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-fb45ad8961a7a258b4ba6efb0ca5ad46f7376f3b14f91465328b3f5d5069aa84 | Three men (Ram, Gopal, and Amit) and their wives (Jaya, Uma, and Sujata) each bought sheep at the market. Each person bought as many sheep as the number of rupees they paid for one sheep. Ram bought 23 more sheep than Uma, Gopal bought 11 more sheep than Jaya, and each man spent 63 rupees more than his wife. Determine ... | [
"Ram is married to Uma, Gopal is married to Jaya, and Amit is married to Sujata."
] | [
"Three men (Ram, Gopal, and Amit) and their wives (Jaya, Uma, and Sujata) each bought sheep at the market. Each person bought as many sheep as the number of rupees they paid for one sheep. Ram bought 23 more sheep than Uma, Gopal bought 11 more sheep than Jaya, and each man spent 63 rupees more than his wife. Deter... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-6492a5139feb4f7d83dad62312fc4ff6c1136236f3196e774a112082e0c317be | Three men and a monkey are shipwrecked on an island. They gather mangoes and put them in a pile. During the night, each man wakes up one by one, splits the pile into three equal parts with one mango left over (which is given to the monkey), hides one part, and combines the remaining two parts back into a pile. The next... | [
"79"
] | [
"Three men and a monkey are shipwrecked on an island. They gather mangoes and put them in a pile. During the night, each man wakes up one by one, splits the pile into three equal parts with one mango left over (which is given to the monkey), hides one part, and combines the remaining two parts back into a pile. The... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff547a24e21aaa6fa7ad6dd5910b7910a8205e822a6f730557f40a9c6ac3b853 | Three men are playing cards. In the first game, the first man loses to the other two men an amount equal to what each of them already owns. In the second game, the second man loses to the other two men an amount equal to what each of them already owns. In the third game, the first and second men each gain from the thir... | [
"First man had \\$39, Second man had \\$21, Third man had \\$12"
] | [
"Three men are playing cards. In the first game, the first man loses to the other two men an amount equal to what each of them already owns. In the second game, the second man loses to the other two men an amount equal to what each of them already owns. In the third game, the first and second men each gain from the... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-606c2b6a2a96b00dc94389707655e9bcfe0401221ef3b4f28996acd4e1457347 | Three men, $A$, $B$, and $C$, walk in the same direction in the circumferences of three concentric circles, starting simultaneously from points where they are at their least distances from each other. $A$ walks his circuit in an even number of hours (greater than four), $B$ walks his circuit in one hour, and $C$ walks ... | [
"\\dfrac{1}{10}"
] | [
"Three men, $A$, $B$, and $C$, walk in the same direction in the circumferences of three concentric circles, starting simultaneously from points where they are at their least distances from each other. $A$ walks his circuit in an even number of hours (greater than four), $B$ walks his circuit in one hour, and $C$ w... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-99c6bded40562555e075b53ef03ced4cbdd0365a4467556d466ed53bdfd4224b | Three men, Atkins, Brown, and Cranby, had to go a journey of forty miles. Atkins could walk one mile an hour, Brown could walk two miles an hour, and Cranby could go in his donkey cart at eight miles an hour. They start at the same time. Cranby drove Atkins a certain distance, and dropping him to walk the remainder, dr... | [
"\\dfrac{415}{41}"
] | [
"Three men, Atkins, Brown, and Cranby, had to go a journey of forty miles. Atkins could walk one mile an hour, Brown could walk two miles an hour, and Cranby could go in his donkey cart at eight miles an hour. They start at the same time. Cranby drove Atkins a certain distance, and dropping him to walk the remainde... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e9ffafc5ef507c2601202868ae02b3b5364afb6e55ef62234964db4a6130f3b7 | Three monkeys randomly toss 12 ping pong balls numbered consecutively from 1 through 12 into three containers labeled A, B, and C. Each ball goes into one of the containers and is equally likely to end up in any one of the containers. Find the number of distinct outcomes in which all containers have the same number of ... | [
"34650"
] | [
"Three monkeys randomly toss 12 ping pong balls numbered consecutively from 1 through 12 into three containers labeled A, B, and C. Each ball goes into one of the containers and is equally likely to end up in any one of the containers. Find the number of distinct outcomes in which all containers have the same numbe... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d274f3bd71dc098894c273d71320c7adc1dba5fba21576fe6d9ba000d1b67167 | Three mothers each with their one child are called for an interview for admission of their child. The principal proposes to meet only one (either child or mother at a time). Find the probability that no child is called before her mother if the calling process is random. | [
"\\(\\frac{1}{8}\\)"
] | [
"Three mothers each with their one child are called for an interview for admission of their child. The principal proposes to meet only one (either child or mother at a time). Find the probability that no child is called before her mother if the calling process is random."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f9e78f1881353d488067f7487bb4b37868b5e7c212d09e2873515967fda78ea1 | Three motorcyclists started their rides at the same time and returned home at the same time. The first motorcyclist travelled twice as much as the second motorcyclist rested, the second motorcyclist travelled three times as much as the third motorcyclist rested, and the third motorcyclist travelled four times as much a... | [
"32"
] | [
"Three motorcyclists started their rides at the same time and returned home at the same time. The first motorcyclist travelled twice as much as the second motorcyclist rested, the second motorcyclist travelled three times as much as the third motorcyclist rested, and the third motorcyclist travelled four times as m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-f5b8921bb10e937864041494d649464325f74812f4e5a9457de97d3382a367a1 | Three movie critics are ranking \(2n+1\) movies, each giving marks from 0 to \(2n\). The sum of all marks given by the three critics for each movie is called the 'total marks' for that movie. Let \(S\) be the set of all distinct total marks for the movies. If each movie is more popular than exactly \(n\) other movies, ... | [
"3n(2n+1)"
] | [
"Three movie critics are ranking \\(2n+1\\) movies, each giving marks from 0 to \\(2n\\). The sum of all marks given by the three critics for each movie is called the 'total marks' for that movie. Let \\(S\\) be the set of all distinct total marks for the movies. If each movie is more popular than exactly \\(n\\) o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d0292786f938124832463039bec6dac45afb45666acc0e147c7b794d3d89050 | Three mutually and externally disjoint circles centered at \(X_1\), \(X_2\), and \(X_3\) respectively are on the same plane. The two internal common tangents of Circle 2 and Circle 3, Circle 3 and Circle 1, and Circle 1 and Circle 2 meet at points \(P\), \(Q\), and \(R\) respectively. If the coordinates of \(X_1\), \(X... | [
"\\left( \\dfrac{32}{11}, \\dfrac{20}{11} \\right)"
] | [
"Three mutually and externally disjoint circles centered at \\(X_1\\), \\(X_2\\), and \\(X_3\\) respectively are on the same plane. The two internal common tangents of Circle 2 and Circle 3, Circle 3 and Circle 1, and Circle 1 and Circle 2 meet at points \\(P\\), \\(Q\\), and \\(R\\) respectively. If the coordinate... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-295eca1367f2008e064180be44c618d59320b2f8e2a162bdc86b82bb3c5bc197 | Three mutually distinct real numbers \(a, b, c\) satisfy the equations: \[ a = (b-2)c, \] \[ b = (c-2)a, \] \[ c = (a-2)b. \] Compute the product \(abc\). | [
"3"
] | [
"Three mutually distinct real numbers \\(a, b, c\\) satisfy the equations:\n\\[ a = (b-2)c, \\]\n\\[ b = (c-2)a, \\]\n\\[ c = (a-2)b. \\]\nCompute the product \\(abc\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-b6f134b3cc48465ac654f2028683e1079bfb3f9fad2c21effcfc918f3c37a005 | Three mutually perpendicular rays \(O_x, O_y, O_z\) and three points \(A, B, C\) on \(O_x, O_y, O_z\), respectively. A variable sphere through \(A, B, C\) meets \(O_x, O_y, O_z\) again at \(A', B', C'\), respectively. Let \(M\) and \(M'\) be the centroids of triangles \(ABC\) and \(A'B'C'\). Find the locus of the midpo... | [
"6ax - a^2 = 6by - b^2 = 6cz - c^2"
] | [
"Three mutually perpendicular rays \\(O_x, O_y, O_z\\) and three points \\(A, B, C\\) on \\(O_x, O_y, O_z\\), respectively. A variable sphere through \\(A, B, C\\) meets \\(O_x, O_y, O_z\\) again at \\(A', B', C'\\), respectively. Let \\(M\\) and \\(M'\\) be the centroids of triangles \\(ABC\\) and \\(A'B'C'\\). Fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c74ea55727c23dae39a1848801ac188335f4c2a3f910656051ed4cc420927857 | Three mutually tangent spheres each with radius 5 sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length 6, has three congruent isosceles triangles for vertical faces, and has height 12. The base of the pyramid is parallel to the plane, and the vertex of the pyramid ... | [
"\\( m + n = 44 + 7 = 51 \\)."
