pB, SE); label("$C$", pC, NW); label("$D$", pD, NE); ... | [
"\\frac{75}{4}"
] | [
"Two right triangles share a side as follows: [asy]\npair pA, pB, pC, pD, pE;\npA = (0, 0);\npB = pA + 6 * dir(0);\npC = pA + 10 * dir(90);\npD = pB + 6 * dir(90);\npE = (6 * pA + 10 * pD) / 16;\ndraw(pA--pB--pC--pA);\ndraw(pA--pB--pD--pA);\nlabel(\"$A$\", pA, SW);\nlabel(\"$B$\", pB, SE);\nlabel(\"$C$\", pC, NW);\... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a871ec75f55fdf2a31504a7a895ca9104f1e056243efe1bd4b19034a9bd8c4c | Two right triangles with sides of lengths 3, 4, and 5 are placed so that their hypotenuses coincide, and so that the interior of the triangles overlap without coinciding. What is the area of the intersection of the interiors of the triangles? | [
"$\\frac{75}{16}$"
] | [
"Two right triangles with sides of lengths 3, 4, and 5 are placed so that their hypotenuses coincide, and so that the interior of the triangles overlap without coinciding. What is the area of the intersection of the interiors of the triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-2c218fc387302900b524c508f3f0912b48d26bcd8f2cced9f7e0c8ba3a28ed5e | Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths... | [
"\\(\\frac{28}{3}\\)"
] | [
"Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side le... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 24
} | 76 | 0 | 0 |
omr-ws-4d3e1c7646beb477bcbd5b415faff6beebaceda43fe3eaf7face2f37b0d9106f | Two right-angled triangles are given such that the incircle of the first triangle is equal to the circumcircle of the second triangle. Let \( S \) and \( S' \) be the areas of the first and second triangles, respectively. Calculate the minimum possible value of \( \frac{S}{S'} \). | [
"3 + 2\\sqrt{2}"
] | [
"Two right-angled triangles are given such that the incircle of the first triangle is equal to the circumcircle of the second triangle. Let \\( S \\) and \\( S' \\) be the areas of the first and second triangles, respectively. Calculate the minimum possible value of \\( \\frac{S}{S'} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-732e99ae924f972b3d9f44c6ccc94be2527521995d4cd99ac685fed3e16c15fa | Two right-angled triangles are given such that the incircle of the first triangle is equal to the circumcircle of the second triangle. Let \( S \) be the area of the first triangle and \( S' \) be the area of the second triangle. Calculate the minimum possible value of \(\frac{S}{S'}\). | [
"3 + 2\\sqrt{2}"
] | [
"Two right-angled triangles are given such that the incircle of the first triangle is equal to the circumcircle of the second triangle. Let \\( S \\) be the area of the first triangle and \\( S' \\) be the area of the second triangle. Calculate the minimum possible value of \\(\\frac{S}{S'}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57b86453bac81adc349e4b2edb06476fabced8320aad977e573a4d985fd0eff4 | Two right-angled triangles are such that the incircle of one triangle is equal in size to the circumcircle of the other. If \( s_1 \) is the area of the first triangle and \( s_2 \) is the area of the second triangle, find the minimum value of \( \frac{s_1}{s_2} \). | [
"3 + 2\\sqrt{2}"
] | [
"Two right-angled triangles are such that the incircle of one triangle is equal in size to the circumcircle of the other. If \\( s_1 \\) is the area of the first triangle and \\( s_2 \\) is the area of the second triangle, find the minimum value of \\( \\frac{s_1}{s_2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ccf8f4dbee6bd9e4e21d6cda4444e1e5a98a4bd85d536d33aad0a3e400cd291a | Two right-angled triangles, \( \triangle ABC \) and \( \triangle A'B'C' \), are inscribed in a circle with center \( O \) (with \( AB \) and \( A'B' \) as diameters). Let \( AC \cap A'C' = P \), \( BC \cap B'C' = Q \), and \( BB' \cap CC' = R \). If the coordinates of \( O \) are \( (0,0) \), \( A = (-1,0) \), \( B = (... | [
"-1"
] | [
"Two right-angled triangles, \\( \\triangle ABC \\) and \\( \\triangle A'B'C' \\), are inscribed in a circle with center \\( O \\) (with \\( AB \\) and \\( A'B' \\) as diameters). Let \\( AC \\cap A'C' = P \\), \\( BC \\cap B'C' = Q \\), and \\( BB' \\cap CC' = R \\). If the coordinates of \\( O \\) are \\( (0,0) \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56cf74a2da6dc3789589ebd25d4152edab183b0e1954a92581c370f4d0b60d2e | Two rivers flow into a lake. A motor-boat leaves the landing-stage $A$ situated on the first river, goes 24 km downstream to reach the lake, sails along the lake for two hours, and then goes 32 km along the second river and reaches the landing-stage $B$. It took the motor-boat 8 hours to cover the whole path from $A$ t... | [
"3 km/h and 1 km/h"
] | [
"Two rivers flow into a lake. A motor-boat leaves the landing-stage $A$ situated on the first river, goes 24 km downstream to reach the lake, sails along the lake for two hours, and then goes 32 km along the second river and reaches the landing-stage $B$. It took the motor-boat 8 hours to cover the whole path from ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9dea558f700dab36283e9e73012c235e51b9a743d81a885584fa2404b84cb44b | Two roads cross each other at right angles. A car on the north-south road is 3 miles north of the intersection and is approaching the intersection at 60 miles per hour. Another car traveling 40 miles per hour on the east-west road is 2 miles west of the intersection and headed west. How fast is the distance between the... | [
"\\(-\\frac{100\\sqrt{13}}{13}\\) miles per hour."
