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-a6e038e10b85e1ca0dc2aa5a6d283005555c3141107863c5e8a2d87d2320a236 | Three spheres, $S_1$, $S_2$, and $S_3$, are all externally tangent to each other and tangent to the same plane $P_1$. Spheres $S_1$ and $S_2$ both have a radius of $6$, and the radius of $S_3$ is less than $6$. There exists another plane, $P_2$, that is tangent to all three spheres and forms a $60^\circ$ dihedral angle... | [
"25"
] | [
"Three spheres, $S_1$, $S_2$, and $S_3$, are all externally tangent to each other and tangent to the same plane $P_1$. Spheres $S_1$ and $S_2$ both have a radius of $6$, and the radius of $S_3$ is less than $6$. There exists another plane, $P_2$, that is tangent to all three spheres and forms a $60^\\circ$ dihedral... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-ee59d4f973154144a3c4b8966846723f76f298882c692fabf2c606c7ca44d4e8 | Three spherical balls, each of radius 10 cm, are placed in a hemispherical dish such that the tops of the balls are exactly level with the top of the dish. What is the radius, in centimetres, of the dish? | [
"\\( 10 \\left(1 + \\sqrt{\\frac{7}{3}}\\right) \\)"
] | [
"Three spherical balls, each of radius 10 cm, are placed in a hemispherical dish such that the tops of the balls are exactly level with the top of the dish. What is the radius, in centimetres, of the dish?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-18f7c4566a46ca32bef9c03351081f8b5e43a5fdbe45ae2cfcd5291d206ab4a4 | Three spherical balls, each of radius 15 cm, are placed in a hemispherical bowl such that they touch each other and each one touches the inside of the bowl. The tops of the balls are all perfectly level with the top of the bowl. What is the radius of the bowl? | [
"\\(15 + 10\\sqrt{3}\\)"
] | [
"Three spherical balls, each of radius 15 cm, are placed in a hemispherical bowl such that they touch each other and each one touches the inside of the bowl. The tops of the balls are all perfectly level with the top of the bowl. What is the radius of the bowl?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-479873100108300f0c76f8386aff7ca05537a567ac82491a9f310ca609e1b520 | Three spherical melons, each with a radius of 9 cm, are placed on a flat table such that each melon touches the other two. What is the radius of the largest orange that can sit on the table in the space between the three melons? | [
"3"
] | [
"Three spherical melons, each with a radius of 9 cm, are placed on a flat table such that each melon touches the other two. What is the radius of the largest orange that can sit on the table in the space between the three melons?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-2319c8e5d5c1e729fdb2348abdb6b0831688ebd7993242d6cac4babaf629ad31 | Three sports cars of different makes started simultaneously in a motor-rally. The crew of the first car lost three hours on repairs during the rally and, as a result, they finished an hour later than the second crew. Find the speeds of the cars if it is known that the ratio of the speed of the second car to that of the... | [
"90 km/h, 75 km/h, 60 km/h"
] | [
"Three sports cars of different makes started simultaneously in a motor-rally. The crew of the first car lost three hours on repairs during the rally and, as a result, they finished an hour later than the second crew. Find the speeds of the cars if it is known that the ratio of the speed of the second car to that o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee5c94d56c05c6bf74b22b34ed8bc1ccce29f53c7a7127f85963e844ab4c82ad | Three sprinters \(A\), \(B\), and \(C\) sprint from points \(P\) to \(Q\) and back again. The time interval between the starting times of the three sprinters \(A\), \(B\), and \(C\) was \(5\) seconds each. Thus \(C\) started \(10\) seconds after \(A\), while \(B\) started \(5\) seconds after \(A\). The three sprinters ... | [
"10"
] | [
"Three sprinters \\(A\\), \\(B\\), and \\(C\\) sprint from points \\(P\\) to \\(Q\\) and back again. The time interval between the starting times of the three sprinters \\(A\\), \\(B\\), and \\(C\\) was \\(5\\) seconds each. Thus \\(C\\) started \\(10\\) seconds after \\(A\\), while \\(B\\) started \\(5\\) seconds ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3befd18b0152dcce122dd871a1c50bbad0f8a0f284647f04fb442f419f1c5e8 | Three squares $ABB_1B_2$, $BCC_1C_2$, and $CAA_1A_2$ are constructed in the exterior of a triangle $ABC$. In the exterior of these squares, another three squares $A_1B_2B_3B_4$, $B_1C_2C_3C_4$, and $C_1A_2A_3A_4$ are constructed. Let the area of $\triangle ABC$ be $K$. Calculate the area of the triangle with sides $C_3... | [
"7K"
] | [
"Three squares $ABB_1B_2$, $BCC_1C_2$, and $CAA_1A_2$ are constructed in the exterior of a triangle $ABC$. In the exterior of these squares, another three squares $A_1B_2B_3B_4$, $B_1C_2C_3C_4$, and $C_1A_2A_3A_4$ are constructed. Let the area of $\\triangle ABC$ be $K$. Calculate the area of the triangle with side... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-42e82cc74daf4a104006dc8eb465d2ea8dfdb92b310f5f1e01c0c8b8190afd39 | Three squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. If $AB = 5$, $BC = 12$, and $CA = 13$, calculate the length of segment $AP$. | [
"\\sqrt{314}"
] | [
"Three squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. If $AB = 5$, $BC = 12$, and $CA = 13$, calculate the length of segment $AP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-ea9086fa6f2b2192ba70f64d10352f3b110625dec7bc4aff7142312d1e691306 | Three squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. If $AP = 10\sqrt{2}$, find the length of $AQ$. | [
"10\\sqrt{2}"
] | [
"Three squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. If $AP = 10\\sqrt{2}$, find the length of $AQ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9ecad507bd7d93bf8375d9ed28b5435d7e362c695dbbb6a69ca48f734334bc83 | Three squares are in the Cartesian plane with their lower left vertices at $(0,0)$, $(3,1)$, and $(7,2)$ respectively. Each square has a side length of 2 and their sides are parallel to either the x or y axes. Find a line through the origin such that the area below the line and inside the squares is $\frac{1}{3}$ of th... | [
"\\( y = \\frac{5}{13}x \\)"
] | [
"Three squares are in the Cartesian plane with their lower left vertices at $(0,0)$, $(3,1)$, and $(7,2)$ respectively. Each square has a side length of 2 and their sides are parallel to either the x or y axes. Find a line through the origin such that the area below the line and inside the squares is $\\frac{1}{3}$... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-3613f3365e83dd4cc642a13703f1e67a15a65c0c380a0126a53b29ed12a16964 | Three squares with areas $4$, $9$, and $36$ are inscribed in a triangle. Find the area of the triangle. | [
"72"
] | [
"Three squares with areas $4$, $9$, and $36$ are inscribed in a triangle. Find the area of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-bfb5da33d4356d1d8bd85889de1fe1fd71967fdcff9a02a6892e5758df84e257 | Three squares with sides of lengths $1$, $2$, and $x$ are drawn adjacent to each other. A straight line divides the $2 \times 2$ square into two parts of equal area. Find the value of $x$. | [
"3"
] | [
"Three squares with sides of lengths $1$, $2$, and $x$ are drawn adjacent to each other. A straight line divides the $2 \\times 2$ square into two parts of equal area. Find the value of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-f1dbf04e3564d313ed74426add266a59d238d09435e71c7e3b3690e3cc694a10 | Three standard dice are rolled. Let \( P(k) \) denote the probability that the sum of the outcomes is \( k \) where \( 9 \le k \le 14 \), and let \( \mathcal{S} = \sum_{k=9}^{14} P(k) \). \(\mathcal{S}\) can be represented as a fraction \(\frac{x}{y}\) where \( x \) and \( y \) are relatively prime positive integers. F... | [
"3"
] | [