] | [
"Three mutually tangent spheres each with radius 5 sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length 6, has three congruent isosceles triangles for vertical faces, and has height 12. The base of the pyramid is parallel to the plane, and the vertex of the pyr... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-2ebdf64f8be81baeb452a1b332323c481eccdf8be71a5a3ce37485ebed7ced17 | Three mutually tangent spheres of radius 1 rest on a horizontal plane. Another sphere of radius 1 rests on top of them. What is the distance from the plane to the top of the upper sphere? | [
"\\(\\frac{2\\sqrt{6}}{3} + 2\\)"
] | [
"Three mutually tangent spheres of radius 1 rest on a horizontal plane. Another sphere of radius 1 rests on top of them. What is the distance from the plane to the top of the upper sphere?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-73a17ef06e76bcb1d1e584131703c91f22184c12969367da34f539d52adb91f3 | Three mutually tangent spheres with radii of $1$ are tangent to a table, and a cone is tangent to all three spheres with its tip oriented towards the table. If the cone has height $\sqrt{2}$ and its tip is $\tfrac{3+\sqrt{3}}{3}$ units above the table, compute the radius of the cone. | [
"\\(\\sqrt{2} + \\sqrt{3}\\)"
] | [
"Three mutually tangent spheres with radii of $1$ are tangent to a table, and a cone is tangent to all three spheres with its tip oriented towards the table. If the cone has height $\\sqrt{2}$ and its tip is $\\tfrac{3+\\sqrt{3}}{3}$ units above the table, compute the radius of the cone."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-1aeb282a92979965835355aca6fb27ee9b03276180b2847f48cdc43c420662a5 | Three mutually tangent unit spheres (radius 1) are placed on a surface. A soccer ball of unknown radius is placed on top of them. The distance from the surface to the top of the soccer ball is in the form \( 8 + \frac{2 \sqrt{m}}{3} \) where \( m \) is not divisible by any perfect square greater than 1. What is the rem... | [
"3"
] | [
"Three mutually tangent unit spheres (radius 1) are placed on a surface. A soccer ball of unknown radius is placed on top of them. The distance from the surface to the top of the soccer ball is in the form \\( 8 + \\frac{2 \\sqrt{m}}{3} \\) where \\( m \\) is not divisible by any perfect square greater than 1. What... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-f95cd47d3856b2284b3726762c9bc2cdabc45272150d3befbfff9e8141884d0b | Three natural numbers $a$, $b$, and $c$ are given such that each of them is divisible by the greatest common divisor of the remaining two and also divides the least common multiple of the remaining two. If $a \leq b \leq c$, find the value of $\frac{c}{a}$ when $a$, $b$, and $c$ are the smallest possible distinct numbe... | [
"no_valid_answer_found"
] | [
"Three natural numbers $a$, $b$, and $c$ are given such that each of them is divisible by the greatest common divisor of the remaining two and also divides the least common multiple of the remaining two. If $a \\leq b \\leq c$, find the value of $\\frac{c}{a}$ when $a$, $b$, and $c$ are the smallest possible distin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbe79596b5f91e21d233d88a63f532f6190b63485a3eb78ac5296071426b5137 | Three natural numbers greater than or equal to $2$ are written, not necessarily different. A sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the second-to-last one is $b$, and the last one is $c$, the next number $x$ is written such that $$x \cdot c = a + b... | [
"Any natural numbers greater than or equal to 2."
] | [
"Three natural numbers greater than or equal to $2$ are written, not necessarily different. A sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the second-to-last one is $b$, and the last one is $c$, the next number $x$ is written such that $$x \\cdot c =... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93cae44dab2c3bb9f60c9863738aa4f914ac969a2c03c11c80dae3ddff651b7d | Three natural numbers that are consecutive members of an arithmetic progression were written on the board. After erasing the commas separating the numbers, a seven-digit number was formed. What is the largest possible seven-digit number that could result? | [
"9995049"
] | [
"Three natural numbers that are consecutive members of an arithmetic progression were written on the board. After erasing the commas separating the numbers, a seven-digit number was formed. What is the largest possible seven-digit number that could result?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 5,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-86ff4369be893d01dbf6c2a0f82160f10931df4d9424315aaed8d8ed2cdb6211 | Three non-collinear points \( A, B, C \) exist in the same plane with coordinates \( A = (1,1) \), \( B = (2,3) \), and \( C = (3,1) \). Point \( P \) internally divides \( AB \) in the ratio \( t : (1-t) \), and point \( Q \) internally divides \( BC \) in the same ratio \( t : (1-t) \). Point \( R \) divides \( PQ \)... | [
"The locus of \\( R \\) is a parabola given by the parametric equations \\( x = 2t + 1 \\) and \\( y = 3t + 1 - 4t^2 \\)."