] | [
"Two roads cross each other at right angles. A car on the north-south road is 3 miles north of the intersection and is approaching the intersection at 60 miles per hour. Another car traveling 40 miles per hour on the east-west road is 2 miles west of the intersection and headed west. How fast is the distance betwee... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7403c55ff290066988a73dc45e402c4940f1550b4d2df951e70e501d57f5ca2b | Two robbers want to divide an open necklace that has $d$ different kinds of stones, where the number of stones of each kind is even. They decide to cut the necklace in at most $d$ places to achieve an equal division. If the necklace has $2n$ stones in total, with $n$ being a positive integer, find the maximum possible ... | [
"16"
] | [
"Two robbers want to divide an open necklace that has $d$ different kinds of stones, where the number of stones of each kind is even. They decide to cut the necklace in at most $d$ places to achieve an equal division. If the necklace has $2n$ stones in total, with $n$ being a positive integer, find the maximum poss... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-9025f7a22138f7d9ce56aa164b592fa737359438382f1e30fe55485209d7417a | Two robot painters are tasked with painting a $1000$ meter long circular fence. One robot painter paints a randomly selected $720$ meter stretch of the fence, and the other paints a randomly selected $750$ meter stretch. Find the probability that the entire fence is painted after both robots have completed their tasks. | [
"$\\frac{471}{1000}$"
] | [
"Two robot painters are tasked with painting a $1000$ meter long circular fence. One robot painter paints a randomly selected $720$ meter stretch of the fence, and the other paints a randomly selected $750$ meter stretch. Find the probability that the entire fence is painted after both robots have completed their t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7af48136149e648a7b17b7108b186f488612c7a6f8e695659677be938e7d8266 | Two robot painters are tasked with painting a 1000 meter long circular fence. One robot paints a randomly selected 720 meter stretch, and the other paints a randomly selected 750 meter stretch. Find the probability that the entire fence is painted after both robots have completed their tasks. | [
"\\(\\frac{47}{100}\\)"
] | [
"Two robot painters are tasked with painting a 1000 meter long circular fence. One robot paints a randomly selected 720 meter stretch, and the other paints a randomly selected 750 meter stretch. Find the probability that the entire fence is painted after both robots have completed their tasks."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-a3cdc5777837301aa37dd9057cbe8890249c02b6f1fa31076d6da5d7035b1bb9 | Two robots $G$ and $J$ are at diagonally opposite corners of a $5 \times 5$ grid. $G$ is at the upper right corner and $J$ is at the lower left corner. $G$ can only move down and left, while $J$ can only move up and right. Find the probability that the two robots meet at any point on the grid. | [
"\\(\\frac{1}{252}\\)"
] | [
"Two robots $G$ and $J$ are at diagonally opposite corners of a $5 \\times 5$ grid. $G$ is at the upper right corner and $J$ is at the lower left corner. $G$ can only move down and left, while $J$ can only move up and right. Find the probability that the two robots meet at any point on the grid."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 12
} | 32 | 0 | 0 |
omr-ws-31e26df19c78afa3045125f016fb1243320cbdd674fb22acaf605be2ac191a63 | Two robots $R_2$ and $D_2$ are at a point $O$ on an island. $R_2$ can travel at a maximum of $2$ km/hr and $D_2$ at a maximum of $1$ km/hr. One treasure is located at a point $P$ which is $1$ km west of $O$. The second treasure is located at a point $X$ on the straight line through $P$ and $O$ (but not at $O$). Find al... | [
"\\( d \\geq 1 \\text{ km west of } O \\text{ or } d > 1 \\text{ km east of } O \\)"
] | [
"Two robots $R_2$ and $D_2$ are at a point $O$ on an island. $R_2$ can travel at a maximum of $2$ km/hr and $D_2$ at a maximum of $1$ km/hr. One treasure is located at a point $P$ which is $1$ km west of $O$. The second treasure is located at a point $X$ on the straight line through $P$ and $O$ (but not at $O$). Fi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-8e981b85d9cc7f95c63625e2a78df4f95974a74faec5443d5dbd23a8e4b62534 | Two robots G and J are at diagonally opposite corners of a 5×5 grid. G is at the upper right corner and J is at the lower left corner. G can move down and left, while J can move up and right. Find the probability that the two robots meet at any point on the grid. | [
"0"
] | [
"Two robots G and J are at diagonally opposite corners of a 5×5 grid. G is at the upper right corner and J is at the lower left corner. G can move down and left, while J can move up and right. Find the probability that the two robots meet at any point on the grid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e221307493f85e7a2601cce7d46156a9fefaec9d0235e971f4e361bbdc5a0d97 | Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. The first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the lines... | [
"\\(\\sqrt{2} + 1\\)"
] | [
"Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. The first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15a9ec9d7be6440251da4086ed4fe8029a7e1a5e098cb1e40ecab44b67285e7a | Two roofs meet at right angles and make angles $\alpha$ and $\beta$ with the horizontal. The line of intersection of these roofs makes an angle $\theta$ with the horizontal. If $\alpha = 45^\circ$ and $\beta = 60^\circ$, find the value of $\sin(\theta)$. | [
"\\dfrac{1}{2}"
] | [
"Two roofs meet at right angles and make angles $\\alpha$ and $\\beta$ with the horizontal. The line of intersection of these roofs makes an angle $\\theta$ with the horizontal. If $\\alpha = 45^\\circ$ and $\\beta = 60^\\circ$, find the value of $\\sin(\\theta)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-98a0c1c990666a3ed0ae90d870465f76567e56012b22e0f2611cc09950bc0408 | Two rooms each contain countably infinitely many boxes, with each box labeled with a real number \(x_1, x_2, \ldots, x_n, \ldots\). Two mathematicians, after discussing a strategy, are sent to separate rooms and cannot communicate further. Each mathematician can open any number of boxes (possibly infinitely many) to se... | [
"\\infty"
] | [
"Two rooms each contain countably infinitely many boxes, with each box labeled with a real number \\(x_1, x_2, \\ldots, x_n, \\ldots\\). Two mathematicians, after discussing a strategy, are sent to separate rooms and cannot communicate further. Each mathematician can open any number of boxes (possibly infinitely ma... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52f5a2842a825b0b3e189ff095b1227b550a125bf528051b1508927d37880f68 | Two roots of the real quartic equation \(x^4 - 18x^3 + ax^2 + 200x - 1984 = 0\) have a product of -32. Find the value of \(a\). | [
"86"
] | [
"Two roots of the real quartic equation \\(x^4 - 18x^3 + ax^2 + 200x - 1984 = 0\\) have a product of -32. Find the value of \\(a\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-bc13bfb907ea38381a337d21227d469452cedaf3393bc966aae3698e13bddb70 | Two rows of ten pegs are lined up with adjacent pegs spaced 1 unit apart. How many ways can ten rubber bands be looped around the pegs so that each peg contains exactly one rubber band? (Rubber bands cannot stretch more than $\sqrt{2}$ units.) | [
"89"
] | [