"Three standard dice are rolled. Let \\( P(k) \\) denote the probability that the sum of the outcomes is \\( k \\) where \\( 9 \\le k \\le 14 \\), and let \\( \\mathcal{S} = \\sum_{k=9}^{14} P(k) \\). \\(\\mathcal{S}\\) can be represented as a fraction \\(\\frac{x}{y}\\) where \\( x \\) and \\( y \\) are relatively... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91b8c6211d4e7447009e61bfd6aa81b411de6b9e6b9397fb41afc8cedeaf920d | Three standard dice are stacked in a tower so that the numbers on each pair of touching faces add to 5. The number on the top of the tower is even. What is the number on the base of the tower? | [
"5"
] | [
"Three standard dice are stacked in a tower so that the numbers on each pair of touching faces add to 5. The number on the top of the tower is even. What is the number on the base of the tower?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2987ff4781e06084aa93910e6b95d15dffc9c4ea0b514fe6ff9e877bc61a7089 | Three standard, fair, 6-sided dice are tossed simultaneously. What is the probability that the numbers shown on some two of the dice add up to give the number shown on the remaining die? | [
"$\\frac{5}{24}$"
] | [
"Three standard, fair, 6-sided dice are tossed simultaneously. What is the probability that the numbers shown on some two of the dice add up to give the number shown on the remaining die?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-8414bc6bf8576dd79ee66bf7c7607b6046236a2c6e70b3d3c8304ab38b385c5c | Three straight lines $\ell_1$, $\ell_2$, and $\ell_3$ form a triangle and divide the plane into 7 parts. Points $M_1$, $M_2$, and $M_3$ lie in one of the angles, vertical to some angle of the triangle. The distances from $M_1$ to the lines $\ell_1$, $\ell_2$, and $\ell_3$ are 7, 3, and 1 respectively. The distances fro... | [
"\\dfrac{55}{41}"
] | [
"Three straight lines $\\ell_1$, $\\ell_2$, and $\\ell_3$ form a triangle and divide the plane into 7 parts. Points $M_1$, $M_2$, and $M_3$ lie in one of the angles, vertical to some angle of the triangle. The distances from $M_1$ to the lines $\\ell_1$, $\\ell_2$, and $\\ell_3$ are 7, 3, and 1 respectively. The di... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-da4bfc41822c7f57f697cc1619dfd2668508613f21837bbd36239bb081c9e4e1 | Three straight lines are drawn on the plane. It is known that for any numbers $a$, $b$, and $c$, if there is a point with distances to these straight lines: $a$ to the first, $b$ to the second, and $c$ to the third, then there is also a point with distances: $b$ to the first, $c$ to the second, and $a$ to the third. De... | [
"Three concurrent lines."
] | [
"Three straight lines are drawn on the plane. It is known that for any numbers $a$, $b$, and $c$, if there is a point with distances to these straight lines: $a$ to the first, $b$ to the second, and $c$ to the third, then there is also a point with distances: $b$ to the first, $c$ to the second, and $a$ to the thir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2756fa9638cb4095cf8e0c4f5a58d3020cc61c1e19d5402deecbc07dd839539 | Three straight lines are drawn through a point \( P \) lying in the interior of \( \triangle ABC \) and parallel to its sides. The areas of the three resulting triangles with \( P \) as the vertex are \( s_1, s_2, \) and \( s_3 \). Find the area of \( \triangle ABC \) in terms of \( s_1, s_2, \) and \( s_3 \). | [
"(\\sqrt{s_1} + \\sqrt{s_2} + \\sqrt{s_3})^2"
] | [
"Three straight lines are drawn through a point \\( P \\) lying in the interior of \\( \\triangle ABC \\) and parallel to its sides. The areas of the three resulting triangles with \\( P \\) as the vertex are \\( s_1, s_2, \\) and \\( s_3 \\). Find the area of \\( \\triangle ABC \\) in terms of \\( s_1, s_2, \\) an... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a1f29c6277e69c22d36eb3241afe2e67396c63d699bdf4e2f5f035323fd5408 | Three straws are chosen randomly from a set of 9 straws whose lengths are 1 inch, 2 inches, 3 inches, ..., 9 inches. What is the probability that the three straws, placed end to end, can be used to form a triangle? | [
"\\(\\frac{17}{42}\\)"
] | [
"Three straws are chosen randomly from a set of 9 straws whose lengths are 1 inch, 2 inches, 3 inches, ..., 9 inches. What is the probability that the three straws, placed end to end, can be used to form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-495073c43aca43b6210e3b8f698e0298ae4ec8cbacb3093192a34137ab03fc60 | Three students $A$, $B$, and $C$ are traveling from a location on National Highway No. 5 to Hanoi. Initially, $A$ takes $B$ on a motorcycle, and $C$ rides a bicycle. After one and a half hours, $B$ switches to a bicycle and continues the trip, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the travel on ... | [
"100 km"
] | [
"Three students $A$, $B$, and $C$ are traveling from a location on National Highway No. 5 to Hanoi. Initially, $A$ takes $B$ on a motorcycle, and $C$ rides a bicycle. After one and a half hours, $B$ switches to a bicycle and continues the trip, while $A$ returns to pick up $C$. Upon meeting, $C$ continues the trave... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-fbe26514f4c137214ed6da27dc8990def5c98cd4007f6b67764668ab789810a2 | Three students each randomly choose a prime number less than $100$. The teacher then multiplies the three chosen primes together. What is the probability that the product is divisible by $2021$? | [
"\\(\\frac{6}{25^2}\\)"
] | [
"Three students each randomly choose a prime number less than $100$. The teacher then multiplies the three chosen primes together. What is the probability that the product is divisible by $2021$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 27
} | 33 | 0 | 0 |
omr-ws-2f9225957dfb9bb99e961153df8c389f2a7f35bcdff6a62bbf5438b263e77823 | Three students went to a library where there are 9 distinct books: 3 are Big books and 6 are Small books. Each student must choose either one Big book or two Small books. How many ways can the students make their selections? | [
"1176"
] | [
"Three students went to a library where there are 9 distinct books: 3 are Big books and 6 are Small books. Each student must choose either one Big book or two Small books. How many ways can the students make their selections?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d15ac0cbb70a14fb02013712cb27aa0615a89b589569ae0afc6607319b8c5752 | Three students write three two-digit perfect squares next to each other on a whiteboard. The resulting 6-digit number is also a perfect square. Find all possible values of this 6-digit number. | [
"166464, 646416"
] | [
"Three students write three two-digit perfect squares next to each other on a whiteboard. The resulting 6-digit number is also a perfect square. Find all possible values of this 6-digit number."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-475f8a74b237938917cbd4f074d936e5ac5c3d4adaab34105fca401b72596af0 | Three students write three two-digit squares next to each other on a blackboard. The resulting 6-digit number is also a square. Find this 6-digit square number. | [
"\\(166464\\) or \\(646416\\)"
] | [
"Three students write three two-digit squares next to each other on a blackboard. The resulting 6-digit number is also a square. Find this 6-digit square number."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2f7e02c20435ada51573aeb82e4780334a3d82da44dd51097c80c5a17738f2b1 | Three students, Xena, Yolanda, and Zachary, generate random numbers in the interval [0, 2]. Find the probability that the sum of their numbers is at least 1 but no more than 2. | [
"\\(\\frac{7}{48}\\)"
] | [
"Three students, Xena, Yolanda, and Zachary, generate random numbers in the interval [0, 2]. Find the probability that the sum of their numbers is at least 1 but no more than 2."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-0c830b40a79147e8d58cbfd7690edc75f01e749c8c4a34c23081289bc8801563 | Three tangents of a circle intersect at points \(A\), \(B\), and \(C\). If the radius of the circle is \(r\), find the perimeter of triangle \(ABC\) in terms of \(r\). | [