] | [
"Three non-collinear points \\( A, B, C \\) exist in the same plane with coordinates \\( A = (1,1) \\), \\( B = (2,3) \\), and \\( C = (3,1) \\). Point \\( P \\) internally divides \\( AB \\) in the ratio \\( t : (1-t) \\), and point \\( Q \\) internally divides \\( BC \\) in the same ratio \\( t : (1-t) \\). Point... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 14
} | 26 | 0 | 0 |
omr-ws-164b624bdfb09b77e76407a49eb950c3ec53d1b1dc7d01184993cf637f7e5ad7 | Three non-collinear points are randomly picked from the sides of a square such that they form a triangle. What is the probability that the circumcircle of the triangle has a greater area than the square? | [
"\\dfrac{1}{4}"
] | [
"Three non-collinear points are randomly picked from the sides of a square such that they form a triangle. What is the probability that the circumcircle of the triangle has a greater area than the square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-3bc374b4bab03fbe9b9cc920e75f42eabf8bc5889bae1c5ba8b8807b4905a255 | Three non-coplanar circles are given in the Euclidean space so that they are pairwise tangent. Let $r_1$, $r_2$, and $r_3$ be the radii of these circles, and let $R$ be the radius of a sphere that passes through all three circles. If $r_1 = 2$, $r_2 = 3$, and $r_3 = 6$, find the value of $R$. | [
"12"
] | [
"Three non-coplanar circles are given in the Euclidean space so that they are pairwise tangent. Let $r_1$, $r_2$, and $r_3$ be the radii of these circles, and let $R$ be the radius of a sphere that passes through all three circles. If $r_1 = 2$, $r_2 = 3$, and $r_3 = 6$, find the value of $R$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5601c2c60e1c6de7d136a06756362849e925b32b6c8e18f3199e545430839242 | Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^\circ$. Thus, the three polygons form a new polygon with $A$ as an interior point. What is the largest ... | [
"21"
] | [
"Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^\\circ$. Thus, the three polygons form a new polygon with $A$ as an interior point. What is the la... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-109eee692903a2fc433fcf157f83260f04af13f9e27dccb639b767b903b076d5 | Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ such that the sum of the three interior angles at $A$ is $360^{\circ}$. What is the largest possible perimeter of the new polygon formed with $A$ as an interior point? | [
"21"
] | [
"Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ such that the sum of the three interior angles at $A$ is $360^{\\circ}$. What is the largest possible perimeter of the new polygon formed with $A$ as an interior point?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-429386950704668102e07bc10a84a573f4d213c3d6761385618d9f0b440d28cb | Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length 1. The polygons meet at a point \( A \) in such a way that the sum of the three interior angles at \( A \) is \( 360^\circ \). Thus, the three polygons form a new polygon with \( A \) as an interior point. What i... | [
"21"
] | [
"Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length 1. The polygons meet at a point \\( A \\) in such a way that the sum of the three interior angles at \\( A \\) is \\( 360^\\circ \\). Thus, the three polygons form a new polygon with \\( A \\) as an interior... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-6abfdaca24b1aafaa63631e049f7ffa167bc5f66c76b4dfea0bfa06676f13154 | Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length 1. The polygons meet at point A such that the sum of the three interior angles at A is 360 degrees, forming a new polygon with A as an interior point. What is the largest possible perimeter of this new polygon? | [
"21"
] | [
"Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length 1. The polygons meet at point A such that the sum of the three interior angles at A is 360 degrees, forming a new polygon with A as an interior point. What is the largest possible perimeter of this new polyg... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 5
} | 27 | 0 | 0 |
omr-ws-d6a5bbc5365ce1dbca805a7c91453d34dbe62688fbcef9d36868791a31e37a36 | Three non-overlapping squares of integer side lengths each have one vertex at the origin and sides parallel to the coordinate axes. Together, the three squares enclose a region whose area is $41$. Compute the largest possible perimeter of the region. | [
"32"
] | [
"Three non-overlapping squares of integer side lengths each have one vertex at the origin and sides parallel to the coordinate axes. Together, the three squares enclose a region whose area is $41$. Compute the largest possible perimeter of the region."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 3
} | 6 | 0 | 0 |
omr-ws-bcc6f693a4625968ddc6103cfa294e14a016f7e66faba33d18cd9570af8a5677 | Three non-zero real numbers \(a, b, c\) satisfy \(a + b + c = 0\) and \(a^4 + b^4 + c^4 = 128\). Determine all possible values of \(ab + bc + ca\). | [
"-8"
] | [
"Three non-zero real numbers \\(a, b, c\\) satisfy \\(a + b + c = 0\\) and \\(a^4 + b^4 + c^4 = 128\\). Determine all possible values of \\(ab + bc + ca\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9219d7031ab7298753b461feaf66a6d5b3f5131666d04f38ad5d8e3b5438f81 | Three nonnegative reals \( x, y, z \) satisfy \( x + y + z = 12 \) and \( xy + yz + zx = 21 \). Find the maximum value of \( xyz \). | [
"10"
] | [
"Three nonnegative reals \\( x, y, z \\) satisfy \\( x + y + z = 12 \\) and \\( xy + yz + zx = 21 \\). Find the maximum value of \\( xyz \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-2de6a70afa48596becd9bbd2137148c5ccd63f6d42ae459890584a9750f23cd9 | Three nonnegative reals \( x, y, z \) satisfy \( x + y + z = 15 \) and \( xy + yz + zx = 28 \). Find the maximum of \( xyz \). | [
"\\dfrac{94\\sqrt{141} - 990}{9}"
] | [
"Three nonnegative reals \\( x, y, z \\) satisfy \\( x + y + z = 15 \\) and \\( xy + yz + zx = 28 \\). Find the maximum of \\( xyz \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-aa606052dee6022e3b223c4e7ca84862a303384599240fcfa140c8534fb4acad | Three nonnegative reals \(x, y, z\) satisfy \(x + y + z = 12\) and \(xy + yz + zx = 21\). Find the maximum of \(xyz\). | [
"10"
] | [
"Three nonnegative reals \\(x, y, z\\) satisfy \\(x + y + z = 12\\) and \\(xy + yz + zx = 21\\). Find the maximum of \\(xyz\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd063cf892a68662ba13f2d56aa3b5a8fd26848bc7ec1d6593ae503df05cf7dd | Three nonzero real numbers form an arithmetic progression and the squares of these numbers taken in the same order constitute a geometric progression. Find all possible common ratios of the geometric progression. | [
"$3-2\\sqrt 2$, 1, $3+2\\sqrt 2$"
] | [
"Three nonzero real numbers form an arithmetic progression and the squares of these numbers taken in the same order constitute a geometric progression. Find all possible common ratios of the geometric progression."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abe0e01e47b5afb4b0825b87397b7799e5186a7f77e0a9fa25e78c064760068a | Three normal dice are piled on top of each other so that the total number of pips on two faces placed together is always 5. One of the visible faces on the bottom die has one pip. How many pips are on the top face of the top die? | [
"6"
] | [
"Three normal dice are piled on top of each other so that the total number of pips on two faces placed together is always 5. One of the visible faces on the bottom die has one pip. How many pips are on the top face of the top die?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 7
} | 38 | 0 | 0 |