"Two rows of ten pegs are lined up with adjacent pegs spaced 1 unit apart. How many ways can ten rubber bands be looped around the pegs so that each peg contains exactly one rubber band? (Rubber bands cannot stretch more than $\\sqrt{2}$ units.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-0a1f6487592e307b0a4cf1e223be0417bdfb6460cf0d9ab00b0c850bab856bf4 | Two runners start at opposite corners of a square with side length $1$ km and run along the diagonals of the square at constant speeds. If the runners meet for the first time after $t$ hours, find the value of $t$ such that the distance between them is exactly $\frac{1}{2}$ km. | [
"\\dfrac{\\sqrt{2}}{4}"
] | [
"Two runners start at opposite corners of a square with side length $1$ km and run along the diagonals of the square at constant speeds. If the runners meet for the first time after $t$ hours, find the value of $t$ such that the distance between them is exactly $\\frac{1}{2}$ km."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-032f241949d901a2a24e8046e330e402b8461bbc489af9f98246f08f96f83c6d | Two runners start at opposite corners of a square with side length $1$ km and run along the diagonals of the square at constant speeds. Let $d(t)$ represent the distance between the runners at time $t$. Find the minimum value of $d(t)$ as $t$ approaches infinity. | [
"0"
] | [
"Two runners start at opposite corners of a square with side length $1$ km and run along the diagonals of the square at constant speeds. Let $d(t)$ represent the distance between the runners at time $t$. Find the minimum value of $d(t)$ as $t$ approaches infinity."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-a25f7d61591c9eb3f585e99cf31e6d5043fc3f45285ec33aff2ca1f2dfd8f95f | Two runners start running along a circular track of unit length from the same starting point and in the same direction, with constant speeds \( v_1 \) and \( v_2 \) respectively, where \( v_1 \) and \( v_2 \) are two distinct relatively prime natural numbers. They continue running until they simultaneously reach the st... | [
"0"
] | [
"Two runners start running along a circular track of unit length from the same starting point and in the same direction, with constant speeds \\( v_1 \\) and \\( v_2 \\) respectively, where \\( v_1 \\) and \\( v_2 \\) are two distinct relatively prime natural numbers. They continue running until they simultaneously... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-ddbb9fd15a81f8a81e4d027ab4a6393ff879b1fac843219a91a65fefb6bcf08b | Two runners start running along a circular track of unit length from the same starting point and in the same direction, with constant speeds \( v_1 \) and \( v_2 \) respectively, where \( v_1 \) and \( v_2 \) are two distinct relatively prime natural numbers. They continue running until they simultaneously reach the st... | [
"\\dfrac{1}{3}"
] | [
"Two runners start running along a circular track of unit length from the same starting point and in the same direction, with constant speeds \\( v_1 \\) and \\( v_2 \\) respectively, where \\( v_1 \\) and \\( v_2 \\) are two distinct relatively prime natural numbers. They continue running until they simultaneously... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dde1eaf9c537e6fb7338e16e6cea13b6385b34612bd1dd7b8b9d5af678da6883 | Two runners start running along the diagonals of a square with a side length of $1$ km. The runners start at opposite corners and run towards each other. Let $a$ be a positive number less than $500$ meters. If at some moment the distance between the runners is $a$ meters, calculate the minimum possible distance in mete... | [
"\\sqrt{2(500^2 - a^2)}"
] | [
"Two runners start running along the diagonals of a square with a side length of $1$ km. The runners start at opposite corners and run towards each other. Let $a$ be a positive number less than $500$ meters. If at some moment the distance between the runners is $a$ meters, calculate the minimum possible distance in... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf7d128505b7079cee0d19920f4a2879f1d5059b7a068cc8246d35aacbe11aa1 | Two runners started a race simultaneously. Initially, they ran on the street toward the stadium and then completed 3 laps on the stadium. Both runners maintained their own constant speeds throughout the race. During the race, the first runner passed the second runner exactly twice. If the distance of the street section... | [
"\\dfrac{d + 3l}{d + l}"
] | [
"Two runners started a race simultaneously. Initially, they ran on the street toward the stadium and then completed 3 laps on the stadium. Both runners maintained their own constant speeds throughout the race. During the race, the first runner passed the second runner exactly twice. If the distance of the street se... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-cf4753e0fc64edc58843a0542b24850174db04d1841a491ab3eaee86809b3a33 | Two runners, A and B, run around a circular track in the same direction. Runner A completes a lap every 78 seconds, and runner B completes a lap every 91 seconds. If runner A has just passed runner B, when will runner A overtake runner B again? | [
"546"
] | [
"Two runners, A and B, run around a circular track in the same direction. Runner A completes a lap every 78 seconds, and runner B completes a lap every 91 seconds. If runner A has just passed runner B, when will runner A overtake runner B again?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-dfa07a0f1e4828fd6d5af48f237395bad1d001067ac1218562a2e1bfe6076026 | Two sacks contain 140 kg of flour. Each will contain one and the same amount, if we take 12.5\% of the flour of the first sack and put it into the second. How many kilograms of flour does each sack contain? | [
"The first sack contains 80 kg, the second, 60 kg"
] | [
"Two sacks contain 140 kg of flour. Each will contain one and the same amount, if we take 12.5\\% of the flour of the first sack and put it into the second. How many kilograms of flour does each sack contain?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8940e2a7f6a45a23cbc45f42f1aa1069d6ee0f1a50f240174fe15cda4023bfe6 | Two samples have sizes 40 and 50 with mean deviations of 19 and 8, respectively. One of the sample means is 53. Find the mean deviation of the combined sample of size 90. | [
"\\dfrac{116}{9}"
] | [
"Two samples have sizes 40 and 50 with mean deviations of 19 and 8, respectively. One of the sample means is 53. Find the mean deviation of the combined sample of size 90."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72cda82ff1ffed7b38731520ca83ae7f638eccf555f29000efdbc1f660996ef2 | Two schoolboys left their house at the same time and walked to school at the same speed. Three minutes later one of them remembered that he had forgotten a necessary book and ran home at a speed exceeding the initial speed by 60 m/min. He took the book and ran to school at the same speed. He caught up with his friend, ... | [
"40 m/min"
] | [
"Two schoolboys left their house at the same time and walked to school at the same speed. Three minutes later one of them remembered that he had forgotten a necessary book and ran home at a speed exceeding the initial speed by 60 m/min. He took the book and ran to school at the same speed. He caught up with his fri... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcfe3f5a565fd2f2f859e5400a029316b9667ca68bdf5bf03f77ae5e2ca834f2 | Two scientists $A$ and $B$ take turns eating pizzas from two pizzerias, each initially containing 2010 pizzas. On each turn, a scientist can eat any number of pizzas from one pizzeria or exactly one pizza from each pizzeria. The scientist who eats the last pizza wins. Assuming both use the best possible strategy, who w... | [
"B wins."