"6r\\sqrt{3}"
] | [
"Three tangents of a circle intersect at points \\(A\\), \\(B\\), and \\(C\\). If the radius of the circle is \\(r\\), find the perimeter of triangle \\(ABC\\) in terms of \\(r\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82cc4aebc969108e8dd13b574425920c4c1509e00343df26f72ac6da539092c3 | Three teams, the red team, the blue team, and the gold team, each consisting of two runners, compete in a two-mile relay race. Each runner runs one mile of the relay. The red team's two runners run the mile at 9 mph and 11 mph respectively. The blue team's runners run the mile at 8 mph and 13 mph respectively. Both gol... | [
"Gold, Blue, Red"
] | [
"Three teams, the red team, the blue team, and the gold team, each consisting of two runners, compete in a two-mile relay race. Each runner runs one mile of the relay. The red team's two runners run the mile at 9 mph and 11 mph respectively. The blue team's runners run the mile at 8 mph and 13 mph respectively. Bot... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-8f9f9562ea90e11619a5ebbcc4a4ad11ad29edf85721d7e00734bbe02b887084 | Three television towers are located on a flat surface. From three distinct points \(A\), \(B\), and \(C\), a person can see the tips of any two towers overlapping. If the coordinates of points \(A\), \(B\), and \(C\) are \((1, 2)\), \((4, 6)\), and \((7, k)\) respectively, find the value of \(k\) that ensures points \(... | [
"10"
] | [
"Three television towers are located on a flat surface. From three distinct points \\(A\\), \\(B\\), and \\(C\\), a person can see the tips of any two towers overlapping. If the coordinates of points \\(A\\), \\(B\\), and \\(C\\) are \\((1, 2)\\), \\((4, 6)\\), and \\((7, k)\\) respectively, find the value of \\(k\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4888cd430ad7aa1660e44677c59663a6a278e60017fad422b4fe5732587cfce9 | Three tiles are marked $X$ and two other tiles are marked $O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $XOXOX$? | [
"$\\frac{1}{10}$"
] | [
"Three tiles are marked $X$ and two other tiles are marked $O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $XOXOX$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-b2f83775bfa0d2d3ebc56f213973ac3720c93a3e9454e527b88c6aedf99ec561 | Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX? | [
"\\frac{1}{10}"
] | [
"Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a7e47500468eda5f3a41fcb0b07deaa265bc49c7b6142b20f5a53b9232915db | Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. Find the sum of the squares of their ages. | [
"42"
] | [
"Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. Find the sum of the squares of their ages."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-51d7062249b871b0d24b1a7b92e8aa2c83ea03983de2152dab8f24745adc9231 | Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. What is the sum of the squares of their ages? | [
"42"
] | [
"Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. What is the sum of the squares of their ages?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-828a6f6b85208705eb38acdd0fcedd6f15019827fcf92714106b991a3b31ed21 | Three tokens \( A \), \( B \), and \( C \) are placed at the vertices of an equilateral triangle of side length \( n \), which is divided into smaller equilateral triangles of side length 1. Initially, all lines are painted blue. The tokens move along the lines, painting them red, following these rules: 1. The tokens m... | [
"45"
] | [
"Three tokens \\( A \\), \\( B \\), and \\( C \\) are placed at the vertices of an equilateral triangle of side length \\( n \\), which is divided into smaller equilateral triangles of side length 1. Initially, all lines are painted blue. The tokens move along the lines, painting them red, following these rules:\n1... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8881bb7a3e8e320fe05c887e410ff16bebedb395c98d55b217487606a1d71053 | Three tortoises are at the vertices of an equilateral triangle with side length 100 meters. Tortoise A moves towards Tortoise B, Tortoise B towards Tortoise C, and Tortoise C towards Tortoise A, each moving at a speed of 1 meter per second. Where will they meet, and after what time will they meet? | [
"\\(\\frac{200 \\sqrt{3}}{3}\\) seconds"
] | [
"Three tortoises are at the vertices of an equilateral triangle with side length 100 meters. Tortoise A moves towards Tortoise B, Tortoise B towards Tortoise C, and Tortoise C towards Tortoise A, each moving at a speed of 1 meter per second. Where will they meet, and after what time will they meet?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31f9f6ed55e42affefada77c641735573431053fd9c7c1eccbd982d0105bfbfc | Three tourists travel from city $A$ to city $B$, a distance of 30 km. They have two bicycles: a racing bike that allows them to travel at 30 km/h and a tourist bike that allows them to travel at 20 km/h. Each tourist can also walk at a speed of 6 km/h. Bicycles can be left on the road for others to use. What is the sho... | [
"\\(\\frac{8}{3}\\) hours."
] | [
"Three tourists travel from city $A$ to city $B$, a distance of 30 km. They have two bicycles: a racing bike that allows them to travel at 30 km/h and a tourist bike that allows them to travel at 20 km/h. Each tourist can also walk at a speed of 6 km/h. Bicycles can be left on the road for others to use. What is th... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5cdeb47ea216ea96fe949e0e96323165d52eb587880c39b0fce623285b9259ca | Three towns \( X, Y, \) and \( Z \) lie on a plane with coordinates \( (0,0), (200,0) \), and \( (0,300) \) respectively. There are 100, 200, and 300 students in towns \( X, Y, \) and \( Z \) respectively. A school is to be built on a grid point \( (x, y) \) where \( x \) and \( y \) are both integers, such that the ov... | [
"300"
] | [
"Three towns \\( X, Y, \\) and \\( Z \\) lie on a plane with coordinates \\( (0,0), (200,0) \\), and \\( (0,300) \\) respectively. There are 100, 200, and 300 students in towns \\( X, Y, \\) and \\( Z \\) respectively. A school is to be built on a grid point \\( (x, y) \\) where \\( x \\) and \\( y \\) are both int... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-f56bfdfdf77d2f78783b231c0bf27d205f4e788b758521ed5ee3d44301a177ed | Three travelers $A$, $B$ and $C$ are crossing a water obstacle $s$ km wide. A is swimming at a speed of $v$ km/h, and $B$ and $C$ are in a motor boat going at $v_1$ km/h. Some time after the start $C$ decides to swim the rest of the distance, his speed being equal to that of $A$. At this moment $B$ decides to pick up $... | [
"\\[\nT = \\frac{s}{v_1} \\cdot \\frac{v + 3v_1}{3v + v_1}\n\\]"
] | [
"Three travelers $A$, $B$ and $C$ are crossing a water obstacle $s$ km wide. A is swimming at a speed of $v$ km/h, and $B$ and $C$ are in a motor boat going at $v_1$ km/h. Some time after the start $C$ decides to swim the rest of the distance, his speed being equal to that of $A$. At this moment $B$ decides to pick... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13cfa8f7abbdf76eee6c07002934d3254e3468b0a637310f31246892e650958b | Three triangles $DBC$, $EAC$, and $FAB$ are constructed outwardly on the sides of triangle $ABC$. The angles $\angle D$, $\angle E$, and $\angle F$ satisfy $\angle D + \angle E + \angle F = 180^\circ$. Let $P$ be the point of intersection of the circumcircles of triangles $DBC$ and $EAC$. If the circumcircle of triangl... | [
"180^\\circ"
] | [
"Three triangles $DBC$, $EAC$, and $FAB$ are constructed outwardly on the sides of triangle $ABC$. The angles $\\angle D$, $\\angle E$, and $\\angle F$ satisfy $\\angle D + \\angle E + \\angle F = 180^\\circ$. Let $P$ be the point of intersection of the circumcircles of triangles $DBC$ and $EAC$. If the circumcircl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-0edf68d8d68e18deb8648087fa9f65c08be7a04b47d24841a708e0194f0aba2f | Three trolls have divided $n$ pancakes among themselves such that: - Each troll has a positive integer number of pancakes. - The greatest common divisor of the number of pancakes held by any two trolls is bigger than $1$. - The three greatest common divisors obtained in this way are all distinct. What is the smallest p... | [