omr-ws-f0fbabdb1b077b566a435114c34bb9a7061fd620d0c47fe6f085585c27deda05 | Three normals are drawn to the parabola $y^2 = 4ax\cos\alpha$ from any point on the line $y = b\sin\alpha$. Find the locus of the orthocenter of the triangle formed by the corresponding tangents at the points of contact of the normals. | [
"\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1"
] | [
"Three normals are drawn to the parabola $y^2 = 4ax\\cos\\alpha$ from any point on the line $y = b\\sin\\alpha$. Find the locus of the orthocenter of the triangle formed by the corresponding tangents at the points of contact of the normals."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-799c5c607d821de7932c926a3fc85f20d9813f19584b521fd386b4727782f23b | Three numbers $a, b, c$ are generated uniformly in the interval $[0, 1]$. What is the probability that the line with equation $a x + b y = c$ intersects the unit circle $x^2 + y^2 = 1$? | [
"1 - \\dfrac{\\pi}{12}"
] | [
"Three numbers $a, b, c$ are generated uniformly in the interval $[0, 1]$. What is the probability that the line with equation $a x + b y = c$ intersects the unit circle $x^2 + y^2 = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f613647bac279ee4d5f2852ed545d15fbb1e12fbfd96dc2aaff36ff6f7c81c5 | Three numbers $a_1, a_2, a_3$ are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers $b_1, b_2, b_3$ are then drawn randomly and without replacement from the remaining set of 997 numbers. Compute the probability that, after a suitable rotation, a brick of dimensions $a_1... | [
"\\(\\frac{1}{4}\\)"
] | [
"Three numbers $a_1, a_2, a_3$ are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 1000\\}$. Three other numbers $b_1, b_2, b_3$ are then drawn randomly and without replacement from the remaining set of 997 numbers. Compute the probability that, after a suitable rotation, a brick of dimensi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-59cd94ac80b4bcb1e23ad824c6827b34de6e3b055d5fc858ad3b242b6d3e0a07 | Three numbers $x, y, z$ are selected at random, with replacement, from the set $\{-3,-2,-1, 1, 2, 3\}$. What is the probability that all three of the numbers $\frac{x}{y},\frac{y}{z},\frac{2z}{x}$ are integers? | [
"\\dfrac{5}{27}"
] | [
"Three numbers $x, y, z$ are selected at random, with replacement, from the set $\\{-3,-2,-1, 1, 2, 3\\}$. What is the probability that all three of the numbers $\\frac{x}{y},\\frac{y}{z},\\frac{2z}{x}$ are integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-17745e47e3878cf7a0df706527603e85b9aea4eddf7695402ef649b87b01e0f2 | Three numbers \( N, n, r \) are such that the digits of \( N, n, r \) taken together are formed by \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \) without repetition. If \( N = n^2 - r \), find all possible combinations of \( N, n, r \). | [
"\\( N = 1397, n = 45, r = 628 \\)"
] | [
"Three numbers \\( N, n, r \\) are such that the digits of \\( N, n, r \\) taken together are formed by \\( 1, 2, 3, 4, 5, 6, 7, 8, 9 \\) without repetition. If \\( N = n^2 - r \\), find all possible combinations of \\( N, n, r \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-26c281f94cac31557724fce153466edf98ac93198d0a43222856f3fffc41217f | Three numbers \( x, y, \) and \( z \) are chosen randomly from the range \([0, 2021]\). What is the probability that the square of one of the numbers is greater than the sum of the squares of the other two numbers? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"Three numbers \\( x, y, \\) and \\( z \\) are chosen randomly from the range \\([0, 2021]\\). What is the probability that the square of one of the numbers is greater than the sum of the squares of the other two numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-adcdbb9cf627145c3d057dde177819ab6ab85c1e4c3d2d7845f423f337448ad8 | Three numbers \( x, y, z \in \mathbb{N} \) are chosen at random such that \( 1 \le x, y, z \le 10 \). What is the probability that their product is divisible by 9? | [
"\\(\\frac{363}{1000}\\)"
] | [
"Three numbers \\( x, y, z \\in \\mathbb{N} \\) are chosen at random such that \\( 1 \\le x, y, z \\le 10 \\). What is the probability that their product is divisible by 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-c6424cdb9f73a9627333068f08e3ed46a692c0be61ffe29bd4fbccdfd11e2209 | Three numbers \( x_1, x_2, x_3 \) are chosen from the set \(\{1, 2, \ldots, 2000\}\) and removed without replacement. Then three numbers \( y_1, y_2, y_3 \) are chosen. What is the probability that \( y_i > x_i \) for all \( i \)? | [
"\\dfrac{1}{8}"
] | [
"Three numbers \\( x_1, x_2, x_3 \\) are chosen from the set \\(\\{1, 2, \\ldots, 2000\\}\\) and removed without replacement. Then three numbers \\( y_1, y_2, y_3 \\) are chosen. What is the probability that \\( y_i > x_i \\) for all \\( i \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-73738ffc5bf37d13f8e0261648a10027d4f8e434e92f6c978c12605ead69ebb3 | Three numbers \(a < b < c\) are chosen from the set \(\{1, 18, 27, 44, 67, 87, 98, 103, 312\}\). What is the expected value of \(2a + 9b - c\)? | [
"\\dfrac{9670}{21}"
] | [
"Three numbers \\(a < b < c\\) are chosen from the set \\(\\{1, 18, 27, 44, 67, 87, 98, 103, 312\\}\\). What is the expected value of \\(2a + 9b - c\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a597776233339d3db11ef58f018ca2ec5ccc57f1b656526a97c0a1d390a1913 | Three numbers \(a, b, c\), none zero, form an arithmetic progression. Increasing \(a\) by 1 or increasing \(c\) by 2 results in a geometric progression. Then \(b\) equals: | [
"12"
] | [
"Three numbers \\(a, b, c\\), none zero, form an arithmetic progression. Increasing \\(a\\) by 1 or increasing \\(c\\) by 2 results in a geometric progression. Then \\(b\\) equals:"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71a0633ce6abe6f981e15940e5db6d20ec77f684f75343f9e418a9ee9bf76d87 | Three numbers \(a\), \(b\), and \(c\) are chosen uniformly and independently at random from the interval \([0,1]\) and arranged such that \(a \leq b \leq c\). What is the probability that \(4a + 3b + 2c \leq 1\)? | [
"\\dfrac{1}{90}"
] | [
"Three numbers \\(a\\), \\(b\\), and \\(c\\) are chosen uniformly and independently at random from the interval \\([0,1]\\) and arranged such that \\(a \\leq b \\leq c\\). What is the probability that \\(4a + 3b + 2c \\leq 1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-edd5be7f6aee8220b4322498f3d4f89b41bdb56dcb250676cc32c6863a5ea347 | Three numbers \(a\), \(b\), and \(c\) are in arithmetic progression (AP). A 9-digit number is formed using each of these digits exactly three times, such that at least three consecutive digits are in AP. Find the number of such 9-digit numbers. | [
"1668"
] | [
"Three numbers \\(a\\), \\(b\\), and \\(c\\) are in arithmetic progression (AP). A 9-digit number is formed using each of these digits exactly three times, such that at least three consecutive digits are in AP. Find the number of such 9-digit numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-e4940c5188d431e5b1af3b0792aebd644e8911a92c19503e4231d49c7758f682 | Three numbers \(a\), \(b\), and \(c\) are selected from the interval \([0, 1]\) with \(a \ge b \ge c\). Find the probability that \(4a + 3b + 2c \ge 1\). | [
"\\(\\frac{251}{252}\\)"
] | [
"Three numbers \\(a\\), \\(b\\), and \\(c\\) are selected from the interval \\([0, 1]\\) with \\(a \\ge b \\ge c\\). Find the probability that \\(4a + 3b + 2c \\ge 1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-30a678878124fe3c99e7b1262ffb890689790f49d05b42703dbb49b7216c04c3 | Three numbers \(a\), \(b\), and \(c\) satisfy the equations \(ab + c = 10\), \(bc + a = 11\), and \(ca + b = 14\). Find the value of \((a - b)(b - c)(c - a)(a - 1)(b - 1)(c - 1)\). | [
