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-5aeb9f63703febb35d0de0b0256b9b36b2dd8dbacb057ff0e19ac74978bcaee3 | A plane is missing and is presumed to have an equal probability of going down in any of three regions. If the plane is actually down in region \(i\), let \(1 - A_i\) denote the probability that the plane will be found upon a search of the \(i\)-th region, where \(i = 1, 2, 3\). What is the conditional probability that ... | [
"\\(\\frac{A_1}{A_1 + A_2 + A_3}\\)"
] | [
"A plane is missing and is presumed to have an equal probability of going down in any of three regions. If the plane is actually down in region \\(i\\), let \\(1 - A_i\\) denote the probability that the plane will be found upon a search of the \\(i\\)-th region, where \\(i = 1, 2, 3\\). What is the conditional prob... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1458a2b585e45cb035aaab7fa1c159a52ebf12254599434f0f6978dfd4de681 | A plane is partitioned into triangles. Let $\mathcal{T}_0$ denote the set of vertices of these triangles. Consider a triangle $ABC$ with vertices $A, B, C \in \mathcal{T}_0$ and let $\theta$ be the smallest angle of $\triangle ABC$. Assume that no point of $\mathcal{T}_0$ lies inside the circumcircle of $\triangle ABC$... | [
"7"
] | [
"A plane is partitioned into triangles. Let $\\mathcal{T}_0$ denote the set of vertices of these triangles. Consider a triangle $ABC$ with vertices $A, B, C \\in \\mathcal{T}_0$ and let $\\theta$ be the smallest angle of $\\triangle ABC$. Assume that no point of $\\mathcal{T}_0$ lies inside the circumcircle of $\\t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-6dc5c2649d8d23dcc3045067e6c2240b343b74c33c5cb81b4eb72dc2bb15b8ec | A plane is tiled with regular hexagons of side length 1. Let $A$ be a fixed hexagon vertex. Find the number of paths $P$ such that: 1. One endpoint of $P$ is $A$, 2. The other endpoint of $P$ is a hexagon vertex, 3. $P$ lies along hexagon edges, 4. $P$ has length 60, and 5. There is no shorter path along hexagon edges ... | [
"2^{60}"
] | [
"A plane is tiled with regular hexagons of side length 1. Let $A$ be a fixed hexagon vertex. Find the number of paths $P$ such that:\n1. One endpoint of $P$ is $A$,\n2. The other endpoint of $P$ is a hexagon vertex,\n3. $P$ lies along hexagon edges,\n4. $P$ has length 60, and\n5. There is no shorter path along hexa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-b9002e96e681b6bea23493520dc0195cd58832ef1b7aee799be184248669e0f8 | A plane is traveling at a constant speed between two countries. If its speed is increased by 100 km/h, it arrives one hour earlier than usual. If its speed is decreased by 200 km/h, it arrives 3 hours later than usual. What is the distance between the two countries? | [
"7200"
] | [
"A plane is traveling at a constant speed between two countries. If its speed is increased by 100 km/h, it arrives one hour earlier than usual. If its speed is decreased by 200 km/h, it arrives 3 hours later than usual. What is the distance between the two countries?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-613c2b0ae98d98896520a1b2c8ae2d2562bb71a96b75bcf8d556acda681f48c9 | A plane is traveling directly at you at a speed of 200 mph and an altitude of 3000 feet. You hear the sound at what seems to be an angle of inclination of 20 degrees. At what angle should you actually look in order to see the plane? Assume the speed of sound is 1100 ft/sec. | [
"27^\\circ"
] | [
"A plane is traveling directly at you at a speed of 200 mph and an altitude of 3000 feet. You hear the sound at what seems to be an angle of inclination of 20 degrees. At what angle should you actually look in order to see the plane? Assume the speed of sound is 1100 ft/sec."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-68eb2315ad2f42609889fea5ddf6ab2ac2054ccf7e0f95085e7e9e6208b219bc | A plane is uniquely determined by three non-collinear points. What is the maximum possible number of planes that can be determined by 12 points in space? | [
"220"
] | [
"A plane is uniquely determined by three non-collinear points. What is the maximum possible number of planes that can be determined by 12 points in space?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-945cb32cb35c2796a0ab434bdc1a7f726fcf203896ab76773dbe0187177ff72d | A plane leaves an airport from a runway with a bearing of N40°E and flies for 1/4 of a mile. The pilot then turns 90 degrees and flies for 2 miles. What is the bearing of the plane from the airport after these maneuvers? | [
"\\( 122° 52' \\)"
] | [
"A plane leaves an airport from a runway with a bearing of N40°E and flies for 1/4 of a mile. The pilot then turns 90 degrees and flies for 2 miles. What is the bearing of the plane from the airport after these maneuvers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 18
} | 36 | 0 | 0 |
omr-ws-c02c26080f4ec93936be5b8b8874088efa1efc1d01055b440aef5eefd418b357 | A plane leaves an airport from runway 4 RIGHT, whose bearing is N40°E. After flying for 1/2 mile, the pilot turns 90 degrees and heads toward the southeast. After flying 1 mile in this direction, what bearing should the control tower use to locate the plane? | [
"S76.6°E"
] | [
"A plane leaves an airport from runway 4 RIGHT, whose bearing is N40°E. After flying for 1/2 mile, the pilot turns 90 degrees and heads toward the southeast. After flying 1 mile in this direction, what bearing should the control tower use to locate the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 5
} | 26 | 0 | 0 |
omr-ws-e899d5278ed3996444d9960c36447df52c3724d4914d3f45a5eedc7936208618 | A plane maintains an airspeed of 550 miles per hour in a southwestern direction. The velocity of the wind is a constant 80 miles per hour from the west. Find the actual speed and direction of the plane. | [
"51.5^\\circ"
] | [
"A plane maintains an airspeed of 550 miles per hour in a southwestern direction. The velocity of the wind is a constant 80 miles per hour from the west. Find the actual speed and direction of the plane."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-4aae2b6182c07d5c3dda2ab673bc4429093abb754f390d9982a0459fdbe0d2f9 | A plane passes through the point $(1, -2, 1)$ and is perpendicular to the planes $2x - 2y + z = 0$ and $x - y + 2z = 4$. What is the distance of this plane from the point $(1, 2, 2)$? | [
"\\( 2\\sqrt{2} \\)"
] | [
"A plane passes through the point $(1, -2, 1)$ and is perpendicular to the planes $2x - 2y + z = 0$ and $x - y + 2z = 4$. What is the distance of this plane from the point $(1, 2, 2)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4ffa2d5ecd164d2c75b6bc49c7787126650b61a22ae145d2a0453c3cbb77e57b | A plane point set with $3n$ points is called a "weakly compact set" if the distance between any two points in any $n + 1$ points is $1$. Find the minimum number of pairs of points whose distance is $1$ in a weakly compact set. | [
"\\(3n\\)"
] | [
"A plane point set with $3n$ points is called a \"weakly compact set\" if the distance between any two points in any $n + 1$ points is $1$. Find the minimum number of pairs of points whose distance is $1$ in a weakly compact set."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-30bfd8230f09a352c81c17df620c7b2ee52d31cad5bc3ef57c1f059a9f384f51 | A plane randomly cuts a sphere into two parts. Two points are then randomly chosen inside the sphere. What is the probability that the two points are on the same side of the plane? | [
"\\(\\frac{1}{2}\\)"
] | [
"A plane randomly cuts a sphere into two parts. Two points are then randomly chosen inside the sphere. What is the probability that the two points are on the same side of the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efc92ca76826eaad21f8822a08d46f7b03ecffad029fdeebbb86eb758eca28e7 | A plane section of a rectangular parallelepiped forms a hexagon, which is covered by a rectangle Q of dimensions \(a \times b\). If the dimensions of the rectangular parallelepiped are \(x\), \(y\), and \(z\), find the minimum possible value of \(\frac{a}{x} + \frac{b}{y} + \frac{a}{z} + \frac{b}{z}\). | [
"4"
] | [
"A plane section of a rectangular parallelepiped forms a hexagon, which is covered by a rectangle Q of dimensions \\(a \\times b\\). If the dimensions of the rectangular parallelepiped are \\(x\\), \\(y\\), and \\(z\\), find the minimum possible value of \\(\\frac{a}{x} + \\frac{b}{y} + \\frac{a}{z} + \\frac{b}{z}\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-debfb7d6954be9e0c85ef7cda48d705c260097890a55778af4053d670d9e5c46 | A plane takes a round trip flight from town A to town B and back to town A. With no wind, the total trip takes 2 hours and 48 minutes. When there is a 100 mph wind blowing from town A to town B, the round trip takes 5 hours. How far is it from town A to town B? | [
"\\( \\frac{700}{\\sqrt{11}} \\) miles."