] | [
"Two scientists $A$ and $B$ take turns eating pizzas from two pizzerias, each initially containing 2010 pizzas. On each turn, a scientist can eat any number of pizzas from one pizzeria or exactly one pizza from each pizzeria. The scientist who eats the last pizza wins. Assuming both use the best possible strategy, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-762a0f6c8e3bea0cf8e3d5bfb79d7c5ef319445b38d352a6a8509d8bc8647cd6 | Two secant circles \( C1 \) and \( C2 \) have a common tangent which meets \( C1 \) at \( P \) and \( C2 \) at \( Q \). The two circles intersect at \( M \) and \( N \), where \( N \) is closer to \( PQ \) than \( M \). Line \( PN \) meets circle \( C2 \) a second time at \( A \). If the length of \( PQ \) is 10 units ... | [
"no_valid_answer_found"
] | [
"Two secant circles \\( C1 \\) and \\( C2 \\) have a common tangent which meets \\( C1 \\) at \\( P \\) and \\( C2 \\) at \\( Q \\). The two circles intersect at \\( M \\) and \\( N \\), where \\( N \\) is closer to \\( PQ \\) than \\( M \\). Line \\( PN \\) meets circle \\( C2 \\) a second time at \\( A \\). If th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-179e31fb2ecb61f8a2d1e7e8f382842bad34c1401e7396dd6480e2145ca11767 | Two secant lines are drawn from a point outside a circle. The internal segment (the chord) of the first secant is equal to 47 m, and the external one, to 9 m; the internal segment of the second secant exceeds its external segment by 72 m. Determine the length of the second secant line. | [
"$AE=84$ m"
] | [
"Two secant lines are drawn from a point outside a circle. The internal segment (the chord) of the first secant is equal to 47 m, and the external one, to 9 m; the internal segment of the second secant exceeds its external segment by 72 m. Determine the length of the second secant line."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19106fb7435c3c298830cacf9f5591765831320e3f16dc0ef0e2f2ddac2d21d8 | Two secants $AKC$ and $ALB$ are drawn from a point $A$, which is at the distance of 5 cm from the center of a circle of the radius of 3 cm, the angle between the secants being equal to $30^\circ$ ($K$, $C$, $L$, and $B$ are the points of intersection of the secants and the circle). Find the area of $\triangle AKL$ if t... | [
"$\\frac{8}{5}$ cm$^2$"
] | [
"Two secants $AKC$ and $ALB$ are drawn from a point $A$, which is at the distance of 5 cm from the center of a circle of the radius of 3 cm, the angle between the secants being equal to $30^\\circ$ ($K$, $C$, $L$, and $B$ are the points of intersection of the secants and the circle). Find the area of $\\triangle AK... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2d1aa4a7098e3f8050c3abefc06ce4588e1868e5b09482f5b39de93538682a6 | Two secants to the circle $\omega$ pass through the point $S$ lying outside $\omega$. Let $A$ and $B$, $C$ and $D$ be the intersection points with $SA < SB$ and $SC < SD$. Denote by $T$ the intersection point of the chords $AD$ and $BC$. Let $X$ be the intersection point of the two tangent lines passing through $B$ and... | [
"\\dfrac{1}{2}"
] | [
"Two secants to the circle $\\omega$ pass through the point $S$ lying outside $\\omega$. Let $A$ and $B$, $C$ and $D$ be the intersection points with $SA < SB$ and $SC < SD$. Denote by $T$ the intersection point of the chords $AD$ and $BC$. Let $X$ be the intersection point of the two tangent lines passing through ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-bf07b4794a1ec68936de5d02024d0467616f68d106f936fd068436cebad49d6a | Two sections of a coal mine were in operation when a third was opened. The third section raised the output of the mine by 1.5 times. If the first and third sections together produce for four months as much coal as the second section does in a year, what is the percentage output of the second section in terms of the fir... | [
"60\\%"
] | [
"Two sections of a coal mine were in operation when a third was opened. The third section raised the output of the mine by 1.5 times. If the first and third sections together produce for four months as much coal as the second section does in a year, what is the percentage output of the second section in terms of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d04519435e24123c97cd4d5d45ed668007881aba8d019a8449ba08744a61f9f5 | Two sections of a coal mine were operating for some time, then a third section went into operation. As a result, the productivity of the mine increased by one and a half times. What is the percentage of the productivity of the second section as compared to that of the first if it is known that the first and the third s... | [
"60\\%"
] | [
"Two sections of a coal mine were operating for some time, then a third section went into operation. As a result, the productivity of the mine increased by one and a half times. What is the percentage of the productivity of the second section as compared to that of the first if it is known that the first and the th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d5535459711e64991f7246be3d21d76b03cb27e7d198565ed7080da67461cea | Two semicircles are inscribed in a square with each side measuring 4 units. The area of each semicircle is \(8\pi\). Find the area of the shaded region within the square but outside the semicircles. | [
"\\(2\\pi - 4\\)"
] | [
"Two semicircles are inscribed in a square with each side measuring 4 units. The area of each semicircle is \\(8\\pi\\). Find the area of the shaded region within the square but outside the semicircles."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a350d87b86945a5e84a1a5f5b57fc77830e31c706a051613c50bd432a0e17866 | Two semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to a horizontal line. If \( PQ = QR = RS = 24 \), find the length of the radius \( r \) of the middle circle. | [
"15"
] | [
"Two semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to a horizontal line. If \\( PQ = QR = RS = 24 \\), find the length of the radius \\( r \\) of the middle circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2dc9cf105e1e34f8ef1f60dce65759237159d9e4c3744437de47a90e290bfe9 | Two semicircles of radius $3$ are inside a semicircle of radius $6$. A circle of radius $r$ is tangent to the three semicircles. What is the value of $r$? | [