"-12"
] | [
"Three numbers \\(a\\), \\(b\\), and \\(c\\) satisfy the equations \\(ab + c = 10\\), \\(bc + a = 11\\), and \\(ca + b = 14\\). Find the value of \\((a - b)(b - c)(c - a)(a - 1)(b - 1)(c - 1)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-6324abf466340a3088b84844a94ffbf40d10fd288ed299bc2b33147c3e8f79c2 | Three numbers \(a\), \(b\), and \(c\), none of them zero, form an arithmetic progression. Increasing \(a\) by 1 or increasing \(c\) by 2 results in a geometric progression. Find the value of \(b\). | [
"12"
] | [
"Three numbers \\(a\\), \\(b\\), and \\(c\\), none of them zero, form an arithmetic progression. Increasing \\(a\\) by 1 or increasing \\(c\\) by 2 results in a geometric progression. Find the value of \\(b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ceea6674d231833dee3f9e4e4480e23aed998c2794c53cbc258514290fc7723 | Three numbers \(a\), \(b\), and \(c\), none zero, form an arithmetic progression. Increasing \(a\) by 1 or increasing \(c\) by 2 results in a geometric progression. Find \(b\). | [
"12"
] | [
"Three numbers \\(a\\), \\(b\\), and \\(c\\), none zero, form an arithmetic progression. Increasing \\(a\\) by 1 or increasing \\(c\\) by 2 results in a geometric progression. Find \\(b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09dc293d0503109cd7a8b67fb39414d6e404471ed92ea3a1ee4253e623bfd0df | Three numbers \(a_1, a_2, a_3\) are drawn at random, without replacement, from the set \(\{1, 2, 3, \ldots, 1000\}\). Three other numbers \(b_1, b_2, b_3\) are then drawn randomly and without replacement from the remaining set of 997 numbers. Find the probability that a brick with dimensions \(a_1 \times a_2 \times a_3... | [
"\\(\\frac{1}{4}\\)"
] | [
"Three numbers \\(a_1, a_2, a_3\\) are drawn at random, without replacement, from the set \\(\\{1, 2, 3, \\ldots, 1000\\}\\). Three other numbers \\(b_1, b_2, b_3\\) are then drawn randomly and without replacement from the remaining set of 997 numbers. Find the probability that a brick with dimensions \\(a_1 \\time... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-cd68cdbe3d50a73203644fc5be67a615d73c77a011dc287f133907d5f4f35b2c | Three numbers \(a_1, a_2, a_3\) are drawn randomly and without replacement from the set \(\{1, 2, 3, \ldots, 1000\}\). Three other numbers \(b_1, b_2, b_3\) are then drawn randomly and without replacement from the remaining set of 997 numbers. Let \(P\) be the probability that, after suitable rotation, a brick of dimen... | [
"5"
] | [
"Three numbers \\(a_1, a_2, a_3\\) are drawn randomly and without replacement from the set \\(\\{1, 2, 3, \\ldots, 1000\\}\\). Three other numbers \\(b_1, b_2, b_3\\) are then drawn randomly and without replacement from the remaining set of 997 numbers. Let \\(P\\) be the probability that, after suitable rotation, ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-575070d43cdba3b2511d108071cecbf0bf4f4e67ea4712ef6dd249b1ec99c4cf | Three numbers \(a_1, a_2, a_3\) are drawn randomly and without replacement from the set \(\{1, 2, 3, \ldots, 1000\}\). Three other numbers \(b_1, b_2, b_3\) are then drawn randomly and without replacement from the remaining set of 997 numbers. Let \(p\) be the probability that, after a suitable rotation, a brick of dim... | [
"5\n\nExplanation: The probability \\( p \\) is found to be \\( \\frac{1}{4} \\). When written as a fraction in lowest terms, the numerator is 1 and the denominator is 4. The sum of the numerator and denominator is \\( 1 + 4 = 5 \\)."
] | [
"Three numbers \\(a_1, a_2, a_3\\) are drawn randomly and without replacement from the set \\(\\{1, 2, 3, \\ldots, 1000\\}\\). Three other numbers \\(b_1, b_2, b_3\\) are then drawn randomly and without replacement from the remaining set of 997 numbers. Let \\(p\\) be the probability that, after a suitable rotation... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 43
} | 43 | 0 | 0 |
omr-ws-2bb3aa722e25516bab00d478f993f0b8774ffd8dbc1fbcf948c9ac8cbfc5244d | Three numbers \(a_1\), \(a_2\), and \(a_3\) are chosen without replacement from the set \(\{1, 2, 3, \ldots, 99, 100\}\). Three other numbers \(b_1\), \(b_2\), and \(b_3\) are chosen without replacement from the remaining 97 numbers. What is the probability that after some rotations, a brick with dimensions \(a_1 \time... | [
"\\dfrac{1}{4}"
] | [
"Three numbers \\(a_1\\), \\(a_2\\), and \\(a_3\\) are chosen without replacement from the set \\(\\{1, 2, 3, \\ldots, 99, 100\\}\\). Three other numbers \\(b_1\\), \\(b_2\\), and \\(b_3\\) are chosen without replacement from the remaining 97 numbers. What is the probability that after some rotations, a brick with ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71017471f9dddb3e86253152c7118c90778968a785f3442c87b51eca39507229 | Three numbers are chosen at random between $0$ and $1$. What is the probability that the difference between the greatest and the least is less than $\frac{1}{3}$? | [
"\\dfrac{7}{27}"
] | [
"Three numbers are chosen at random between $0$ and $1$. What is the probability that the difference between the greatest and the least is less than $\\frac{1}{3}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffbeef53cefb8c6d598ac60ccc80114d068db01665a38b3edbe9720a67a66ebf | Three numbers are chosen at random between $0$ and $1$. What is the probability that the greatest minus the least is less than $\frac{1}{3}$? | [
"\\(\\frac{7}{27}\\)"
] | [
"Three numbers are chosen at random between $0$ and $1$. What is the probability that the greatest minus the least is less than $\\frac{1}{3}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-5aa190693b48043e6c8d40fb44f318a8ffc3895add1622525c096c842fc66d4d | Three numbers are chosen at random between $0$ and $2$. What is the probability that the difference between the greatest and least number is less than $\frac{1}{4}$? | [
"\\dfrac{11}{256}"
] | [
"Three numbers are chosen at random between $0$ and $2$. What is the probability that the difference between the greatest and least number is less than $\\frac{1}{4}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-efab6b07468fe6bdfe4ec60eb56909f2f8f3840eb6cc2888857bec168c0f41b4 | Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least is less than 1/3? | [
"\\dfrac{7}{27}"
] | [
"Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least is less than 1/3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8106438063cbbe54dcd6ae10ef23669fd657eb81560953ea8a97982a959bc34c | Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least is less than \( \frac{1}{3} \)? | [
"\\dfrac{7}{27}"
] | [
"Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least is less than \\( \\frac{1}{3} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c2f415d42a11a1e44ee8777c63baf022b7772d0a6006b72d798fbfc71d68a63 | Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least of these numbers is less than $\frac{1}{3}$? | [
"\\(\\frac{7}{27}\\)"
] | [
"Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least of these numbers is less than $\\frac{1}{3}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-29da03fa6b336c3f3723bbe110c22d93e3b188e6e174c905e148d6120d38f48d | Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least of these numbers is less than 1/3? | [
"7/27",
"\\(\\frac{7}{27}\\)"
] | [
"Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least of these numbers is less than 1/3?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 62,
"tir": 61
} | 123 | 0 | 0 |
omr-ws-e4eae85d76652dedd493f8c4279f3f7b5e9bdd23bc62f297e0110b4dda8a67c2 | Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least of these numbers is less than one third? | [
"\\(\\frac{7}{27}\\)"