"22"
] | [
"Three trolls have divided $n$ pancakes among themselves such that:\n- Each troll has a positive integer number of pancakes.\n- The greatest common divisor of the number of pancakes held by any two trolls is bigger than $1$.\n- The three greatest common divisors obtained in this way are all distinct.\n\nWhat is the... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-7f6cf86af9692d96052baea28fb7b1bc01056ac07b0afd7e82dd2a2987b4bd5f | Three tubes are connected to the same pool. If you open one of the tubes, and then one hour later do the same thing with the second tube, and then one hour later do the same thing with the third, the pool will be full in less than 4 hours. Additionally, each tube will have filled the pool exactly one third. Calculate t... | [
"\\dfrac{36}{13}"
] | [
"Three tubes are connected to the same pool. If you open one of the tubes, and then one hour later do the same thing with the second tube, and then one hour later do the same thing with the third, the pool will be full in less than 4 hours. Additionally, each tube will have filled the pool exactly one third. Calcul... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b25a5d1498f7b5b510155e542fa4a0aca1a638869e02879d63431cae25dc4980 | Three tug boats are pulling a barge with forces of 3 tons, 8 tons, and 13 tons along their respective tow cables. The angles of the tugs from the axis of the barge are 15 degrees, 75 degrees, and -60 degrees, respectively. Find the magnitude and direction of the resultant force. | [
"Magnitude = 11.8 tons, Direction = -13.4 degrees"
] | [
"Three tug boats are pulling a barge with forces of 3 tons, 8 tons, and 13 tons along their respective tow cables. The angles of the tugs from the axis of the barge are 15 degrees, 75 degrees, and -60 degrees, respectively. Find the magnitude and direction of the resultant force."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 33
} | 44 | 0 | 0 |
omr-ws-e608c1291cb97ca552436ecee819a94bdf127728d3b5e78cb72b82036b640eb7 | Three two-dimensional subspaces are chosen in four-dimensional space. It is known that none of them is contained in the sum of the other two. Find the dimension of the intersection of all three subspaces. | [
"1"
] | [
"Three two-dimensional subspaces are chosen in four-dimensional space. It is known that none of them is contained in the sum of the other two. Find the dimension of the intersection of all three subspaces."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8998807b0263160f9108acf10fa7a8c7b96f030ea527ae6386ae998472fffa9 | Three uniform tetrahedra (triangular pyramids) each have one face black, another white, another red, and the other green. When tossed on a table, three faces of each tetrahedron can be seen. What is the probability that 3 white faces and only 1 green face can be seen? | [
"\\dfrac{3}{32}"
] | [
"Three uniform tetrahedra (triangular pyramids) each have one face black, another white, another red, and the other green. When tossed on a table, three faces of each tetrahedron can be seen. What is the probability that 3 white faces and only 1 green face can be seen?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de171ba7019aebf50151ac0c59eaedab2ee500a1c4c7d2dfe4088301122e63f5 | Three unit balls (each with radius 1) touch a plane and each other in three-dimensional space. Determine the radius of a ball that touches the same plane and all three unit balls. | [
"\\dfrac{1}{3}"
] | [
"Three unit balls (each with radius 1) touch a plane and each other in three-dimensional space. Determine the radius of a ball that touches the same plane and all three unit balls."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1a31bee7e8c641c2d440bb6ba40ae46e9338f247962025a2b9177ba4333af47 | Three unit circles $\omega_1, \omega_2,$ and $\omega_3$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds the three circles in such a way that each of its four sides is tangent to at least one of $\omega_1, \omega_2,$ and $\omega_3.$ Find the side length... | [
"\\(\\frac{8+\\sqrt6+\\sqrt2}{4}\\)"
] | [
"Three unit circles $\\omega_1, \\omega_2,$ and $\\omega_3$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds the three circles in such a way that each of its four sides is tangent to at least one of $\\omega_1, \\omega_2,$ and $\\omega_3.$ Find the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-6f3d0fca6c0be0e7329246f3dd98bce451aa132f52c60f264bb7b9c2f1234726 | Three unit circles are mutually tangent and enclose a triangular-like region R. Find the area of R. | [
"\\(\\sqrt{3} - \\frac{\\pi}{2}\\)"
] | [
"Three unit circles are mutually tangent and enclose a triangular-like region R. Find the area of R."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-fea41742a9f8dc159118c699dacf47883e101a7bf566fef1933efb220efa97da | Three unit circles have a common point $O$. The other points of (pairwise) intersection are $A, B,$ and $C$. Let $D$ be the center of the circle passing through $A, B,$ and $C$. Calculate the distance $OD$. | [
"1"
] | [
"Three unit circles have a common point $O$. The other points of (pairwise) intersection are $A, B,$ and $C$. Let $D$ be the center of the circle passing through $A, B,$ and $C$. Calculate the distance $OD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce6f7a5a82dea49e884107412ae9c7c06b1947976c805d616572cb300db1318d | Three unit squares are arranged such that two are side-by-side and the third one is on top of the longer side of the 2 x 1 rectangle formed, creating a symmetric figure. Find the smallest radius of a circle that can contain all three squares. | [
"\\(\\frac{5\\sqrt{17}}{16}\\)"
] | [
"Three unit squares are arranged such that two are side-by-side and the third one is on top of the longer side of the 2 x 1 rectangle formed, creating a symmetric figure. Find the smallest radius of a circle that can contain all three squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 21
} | 68 | 0 | 0 |
omr-ws-da831960238ecd690e2f7a2337d0808b738ccb31583b462ad048d396e1c87023 | Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube? | [
"294"
] | [
"Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube?",
"Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube?\n"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 18
} | 50 | 0 | 0 |
omr-ws-e7046bfb6d6181ac77ed853fbf42a0ec473cc44e4de6cf88cc8c97f5335ef1c2 | Three vertices of a cube are connected to form a triangle. Compute the maximum possible perimeter of this triangle if the edge of the cube is \( T \). | [
"\\( 3T\\sqrt{2} \\)"
] | [
"Three vertices of a cube are connected to form a triangle. Compute the maximum possible perimeter of this triangle if the edge of the cube is \\( T \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-608093d47fefb2602a8f933505f4c6e8629b6df5c1d5a3f4592064398e60ff63 | Three vertices of a parallelogram are (1, 2), (5, 3), and (2, 7). If the fourth vertex lies in the first quadrant, what is the sum of its x and y coordinates? | [
"14"
] | [
"Three vertices of a parallelogram are (1, 2), (5, 3), and (2, 7). If the fourth vertex lies in the first quadrant, what is the sum of its x and y coordinates?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-7550914feac53d973beea118da672113a95fefe83be6eacc5bc0fa3ab3209c81 | Three vertices of a regular $(2n+1)$-sided polygon are chosen at random. Find the probability that the center of the polygon lies inside the resulting triangle. | [
"\\(\\frac{n+1}{2(2n-1)}\\)"
] | [
"Three vertices of a regular $(2n+1)$-sided polygon are chosen at random. Find the probability that the center of the polygon lies inside the resulting triangle."