"2"
] | [
"Two semicircles of radius $3$ are inside a semicircle of radius $6$. A circle of radius $r$ is tangent to the three semicircles. What is the value of $r$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1f8e4517573cc8c9e7f4504bc0dfc91cf3e0d18dc5d002ce45d14498ea1a7bf | Two semicircles with diameters $AB$ and $BC$ satisfy $AB=3BC$ and $A-B-C$ (i.e., $A$, $B$, and $C$ are collinear in this order) and are on the same side of line $AC$. Let $a$ and $c$ be tangent lines at points $A$ and $C$, respectively. Let $\Gamma$ be a circle tangent to $a$, $c$, and the larger semicircle. If the rad... | [
"\\dfrac{4}{3}"
] | [
"Two semicircles with diameters $AB$ and $BC$ satisfy $AB=3BC$ and $A-B-C$ (i.e., $A$, $B$, and $C$ are collinear in this order) and are on the same side of line $AC$. Let $a$ and $c$ be tangent lines at points $A$ and $C$, respectively. Let $\\Gamma$ be a circle tangent to $a$, $c$, and the larger semicircle. If t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c341a1f246870c286f1c3c97bbf535c8047dc1ab31fc99e8f566a52ce56b1257 | Two separate circles have radii of 4 and 6, and the distance between their centers is 24. Find the length of their internal common tangent. | [
"2\\sqrt{119}"
] | [
"Two separate circles have radii of 4 and 6, and the distance between their centers is 24. Find the length of their internal common tangent."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4a26c3afcbc90a9ecb710becfdc254f6a8dc9f550ad976db344497f71a11430 | Two sequences \( P(m,n) \) and \( Q(m,n) \) are defined as follows: - \( P(m,0) = 1 \) for \( m \geq 0 \) - \( P(0,n) = 0 \) for \( n \geq 1 \) - \( P(m,n) = 0 \) for \( m,n < 0 \) - \( P(m,n) = \sum_{j=0}^{n} P(m-1,j) \) for \( m \geq 1 \) - \( Q(m,n) = P(m-1,n) + P(m-1,n-1) + P(m-1,n-2) \) for \( m \geq 1 \) Express ... | [
"\\( Q(m,n) = \\binom{n+m-1}{m-1} - \\binom{n+m-4}{m-1} \\) for \\( m, n > 1 \\)."
] | [
"Two sequences \\( P(m,n) \\) and \\( Q(m,n) \\) are defined as follows:\n- \\( P(m,0) = 1 \\) for \\( m \\geq 0 \\)\n- \\( P(0,n) = 0 \\) for \\( n \\geq 1 \\)\n- \\( P(m,n) = 0 \\) for \\( m,n < 0 \\)\n- \\( P(m,n) = \\sum_{j=0}^{n} P(m-1,j) \\) for \\( m \\geq 1 \\)\n- \\( Q(m,n) = P(m-1,n) + P(m-1,n-1) + P(m-1,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 9
} | 53 | 0 | 0 |
omr-ws-ae6cf2fb7c45fe0b2f69274e152a786806975edbca58dd0bcaa849456bd88bc7 | Two sequences \((a_n)\) and \((b_n)\) are defined inductively as follows: \[ \begin{cases} a_0 = 3, \quad b_0 = -3 \\ a_n = 3a_{n-1} + 2b_{n-1}, \quad n \geq 1 \\ b_n = 4a_{n-1} + 3b_{n-1}, \quad n \geq 1 \end{cases} \] Find all natural numbers \(n\) such that \(\prod_{k=0}^{n}(b_k^2 + 9)\) is a perfect square. | [
"n \\text{ is odd}"
] | [
"Two sequences \\((a_n)\\) and \\((b_n)\\) are defined inductively as follows:\n\\[\n\\begin{cases}\na_0 = 3, \\quad b_0 = -3 \\\\\na_n = 3a_{n-1} + 2b_{n-1}, \\quad n \\geq 1 \\\\\nb_n = 4a_{n-1} + 3b_{n-1}, \\quad n \\geq 1\n\\end{cases}\n\\]\nFind all natural numbers \\(n\\) such that \\(\\prod_{k=0}^{n}(b_k^2 +... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-23e63651f1328b1364fc25076c215bba7561f7ee22f5300434506fff19389343 | Two sequences of integers, \( a_1, a_2, a_3, \ldots \) and \( b_1, b_2, b_3, \ldots \), satisfy the equation \[ (a_n - a_{n-1})(a_n - a_{n-2}) + (b_n - b_{n-1})(b_n - b_{n-2}) = 0 \] for each integer \( n \) greater than 2. Let \( k \) be the smallest positive integer such that \( a_k = a_{k+2008} \). Find the value of... | [
"1004"
] | [
"Two sequences of integers, \\( a_1, a_2, a_3, \\ldots \\) and \\( b_1, b_2, b_3, \\ldots \\), satisfy the equation\n\\[ (a_n - a_{n-1})(a_n - a_{n-2}) + (b_n - b_{n-1})(b_n - b_{n-2}) = 0 \\]\nfor each integer \\( n \\) greater than 2. Let \\( k \\) be the smallest positive integer such that \\( a_k = a_{k+2008} \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-342e7cc18dbc6f79317a09da9dbeab272799a214075770b4e392c897a98c0f5e | Two sequences of integers, \(a_{1}, a_{2}, a_{3}, \ldots\) and \(b_{1}, b_{2}, b_{3}, \ldots\) satisfy the equation \[ (a_{n} - a_{n-1})(a_{n} - a_{n-2}) + (b_{n} - b_{n-1})(b_{n} - b_{n-2}) = 0 \] for each integer \(n\) greater than 2. Find the smallest positive integer \(k\) such that \(a_{k} = a_{k+2008}\). | [
"2"
] | [
"Two sequences of integers, \\(a_{1}, a_{2}, a_{3}, \\ldots\\) and \\(b_{1}, b_{2}, b_{3}, \\ldots\\) satisfy the equation\n\n\\[\n(a_{n} - a_{n-1})(a_{n} - a_{n-2}) + (b_{n} - b_{n-1})(b_{n} - b_{n-2}) = 0\n\\]\n\nfor each integer \\(n\\) greater than 2. Find the smallest positive integer \\(k\\) such that \\(a_{k... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3a47a85b407df4582495b0ba009731e800a1e9a22e375a7ac365f8dd34af815 | Two sequences of positive reals, \( \left(x_n\right) \) and \( \left(y_n\right) \), satisfy the relations \( x_{n+2} = x_n + x_{n+1}^2 \) and \( y_{n+2} = y_n^2 + y_{n+1} \) for all natural numbers \( n \). Given that \( x_1 = 2 \), \( x_2 = 3 \), \( y_1 = 1.5 \), and \( y_2 = 2.5 \), find the smallest natural number \... | [
"1"
] | [
"Two sequences of positive reals, \\( \\left(x_n\\right) \\) and \\( \\left(y_n\\right) \\), satisfy the relations \\( x_{n+2} = x_n + x_{n+1}^2 \\) and \\( y_{n+2} = y_n^2 + y_{n+1} \\) for all natural numbers \\( n \\). Given that \\( x_1 = 2 \\), \\( x_2 = 3 \\), \\( y_1 = 1.5 \\), and \\( y_2 = 2.5 \\), find th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0969ead098c2db2e34b62978593353f4efe4e0d8df1df13f682e0e074f292636 | Two sets \( S_1 \) and \( S_2 \), which are not necessarily distinct, are each selected randomly and independently from each other among the 512 subsets of \( S = \{1, 2, \ldots, 9\} \). Let \( \sigma(X) \) denote the sum of the elements of set \( X \). Note that \( \sigma(\emptyset) = 0 \). If \( S_1 \cup S_2 \) stand... | [
"5"
] | [