] | [
"Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the least of these numbers is less than one third?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-af761af4b9f01a434e77a85cefc49011458592f2e8f10df12b2368f69b510f2f | Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the smallest of these numbers is less than 1/3? | [
"\\dfrac{7}{27}"
] | [
"Three numbers are chosen at random between 0 and 1. What is the probability that the difference between the greatest and the smallest of these numbers is less than 1/3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04ce51cd18365b74754da80cee4798a979e26a369bfeaf755f7d63217900cfa2 | Three numbers are chosen at random between 0 and 2. What is the probability that the difference between the greatest and the least of these numbers is less than 1/4? | [
"\\(\\frac{11}{256}\\)"
] | [
"Three numbers are chosen at random between 0 and 2. What is the probability that the difference between the greatest and the least of these numbers is less than 1/4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 27
} | 40 | 0 | 0 |
omr-ws-ab15b8f64b9e41e30e31356275f4c9d0d4314016785f8b7d370122d0372c956b | Three numbers are chosen at random between \(0\) and \(2\). What is the probability that the difference between the greatest and least is less than \(1/4\)? | [
"\\dfrac{11}{256}"
] | [
"Three numbers are chosen at random between \\(0\\) and \\(2\\). What is the probability that the difference between the greatest and least is less than \\(1/4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01767fa1b2d3119504a347b5faaad5b08d9671932a47430c40a55a5ba49292f4 | Three numbers are chosen at random from the set $\{0, 1, 2, \ldots, 29\}$ without replacement. What is the probability that the sum of these numbers is $30$? | [
"$\\frac{15}{812}$"
] | [
"Three numbers are chosen at random from the set $\\{0, 1, 2, \\ldots, 29\\}$ without replacement. What is the probability that the sum of these numbers is $30$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-807af6dc1907f08f213991a34066d36af9f5a1a29a55bf6eb720cc066e487bf8 | Three numbers are chosen at random in the interval \((0, 1)\). What is the probability that the difference between the least and greatest numbers (the numbers can be the same) is less than \(\frac{1}{3}\)? | [
"\\dfrac{7}{27}"
] | [
"Three numbers are chosen at random in the interval \\((0, 1)\\). What is the probability that the difference between the least and greatest numbers (the numbers can be the same) is less than \\(\\frac{1}{3}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-81ef7b1e7a04f874afb6480f8093d602571c11c7160d7ef39d0490a8b7ccf855 | Three numbers are chosen independently at random, one from each of the three intervals $[0, L_1]$, $[0, L_2]$, and $[0, L_3]$. If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen. | [
"\\int_{0}^{\\min(L_1, L_2, L_3)} \\left(1 - \\frac{x}{L_1}\\right)\\left(1 - \\frac{x}{L_2}\\right)\\left(1 - \\frac{x}{L_3}\\right) \\, dx"
] | [
"Three numbers are chosen independently at random, one from each of the three intervals $[0, L_1]$, $[0, L_2]$, and $[0, L_3]$. If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-599ca75eb7a39b7dee79b9cdeaccef30cc8353267d9132578e4fe89cf134285e | Three numbers are chosen randomly from the set $\{1, 3, 3^2, \ldots, 3^n\}$ without replacement. Let $P(n)$ be the probability that these three numbers form an increasing geometric progression. Calculate $P(9) + P(10)$. | [
"\\dfrac{7}{22}"
] | [
"Three numbers are chosen randomly from the set $\\{1, 3, 3^2, \\ldots, 3^n\\}$ without replacement. Let $P(n)$ be the probability that these three numbers form an increasing geometric progression. Calculate $P(9) + P(10)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85871eef5aab12643de4f79b661d9068d1014dfdd95cfdf5ed3bff51fd90e419 | Three numbers are chosen, each between 0 and 2. What is the probability that the sum of the numbers is between 2 and 3? | [
"\\dfrac{1}{3}"
] | [
"Three numbers are chosen, each between 0 and 2. What is the probability that the sum of the numbers is between 2 and 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f30886021c02fca1ac1b8c4f5461d0be732ea606e48cd63c6248e19146eb5d42 | Three numbers are in arithmetic progression and their sum is $k$ (where $k \in \mathbb{R}$). If the first number remains the same and $\frac{k}{6}$ is added to the third number, the resulting sequence forms a geometric progression. Find the three numbers. | [
"\\(\\left(\\frac{k}{6}, \\frac{k}{3}, \\frac{k}{2}\\right)\\) or \\(\\left(\\frac{2k}{3}, \\frac{k}{3}, 0\\right)\\)"
] | [
"Three numbers are in arithmetic progression and their sum is $k$ (where $k \\in \\mathbb{R}$). If the first number remains the same and $\\frac{k}{6}$ is added to the third number, the resulting sequence forms a geometric progression. Find the three numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 14
} | 58 | 0 | 0 |
omr-ws-d8316b51ae4e969f2bbe06b860d477d75f8597d29390357373c931d8efaac0d9 | Three numbers are selected at random without replacement from the set \(\{1, 2, 3, \ldots, n\}\). Find the conditional probability that the third number lies between the first two, given that the first number is known to be less than the second. | [
"\\(\\frac{1}{3}\\)"
] | [
"Three numbers are selected at random without replacement from the set \\(\\{1, 2, 3, \\ldots, n\\}\\). Find the conditional probability that the third number lies between the first two, given that the first number is known to be less than the second."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-1bbb759efcaec80391898aadcc1bedaf8acec7f9a972523ed67adba43da20e64 | Three numbers are selected from the set \(\{1, 2, \ldots, 100\}\). What is the probability that the sum of the selected numbers is divisible by 3? | [
"\\(\\frac{817}{2450}\\)"
] | [
"Three numbers are selected from the set \\(\\{1, 2, \\ldots, 100\\}\\). What is the probability that the sum of the selected numbers is divisible by 3?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-a55797cf80b40dc6e553b59362ccd530f65c91e2947b2bc5ac832db722cf5cdc | Three numbers form a geometric progression. If the third term is decreased by 64, then the three numbers thus obtained will constitute an arithmetic progression. If then the second term of this arithmetic progression is decreased by 8, a geometric progression will be formed again. Determine these numbers. | [
"(1) $\\frac{4}{9};\\frac{52}{9};\\frac{676}{9}$, (2) $4;20;100$"
] | [
"Three numbers form a geometric progression. If the third term is decreased by 64, then the three numbers thus obtained will constitute an arithmetic progression. If then the second term of this arithmetic progression is decreased by 8, a geometric progression will be formed again. Determine these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc6a18b679da4683a2dc336263e1c97f8aa15440baf6dedc3e4e14526b6e47ee | Three numbers form a geometric progression. The sum of these numbers is equal to 126, and their product, to 13,824. Find these numbers. | [
"$6;24;96$"
] | [
"Three numbers form a geometric progression. The sum of these numbers is equal to 126, and their product, to 13,824. Find these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e4a02761a863cf83fbd3efb52e1b5ffae918ca014ded38d1ca1f5cd096e4d63 | Three numbers form an arithmetic sequence with a common difference of $11$. If the first number is decreased by $6$, the second is decreased by $1$, and the third is doubled, the resulting numbers are in geometric sequence. Determine the original numbers which form the arithmetic sequence. | [
"\\(14, 25, 36\\) or \\(-26, -15, -4\\)."