] | [
"A plane takes a round trip flight from town A to town B and back to town A. With no wind, the total trip takes 2 hours and 48 minutes. When there is a 100 mph wind blowing from town A to town B, the round trip takes 5 hours. How far is it from town A to town B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 15
} | 35 | 0 | 0 |
omr-ws-ce22dd39f02d15d264ef4e40096f037c3dbbbc2ccb43df132a7b92057d421f20 | A plane takes off from Chicago's O'Hare Airport on runway 2 LEFT, which has a bearing of N20°E. After flying for 1 mile, the pilot turns 90 degrees and heads toward the northwest. After flying 2 miles in this direction, what bearing should the control tower use to locate the plane? | [
"N43.434949°W or W46.5650511°N"
] | [
"A plane takes off from Chicago's O'Hare Airport on runway 2 LEFT, which has a bearing of N20°E. After flying for 1 mile, the pilot turns 90 degrees and heads toward the northwest. After flying 2 miles in this direction, what bearing should the control tower use to locate the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 26
} | 33 | 0 | 0 |
omr-ws-21b66142ee1b6df107a991c6e3b86b29e420c8a293860c5887fb4aaa27b7ed30 | A plane traveling at a constant rate can cover a 1000 kilometer round-trip distance from A to B and back in 10 hours when there is no wind. If the wind blows from A toward B at $ k$ km/hr, it adds 25 minutes to the round trip. Find $ k$. | [
"20"
] | [
"A plane traveling at a constant rate can cover a 1000 kilometer round-trip distance from A to B and back in 10 hours when there is no wind. If the wind blows from A toward B at $ k$ km/hr, it adds 25 minutes to the round trip. Find $ k$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-87d787cf5fcf308223140ceb4ff24c840bbf49a32cce7d7ce09efc426cd5f8a3 | A plane varies so that it includes a cone of constant volume equal to $\pi a^3 / 3$ with the surface given by the equation $2xy = z^2$ in rectangular coordinates. Find the equation of the envelope of the various positions of this plane. State the result so that it applies to a general cone (conic surface) of the second... | [
"2xy = z^2 + a^2"
] | [
"A plane varies so that it includes a cone of constant volume equal to $\\pi a^3 / 3$ with the surface given by the equation $2xy = z^2$ in rectangular coordinates. Find the equation of the envelope of the various positions of this plane. State the result so that it applies to a general cone (conic surface) of the ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-e46a9c5fc6b0bb73d95cdb0615fcdf18b701cbcf2ada9946eda37cd2c3566c36 | A plane's heading is 35 degrees, and its air speed is 305 miles per hour. If the wind is blowing from a heading of 20 degrees, with a wind speed of 60 miles per hour, what is the ground speed, drift angle, and direction of the airplane? | [
"Ground Speed: \\( 305\\cos{35} + 60\\cos{20} \\), Drift Angle: \\( 35 - \\tan^{-1}\\left(\\frac{305\\sin{35} - 60\\sin{20}}{305\\cos{35} + 60\\cos{20}}\\right) \\), Direction: \\( \\tan^{-1}\\left(\\frac{305\\sin{35} - 60\\sin{20}}{305\\cos{35} + 60\\cos{20}}\\right) \\)"
] | [
"A plane's heading is 35 degrees, and its air speed is 305 miles per hour. If the wind is blowing from a heading of 20 degrees, with a wind speed of 60 miles per hour, what is the ground speed, drift angle, and direction of the airplane?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-9860386155b7f4de2f93b9686406a2cc165ae6e757d54f2d96ef0ad74fb5a28e | A plane, drawn through the vertex of a cone and at an angle $\varphi $ to the base, cuts off the circle of the base an arc $\alpha $; the distance between the plane and the center of the base is equal to $a$. Find the volume of the cone. | [
"$V=\\frac{\\pi a^3}{3\\sin^2 \\varphi \\cos\\varphi \\cos^2\\frac{\\alpha }{2}}$"
] | [
"A plane, drawn through the vertex of a cone and at an angle $\\varphi $ to the base, cuts off the circle of the base an arc $\\alpha $; the distance between the plane and the center of the base is equal to $a$. Find the volume of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1c8fa73db61d412be4760df87dbb1f46d345e9217c41a4d681cd5496b1d29d2 | A plane, drawn through the vertex of a cone, cuts the base along a chord whose length is equal to the radius of the base. Find the ratio of the volumes of the resulting parts of the cone. | [
"$\\frac{2\\pi -3\\sqrt 3}{10\\pi +3\\sqrt 3}$"
] | [
"A plane, drawn through the vertex of a cone, cuts the base along a chord whose length is equal to the radius of the base. Find the ratio of the volumes of the resulting parts of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89bac8c4a816c85bdc8e24d998865892e44cc86232f357740b6ca4e888db0629 | A plane, which contains an edge, divides a regular tetrahedron into two bodies with volume ratio $3:5$. Determine the measures of the angles at which the secant plane divides the dihedral angle of the tetrahedron. | [
"\\(\\arccos \\frac{7\\sqrt{11}}{33}, \\arccos \\frac{3\\sqrt{11}}{11}\\)"
] | [
"A plane, which contains an edge, divides a regular tetrahedron into two bodies with volume ratio $3:5$. Determine the measures of the angles at which the secant plane divides the dihedral angle of the tetrahedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 15
} | 17 | 0 | 0 |
omr-ws-4c715007c126ef3d936dd8349381abefdfe2860fd4926d1c289f65c335e05c13 | A planet is called illuminated if every point on its surface is illuminated. Determine a) what is the smallest number of point suns that can illuminate one globular planet, (b) what is the smallest number of point suns that can illuminate $n$ globular planets. Both suns and planets can be placed in the most favorable p... | [
"a) 4, b) 4"
] | [
"A planet is called illuminated if every point on its surface is illuminated. Determine\na) what is the smallest number of point suns that can illuminate one globular planet,\n(b) what is the smallest number of point suns that can illuminate $n$ globular planets.\nBoth suns and planets can be placed in the most fav... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-46faec144a7fb0eda966333fd201b608d96c11c6ca55dccac99875a864de465a | A plant from one seed can have either 1 or 2 flowers in the first winter with probabilities $\frac{1}{2}$ and $\frac{1}{4}$ respectively, otherwise it dies without blooming. Each flower produces 2 or 3 seeds with equal probability. During the summer, the plant dies, but each seed survives independently with probability... | [
"\\(0.75\\)"
] | [
"A plant from one seed can have either 1 or 2 flowers in the first winter with probabilities $\\frac{1}{2}$ and $\\frac{1}{4}$ respectively, otherwise it dies without blooming. Each flower produces 2 or 3 seeds with equal probability. During the summer, the plant dies, but each seed survives independently with prob... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ebc81addb98e160d27f1ad97b06183c8032795c5e50c8fa9d32636db1a184c3 | A plant from one seed can have either 1 or 2 flowers in the first winter with probabilities $\frac{1}{2}$ and $\frac{1}{4}$ respectively, otherwise it dies without blooming. Each flower produces 2 or 3 seeds with equal probability. During the summer, the plant dies, but each seed survives independently with probability... | [
"\\frac{1}{4} + \\frac{1}{4}\\left(\\left(\\frac{4 + s}{5}\\right)^2 + \\left(\\frac{4 + s}{5}\\right)^3\\right) + \\frac{1}{16}\\left(\\left(\\frac{4 + s}{5}\\right)^2 + \\left(\\frac{4 + s}{5}\\right)^3\\right)^2"
] | [
"A plant from one seed can have either 1 or 2 flowers in the first winter with probabilities $\\frac{1}{2}$ and $\\frac{1}{4}$ respectively, otherwise it dies without blooming. Each flower produces 2 or 3 seeds with equal probability. During the summer, the plant dies, but each seed survives independently with prob... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-c94b75d565bc5ad056c5d391691cf2ae57de316450aed97f1b193eb7d1c99ad9 | A plant wound itself exactly 5 times around a pole with a height of 1 meter and a circumference of 15 cm. As it climbed, its height increased at a constant rate. What is the length of the plant? | [