"Two sets \\( S_1 \\) and \\( S_2 \\), which are not necessarily distinct, are each selected randomly and independently from each other among the 512 subsets of \\( S = \\{1, 2, \\ldots, 9\\} \\). Let \\( \\sigma(X) \\) denote the sum of the elements of set \\( X \\). Note that \\( \\sigma(\\emptyset) = 0 \\). If \... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8a8c3d096fefd9586c3a54b1b82f2fb0d189efadbd0b8b46947101ff86e1ef4 | Two sets of cards, each numbered from 1 to 6, are held by persons P and Q. Each person randomly selects one card from their set. Points are awarded as follows: - If the numbers on the chosen cards are different, the person with the larger number gets that number as their point, and the other person gets 0 points. - If ... | [
"\\(\\frac{35}{18}\\)"
] | [
"Two sets of cards, each numbered from 1 to 6, are held by persons P and Q. Each person randomly selects one card from their set. Points are awarded as follows:\n- If the numbers on the chosen cards are different, the person with the larger number gets that number as their point, and the other person gets 0 points.... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e7bc362487e3751bb8472b45cb1b87232996e6c1720736bb27c999db1999d86a | Two sets of segments $\cal A$ and $\cal B$ are given on the line. The set $\cal A$ contains $2m-1$ segments, each two of which have a common point. Moreover, each segment from $\cal A$ contains at least two non-intersecting segments from $\cal B$. Let $f(m)$ be the minimum number of segments from $\cal A$ that contain ... | [
"3"
] | [
"Two sets of segments $\\cal A$ and $\\cal B$ are given on the line. The set $\\cal A$ contains $2m-1$ segments, each two of which have a common point. Moreover, each segment from $\\cal A$ contains at least two non-intersecting segments from $\\cal B$. Let $f(m)$ be the minimum number of segments from $\\cal A$ th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-ae40ecb36fdcbb7b57893a1e0d6887b3cc3bb556c0854b8327b8bf35a3cee76c | Two sets of three distinct integers are selected randomly from the set of the first 2014 positive integers. Let these sets be $\{ A_{1}, A_{2}, A_{3} \}$ and $\{ B_{1}, B_{2}, B_{3} \}$. Find the probability that the triangles with side lengths $A_1, A_2, A_3$ and $B_1, B_2, B_3$ are similar. | [
"\\dfrac{6}{\\dbinom{2014}{3}}"
] | [
"Two sets of three distinct integers are selected randomly from the set of the first 2014 positive integers. Let these sets be $\\{ A_{1}, A_{2}, A_{3} \\}$ and $\\{ B_{1}, B_{2}, B_{3} \\}$. Find the probability that the triangles with side lengths $A_1, A_2, A_3$ and $B_1, B_2, B_3$ are similar."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-28129e7003e6b977de24cd9bfb8df98f305884c47a0a4ffac47a479ccabb4172 | Two shelters are located at points A(2,1) and B(3,4). Two rectilinear paths from a lake form a 90° angle at the lake. If the lake is on the line \(x - y = 2\), determine the coordinates of the lake. | [
"\\((4,2)\\) and \\((3,1)\\)"
] | [
"Two shelters are located at points A(2,1) and B(3,4). Two rectilinear paths from a lake form a 90° angle at the lake. If the lake is on the line \\(x - y = 2\\), determine the coordinates of the lake."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8016d6326e4a47d5a48bf5b11f8f1d7c5f6d1cc29d49504adce20cfde3383381 | Two ships are separated by a distance \( \lambda \) on a straight coastline. Ship 1 moves perpendicular to the coastline at a constant speed, while Ship 2 moves with the same constant speed such that its velocity vector is always aligned towards the position of Ship 1. After sufficient time, both ships will move along ... | [
"\\( x = \\frac{\\lambda}{2} \\)"
] | [
"Two ships are separated by a distance \\( \\lambda \\) on a straight coastline. Ship 1 moves perpendicular to the coastline at a constant speed, while Ship 2 moves with the same constant speed such that its velocity vector is always aligned towards the position of Ship 1. After sufficient time, both ships will mov... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-4e18ed81d44a4a36cad7cac3401e8c70d850c80056e36a677f0533048fde4222 | Two ships leave a port at 9 AM. One travels at a bearing of N 53° W at 12 mph, and the other travels at a bearing of S 67° W at 16 mph. How far apart are the ships at noon that day? | [
"60 miles"
] | [
"Two ships leave a port at 9 AM. One travels at a bearing of N 53° W at 12 mph, and the other travels at a bearing of S 67° W at 16 mph. How far apart are the ships at noon that day?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb1fc2190db5aed1cdf12ea3ee9e93c3849fe6b83155ad8deca24f1a45f64411 | Two ships leave port at noon. One ship's speed is 15 km/h with a heading of 75 degrees, and the other ship's speed is 24 km/h with a heading of 155 degrees. What is the bearing of each ship from the other at 3 P.M.? | [
"80 degrees counterclockwise (from ship 1 to ship 2) and 80 degrees clockwise (from ship 2 to ship 1)"
] | [
"Two ships leave port at noon. One ship's speed is 15 km/h with a heading of 75 degrees, and the other ship's speed is 24 km/h with a heading of 155 degrees. What is the bearing of each ship from the other at 3 P.M.?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 11
} | 24 | 0 | 0 |
omr-ws-77c90445ab1a40ebaa19d7ba9a3ad8ff4491a0458401add342b4b46295533354 | Two ships leave port at the same time. One travels at 20 mph at a bearing of N32°W, and the second ship travels at 30 mph at a bearing of S28°W. Find the distance in miles between the ships after 2 hours. | [
"\\( 20\\sqrt{19} \\)"
] | [
"Two ships leave port at the same time. One travels at 20 mph at a bearing of N32°W, and the second ship travels at 30 mph at a bearing of S28°W. Find the distance in miles between the ships after 2 hours."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 3
} | 10 | 0 | 0 |
omr-ws-50699cc08da01c8ce186d0e0e0a326f39378a9be509ea8fc65d6c1785195536b | Two ships meet, one going off to the south and the other, to the west. Two hours after their encounter, they are 60 km apart. Find the speed of each ship, if it is known that the speed of one of them is 6 km/h higher than that of the other. | [
"The speed of the first ship is equal to 18 km/h, that of the second, tn 24 km/h"
] | [