] | [
"Three numbers form an arithmetic sequence with a common difference of $11$. If the first number is decreased by $6$, the second is decreased by $1$, and the third is doubled, the resulting numbers are in geometric sequence. Determine the original numbers which form the arithmetic sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 23
} | 30 | 0 | 0 |
omr-ws-b334b75eb4b3a56b458f8e9adaf650f7d6a4dbc703a98caa00c5305768b133b6 | Three numbers form an arithmetic sequence with a common difference of 11. If the first number is decreased by 6, the second number is decreased by 1, and the third number is doubled, the resulting numbers form a geometric sequence. Determine the original three numbers in the arithmetic sequence. | [
"\\(14, 25, 36\\) or \\(-26, -15, -4\\)"
] | [
"Three numbers form an arithmetic sequence with a common difference of 11. If the first number is decreased by 6, the second number is decreased by 1, and the third number is doubled, the resulting numbers form a geometric sequence. Determine the original three numbers in the arithmetic sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-fac6e4f5642d71ceb8b67b42cc87eeeb8061c190b0913f960fbfd36d3e276591 | Three numbers form an arithmetic sequence. If we add 8 to the first number, the resulting sequence becomes a geometric sequence with a sum of 26. Find the original three numbers. | [
"\\((10, 6, 2)\\) or \\((-6, 6, 18)\\)"
] | [
"Three numbers form an arithmetic sequence. If we add 8 to the first number, the resulting sequence becomes a geometric sequence with a sum of 26. Find the original three numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-bbca46f22af4638631723a855743d6eb86ea4a34de9a8544dae7b012c23b4025 | Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area? | [
"\\(\\frac{1}{2}\\)"
] | [
"Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 58
} | 89 | 0 | 0 |
omr-ws-ce4a717a548c47651fe2f5837345b653e5b6324ddccee63b19bce205ccde0f49 | Three numbers in the interval \([0,1]\) are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area? | [
"\\(\\frac{1}{2}\\)"
] | [
"Three numbers in the interval \\([0,1]\\) are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-74fedb799bb7a413ca2aff4ee387dff5260fefcd888999bfc96cab565d0974e5 | Three numbers the third of which is 12 are three successive terms of a geometric progression. If we take 9 instead of 12, then the three numbers become three successive terms of an arithmetic progression. Find the numbers. | [
"$(3,6,12);\\ (27,18,12)$"
] | [
"Three numbers the third of which is 12 are three successive terms of a geometric progression. If we take 9 instead of 12, then the three numbers become three successive terms of an arithmetic progression. Find the numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbf4ff0fb391c7f3c1cfc38346184b71e4d1f0cdf12009c2e303afa31de11047 | Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of ... | [
"5",
"5\n\nExplanation: The probability \\( p \\) is \\(\\frac{1}{4}\\). When written as a fraction in lowest terms, the sum of the numerator and denominator is \\( 1 + 4 = 5 \\)."
] | [
"Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 1000\\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 8
} | 39 | 0 | 0 |
omr-ws-84369a2fb4aab1a2b0aab1ebc6d77050286cfe9e25f76e894362bc4a757755a2 | Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3,\ldots, 1000\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after suitable rotation, a brick of dimensio... | [
"005"
] | [
"Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\\{1, 2, 3,\\ldots, 1000\\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after suitable rotation, a brick of ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-a2be2c7ea31565da107f980f73a8a3e12f60f12351606f106c7d9b7481cac3cd | Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}\,$. Three other numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p\,$ be the probability that, after a suitable rotat... | [
"5"
] | [
"Three numbers, $a_1\\,$, $a_2\\,$, $a_3\\,$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 1000\\}\\,$. Three other numbers, $b_1\\,$, $b_2\\,$, $b_3\\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p\\,$ be the probability that, after ... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-2fecca5f766fbad1fb9ae679a297fa78c1db92103915e42fbc52f482ba1f6334 | Three numbers, $x$, $y$, and $z$, are chosen at random between $0$ and $2$. What is the probability that the difference between the greatest and the least of these numbers is less than $\frac{1}{4}$? | [
"\\(\\frac{11}{256}\\)"
] | [
"Three numbers, $x$, $y$, and $z$, are chosen at random between $0$ and $2$. What is the probability that the difference between the greatest and the least of these numbers is less than $\\frac{1}{4}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 27
} | 49 | 0 | 0 |
omr-ws-9825d673121459713aa70f625d9b0ac27cbfba20a32d16df99a81cd703dfe6b2 | Three numbers, \(a_1, a_2, a_3\), are drawn randomly and without replacement from the set \(\{1, 2, 3, \dots, 1000\}\). Three other numbers, \(b_1, b_2, b_3\), are then drawn randomly and without replacement from the remaining set of 997 numbers. Let \(p\) be the probability that, after a suitable rotation, a brick of ... | [
"5"
] | [
"Three numbers, \\(a_1, a_2, a_3\\), are drawn randomly and without replacement from the set \\(\\{1, 2, 3, \\dots, 1000\\}\\). Three other numbers, \\(b_1, b_2, b_3\\), are then drawn randomly and without replacement from the remaining set of 997 numbers. Let \\(p\\) be the probability that, after a suitable rotat... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-dd37b174c84c69fd292e61dcb807ccee5e5d1b017b973f7da5bff9f31f42d1e3 | Three numbers, whose sum is equal to 93, constitute a geometric progression. They may also be considered as the first, second and seventh terms of an arithmetic progression. Find these numbers. | [
"(1) $3;15;75$, (2) $31;31;31$"
] | [
"Three numbers, whose sum is equal to 93, constitute a geometric progression. They may also be considered as the first, second and seventh terms of an arithmetic progression. Find these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-677a6b948e92f7880fbf4b757332765145647edbdc17723ab80ef76ecfd54f07 | Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is t... | [
"525"
] | [
"Three of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polyg... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-4fceeccfe7853f81f7e8f7fab0abab6cd44ce0f1aa93fc7ad3a18e457cc80d21 | Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is t... | [
"525"
] | [
"Three of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the poly... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 23,
"tir": 5
} | 28 | 0 | 0 |
omr-ws-c977badd2c66c7da0a06e21f16f5cad151b1aa4bfff57efa18d92abff465ca2c | Three of the four endpoints of the axes of an ellipse are, in some order, \[(-2, 4), \; (3, -2), \; (8, 4).\]Find the distance between the foci of the ellipse. | [