"125 cm"
] | [
"A plant wound itself exactly 5 times around a pole with a height of 1 meter and a circumference of 15 cm. As it climbed, its height increased at a constant rate. What is the length of the plant?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-424b6a986b5c148be12c908c01b5f802bda6a6d3d2b31c215f4f1c4cb83b2bfe | A plantation owner has 3000 bananas that need to be transported 1000 km across a desert by camel. The camel can carry a maximum of 1000 bananas at a time and eats one banana per kilometer traveled. What is the largest number of bananas that can be delivered to the market? | [
"\\(\\frac{1600}{3}\\)"
] | [
"A plantation owner has 3000 bananas that need to be transported 1000 km across a desert by camel. The camel can carry a maximum of 1000 bananas at a time and eats one banana per kilometer traveled. What is the largest number of bananas that can be delivered to the market?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-85e7be46434738ecd6e7e3ec24e6c1af0b7ec5133407ce9f1f77486fdce336c7 | A plantation owner has 3000 bananas that need to be transported 1000 miles across a desert by camel. The camel can carry a maximum of 1000 bananas at any time and eats one banana per mile traveled. What is the largest number of bananas that can be delivered to the market? | [
"\\( 533.\\overline{3} \\)"
] | [
"A plantation owner has 3000 bananas that need to be transported 1000 miles across a desert by camel. The camel can carry a maximum of 1000 bananas at any time and eats one banana per mile traveled. What is the largest number of bananas that can be delivered to the market?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 43
} | 43 | 0 | 0 |
omr-ws-682da849da1e3ba687c2401678051950f9db1bd44468ebe502b6499a82ded50e | A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, such that the distance between their centers is 60 mm. Calculate the radius of the hole. | [
"25.3"
] | [
"A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, such that the distance between their centers is 60 mm. Calculate the radius of the hole."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-011a19027d941d3fa461f15302cae4d962c096de0c0591d8dceb0b6e2fd791f0 | A platoon 50 meters long is marching with constant speed. A soldier from the last row runs to the first row to deliver a letter and then returns to his original position at the last row. By the time he returns, the platoon has moved 50 meters forward. How much distance does the messenger soldier run? | [
"\\( 50\\sqrt{2} \\) meters\n\nExplanation:\nFrom the forum discussions, the key equation derived is:\n\n\\[\n\\frac{50}{x-y} + \\frac{50}{x+y} = \\frac{50}{y}\n\\]\n\nBy simplifying this equation, we can find the ratio \\( \\frac{x}{y} \\). Multiplying through by \\( y(x-y)(x+y) \\) to clear the denominators, we g... | [
"A platoon 50 meters long is marching with constant speed. A soldier from the last row runs to the first row to deliver a letter and then returns to his original position at the last row. By the time he returns, the platoon has moved 50 meters forward. How much distance does the messenger soldier run?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13bb83f277603c7a8dc07e4eb58192ff347f7cb178bb4a3aaae7641b00582e10 | A plausible model is $p_j = \theta \left(\frac{2}{3}\right)^{j-1}$. For each numerical $\hat{p_j}$ value found, write down a corresponding expression for $log(\hat{p_j})$ using $log(X_n) = a + \frac{bZ}{\sqrt{n}}$ where $a = log(p)$ and $b = \sqrt{\frac{1-p}{p}}$. | [
"\\log(\\hat{p_j}) = \\log(\\theta) + (j-1)\\log\\left(\\frac{2}{3}\\right) + \\sqrt{\\frac{1 - \\theta \\left(\\frac{2}{3}\\right)^{j-1}}{\\theta \\left(\\frac{2}{3}\\right)^{j-1}}} \\cdot \\frac{Z}{\\sqrt{n}}"
] | [
"A plausible model is $p_j = \\theta \\left(\\frac{2}{3}\\right)^{j-1}$. For each numerical $\\hat{p_j}$ value found, write down a corresponding expression for $log(\\hat{p_j})$ using $log(X_n) = a + \\frac{bZ}{\\sqrt{n}}$ where $a = log(p)$ and $b = \\sqrt{\\frac{1-p}{p}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-664fbce17425cd224c49d0457576d335db6e83c8f72fa2ef43ab1ca93b4e33f0 | A play has two different male roles, two different female roles and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned? | [
"25200"
] | [
"A play has two different male roles, two different female roles and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3405015a0adc6c5380635064e980200ff184e3f56855cb537bd4e3933c7380da | A player (P) tosses a coin, and the house agrees to pay the player \( x \) if P gets a head on the first toss, \( x^2 \) if P gets a head on the second toss, \( x^3 \) if P gets a head on the third toss, and so on. Find the mathematical expectation of P when: a) \( x = 1 \) b) \( x = 1.99 \) c) \( x = 2 \) Write down a... | [
"\\infty"
] | [
"A player (P) tosses a coin, and the house agrees to pay the player \\( x \\) if P gets a head on the first toss, \\( x^2 \\) if P gets a head on the second toss, \\( x^3 \\) if P gets a head on the third toss, and so on. Find the mathematical expectation of P when:\na) \\( x = 1 \\)\nb) \\( x = 1.99 \\)\nc) \\( x ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ee6b6d9c3b8dac3cc519f958e7c6b5ded05948eb3212a43ec467b50fb9ac993 | A player chooses a primary weapon and a side arm, along with three perks and one of the three grenade types. How many different combinations are possible? | [
"7260"
] | [
"A player chooses a primary weapon and a side arm, along with three perks and one of the three grenade types. How many different combinations are possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc405dd688d62b4293331feaaac938847f31ce6abf9d4363f28246259275a7d1 | A player chooses one of the numbers $1$ through $4$. Two regular four-sided (tetrahedral) dice are rolled, with the sides numbered $1$ through $4$. If the chosen number appears on the bottom of exactly one die, the player wins $ \$1$. If the chosen number appears on the bottom of both dice, the player wins $ \$2$. If t... | [
"$-\\frac{1}{16}$"
] | [
"A player chooses one of the numbers $1$ through $4$. Two regular four-sided (tetrahedral) dice are rolled, with the sides numbered $1$ through $4$. If the chosen number appears on the bottom of exactly one die, the player wins $ \\$1$. If the chosen number appears on the bottom of both dice, the player wins $ \\$2... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e6d6c9b5ca04df6a8ecf43a2bbc43e60fc6d271f8629063bb9b8807d77c4608c | A player chooses one of the numbers 1 through 4. Two regular four-sided (tetrahedral) dice are rolled, with the sides numbered 1 through 4. If the chosen number appears on the bottom of exactly one die, the player wins $1. If the chosen number appears on the bottom of both dice, the player wins $2. If the chosen number... | [
"\\(-\\frac{1}{16}\\)"
] | [
"A player chooses one of the numbers 1 through 4. Two regular four-sided (tetrahedral) dice are rolled, with the sides numbered 1 through 4. If the chosen number appears on the bottom of exactly one die, the player wins $1. If the chosen number appears on the bottom of both dice, the player wins $2. If the chosen n... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-21fdcf7121807c52a327b97dec455d4ac3000b868df95ec7a09aa706b9469334 | A player chooses two different primary weapons (no side arms), along with two additional perks and one of the three grenade types. How many different combinations are possible? | [
"320166"
] | [