"Two ships meet, one going off to the south and the other, to the west. Two hours after their encounter, they are 60 km apart. Find the speed of each ship, if it is known that the speed of one of them is 6 km/h higher than that of the other."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88dd835df00e05354b9d13e6c5f069d91f702af914c3ef5560b9e3d9d739336e | Two ships move with constant speeds and directions. At noon, 2pm, and 3pm, the distance between them was respectively 5, 7, and 2 miles. What was the distance between them at 1pm? | [
"$\\sqrt{56}$"
] | [
"Two ships move with constant speeds and directions. At noon, 2pm, and 3pm, the distance between them was respectively 5, 7, and 2 miles. What was the distance between them at 1pm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-25ed14230ed2e0effeb6119d71fb0f3b41f3a69bea43e3f8b2d48a2fb5c6e680 | Two ships sail on the sea with constant speeds and fixed directions. At $9:00$, the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance? | [
"The ships were the smallest distance from each other at $10:20$, and that distance was $12$ miles."
] | [
"Two ships sail on the sea with constant speeds and fixed directions. At $9:00$, the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 21
} | 67 | 0 | 0 |
omr-ws-9ff77aa5033a944743a69f836805c66cdaab5d642e253e2c5fbf9db957b4e156 | Two ships sail on the sea with constant speeds and fixed directions. At 9:00, the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance? | [
"12"
] | [
"Two ships sail on the sea with constant speeds and fixed directions. At 9:00, the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 27
} | 27 | 0 | 0 |
omr-ws-fc52d1ce74f2f758a646f643b43b357e8be783a6527b986b564621239d8c974b | Two ships sail towards each other in a fog at the same speed $v_0$. At the distanced 4 km between them the captains reverse motion for a certain time period with an acceleration of 0.1 m/s$^2$. What is the greatest speed of the ships at which they will not collide? | [
"$v_0=20$ m/s"
] | [
"Two ships sail towards each other in a fog at the same speed $v_0$. At the distanced 4 km between them the captains reverse motion for a certain time period with an acceleration of 0.1 m/s$^2$. What is the greatest speed of the ships at which they will not collide?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bbcb94c63fc0017d224fd3a08d2103a3fee46279e050382ca73b86bf4345037f | Two ships, S and T, are moving in straight lines at constant velocities. At 10:00, they are 5 km apart; at 11:00, they are 4 km apart; and at 13:00, they are 10 km apart. At 7:00 the next day, S is due west of T. Find the speed and direction of both ships. | [
"\\sqrt{17} \\text{ km/h} \\text{ northeast}"
] | [
"Two ships, S and T, are moving in straight lines at constant velocities. At 10:00, they are 5 km apart; at 11:00, they are 4 km apart; and at 13:00, they are 10 km apart. At 7:00 the next day, S is due west of T. Find the speed and direction of both ships."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-55f1a41b7d888e888ea917784387453768ce555691571bf1f5d578f4ccca10a9 | Two shooters, $A$ and $B$, take turns shooting at a target. The first one to hit wins. $A$ hits with probability $p$, and $B$ hits with probability $q$. $A$ starts shooting and we know that $B$ wins. Find the expected value of the number of shots fired. | [
"\\(\\frac{2}{1 - (1-p)(1-q)}\\)"
] | [
"Two shooters, $A$ and $B$, take turns shooting at a target. The first one to hit wins. $A$ hits with probability $p$, and $B$ hits with probability $q$. $A$ starts shooting and we know that $B$ wins. Find the expected value of the number of shots fired."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-d247b870e4d58085f70997cf9f8025b30b44e498e294795fbd903631d20a4532 | Two shooters, Alex and Bob, take turns shooting at a target in the order ABBAABBBAAABBBBAAAABBBBBAAAAAA.... The probability that Alex hits the target is 0.6 and the probability that Bob hits the target is 0.3. All shots are independent. What is the probability that Alex will hit the target first? | [
"\\dfrac{5}{6}"
] | [
"Two shooters, Alex and Bob, take turns shooting at a target in the order ABBAABBBAAABBBBAAAABBBBBAAAAAA.... The probability that Alex hits the target is 0.6 and the probability that Bob hits the target is 0.3. All shots are independent. What is the probability that Alex will hit the target first?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40f0ce3ca8dc379726ef905dea0678131681c2f8ba04bac032ff4b5acee15d69 | Two shuffled decks of one hundred cards numbered one to one hundred are dealt simultaneously one at a time. Find the probability that on none of the one hundred deals do the numbers on the two cards dealt differ by more than one. | [
"$\\frac{F_{100}}{100!}$"
] | [
"Two shuffled decks of one hundred cards numbered one to one hundred are dealt simultaneously one at a time. Find the probability that on none of the one hundred deals do the numbers on the two cards dealt differ by more than one."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ae8b6b728a4108e9827ea60b2e0f04b6b30f6049f785fb93c111660f52209db | Two siblings, Andy and Alexa, bake 24 cookies. Alexa eats some number of cookies that is a positive multiple of the number of cookies eaten by her brother. If the siblings finish all 24 cookies, then what is the maximum number of cookies that the brother, Andy, could have eaten? | [
"12"
] | [
"Two siblings, Andy and Alexa, bake 24 cookies. Alexa eats some number of cookies that is a positive multiple of the number of cookies eaten by her brother. If the siblings finish all 24 cookies, then what is the maximum number of cookies that the brother, Andy, could have eaten?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a57ea2b7d1bad90561a57fd5e51feec281db9d5ecc4efadfb2b0d4c42cafb31 | Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. If $AP=4PN$, calculate the ratio of the length of $AB$ to the length of $CD$. | [
"1"
] | [
"Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. If $AP=4PN$, calculate the ratio of the length of $AB$ to the length of $CD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a4e7f12bccce080739682a01af975482b80867d12dd7fc2c627a73be508b42e | Two sides of a regular polygon with $n$ sides, when extended, meet at an angle of $28$ degrees. What is the smallest possible value of $n$? | [