"2 \\sqrt{11}"
] | [
"Three of the four endpoints of the axes of an ellipse are, in some order, \\[(-2, 4), \\; (3, -2), \\; (8, 4).\\]Find the distance between the foci of the ellipse."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-789b4eec56c22677ad0778df5717df5f1f486db25f61233857eec78f5b2db708 | Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 9$? Express your answer in terms of $\pi$. | [
"\\frac94\\pi"
] | [
"Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 9$? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-591210fe1bf3cede662e87f929c1d1995164e8eed95f4c0c69ef51b42dfa8a79 | Three of the roots of the equation \(x^4 - px^3 + qx^2 - rx + s = 0\) are \(\tan A\), \(\tan B\), and \(\tan C\), where \(A\), \(B\), and \(C\) are angles of a triangle. Determine the fourth root as a function only of \(p\), \(q\), \(r\), and \(s\). | [
"\\(\\rho = \\dfrac {r-p} {q-s-1}\\)"
] | [
"Three of the roots of the equation \\(x^4 - px^3 + qx^2 - rx + s = 0\\) are \\(\\tan A\\), \\(\\tan B\\), and \\(\\tan C\\), where \\(A\\), \\(B\\), and \\(C\\) are angles of a triangle. Determine the fourth root as a function only of \\(p\\), \\(q\\), \\(r\\), and \\(s\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 4,
"tir": 3
} | 44 | 0 | 0 |
omr-ws-c49b883285480c8a766f465e6096c64f6f4bd0e48585e32c331592e360967daa | Three of the sides of an isosceles trapezoid have lengths $1, 10, 20$. Find the sum of all possible values of the fourth side. | [
"30"
] | [
"Three of the sides of an isosceles trapezoid have lengths $1, 10, 20$. Find the sum of all possible values of the fourth side."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e4ef8b8217b0195b41a6464dba93ee30af8835cd18f0c7eef0b5482b8109be73 | Three of the vertices of parallelogram $ABCD$ are $A = (3,-1,2),$ $B = (1,2,-4),$ and $C = (-1,1,2).$ Find the coordinates of $D.$ | [
"(1,-2,8)"
] | [
"Three of the vertices of parallelogram $ABCD$ are $A = (3,-1,2),$ $B = (1,2,-4),$ and $C = (-1,1,2).$ Find the coordinates of $D.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70e103b2eeec1e259983ebf56d514692398bbac53e6d78a03db82029dc7d06f4 | Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, then centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed... | [
"\\(\\sqrt{2} + \\frac{1}{2}\\)"
] | [
"Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, then centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were p... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-d6b2a576e1ec6b86ed36424c10e12f92bedcb43a97b47707a0c2c1718ce58612 | Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated \(45^\circ\), then centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point \(B\) from the line on which the bases of the original squares were pl... | [
"\\(\\sqrt{2} + \\frac{1}{2}\\)"
] | [
"Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated \\(45^\\circ\\), then centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point \\(B\\) from the line on which the bases of the original squar... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-f3ee0cc4f398a36b1f97f9dad10c75aa701e90e5d0281a3d5c3112015e53d166 | Three one-inch squares are placed with their bases on a line. The center square is lifted out, rotated 45 degrees, and then centered and lowered until it touches both adjoining squares. How many inches is the point B from the line on which the bases of the original squares were placed? | [
"\\(\\sqrt{2} + \\frac{1}{2}\\)"
] | [
"Three one-inch squares are placed with their bases on a line. The center square is lifted out, rotated 45 degrees, and then centered and lowered until it touches both adjoining squares. How many inches is the point B from the line on which the bases of the original squares were placed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 5
} | 10 | 0 | 0 |
omr-ws-f996543957dd3b9bd1483c65773fd061373c6d9c16370991c91d08b86040ebcc | Three pairs of twin women are each married to twin men in such a way that each woman's twin sister is married to her husband's twin brother. If all twelve of them enter a mixed double's tournament, in how many ways can they be arranged as six mixed pairs (i.e., in each pair there is one woman and one man) so that no on... | [
"80"
] | [
"Three pairs of twin women are each married to twin men in such a way that each woman's twin sister is married to her husband's twin brother. If all twelve of them enter a mixed double's tournament, in how many ways can they be arranged as six mixed pairs (i.e., in each pair there is one woman and one man) so that ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-daf5a11383a42f29017da6346fe9d1ba6dbbda0bc4a3c80f8242451dd13924b8 | Three pairs of twin women are each married to twin men in such a way that each woman's twin sister is married to her husband's twin brother. If all twelve of them enter a mixed doubles tennis tournament, in how many ways can they be arranged as six mixed pairs (i.e., in each pair there is one woman and one man) so that... | [
"80"
] | [
"Three pairs of twin women are each married to twin men in such a way that each woman's twin sister is married to her husband's twin brother. If all twelve of them enter a mixed doubles tennis tournament, in how many ways can they be arranged as six mixed pairs (i.e., in each pair there is one woman and one man) so... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-8c257332c165d0fd0edce83aba7c9244865a083fac49a168417b9c5a38f37903 | Three pairs of twin women are each married to twin men in such a way that each woman's twin sister is married to her husband's twin brother. If all twelve of them enter a mixed doubles tennis tournament, in how many ways can they be arranged as six mixed pairs so that no one is paired with their spouse or their twin's ... | [
"80"
] | [
"Three pairs of twin women are each married to twin men in such a way that each woman's twin sister is married to her husband's twin brother. If all twelve of them enter a mixed doubles tennis tournament, in how many ways can they be arranged as six mixed pairs so that no one is paired with their spouse or their tw... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 17
} | 30 | 0 | 0 |
omr-ws-67c35e0e43829bd6732b6d328916c844474e5a08ba7d9bc4699a7fb424c6c550 | Three pairwise intersecting rays are given. Points begin to move along each ray with constant speed. It is known that these three points always form a triangle, and the center of the circumscribed circle of this triangle moves uniformly in a straight line. If the initial positions of the points are at distances $a$, $b... | [
"\\dfrac{2}{3}"
] | [
"Three pairwise intersecting rays are given. Points begin to move along each ray with constant speed. It is known that these three points always form a triangle, and the center of the circumscribed circle of this triangle moves uniformly in a straight line. If the initial positions of the points are at distances $a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.