"A player chooses two different primary weapons (no side arms), along with two additional perks and one of the three grenade types. How many different combinations are possible?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-676ba415d698a2007ed6a298a94cfe5a7ebaa8243621cec69593dd11c83e4a84 | A player flips 100 coins and scores \( x(100-x) \) points for flipping \( x \) heads. What is the expected value of the points the player gets? | [
"2475"
] | [
"A player flips 100 coins and scores \\( x(100-x) \\) points for flipping \\( x \\) heads. What is the expected value of the points the player gets?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-111fba4e7a133c911b96cc77d5aa81fa8cd30b2da3a52d03b4fb55e63eb41fd0 | A player is dealt 4 cards from a standard deck. He wins if 2 of the cards have the same suit and 2 of the cards have the same rank. What is his probability of winning? | [
"\\( \\frac{51480}{\\binom{52}{4}} \\)"
] | [
"A player is dealt 4 cards from a standard deck. He wins if 2 of the cards have the same suit and 2 of the cards have the same rank. What is his probability of winning?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 17
} | 21 | 0 | 0 |
omr-ws-2589d1ba16551ef4d7e7e802a30933afff35088b76cb15f6f3a633c1f2eb8b33 | A player repeatedly throws a die until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$. Find the value of $\lim_{n \to \infty} p(n)$. | [
"\\(0\\)"
] | [
"A player repeatedly throws a die until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$. Find the value of $\\lim_{n \\to \\infty} p(n)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fbd8f8c5ae2e224a298bd0030128ed8f267df5af4ab60f18022b7f0c658a04f6 | A player scored 23, 14, 11, and 20 points in the sixth, seventh, eighth, and ninth basketball games of the season, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than 18, what is the least number of points she coul... | [
"29"
] | [
"A player scored 23, 14, 11, and 20 points in the sixth, seventh, eighth, and ninth basketball games of the season, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than 18, what is the least number of points she... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-afa5e0ef07f538046f00cbdbb201fad9ee73fa23a93b6b211caee96141d615fe | A player scored a hole in one on an 18-hole golf course. Six members gave the following clues about which hole it was: - A: "It was an odd number." - B: "It had two digits." - C: "It was a prime number." - D: "It was divisible by 5." - E: "It was a single-digit number." - F: "It was between 5 and 15." One of them lied.... | [
"5"
] | [
"A player scored a hole in one on an 18-hole golf course. Six members gave the following clues about which hole it was:\n- A: \"It was an odd number.\"\n- B: \"It had two digits.\"\n- C: \"It was a prime number.\"\n- D: \"It was divisible by 5.\"\n- E: \"It was a single-digit number.\"\n- F: \"It was between 5 and ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd44c4ee36c1815dc9a15e21f4c1d7322b8097490f7b66144b77826fb70ede6d | A player starts with \( k \) coins where \( 0 < k < 100 \). At each step, the player either loses one coin with a probability of \( \frac{1}{3} \) or wins one coin. What is the probability that the player wins 100 coins before losing all their coins? | [
"\\( 1 - \\frac{1}{2^k} \\)"
] | [
"A player starts with \\( k \\) coins where \\( 0 < k < 100 \\). At each step, the player either loses one coin with a probability of \\( \\frac{1}{3} \\) or wins one coin. What is the probability that the player wins 100 coins before losing all their coins?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-a51f8bc1a3b27552953c52893d88c3eb03d9689708b88d30675602b5060bd028 | A player tosses a penny with a diameter of 3/4 inch onto a table ruled in 1-inch squares. If the penny falls entirely inside a square, the player wins 5 cents. What is the probability that the penny lands entirely within a square? | [
"\\(\\frac{1}{16}\\)"
] | [
"A player tosses a penny with a diameter of 3/4 inch onto a table ruled in 1-inch squares. If the penny falls entirely inside a square, the player wins 5 cents. What is the probability that the penny lands entirely within a square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-3f657b71d3cc409ca6766d2b12e6fbb930af43743e0453f2fda1d5ff529d0b65 | A player tosses an unbiased coin and scores two points for every head and one point for every tail. If \( P(n) \) denotes the probability that the player's score is exactly \( n \) points, find \( P(n) \). | [
"The probability \\( P(n) \\) that the player's score is exactly \\( n \\) points is given by:\n\\[ P(n) = \\sum_{H=0}^{\\lfloor \\frac{n}{2} \\rfloor} \\binom{n - H}{H} \\left(\\frac{1}{2}\\right)^{n - H} \\]"
] | [
"A player tosses an unbiased coin and scores two points for every head and one point for every tail. If \\( P(n) \\) denotes the probability that the player's score is exactly \\( n \\) points, find \\( P(n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-05665978469900c91fc59e356de6b714235a47b2047dd0aa7dc5a9dbb01366f1 | A player wants to play at most $132$ games in exactly $77$ days, with at least $1$ game played each day. Let $g_n$ represent the total number of games played by the end of day $n$. Find the number of distinct values $k$ such that $g_k - g_{k-21} = 21$ for some $k$, where $22 \leq k \leq 77$. | [
"56"
] | [
"A player wants to play at most $132$ games in exactly $77$ days, with at least $1$ game played each day. Let $g_n$ represent the total number of games played by the end of day $n$. Find the number of distinct values $k$ such that $g_k - g_{k-21} = 21$ for some $k$, where $22 \\leq k \\leq 77$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fff6e94ec242eac8f62e37babe27bf1835d119bb75961a116e05ddded7c5392 | A player wins an amount uniformly distributed in the interval $[0,1]$ at each turn and stops when the cumulative total is at least $1$. What is the expected number of turns that the game lasts? | [
"\\( e \\)"
] | [
"A player wins an amount uniformly distributed in the interval $[0,1]$ at each turn and stops when the cumulative total is at least $1$. What is the expected number of turns that the game lasts?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-487f5bc6e2a4ba9347859777c3e24119580f1e7f5a33c8f47e203963f293a18b | A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a 3rd grader is in the middle swing there must be 4th graders on that person’s left and right. And if there is a 4th grader in the middle, there must be 3rd graders on t... | [
"12"
] | [
"A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a 3rd grader is in the middle swing there must be 4th graders on that person’s left and right. And if there is a 4th grader in the middle, there must be 3rd graders... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-21bf49bf1a383785d9103fc4abe079e1cc18d47ca249288e2b00027cd6a19cff | A playoff series between the Hawks and the Knights continues until one of the two teams has won three games, and therefore may take as many as five games. The probability that the Hawks defeat the Knights in any given game is the same for each game. The probability that the series takes only three games is equal to the... | [
"0.357"
] | [
"A playoff series between the Hawks and the Knights continues until one of the two teams has won three games, and therefore may take as many as five games. The probability that the Hawks defeat the Knights in any given game is the same for each game. The probability that the series takes only three games is equal t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fa052f399af5d8e82ff192af98b789ab6b23c30cf71a5207607f9ade9302e94 | A playoff series between two teams proceeds one game at a time until one team has won 5 games. What is the probability that the series lasts 9 games if each team is equally likely to win each game? | [