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-f0559139b69f86934e1a9b7ffbc164ea831c0f39786f02840c84d31f4ef3b9cd | Three vertices of a regular $90$-gon are chosen in order, and an angle $\theta$ of the triangle they determine is chosen randomly. The probability that $\theta$ is acute can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"311"
] | [
"Three vertices of a regular $90$-gon are chosen in order, and an angle $\\theta$ of the triangle they determine is chosen randomly. The probability that $\\theta$ is acute can be expressed in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 7
} | 13 | 0 | 0 |
omr-ws-52011feefff22171b881c68cdc44d7f2462714e41f5095f229974c5bb79143da | Three vertices of a triangle are \( (0, 0) \), \( (4046, 0) \), and \( (2023, 2023\sqrt{3}) \). Let the incenter be \( A \), the orthocenter \( B \), the circumcenter \( C \), and the centroid \( D \). Find \( \sum_{n=1}^{1000} \frac{AB^2 + BC^2 + AB^3 + AD^3 - AC^9}{n^2 - 2023n + 2023} \). | [
"0"
] | [
"Three vertices of a triangle are \\( (0, 0) \\), \\( (4046, 0) \\), and \\( (2023, 2023\\sqrt{3}) \\). Let the incenter be \\( A \\), the orthocenter \\( B \\), the circumcenter \\( C \\), and the centroid \\( D \\). Find \\( \\sum_{n=1}^{1000} \\frac{AB^2 + BC^2 + AB^3 + AD^3 - AC^9}{n^2 - 2023n + 2023} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-192136b39ed5fdbe092be3af11cd8ec64d4abb7fe4f5fff2d1669083d280ac2e | Three vertices of parallelogram PQRS are P(-3, -2), Q(1, -5), and R(9, 1), with P and Q being diagonally opposite. What are the coordinates of S? | [
"(-11, -8)"
] | [
"Three vertices of parallelogram PQRS are P(-3, -2), Q(1, -5), and R(9, 1), with P and Q being diagonally opposite. What are the coordinates of S?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e02a4f438e6e31a39ed4c5e1623bea22ad060a1cab0f35a418c9d50bae76e380 | Three vertices of rectangle \( PQRS \) are \( P(-8, -1) \), \( Q(k, k) \), and \( R(14, 2) \). Determine all possible coordinates \( (x, y) \) of vertex \( S \). | [
"\\((-7, -4)\\) and \\(\\left(\\frac{17}{2}, \\frac{23}{2}\\right)\\)"
] | [
"Three vertices of rectangle \\( PQRS \\) are \\( P(-8, -1) \\), \\( Q(k, k) \\), and \\( R(14, 2) \\). Determine all possible coordinates \\( (x, y) \\) of vertex \\( S \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df07ed691442c84540d499b4a485a10157acb10b96bfd4da44581af7e9994357 | Three vessels contain water. If $1/ 3$ of the water of the first vessel is poured into the second, and then $1/4$ of the water now in the second vessel is poured into the third, and, finally, $1/ 10$ of the water now in the third vessel is poured into the first, then each vessel will contain 9 liters. How much water wa... | [
"12 liters; 8 liters, 7 liters"
] | [
"Three vessels contain water. If $1/ 3$ of the water of the first vessel is poured into the second, and then $1/4$ of the water now in the second vessel is poured into the third, and, finally, $1/ 10$ of the water now in the third vessel is poured into the first, then each vessel will contain 9 liters. How much wat... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef98a2691a642cdbe0a8d00fed1969d8bfa78ff30a167b8ce99e75f1b604245a | Three water pipes, each with an outside diameter of 60 cm, are to be enclosed within a ceramic-coated triangular casing. If the cross-section of the casing is an equilateral triangle, what is the smallest possible length for a side of the triangle? | [
"120 cm"
] | [
"Three water pipes, each with an outside diameter of 60 cm, are to be enclosed within a ceramic-coated triangular casing. If the cross-section of the casing is an equilateral triangle, what is the smallest possible length for a side of the triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-282cc7414664cfbbc5b74fad11bb4546ee1e114936cdaeaa19fb586c85c55357 | Three women and seven men are to be split into two 5-person groups such that each group has at least one woman and at least one man. In how many ways can the groups be formed? | [
"105"
] | [
"Three women and seven men are to be split into two 5-person groups such that each group has at least one woman and at least one man. In how many ways can the groups be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-cde5f0bd04d224a0017ec6f79dc6dd3e97fff485a0a77f0f485ad88a6c07f099 | Three women, A, B, and C, are equally skilled at cleaning plates. In a month, 4 plates were broken. What is the probability that woman A broke 3 plates and woman B broke 1 plate? | [
"\\dfrac{4}{81}"
] | [
"Three women, A, B, and C, are equally skilled at cleaning plates. In a month, 4 plates were broken. What is the probability that woman A broke 3 plates and woman B broke 1 plate?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d5374d71bdd1b2cf38ddcda2989cf153ed80f4d016fd7281a7f969fc98a2466 | Three workers can do some work. The second and third workers together can complete it twice as fast as the first worker, and the first and third workers together can complete it three times faster than the second worker. At what time can the first and second workers together complete the job faster than the third worke... | [
"\\dfrac{5}{7}"
] | [
"Three workers can do some work. The second and third workers together can complete it twice as fast as the first worker, and the first and third workers together can complete it three times faster than the second worker. At what time can the first and second workers together complete the job faster than the third ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-1a0010904e7d17e902e34c03fda8e26b7008a3f6b4682f527fae0f61b5a347cf | Three workers have to make 80 identical parts. Together they manufacture 20 parts per hour. Initially, the first worker got to work and made 20 parts in over three hours. The remaining parts were made by the second and third workers. It took them 8 hours to complete this job together. How much time would it have taken ... | [
"16 hours"
] | [
"Three workers have to make 80 identical parts. Together they manufacture 20 parts per hour. Initially, the first worker got to work and made 20 parts in over three hours. The remaining parts were made by the second and third workers. It took them 8 hours to complete this job together. How much time would it have t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-577979816d107a427e4e03917f1057d5107ad1eec6564b23ac3fa87069c5b2bd | Three workers received a total of 4080 roubles for a job. The sums received by the first and the second workers stand in a ratio of $7\frac{1}{2}$ to $1\frac{3}{4}$. The money received by the third worker is $43\frac{1}{3}\%$ that of the first. What was each worker paid? | [
"2448 roub., 571 roub. 20 kop., 1060 roub. 80 kop"
] | [
"Three workers received a total of 4080 roubles for a job. The sums received by the first and the second workers stand in a ratio of $7\\frac{1}{2}$ to $1\\frac{3}{4}$. The money received by the third worker is $43\\frac{1}{3}\\%$ that of the first. What was each worker paid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2621e6485dc23fba16c8bce79de4dc5c7afa515a7b07f7baecc4dcfabf8e6f8c | Three workers together can complete a task in $t$ hours. The first of them, working alon, can do the job twice as fast as the third and one hour faster than the second. How long would it take each worker to do the job individually? | [
"$x=\\frac{1}{4}\\left(5t-2+\\sqrt {25t^2+4t+4}\\right)$"
] | [
"Three workers together can complete a task in $t$ hours. The first of them, working alon, can do the job twice as fast as the third and one hour faster than the second. How long would it take each worker to do the job individually?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eeb4c551188626aff3340e2d72d5d5d2ba5f9edc2100afd92daaf867d2c658b3 | Three years ago, Ann was 1.5 times as old as Roy was when Ann was a year younger than Roy is now. When Roy is 5 years older than Ann will be when Roy is as old as Ann is now, Ann will be twice as old as Roy is now. How old are Ann and Roy? | [
"Ann is 21 and Roy is 17."