"45"
] | [
"Two sides of a regular polygon with $n$ sides, when extended, meet at an angle of $28$ degrees. What is the smallest possible value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-5b503f44dacfea82b7c4bb50a2e0ef2bbd1d536b547224e490f62e953a074c61 | Two sides of a right triangle have the lengths 4 and 5. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth. | [
"19.2"
] | [
"Two sides of a right triangle have the lengths 4 and 5. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c805f5c93600f27c4ef5f38e62978253d278cb3bfbd146423f8e3cc187c3e12 | Two sides of a scalene triangle ABC measure 3 cm and 5 cm. How many different whole number lengths are possible for the third side? | [
"3"
] | [
"Two sides of a scalene triangle ABC measure 3 cm and 5 cm. How many different whole number lengths are possible for the third side?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-313581541eb52c608d63d1f7013d090c9bc95e0f754ead369cec72014d69a1dd | Two sides of a triangle are $\sqrt{3}$ cm and $\sqrt{2}$ cm. The medians to these sides are perpendicular to each other. Find the length of the third side. | [
"1"
] | [
"Two sides of a triangle are $\\sqrt{3}$ cm and $\\sqrt{2}$ cm. The medians to these sides are perpendicular to each other. Find the length of the third side."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-02dd182f80aa92d47999e025fa9b1e0b0874c1e8d13fdfafca5f848377651531 | Two sides of a triangle are in the ratio $3:5$ and the third side is $18$. If $\Delta$ is the largest possible area of the triangle, then find $\frac{\Delta}{20}$. | [
"7"
] | [
"Two sides of a triangle are in the ratio $3:5$ and the third side is $18$. If $\\Delta$ is the largest possible area of the triangle, then find $\\frac{\\Delta}{20}$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a4e53c7638808020da4f735079b4f06697d9b83b1477966b6c75747e2f3265d | Two sides of a triangle are √3 cm and √2 cm. The medians to these sides are perpendicular to each other. Find the length of the third side. | [
"1"
] | [
"Two sides of a triangle are √3 cm and √2 cm. The medians to these sides are perpendicular to each other. Find the length of the third side."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e2548f83a15ae6b1dda713e20051eab068d655481340aea5129375f00197459 | Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? | [
"12"
] | [
"Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-07c8942e4807218cc2b4bba4fdca8b6e67562a7cc24c35dcf70693adb33d2bf3 | Two sides of a triangle have lengths $4$ and $9$. The median drawn to the third side has a length of $6$. Compute the length of the third side of the triangle. | [
"\\( 5\\sqrt{2} \\)"
] | [
"Two sides of a triangle have lengths $4$ and $9$. The median drawn to the third side has a length of $6$. Compute the length of the third side of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d5dd91fef3dde3d227d481896c1718f59f1fd027913d7aa86365a1be5b51ead4 | Two sides of a triangle have lengths $\sqrt{2}$ cm and $\sqrt{3}$ cm, respectively. The perpendicular height of one side is $2$ cm longer than the perpendicular height of the other side. Find the area of the triangle in $\text{cm}^2$. | [
"\\(3 \\sqrt{2} + 2 \\sqrt{3}\\)"
] | [
"Two sides of a triangle have lengths $\\sqrt{2}$ cm and $\\sqrt{3}$ cm, respectively. The perpendicular height of one side is $2$ cm longer than the perpendicular height of the other side. Find the area of the triangle in $\\text{cm}^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5bff751c1c77c6d1fc684bc97a8507d2df6d5b5163508bdab635f4141cd8a4a2 | Two sides of a triangle have lengths 15 m and 20 m. The angle between them is increasing at a rate of \(\frac{\pi}{90}\) rad/s. At what rate is the length of the third side changing when the angle between the other two sides is \(\frac{\pi}{3}\)? | [
"\\dfrac{\\pi \\sqrt{39}}{39}"
] | [
"Two sides of a triangle have lengths 15 m and 20 m. The angle between them is increasing at a rate of \\(\\frac{\\pi}{90}\\) rad/s. At what rate is the length of the third side changing when the angle between the other two sides is \\(\\frac{\\pi}{3}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3494ab81107ff31f558e629b993fe99578293624866310598e831ed64ecd48a1 | Two sides of a triangle have lengths 4 and 9. The median drawn to the third side has a length of 6. Compute the length of the third side of the triangle. | [
"\\( 5\\sqrt{2} \\)"
] | [
"Two sides of a triangle have lengths 4 and 9. The median drawn to the third side has a length of 6. Compute the length of the third side of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3e7caa861ac6e5c70c3b7a9fa9cf6f0c407fc63c3564536c631ea3fe0b3ec21c | Two sides of a triangle measure 4 cm and 9 cm. What is the greatest integer number of centimeters that could be the third side of this triangle? | [
"12"
] | [
"Two sides of a triangle measure 4 cm and 9 cm. What is the greatest integer number of centimeters that could be the third side of this triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1711b67b75d790492aff7cacce6c289cf4364357ed6d6bf35b5bf39a32a97986 | Two sides of an integer-sided triangle have lengths $18$ and $x$. If there are exactly $35$ possible integer values for $y$ such that $18, x, y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have. | [
"35"
] | [
"Two sides of an integer-sided triangle have lengths $18$ and $x$. If there are exactly $35$ possible integer values for $y$ such that $18, x, y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-c918e664e54fcf901b31bd445f347885e4d4aa76a2881a55083a55fe7b670f12 | Two sides of an isosceles triangle are 10 inches and 20 inches. If the shortest side of a similar triangle is 50 inches, what is the perimeter of the larger triangle? | [
"250\\text{ inches}"
] | [
"Two sides of an isosceles triangle are 10 inches and 20 inches. If the shortest side of a similar triangle is 50 inches, what is the perimeter of the larger triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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