"\\(\\frac{35}{128}\\)"
] | [
"A playoff series between two teams proceeds one game at a time until one team has won 5 games. What is the probability that the series lasts 9 games if each team is equally likely to win each game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-aa5134938b1793113a407a0ce0152e1675c56c9deb5979eed6035a11303957ac | A plot of land is divided into a $10 \times 10$ grid, forming 100 plots, of which 9 have weeds. Each year, a plot becomes infected if it has exactly two infected neighbors (plots sharing a common side). What is the maximum number of plots that can become infected, assuming the initial configuration of infected plots al... | [
"81"
] | [
"A plot of land is divided into a $10 \\times 10$ grid, forming 100 plots, of which 9 have weeds. Each year, a plot becomes infected if it has exactly two infected neighbors (plots sharing a common side). What is the maximum number of plots that can become infected, assuming the initial configuration of infected pl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-705d5dba3940aa89af1a12bb9e566e4503481ce7b5944b4d180fb7a53e461a05 | A plus is a polygon made of 5 congruent squares (one in the center, and 4 built on each side) for 12 total sides, all the same length. The 8 outer points are inscribed in a circle. If the length of one side of a plus is \( p \), what is the ratio of the circle area to the plus area? | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"A plus is a polygon made of 5 congruent squares (one in the center, and 4 built on each side) for 12 total sides, all the same length. The 8 outer points are inscribed in a circle. If the length of one side of a plus is \\( p \\), what is the ratio of the circle area to the plus area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-247ea0bce7e294b934fb60768cce026c8a2ac797d21be82d6e6b16ebe141e0a7 | A point $(3\sqrt{5},d+3)$ is $3d$ units away from the origin. What is the smallest possible value of $d$? | [
"3"
] | [
"A point $(3\\sqrt{5},d+3)$ is $3d$ units away from the origin. What is the smallest possible value of $d$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ca9e82f319e999e28f316256fb86aaceff5bba1f6d2dc069190eba0cdeba32a | A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$? | [
"15"
] | [
"A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb280b8c921dca015118a80c9598b7360b0f4d67a31dbee38461c2069a471621 | A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$? | [
"\\dfrac{7}{8}"
] | [
"A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55f5bcc7da949571b00feb5869d1e014e0ef8346248a3d1619e21201eb5d2ef1 | A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$? | [
"\\dfrac{1}{3}"
] | [
"A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7e86cfb0f587cb9d7aebe25f9863af8f7699b9325b31e2e5e0238e6683a8347 | A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$? | [
"\\frac{1}{8}"
] | [
"A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a66a4bf6ccb76b2a966759f031efb04240dde9b39abb82e2945f968cba9356e4 | A point $(x,y)$ is randomly selected such that $0 \le x \le 3$ and $0 \le y \le 6$. What is the probability that $x+y \le 4$? Express your answer as a common fraction. | [
"\\frac{5}{12}"
] | [
"A point $(x,y)$ is randomly selected such that $0 \\le x \\le 3$ and $0 \\le y \\le 6$. What is the probability that $x+y \\le 4$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c85321361cc6cc20046c1e9717f1843de4ce6427ffdad7ac7a1bf27cfcfadce | A point $(x_0, y_0)$ with integer coordinates is a primitive point of a circle if for some pair of integers $(a, b)$, the line $ax + by = 1$ intersects the circle at $(x_0, y_0)$. How many primitive points are there of the circle centered at $(2, -3)$ with radius $5$? | [
"5"
] | [
"A point $(x_0, y_0)$ with integer coordinates is a primitive point of a circle if for some pair of integers $(a, b)$, the line $ax + by = 1$ intersects the circle at $(x_0, y_0)$. How many primitive points are there of the circle centered at $(2, -3)$ with radius $5$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2062298d59446c39d87d0b310541b67bc1213fe13847af5ddee7d965db7befce | A point $A$ is chosen inside a convex polygon $M$. The polygon $M'$ is obtained from $M$ by a translation vector $\vec{v}$. Denote by $r$ and $r'$ the maximum distances from $A$ to the vertices of the polygons $M$ and $M'$, respectively. If $|\vec{v}| = 10$, find the minimum possible value of $r'$ when $r = 15$. | [
"10"
] | [
"A point $A$ is chosen inside a convex polygon $M$. The polygon $M'$ is obtained from $M$ by a translation vector $\\vec{v}$. Denote by $r$ and $r'$ the maximum distances from $A$ to the vertices of the polygons $M$ and $M'$, respectively. If $|\\vec{v}| = 10$, find the minimum possible value of $r'$ when $r = 15$.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3346e66b44797917704f4f43a0a82ff338ec41894e8a99cfa9f1a88151fec9f | A point $A$ is outside a circle $\mathcal{K}$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle at $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ and $F$... | [
"1"
] | [
"A point $A$ is outside a circle $\\mathcal{K}$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle at $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cd53800f9502a0eecd90c8e5ad8aae025274a6fa5a2102fd683e2da4ec2cd7b7 | A point $A$ is selected at random from an $n \times n$ square grid of points. A second point $B$ is selected at random from those points that are not in the same row or column as the first point. There exists a unique rectangle that has $A$ and $B$ as opposite vertices and whose sides are parallel to the edges of the g... | [
"\\(\\frac{(n+1)^2}{9}\\)"
] | [
"A point $A$ is selected at random from an $n \\times n$ square grid of points. A second point $B$ is selected at random from those points that are not in the same row or column as the first point. There exists a unique rectangle that has $A$ and $B$ as opposite vertices and whose sides are parallel to the edges of... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 8
} | 25 | 0 | 0 |
omr-ws-a4f048cb86d08296d655cd0796c84abad8d6c2bf3119b5f590fa76b7f8a3effa | A point $A$ lies on the common external tangent $t_1$ of two disjoint circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$, respectively. The second tangents from $A$ to $\omega_1$ and $\omega_2$ meet their second common external tangent $t_2$ at points $B$ and $C$. If the excircle of $\triangle ABC$ opposite... | [
"4\\sqrt{6}"
] | [
"A point $A$ lies on the common external tangent $t_1$ of two disjoint circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$, respectively. The second tangents from $A$ to $\\omega_1$ and $\\omega_2$ meet their second common external tangent $t_2$ at points $B$ and $C$. If the excircle of $\\triangle ABC... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-32d6ba53ae475c2c7e027be2df978947f6de4a83542a9cb438f31de7659a09f9 | A point $A$ lies outside of a circle. Let $B$ be any point on the circle and $M$ be the midpoint of $AB$. As $B$ varies, describe the curve traced by the point $M$. | [
"\\text{A circle with radius half of the original circle's radius and center at the midpoint between } A \\text{ and the original circle's center}"
] | [
"A point $A$ lies outside of a circle. Let $B$ be any point on the circle and $M$ be the midpoint of $AB$. As $B$ varies, describe the curve traced by the point $M$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06488cb8a649a38389f7584a2af125009568c91cae9bfba61dc43c7c28477247 | A point $B$ inside a regular hexagon $A_1A_2...A_6$ is given such that $\angle A_2A_1B = \angle A_4A_3B = 50^\circ$. Find $\angle A_1A_2B$. | [
"$20^\\circ$"
] | [