] | [
"Three years ago, Ann was 1.5 times as old as Roy was when Ann was a year younger than Roy is now. When Roy is 5 years older than Ann will be when Roy is as old as Ann is now, Ann will be twice as old as Roy is now. How old are Ann and Roy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-a051c63d3bdc50cab9f2c82dde69be2c6ce261a9a5bd22cdab3bd3c09180d3f6 | Three yellow and two purple butterflies are flying together. Suddenly, they all randomly sit on a wire stretched between two poles. What is the probability that the two purple butterflies do not sit together? Assume the butterflies of the same color are identical. | [
"\\(\\frac{3}{5}\\)"
] | [
"Three yellow and two purple butterflies are flying together. Suddenly, they all randomly sit on a wire stretched between two poles. What is the probability that the two purple butterflies do not sit together? Assume the butterflies of the same color are identical."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-da1753ceed2c9e71a8de8ee8c73a3edf618ef18227a2aeaa1f12eb0209713056 | Three yellow and two purple butterflies are flying together. Suddenly, they all randomly sit on a wire stretched between two poles. What is the probability that the two purple butterflies do not sit together? Assume the butterflies of the same color are not identical. | [
"\\(\\frac{3}{5}\\)"
] | [
"Three yellow and two purple butterflies are flying together. Suddenly, they all randomly sit on a wire stretched between two poles. What is the probability that the two purple butterflies do not sit together? Assume the butterflies of the same color are not identical."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-d9d81cc8db5639828b71977954492516f3cf2797b1c9582750508095e73c6b6d | Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are 92 parrots total on Bird Island, how many of those parrots are blue? | [
"23"
] | [
"Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are 92 parrots total on Bird Island, how many of those parrots are blue?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13c0f75198f2bc58854c7b8576380edd53f1faa80156070eee4c06f28982efde | Through a given point $O$ on the extension of the diameter $AD$ of a semicircle, draw a secant $OBC$ such that the area of quadrilateral $ABCD$ is maximized. | [
"no_valid_answer_found"
] | [
"Through a given point $O$ on the extension of the diameter $AD$ of a semicircle, draw a secant $OBC$ such that the area of quadrilateral $ABCD$ is maximized."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0038cb6030d38c8893b7fc8698bbc7025cd6ea76bd97c6ad4da4fc8d20fc5652 | Through a given point B on a diameter AC of a circle, construct a chord DBE such that the arc CE is three times the length of the arc AD. | [
"DE"
] | [
"Through a given point B on a diameter AC of a circle, construct a chord DBE such that the arc CE is three times the length of the arc AD."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-fed419ea5bc0036d3510830f983240e74c457f7b66ed89dfab20dc0f2cd82e4a | Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths is maximized. | [
"4\\sqrt{R^2 - \\frac{d^2}{2}}"
] | [
"Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths is maximized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-025c38d89f8b10f4c35a846214cfdcc6d364b9539309c940262037f9e8433d11 | Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths is minimized. | [
"The sum of the lengths of the two perpendicular chords is minimized when one chord is the diameter passing through the point and the other is the chord perpendicular to this diameter and passing through the same point."
] | [
"Through a given point inside a circle, construct two perpendicular chords such that the sum of their lengths is minimized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 5
} | 16 | 0 | 0 |
omr-ws-b2f2f74d6c75ec32a750cd64fa6ad6505c4dc536d486d2cde55c8a1007ad80a8 | Through a given point, construct a line such that the two segments determined on it by three given concurrent lines have a given ratio. | [
"\\text{Construct a line through the given point using homothety centered at the concurrency point to achieve the desired ratio.}"
] | [
"Through a given point, construct a line such that the two segments determined on it by three given concurrent lines have a given ratio."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-fda0cede9e8c5d2b735c6f3db2acf1d41d69016aa0d0a6ec00d6baaff88e1310 | Through a given point, draw a line that bisects the area of a given triangle. | [
"\\text{Draw a line through the given point and the midpoint of one of the triangle's sides.}"
] | [
"Through a given point, draw a line that bisects the area of a given triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-334c435820c995217be4d801ac0d036fe3ca5f419cf0524894b3620ced918f1b | Through a point $A$ $n$ rays are drawn which form the angles $\frac{2\pi }{n}$ with each other. From a point $B$ lying on one of the rays at a distance $d$ from $A$ a perpendicular is drawn to the next ray. Then from the foot of this perpendicular a new perpendicular is drawn to the neighboring ray and so on, unlimited... | [
"L=d\\cot\\frac{\\pi }{n}"
] | [
"Through a point $A$ $n$ rays are drawn which form the angles $\\frac{2\\pi }{n}$ with each other. From a point $B$ lying on one of the rays at a distance $d$ from $A$ a perpendicular is drawn to the next ray. Then from the foot of this perpendicular a new perpendicular is drawn to the neighboring ray and so on, un... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11a6730ed4541109cf3bb213544f509ca8c30e5687ccc0e84fd1986d7409f4ef | Through a point $M$ lying on the side $AC$ of a triangle $ABC$ draw a straight line $MN$ cutting from the triangle a part whose area is $\frac{1}{k}$ that of the whole triangle. How many solutions has the problem? | [
"Two solutions for \\( k \\ne 2 \\) and one solution for \\( k = 2 \\)."
] | [
"Through a point $M$ lying on the side $AC$ of a triangle $ABC$ draw a straight line $MN$ cutting from the triangle a part whose area is $\\frac{1}{k}$ that of the whole triangle. How many solutions has the problem?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e28238745b04b6979e5b407fa5f3c2de3f7734c915dab065960224428501fa9b | Through a point $P$ inside a given circle, two perpendicular lines are drawn, intersecting the circle at points $A$ and $B$. Find the locus of the projections of $P$ onto the line segment $AB$. | [
"A circle."