"A point $B$ inside a regular hexagon $A_1A_2...A_6$ is given such that $\\angle A_2A_1B = \\angle A_4A_3B = 50^\\circ$. Find $\\angle A_1A_2B$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42d867f7681093ec861649333e4b1194f8dc5dfa7dcccf5f8bc85a30ab88b1e8 | A point $B$ lies on a chord $AC$ of circle $\omega$. Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet at a point $F$, and the rays $... | [
"\\dfrac{4}{3}"
] | [
"A point $B$ lies on a chord $AC$ of circle $\\omega$. Segments $AB$ and $BC$ are diameters of circles $\\omega_1$ and $\\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet at a point $F$, and t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e14d0f823c17b8b0bd0c1979b1d60e4ffa081a3ebb5c4ed2f18c0df3f648a4f | A point $B$ lies on a chord $AC$ of circle $\omega$. Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F$, and the rays $... | [
"1"
] | [
"A point $B$ lies on a chord $AC$ of circle $\\omega$. Segments $AB$ and $BC$ are diameters of circles $\\omega_1$ and $\\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F$, and t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-74216ff790d58f58a47e1fa73c883e1d8d92e150fcb689b68bbc724fa28d2fab | A point $C$ is marked on a chord $AB$ of a circle $\omega$. Let $D$ be the midpoint of $AC$, and $O$ be the center of the circle $\omega$. The circumcircle of the triangle $BOD$ intersects the circle $\omega$ again at point $E$ and the straight line $OC$ again at point $F$. If the radius of $\omega$ is $r$ and the leng... | [
"\\dfrac{\\pi}{3}"
] | [
"A point $C$ is marked on a chord $AB$ of a circle $\\omega$. Let $D$ be the midpoint of $AC$, and $O$ be the center of the circle $\\omega$. The circumcircle of the triangle $BOD$ intersects the circle $\\omega$ again at point $E$ and the straight line $OC$ again at point $F$. If the radius of $\\omega$ is $r$ and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e59935d6686c88b942e99f4ff26e40ec85098d9deb611356b14f83305b9d4d5 | A point $C$ lies on a line segment $AB$ between $A$ and $B$, and circles are drawn having $AC$ and $CB$ as diameters. A common tangent line to both circles touches the circle with $AC$ as diameter at $P \neq C$ and the circle with $CB$ as diameter at $Q \neq C$. Let $R$ be the intersection point of lines $AP$ and $BQ$.... | [
"r"
] | [
"A point $C$ lies on a line segment $AB$ between $A$ and $B$, and circles are drawn having $AC$ and $CB$ as diameters. A common tangent line to both circles touches the circle with $AC$ as diameter at $P \\neq C$ and the circle with $CB$ as diameter at $Q \\neq C$. Let $R$ be the intersection point of lines $AP$ an... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7c6f6b63553148e0d6a279cc1e069a8f7c18fc77655c3ff21b0646d45a79ea10 | A point $C$ lies on a line segment $AB$ between $A$ and $B$, and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$. Let $R$ be the point where $AP$, $BQ$, and the common tangent to... | [
"(4, 2\\sqrt{6})"
] | [
"A point $C$ lies on a line segment $AB$ between $A$ and $B$, and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \\ne C$ and the circle with $CB$ as diameter at $Q \\ne C$. Let $R$ be the point where $AP$, $BQ$, and the common tan... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbb5eddf985ac4e27120c293d2c87bbb7d85bc4869cc01dfe7743b249452e0aa | A point $D$ is chosen inside a non-isosceles acute-angled triangle $ABC$ such that $\angle DBC = \angle DCB = 20^\circ$. It seemed that $\angle DBA + \angle DAC = 70^\circ$. Find the angle $BAC$. | [
"70^\\circ"
] | [
"A point $D$ is chosen inside a non-isosceles acute-angled triangle $ABC$ such that $\\angle DBC = \\angle DCB = 20^\\circ$. It seemed that $\\angle DBA + \\angle DAC = 70^\\circ$. Find the angle $BAC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-cfdfbd3663e1b3fc257875638542cab655314b382526bfe2e04f8a6b19ff1abe | A point $D$ is chosen inside a scalene triangle $ABC$ such that $\angle ADB = \angle CDB + 90^{\circ}$ and $AC \cdot BD = AD \cdot BC$. Find the value of $\frac{AB \cdot CD}{AC \cdot BD}$. | [
"1"
] | [
"A point $D$ is chosen inside a scalene triangle $ABC$ such that $\\angle ADB = \\angle CDB + 90^{\\circ}$ and $AC \\cdot BD = AD \\cdot BC$. Find the value of $\\frac{AB \\cdot CD}{AC \\cdot BD}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-a8124167717e2c2ff807c2e72c322e182cb6b80f2d14c07344974fb303f393bd | A point $D$ is drawn inside an equilateral triangle $ABC$ such that the distances from $D$ to $A$, $B$, and $C$ are $8$, $7$, and $5$, respectively. Compute the side length of the triangle. | [
"\\(\\sqrt{129}\\)"
] | [
"A point $D$ is drawn inside an equilateral triangle $ABC$ such that the distances from $D$ to $A$, $B$, and $C$ are $8$, $7$, and $5$, respectively. Compute the side length of the triangle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-dcab3e884fd5a5a6be7983d62995297ae95fe377ba85c0e48465159eade12466 | A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\angle ACB$ if $\angle ABC = 40^\circ$. | [
"\\( 40^\\circ \\)"
] | [
"A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\\angle ACB$ if $\\angle ABC = 40^\\circ$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e66a8f8dfcc6592732b58c28e92d1f945d8bc3f8222f178a7e3632c900077be2 | A point $D$ is on side $BC$ of triangle $ABC$ such that $BD=20$ and $DC=14$. A circle $\Gamma$ centered at $D$ is tangent to rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$. Line $l$ is a tangent of $\Gamma$ that is parallel to $BC$, and let $l$ intersect $\overrightarrow{AB}$ at $X$ and $\overrightarrow{AC}$ at $... | [
"27"
] | [
"A point $D$ is on side $BC$ of triangle $ABC$ such that $BD=20$ and $DC=14$. A circle $\\Gamma$ centered at $D$ is tangent to rays $\\overrightarrow{AB}$ and $\\overrightarrow{AC}$. Line $l$ is a tangent of $\\Gamma$ that is parallel to $BC$, and let $l$ intersect $\\overrightarrow{AB}$ at $X$ and $\\overrightarro... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-831e086e3e0f667696fc1b22c34d91f290fae9dbf8f05ba0904982dde95d210b | A point $D$ is selected on the base $AC$ of triangle $ABC$. It is known that the distance between the centers of the circles circumscribed around triangles $ABD$ and $CBD$ is twice the length of $AC$. Find the angle between the lines $AC$ and $DB$. | [
"$\\arcsin \\frac{1}{4}$"
] | [
"A point $D$ is selected on the base $AC$ of triangle $ABC$. It is known that the distance between the centers of the circles circumscribed around triangles $ABD$ and $CBD$ is twice the length of $AC$. Find the angle between the lines $AC$ and $DB$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 12
} | 16 | 0 | 0 |
omr-ws-e2c43f9e5e87b1bfe79d2384ec8ad79f467720d076b283262e7e9b06906d73aa | A point $D$ is taken inside an isosceles triangle $ABC$ with base $AB$ and $\angle C=80^\circ$ such that $\angle DAB=10^\circ$ and $\angle DBA=20^\circ$. Compute $\angle ACD$. | [
"60°"
] | [
"A point $D$ is taken inside an isosceles triangle $ABC$ with base $AB$ and $\\angle C=80^\\circ$ such that $\\angle DAB=10^\\circ$ and $\\angle DBA=20^\\circ$. Compute $\\angle ACD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-3ec4cc7a33ed8c02ddd028514f11c30fb9b12ce55d93487c2c51a7042c618e2b | A point $D$ is taken on side $AB$ of a triangle $ABC$ between $A$ and $B$ so that $AD : AB=\alpha $ (where $\alpha <1$). Take a point $E$ on $BC$ between $B$ and $C$ so that $BE : BC=\beta $ (where $\beta <1$). Through point $E$ draw a line parallel to $AC$ and intersecting side $AB$ at $F$. Find the ratio of the areas... | [
"$(1-\\alpha )/\\beta $"
] | [
"A point $D$ is taken on side $AB$ of a triangle $ABC$ between $A$ and $B$ so that $AD : AB=\\alpha $ (where $\\alpha <1$). Take a point $E$ on $BC$ between $B$ and $C$ so that $BE : BC=\\beta $ (where $\\beta <1$). Through point $E$ draw a line parallel to $AC$ and intersecting side $AB$ at $F$. Find the ratio of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6014d9596c45ff3e00b8cb089e186d34678c8e13a4cfd0427e7843be0831532 | A point $D$ is taken on the side $AC$ in the triangle $ABC$ such that the circles inscribed in the triangles $ABD$ and $BCD$ touch each other. It is known that $| AD | = 2$, $| CD | = 4$, $| BD | = 5$. Find the radii of the circles. | [
"$\\sqrt {\\frac{3}{2}},\\ \\sqrt {\\frac{2}{3}}$"
] | [
"A point $D$ is taken on the side $AC$ in the triangle $ABC$ such that the circles inscribed in the triangles $ABD$ and $BCD$ touch each other. It is known that $| AD | = 2$, $| CD | = 4$, $| BD | = 5$. Find the radii of the circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b22f4651ce92f4dd9af0a7d21ef63bb264be6eecb2b2b8c2df63db8b6847dffa | A point $E$ is located inside a parallelogram $ABCD$ such that $\angle BAE = \angle BCE$. Let $O_1$, $O_2$, $O_3$, and $O_4$ be the centers of the circumcircles of the triangles $ABE$, $ECB$, $CDE$, and $DAE$ respectively. If the radius of the circumcircle of $ABE$ is $5$ and the circumcircles of $ABE$ and $CDE$ are ta... | [
"10"
] | [
"A point $E$ is located inside a parallelogram $ABCD$ such that $\\angle BAE = \\angle BCE$. Let $O_1$, $O_2$, $O_3$, and $O_4$ be the centers of the circumcircles of the triangles $ABE$, $ECB$, $CDE$, and $DAE$ respectively. If the radius of the circumcircle of $ABE$ is $5$ and the circumcircles of $ABE$ and $CDE$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6c86301d22b615bb21b30a6170f62b6a11a3a8e5ed178b279ae62f9b6803ab98 | A point $E$ is selected on the diagonal $BD$ of the square $ABCD$. Points $O_1$ and $O_2$ are the centers of the circles circumscribed around $\triangle ABE$ and $\triangle ADE$, respectively. If the side length of square $ABCD$ is $s$, find the length of segment $AO_1$ in terms of $s$. | [
"\\dfrac{s}{2}"
] | [
"A point $E$ is selected on the diagonal $BD$ of the square $ABCD$. Points $O_1$ and $O_2$ are the centers of the circles circumscribed around $\\triangle ABE$ and $\\triangle ADE$, respectively. If the side length of square $ABCD$ is $s$, find the length of segment $AO_1$ in terms of $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-2bdc4fbde7175966827b2e009a2846dd41f444429292f7adc2691aa06f9f84b1 | A point $E$ is taken inside the parallelogram $ABCD$ such that $AE = BC$. Points $M$ and $N$ are the midpoints of segments $AB$ and $CE$, respectively. Find the angle between lines $DE$ and $MN$. | [
"90^\\circ"
] | [
"A point $E$ is taken inside the parallelogram $ABCD$ such that $AE = BC$. Points $M$ and $N$ are the midpoints of segments $AB$ and $CE$, respectively. Find the angle between lines $DE$ and $MN$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-303187dabf02725ae59d36ff3fa6599047a60a112dde9fd41a9ba7375e8665da | A point $E$ lies in square $ABCD$. The minimum value of $\overline{EA}+\overline{EB}+\overline{EC}$ is $\sqrt{2}+\sqrt{6}$. Find the side length of square $ABCD$. | [
"\\(\\frac{4}{3}\\)"
] | [
"A point $E$ lies in square $ABCD$. The minimum value of $\\overline{EA}+\\overline{EB}+\\overline{EC}$ is $\\sqrt{2}+\\sqrt{6}$. Find the side length of square $ABCD$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae0e6d7f647efdb410a9bd297a6a9ca2ceb3486ec92e404c5676c7a7881b962c | A point $E$ lies on the extension of the side $AD$ of the rectangle $ABCD$ over $D$. The ray $EC$ meets the circumcircle $\omega$ of $ABE$ at the point $F \ne E$. The rays $DC$ and $AF$ meet at $P$. $H$ is the foot of the perpendicular drawn from $C$ to the line $\ell$ going through $E$ and parallel to $AF$. If the len... | [
"no_valid_answer_found"
] | [
"A point $E$ lies on the extension of the side $AD$ of the rectangle $ABCD$ over $D$. The ray $EC$ meets the circumcircle $\\omega$ of $ABE$ at the point $F \\ne E$. The rays $DC$ and $AF$ meet at $P$. $H$ is the foot of the perpendicular drawn from $C$ to the line $\\ell$ going through $E$ and parallel to $AF$. If... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f0968e98b5630e8067f84166d2fc1d00a2cd0600abd835c7476c60cb24700c6 | A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\triangle DBE$ is isosceles and $\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle. | [
"\\(\\sqrt{\\sqrt{5}-2}\\)"
] | [
"A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\\triangle DBE$ is isosceles and $\\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-be0c776e6944237ac4604d66b798628ab406a34a1d14a120398dfe5f0c8e7772 | A point $F$ is taken on the edge $BB_1$ of the cube $ABCDA_1B_1C_1D_1$ such that $|B_1F|=\frac{1}{3}|BB_1|$, a point $E$ is taken on the edge $C_1D_1$ such that $|D_1E|=\frac{1}{3}|C_1D_1|$. What is the largest value that the ratio $\frac{|AP|}{|PQ|}$ can assume, where the point $P$ lies on the ray $DE$ and the point $... | [
"$\\sqrt {\\frac{5}{2}}$"
] | [
"A point $F$ is taken on the edge $BB_1$ of the cube $ABCDA_1B_1C_1D_1$ such that $|B_1F|=\\frac{1}{3}|BB_1|$, a point $E$ is taken on the edge $C_1D_1$ such that $|D_1E|=\\frac{1}{3}|C_1D_1|$. What is the largest value that the ratio $\\frac{|AP|}{|PQ|}$ can assume, where the point $P$ lies on the ray $DE$ and the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a38f084f16c11f3263559d719b0214476f752890a9fb7b081b813281021d73dc | A point $K$ is chosen inside the acute triangle $ABC$ so that $BC$ is the common tangent of the circumcircles of $AKB$ and $AKC$. Let $D$ and $E$ be the intersections of the lines $CK$ and $AB$, and $BK$ and $AC$, respectively. Let $F$ be the intersection of the line $BC$ and the perpendicular bisector of the line segm... | [
"10"
] | [
"A point $K$ is chosen inside the acute triangle $ABC$ so that $BC$ is the common tangent of the circumcircles of $AKB$ and $AKC$. Let $D$ and $E$ be the intersections of the lines $CK$ and $AB$, and $BK$ and $AC$, respectively. Let $F$ be the intersection of the line $BC$ and the perpendicular bisector of the line... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48006d5f13a414c8bc0946b5811af5412de78e3910544e18fc4ac2177585b875 | A point $K$ is inside the parallelogram $ABCD$. Point $M$ is the midpoint of $BC$, and point $P$ is the midpoint of $KM$. Given that $\angle APB = \angle CPD = 90^\circ$, calculate the value of $\frac{AK^2 + DK^2}{AK \cdot DK}$. | [
"2"
] | [
"A point $K$ is inside the parallelogram $ABCD$. Point $M$ is the midpoint of $BC$, and point $P$ is the midpoint of $KM$. Given that $\\angle APB = \\angle CPD = 90^\\circ$, calculate the value of $\\frac{AK^2 + DK^2}{AK \\cdot DK}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf19ea46dd07f09f122ba141e92b1b516fbee93b7724a2a98d68bf37de914a34 | A point $M$ is arbitrarily chosen on the diagonal $AC$ of the rectangle $ABCD$. Let $O_1$ and $O_2$ be the centers of circles circumscribed about triangles $AMD$ and $CMD$, respectively. If $AB = 6$ and $BC = 8$, find the length of segment $O_1O_2$. | [
"no_valid_answer_found"
] | [
"A point $M$ is arbitrarily chosen on the diagonal $AC$ of the rectangle $ABCD$. Let $O_1$ and $O_2$ be the centers of circles circumscribed about triangles $AMD$ and $CMD$, respectively. If $AB = 6$ and $BC = 8$, find the length of segment $O_1O_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8476e0b1660950d7329b8e0fef9f9926b38564875be3ec827df5d7ebea75b219 | A point $M$ is given inside an acute angle $AOB$. Construct with a compass and straightedge on different sides of this angle, points $P$ and $Q$ in such a way that $OP + OQ = OM$ and the sum of segments $MP$ and $MQ$ is the smallest for all such pairs of points $P$ and $Q$. | [
"P \\text{ and } Q"
] | [