] | [
"Through a point $P$ inside a given circle, two perpendicular lines are drawn, intersecting the circle at points $A$ and $B$. Find the locus of the projections of $P$ onto the line segment $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-b3cdff5823ef1707d8f7e0a197b46917a6a12765b6b0d766e823f94db49dc273 | Through a point $X$ inside a square $ABCD$, segments $PQ$ and $EF$ parallel to the sides $AD$ and $AB$ respectively are constructed, with the endpoints on the sides of the square ($P$ on $AB$, $F$ on $AD$). If $S_{ECQX} = 2S_{PXFA}$, determine $\angle EAQ$. | [
"45^\\circ"
] | [
"Through a point $X$ inside a square $ABCD$, segments $PQ$ and $EF$ parallel to the sides $AD$ and $AB$ respectively are constructed, with the endpoints on the sides of the square ($P$ on $AB$, $F$ on $AD$). If $S_{ECQX} = 2S_{PXFA}$, determine $\\angle EAQ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c0728eab53bb31782ae1eb8677927ff4decb6dc051b5a881f9a2de3a96cecd3d | Through a point \( O \) inside a triangle of area \( S \), three lines are drawn such that each side of the triangle intersects two of them. These lines divide the triangle into three triangles with common vertex \( O \) and areas \( S_1, S_2, S_3 \), and three quadrilaterals. If \( S = 18 \), find the minimum possible... | [
"\\dfrac{1}{2}"
] | [
"Through a point \\( O \\) inside a triangle of area \\( S \\), three lines are drawn such that each side of the triangle intersects two of them. These lines divide the triangle into three triangles with common vertex \\( O \\) and areas \\( S_1, S_2, S_3 \\), and three quadrilaterals. If \\( S = 18 \\), find the m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-c05b76a5c77b374c6357d4d583169ccd3a597f1a3efb3e70ef8241da3e704f4c | Through a point \( O \) on the diagonal \( BD \) of a parallelogram \( ABCD \), segments \( MN \) parallel to \( AB \) and \( PQ \) parallel to \( AD \) are drawn, with \( M \) on \( AD \) and \( Q \) on \( AB \). Let \( X \) be the point of intersection of the diagonals \( AO \), \( BP \), and \( DN \) (extended if ne... | [
"no_valid_answer_found"
] | [
"Through a point \\( O \\) on the diagonal \\( BD \\) of a parallelogram \\( ABCD \\), segments \\( MN \\) parallel to \\( AB \\) and \\( PQ \\) parallel to \\( AD \\) are drawn, with \\( M \\) on \\( AD \\) and \\( Q \\) on \\( AB \\). Let \\( X \\) be the point of intersection of the diagonals \\( AO \\), \\( BP ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b030741e1892d163dd04a407d5b911775809c37f52252136b79333aa5b735c24 | Through a point \( P \) exterior to a given circle, a secant intersects the circle at points \( A \) and \( B \), and a tangent touches the circle at point \( C \) on the same side of the diameter through \( P \) as points \( A \) and \( B \). The projection of point \( C \) on the diameter is called \( Q \). If the co... | [
"-1"
] | [
"Through a point \\( P \\) exterior to a given circle, a secant intersects the circle at points \\( A \\) and \\( B \\), and a tangent touches the circle at point \\( C \\) on the same side of the diameter through \\( P \\) as points \\( A \\) and \\( B \\). The projection of point \\( C \\) on the diameter is call... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f8ef841f40155381bb7039b733ded21b6e5f9e204a09de2007b5ff22c8eca13 | Through a point \( P \) in the interior of a fixed triangle \( ABC \), lines \( PL, PM, PN \) are drawn parallel to the medians through \( A, B, C \) respectively to meet \( BC, CA, AB \) at \( L, M, N \) respectively. If \( BL = 3 \), \( BC = 9 \), \( CM = 4 \), and \( CA = 12 \), calculate \( \frac{AN}{AB} \). | [
"\\dfrac{5}{6}"
] | [
"Through a point \\( P \\) in the interior of a fixed triangle \\( ABC \\), lines \\( PL, PM, PN \\) are drawn parallel to the medians through \\( A, B, C \\) respectively to meet \\( BC, CA, AB \\) at \\( L, M, N \\) respectively. If \\( BL = 3 \\), \\( BC = 9 \\), \\( CM = 4 \\), and \\( CA = 12 \\), calculate \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-46f3b00a2a151d6327a5e348539e39ee6748bc5fce93406176e22ea9de401bfd | Through a point \( P \) within a triangle \( ABC \), the lines \( l, m \), and \( n \) perpendicular respectively to \( AP, BP, CP \) are drawn. Let \( l \) intersect the line \( BC \) in \( Q \), \( m \) intersect \( AC \) in \( R \), and \( n \) intersect \( AB \) in \( S \). If the coordinates of \( A, B, \) and \( ... | [
"2x + y + 2 = 0"
] | [
"Through a point \\( P \\) within a triangle \\( ABC \\), the lines \\( l, m \\), and \\( n \\) perpendicular respectively to \\( AP, BP, CP \\) are drawn. Let \\( l \\) intersect the line \\( BC \\) in \\( Q \\), \\( m \\) intersect \\( AC \\) in \\( R \\), and \\( n \\) intersect \\( AB \\) in \\( S \\). If the c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e19f77bdc1282c8f9e6f8960f59ae5358b45a9776fe2c8081f29740b85b679ed | Through a point \(P\) inside a given circle, two perpendicular lines are drawn, meeting the circle at points \(A\) and \(B\). Find the locus of the projections of \(P\) onto \(AB\). | [
"\\left(x - \\frac{h}{2}\\right)^2 + \\left(y - \\frac{k}{2}\\right)^2 = \\frac{2R^2 - h^2 - k^2}{4}"
] | [
"Through a point \\(P\\) inside a given circle, two perpendicular lines are drawn, meeting the circle at points \\(A\\) and \\(B\\). Find the locus of the projections of \\(P\\) onto \\(AB\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c21e1bacdc2404ca541d220ba921185000ca624ab00ff2a8e1c4141d23ca42ac | Through a point in the interior of a sphere, three pairwise perpendicular planes are drawn, dissecting the sphere's surface into eight curvilinear triangles. Alternately coloring these triangles black and white to resemble a checkerboard, calculate the ratio of the total surface area colored black to the total surface ... | [
"\\dfrac{1}{2}"
] | [
"Through a point in the interior of a sphere, three pairwise perpendicular planes are drawn, dissecting the sphere's surface into eight curvilinear triangles. Alternately coloring these triangles black and white to resemble a checkerboard, calculate the ratio of the total surface area colored black to the total sur... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d56bd41987a0087f66520a49a258fcf7076d801cfe57e327f32f313afca580af | Through a point in the interior of a triangle, draw lines parallel to the three sides, partitioning the triangle into three smaller triangles and three parallelograms. Let \( T_1, T_2, T_3 \) be the areas of the three smaller triangles. Let \( P \) be the sum of the areas of the three parallelograms. If \( T_1 = 4 \), ... | [
"72"
] | [
"Through a point in the interior of a triangle, draw lines parallel to the three sides, partitioning the triangle into three smaller triangles and three parallelograms. Let \\( T_1, T_2, T_3 \\) be the areas of the three smaller triangles. Let \\( P \\) be the sum of the areas of the three parallelograms. If \\( T_... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e70b42fab47ac55330232c392bc625009f69531c0f922bf3f768ef4dfde7578 | Through a point inside a triangle, three lines are drawn parallel to the sides of the triangle, splitting it into six parts, three of which are triangles with areas \( S_1 \), \( S_2 \), and \( S_3 \). Find the area of the original triangle. | [
"\\(\\left(\\sqrt{S_1}+\\sqrt{S_2}+\\sqrt{S_3}\\right)^2\\)"
] | [
"Through a point inside a triangle, three lines are drawn parallel to the sides of the triangle, splitting it into six parts, three of which are triangles with areas \\( S_1 \\), \\( S_2 \\), and \\( S_3 \\). Find the area of the original triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-7bae6642eb92f31f689f91d1a8e35792a9364aa42c02f19c080f1e53cbb47e1a | Through a point located on a side of a triangle of area $1$, two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let $s$ be the largest of the areas of these parts. Find the smallest possible value of $s$. | [
"\\(\\frac{4}{9}\\)"
] | [
"Through a point located on a side of a triangle of area $1$, two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let $s$ be the largest of the areas of these parts. Find the smallest possible value of $s$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 16
} | 25 | 0 | 0 |
omr-ws-593afd449291b06318233381913a95d17dd7434d1c1da77ba8c3faeb3c9ca2cd | Through a point lying inside a triangle three straight lines parallel to its sides are drawn. The lines divide the triangle into six parts three of which are triangles with areas $S_1$, $S_2$ and $S_3$, respectively. Find the area of the given triangle. | [
"\\(\\left(\\sqrt{S_1} + \\sqrt{S_2} + \\sqrt{S_3}\\right)^2\\)"
] | [
"Through a point lying inside a triangle three straight lines parallel to its sides are drawn. The lines divide the triangle into six parts three of which are triangles with areas $S_1$, $S_2$ and $S_3$, respectively. Find the area of the given triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f04ad75875dcf4ebaca5dd85fac754ed1f1cef290ee78fd4ca012e63a854e7d2 | Through a point on a diagonal of a cube with edge $a$ a plane is drawn perpendicularly to this diagonal. (1) What polygon is obtained in the section of the faces of the cube by the plane? (2) Find the lengths of the sides of this polygon depending on the distance $x$ from the center of symmetry $O$ of the cube to the c... | [
"The final answer extracted from the solution is:\n\nFor \\( x \\ge \\frac{a\\sqrt{3}}{6} \\), the sections are equilateral triangles with side length \\( MN = \\frac{3}{2}\\sqrt{2}\\, a - x\\sqrt{6} \\).\n\nFor \\( \\frac{a\\sqrt{3}}{6} > x \\ge 0 \\), the section is a hexagon."