"A point $M$ is given inside an acute angle $AOB$. Construct with a compass and straightedge on different sides of this angle, points $P$ and $Q$ in such a way that $OP + OQ = OM$ and the sum of segments $MP$ and $MQ$ is the smallest for all such pairs of points $P$ and $Q$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-72fbf7aa3186bd1ef92684c59bd69377f0438004a620fcf524e9ecce37109f2b | A point $M$ is taken inside an angle $AOB$; $\angle MOA=\alpha $, $\angle MOB=\beta $, $OM=a$ $(\alpha +\beta <\pi )$. Find the radius of a circle passing through $M$ and cutting off, on sides $OA$ and $OB$ of the given angle, chords of length $2a$. | [
"$a\\sqrt {1+4\\frac{\\cos ^2 \\frac{\\alpha -\\beta }{2}\\sin ^2 \\frac{\\alpha +\\beta }{2}} {\\cos ^4 \\frac{\\alpha +\\beta }{2}}}$"
] | [
"A point $M$ is taken inside an angle $AOB$; $\\angle MOA=\\alpha $, $\\angle MOB=\\beta $, $OM=a$ $(\\alpha +\\beta <\\pi )$. Find the radius of a circle passing through $M$ and cutting off, on sides $OA$ and $OB$ of the given angle, chords of length $2a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eea4a65d8af7fd9685bb5408647c3b0aa6bcfb58805a09ca32246999015a3bf4 | A point $M$ is taken on a sphere of radius $R$ and from it are drawn three equal chords $MP$, $MQ$ and $MR$ so that $\angle PMQ=\angle QMR=\angle RMP=\alpha $, Find the lengths of the chords. | [
"$2/3R\\sqrt {3+6\\cos \\alpha }$"
] | [
"A point $M$ is taken on a sphere of radius $R$ and from it are drawn three equal chords $MP$, $MQ$ and $MR$ so that $\\angle PMQ=\\angle QMR=\\angle RMP=\\alpha $, Find the lengths of the chords."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-737b353ea6521f4972a72b1d8af60051bba03747469dca5ac7ef93666416ce6b | A point $M$ lies on the side $AC$ of a triangle $ABC$. The circle $\gamma$ with the diameter $BM$ intersects the lines $AB$ and $BC$ at $P$ and $Q$, respectively. Find the locus of the intersection point of the tangents to $\gamma$ at $P$ and $Q$ when point $M$ varies. | [
"\\text{The line through } C \\text{ perpendicular to } BC"
] | [
"A point $M$ lies on the side $AC$ of a triangle $ABC$. The circle $\\gamma$ with the diameter $BM$ intersects the lines $AB$ and $BC$ at $P$ and $Q$, respectively. Find the locus of the intersection point of the tangents to $\\gamma$ at $P$ and $Q$ when point $M$ varies."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4bf2b3f60dd152cbbe4a1467cbad8dbb3ec2a6409b2a2c9836c72cd3dc5db18 | A point $O$ is chosen inside a convex quadrilateral $ABCD$ not lying on the diagonal $BD$ such that $\angle ODC = \angle CAB$ and $\angle OBC = \angle CAD$. If $\angle ACB = 30^\circ$, find the measure of $\angle OCD$ in degrees. | [
"30"
] | [
"A point $O$ is chosen inside a convex quadrilateral $ABCD$ not lying on the diagonal $BD$ such that $\\angle ODC = \\angle CAB$ and $\\angle OBC = \\angle CAD$. If $\\angle ACB = 30^\\circ$, find the measure of $\\angle OCD$ in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c0ec57eef296e314c9f33aa889e25d96e4c2e05ffe86b5d98dff1384e9e65e2d | A point $O$ is chosen inside a triangle $ABC$ such that the lengths of segments $OA$, $OB$, and $OC$ are $15$, $12$, and $20$, respectively. The feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ form an equilateral triangle. Find the value of the angle $BAC$. | [
"\\( \\sin^{-1} \\left( \\frac{4}{5} \\right) \\) or approximately \\( 53.13^\\circ \\)"
] | [
"A point $O$ is chosen inside a triangle $ABC$ such that the lengths of segments $OA$, $OB$, and $OC$ are $15$, $12$, and $20$, respectively. The feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ form an equilateral triangle. Find the value of the angle $BAC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ec6d22c3abd3b56f8dbcb019bde072b81b9c11966d0e0c7e877353d35393a76 | A point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB + \angle COD = 180^{\circ}$. If $\angle OBC = x$ and $\angle ODC = y$, find the value of $x - y$. | [
"0"
] | [
"A point $O$ is situated inside the parallelogram $ABCD$ such that $\\angle AOB + \\angle COD = 180^{\\circ}$. If $\\angle OBC = x$ and $\\angle ODC = y$, find the value of $x - y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61f89f8764d9fcf9066d10cc3312a99467701e1d9611c713977eec402b4bd674 | A point $O$ is taken on the straight line $t$. 1983 unit vectors are drawn from it so that their endpoints are all on the same side of $t$. Let $\vec{S}$ be the sum of these vectors. Calculate the minimum possible value of $|\vec{S}|$. If the number of vectors is reduced to 1982, calculate the new minimum possible valu... | [
"(1, 0)"
] | [
"A point $O$ is taken on the straight line $t$. 1983 unit vectors are drawn from it so that their endpoints are all on the same side of $t$. Let $\\vec{S}$ be the sum of these vectors. Calculate the minimum possible value of $|\\vec{S}|$. If the number of vectors is reduced to 1982, calculate the new minimum possib... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75b3f75a56b5f02fe17d098b2ea60f23003c068501f2a59eef4dc3ee1beda394 | A point $O$ is taken outside the right angle with vertex $C$ on the extension of its bisector such that $| OC | = \sqrt 2$. A circle of radius 2 with center at the point $O$ is constructed. Find the area of the figure bounded by the sides of the angle and the arc of the circle contained between them. | [
"$1-\\sqrt 3+\\frac{2}{3}\\pi $"
] | [
"A point $O$ is taken outside the right angle with vertex $C$ on the extension of its bisector such that $| OC | = \\sqrt 2$. A circle of radius 2 with center at the point $O$ is constructed. Find the area of the figure bounded by the sides of the angle and the arc of the circle contained between them."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ec324de7c50c7cf1b6533c11a001d75fc1c1ac6b41803da24d85a3e67019324 | A point $O$ lying inside a regular $2n$-gon is connected to its vertices, dividing the $2n$-gon into $2n$ triangles. These triangles are then colored alternately with red and blue colors starting from any triangle. If the area of the $2n$-gon is $A$, find the absolute difference between the sum of the areas of the red ... | [
"0"
] | [
"A point $O$ lying inside a regular $2n$-gon is connected to its vertices, dividing the $2n$-gon into $2n$ triangles. These triangles are then colored alternately with red and blue colors starting from any triangle. If the area of the $2n$-gon is $A$, find the absolute difference between the sum of the areas of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2a9bbc99837d65e396e2245cf5cb36de224a10ae3c31ffb17a1c6add91d26fdd | A point $P$ and a line $k$ are given in the plane. Find the set of all lines $t$ that have the property: rotating the line $t$ around the point $P$ by an angle $a$ and reflecting the resulting line symmetrically with respect to $k$ yields a line parallel to the line $t$. | [
"\\text{All lines forming an angle of } \\frac{a}{2} \\text{ or } \\frac{\\pi}{2} - \\frac{a}{2} \\text{ with line } k"
] | [
"A point $P$ and a line $k$ are given in the plane. Find the set of all lines $t$ that have the property: rotating the line $t$ around the point $P$ by an angle $a$ and reflecting the resulting line symmetrically with respect to $k$ yields a line parallel to the line $t$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-06be9510cc41f2732b592f994d97d87337c629a49faa728c35d99ef9780a1145 | A point $P$ in the interior of a regular hexagon is at distances $8$, $8$, and $16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is the radius of the circumscribed circle of the hexagon, what is the integer closest to $r$? | [
"14"
] | [
"A point $P$ in the interior of a regular hexagon is at distances $8$, $8$, and $16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is the radius of the circumscribed circle of the hexagon, what is the integer closest to $r$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
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