] | [
"Through a point on a diagonal of a cube with edge $a$ a plane is drawn perpendicularly to this diagonal. (1) What polygon is obtained in the section of the faces of the cube by the plane? (2) Find the lengths of the sides of this polygon depending on the distance $x$ from the center of symmetry $O$ of the cube to ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4b9088391738ff74bcec5a2352ad001baf7c703e44a6d4f782eb739d7d49948 | Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is "k" times the area of the square. What is the ratio of the area of the square to ... | [
"4k"
] | [
"Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \"k\" times the area of the square. What is the ratio of the area of the squ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-83e575efaaff72ff1011553c99b68bf2ff9fee63e777467d27dec4b68626a798 | Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. What is the ratio of the area of the other smal... | [
"\\frac{1}{4m}"
] | [
"Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. What is the ratio of the area of the other... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df64b3b563fd6b33175f0d362c529e42ce54a6c1b75077d361a54de5658c2ebd | Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle, dividing the triangle into a square and two smaller right triangles. The area of one of the two smaller right triangles is \( m \) times the area of the square. What is the ratio of the area of the other smaller... | [
"\\(\\frac{1}{4m}\\)"
] | [
"Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle, dividing the triangle into a square and two smaller right triangles. The area of one of the two smaller right triangles is \\( m \\) times the area of the square. What is the ratio of the area of the other ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-afc2e11bb7bd1a816223b13787f65a88434ddc8bc71f6d80e453be36286a0d49 | Through a random point $C_1$ on the edge $DC$ of the regular tetrahedron $ABCD$, a plane parallel to the plane $ABC$ is drawn. This plane intersects the edges $DA$ and $DB$ at points $A_1$ and $B_1$, respectively. Let $H$ be the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$, and $M$ be... | [
"90"
] | [
"Through a random point $C_1$ on the edge $DC$ of the regular tetrahedron $ABCD$, a plane parallel to the plane $ABC$ is drawn. This plane intersects the edges $DA$ and $DB$ at points $A_1$ and $B_1$, respectively. Let $H$ be the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$, and $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87b86b4f1db88ca6f996fd76d0d7e7944984dc68502c8e7af5f6aa9567f41a5a | Through a side of the base of a regular triangular pyramid a plane is drawn perpendicular to the opposite lateral edge. Determine the total surface area of the pyramid, if the plane divides the lateral edge in the ratio $m: n$, and the side of the base is equal to $q$. | [
"$S_{total}=\\frac{q^2\\sqrt 3}{2}\\cdot \\left[1+\\sqrt {\\displaystyle\\frac{3(n+2m)}{n}}\\right]$"
] | [
"Through a side of the base of a regular triangular pyramid a plane is drawn perpendicular to the opposite lateral edge. Determine the total surface area of the pyramid, if the plane divides the lateral edge in the ratio $m: n$, and the side of the base is equal to $q$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7c0097d1ee0729955d67da5112da10b0d38c1af0b3c0215e98ffc24c2c9e5a9 | Through a vertex of a triangle, draw a line such that the product of its distances from the other two vertices is a given value \(k^2\). | [
"2"
] | [
"Through a vertex of a triangle, draw a line such that the product of its distances from the other two vertices is a given value \\(k^2\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-7aa657978b0a276c32e4a39fbbcf500081ac3d1a7a2ef3ceceba9f33cdc6efb7 | Through an internal point \( O \) of \( \Delta ABC \), one draws three lines parallel to each of the sides, intersecting at points \( F \), \( E \), and \( N \). Find the value of \( \frac{|AF|}{|AB|} + \frac{|BE|}{|BC|} + \frac{|CN|}{|CA|} \). | [
"1"
] | [
"Through an internal point \\( O \\) of \\( \\Delta ABC \\), one draws three lines parallel to each of the sides, intersecting at points \\( F \\), \\( E \\), and \\( N \\). Find the value of \\( \\frac{|AF|}{|AB|} + \\frac{|BE|}{|BC|} + \\frac{|CN|}{|CA|} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-39d62dc703cd359effb42a3b636edca324cb5a7137e6f2adde74fab3614c30c7 | Through each edge of a tetrahedron a plane parallel to the opposite edge is drawn. Find the ratio of the volume of the parallelepiped thus formed to the volume of the tetrahedron. | [
"3"
] | [
"Through each edge of a tetrahedron a plane parallel to the opposite edge is drawn. Find the ratio of the volume of the parallelepiped thus formed to the volume of the tetrahedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c0a4f82d4a2b8c16583783f614391b3ed234e343d46a6411a7a5ddd980ba012 | Through one and the same point of a circle two chords (equal to $a$ and $b$) are drawn. The area of the triangle formed by joining their ends is equal to $S$. Determine the radius of the circle. | [
"$R=\\frac{ab\\sqrt {a^2+b^2\\mp 2\\sqrt {a^2b^2-4S^2}}}{4S}$. For $S>\\frac{1}{2}ab$ there is no solution, for $S<\\frac{1}{2}ab$, two solutions (the minus sign if the angle between the chords is acute, the plus one if it is obtuse). At $S=\\frac{1}{2}ab$ we have one solution (the chords are mutually perpendicular... | [
"Through one and the same point of a circle two chords (equal to $a$ and $b$) are drawn. The area of the triangle formed by joining their ends is equal to $S$. Determine the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb17a307a4249b7a609b4478b226ced91b2b91c1b14f3b297b3dcd848bb33aef | Through the center \( I \) of the inscribed circle in triangle \( ABC \), pass the segments \( A_1B_2 \), \( B_1C_2 \), and \( C_1A_2 \), each parallel to sides \( AB \), \( BC \), and \( CA \) respectively. Find the value of the sum \( \frac{A_1B_2}{AB} + \frac{B_1C_2}{BC} + \frac{C_1A_2}{CA} \). | [
"2"
] | [
"Through the center \\( I \\) of the inscribed circle in triangle \\( ABC \\), pass the segments \\( A_1B_2 \\), \\( B_1C_2 \\), and \\( C_1A_2 \\), each parallel to sides \\( AB \\), \\( BC \\), and \\( CA \\) respectively. Find the value of the sum \\( \\frac{A_1B_2}{AB} + \\frac{B_1C_2}{BC} + \\frac{C_1A_2}{CA} ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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