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-4a777787932a057737b8974ff4961a4eb1d441b7c38a3f6aa98812c85d4a5b25 | Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m > n$). What is the greatest possible number of parts they can break the plane into? | [
"2n + 2"
] | [
"Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m > n$). What is the greatest possible number of parts they can break the plane into?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-021fd4eecfb684bc19c9d08b55512a809d778a642e0ba5819a5aa0bfaf9572f4 | Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has $2012$ symmetry planes? | [
"2012"
] | [
"Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has $2012$ symmetry planes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-f7e129d097eee937a4eae4b76510623aaa79557d18de9874ef25541a4c3d2f7d | Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has $2013$ symmetry planes? | [
"2012"
] | [
"Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has $2013$ symmetry planes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-75cf9b70fb3fd44f5940cc46cef7c0b6c1bff4c72fbe06da74853818cb68b040 | Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ symmetry axes. What is the maximal number of symmetry axes of the union of $A$ and $B$, if $B$ has $2013$ symmetry axes? | [
"1"
] | [
"Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ symmetry axes. What is the maximal number of symmetry axes of the union of $A$ and $B$, if $B$ has $2013$ symmetry axes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-4fb44cfff452d06462d1326eae8a58c47f3236637934e73c9f3e36a35bcba55a | Two corridors meet at a right angle. One corridor has a width of 8 meters, and the other has a width of 27 meters. Find the length of the largest pipe that can be carried horizontally from one corridor, around the corner, and into the other corridor. | [
"\\(13^{3/2}\\) or approximately 46.8722 meters"
] | [
"Two corridors meet at a right angle. One corridor has a width of 8 meters, and the other has a width of 27 meters. Find the length of the largest pipe that can be carried horizontally from one corridor, around the corner, and into the other corridor."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-15b149fa45690d0a335b7d2dc93e59a28e3d9ded50d782bbc6967066a2b8d4dc | Two couns (one inside the other) are constructed on one and the same base; the angle between the altitude and the generator of the smaller cone is equal to $\alpha $, and that of the larger cone, to $\beta $. The difference between the altitudes is equal to $h$. Find the volume of the solid bounded by the curved surfac... | [
"$V=\\frac{\\pi h^3}{3(\\cot\\beta -\\cot\\alpha )^2} =\\frac{\\pi h^3\\sin^2 \\alpha \\sin^2 \\beta }{3\\sin^2 (\\alpha -\\beta )}$"
] | [
"Two couns (one inside the other) are constructed on one and the same base; the angle between the altitude and the generator of the smaller cone is equal to $\\alpha $, and that of the larger cone, to $\\beta $. The difference between the altitudes is equal to $h$. Find the volume of the solid bounded by the curved... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38d26e14d40319485a91a90144d1f63d216bca70a09b960d2e84da144e16a8a2 | Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each counterfeit coin is different from the weight of each genuine coin. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8... | [
"\\(\\frac{15}{19}\\)"
] | [
"Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each counterfeit coin is different from the weight of each genuine coin. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remain... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 128,
"tir": 117
} | 245 | 0 | 0 |
omr-ws-8d0fd601445cd6e664f3fbf7e4abd0b40c250f0e4acb0fc2da0f8c780714a22d | Two countries ($A$ and $B$) organize a conference and send an equal number of participants. Some participants have known each other from a previous conference. Let $N$ be the number of ways to choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds: - Each participant from $B$ kno... | [
"no_valid_answer_found"
] | [
"Two countries ($A$ and $B$) organize a conference and send an equal number of participants. Some participants have known each other from a previous conference. Let $N$ be the number of ways to choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds:\n- Each participant from $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ec6242cae3d4a22b4cc9e7beacdb354a2f78c49f7ee2635695cdfeee1fe5eea | Two couriers, Anna and Brian, set off from the depot together to deliver a letter to Connor's house. Each motorbike can travel 300 km on a full tank of fuel, and they can transfer fuel between the bikes at any time. While only one bike is required to deliver the letter, both must return to the depot. What is the greate... | [
"150"
] | [
"Two couriers, Anna and Brian, set off from the depot together to deliver a letter to Connor's house. Each motorbike can travel 300 km on a full tank of fuel, and they can transfer fuel between the bikes at any time. While only one bike is required to deliver the letter, both must return to the depot. What is the g... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8fa544dfa93258485705d8deea631d0b45c0d60f8cf56e5351ab81686a06f99 | Two couriers, Anna and Brian, set off from the depot together to deliver a letter to Connor’s house. Their motorbikes can travel 300 km on a full tank of fuel. They can transfer fuel between the bikes at any time but are not allowed to return to the depot to get more fuel. While only one bike is required to deliver the... | [
"225 km"
] | [
"Two couriers, Anna and Brian, set off from the depot together to deliver a letter to Connor’s house. Their motorbikes can travel 300 km on a full tank of fuel. They can transfer fuel between the bikes at any time but are not allowed to return to the depot to get more fuel. While only one bike is required to delive... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aaa1ed6dee7e3a7ff7c250d5b9dc706ef075352019ac28a5b6971888fe7ebc97 | Two cows play a game on an octahedron where each cow starts at opposite vertices and takes turns moving their piece to an adjacent vertex. The winner is the first cow to move its piece to the vertex occupied by the opponent's piece. If the cows move their pieces randomly, what is the probability that the first cow to m... | [
"\\(\\frac{2}{9}\\)"
] | [
"Two cows play a game on an octahedron where each cow starts at opposite vertices and takes turns moving their piece to an adjacent vertex. The winner is the first cow to move its piece to the vertex occupied by the opponent's piece. If the cows move their pieces randomly, what is the probability that the first cow... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da48891b973ec602fe002c5fd6820533c51c0b8ec1c4fd988cf70d8bf1e834b1 | Two cows play a game on an octahedron where each cow starts at opposite vertices and takes turns moving their piece to an adjacent vertex. The winner is the first player to move their piece to the vertex occupied by the opponent's piece. If the cows move their pieces randomly, what is the probability that the first cow... | [
"\\(\\frac{2}{5}\\)"
] | [
"Two cows play a game on an octahedron where each cow starts at opposite vertices and takes turns moving their piece to an adjacent vertex. The winner is the first player to move their piece to the vertex occupied by the opponent's piece. If the cows move their pieces randomly, what is the probability that the firs... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 7
} | 26 | 0 | 0 |
omr-ws-0718e98c18a761b5f260d449cc44fd482ee214fa1c8169306c937bf8e36ed6a0 | Two cows play a game on an octahedron where each cow starts at opposite vertices. They take turns moving their piece to an adjacent vertex. The winner is the first player who moves their piece to the vertex occupied by the opponent's piece. If the cows move their pieces randomly, what is the probability that the first ... | [
"\\(\\frac{2}{5}\\)"
] | [
"Two cows play a game on an octahedron where each cow starts at opposite vertices. They take turns moving their piece to an adjacent vertex. The winner is the first player who moves their piece to the vertex occupied by the opponent's piece. If the cows move their pieces randomly, what is the probability that the f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 6,
"tir": 4
} | 48 | 0 | 0 |
omr-ws-758aad968efa8ec3e1e7d2c38add78a70e0e78b325a5593323b0bad7e7c58546 | Two cows play a game on an octahedron, starting with their pieces on opposite vertices. They take turns moving their pieces to adjacent vertices. The first cow to move its piece to the vertex occupied by the opponent's piece wins. If the cows move their pieces randomly, what is the probability that the first cow to mov... | [
"\\(\\frac{4}{7}\\)"
] | [
"Two cows play a game on an octahedron, starting with their pieces on opposite vertices. They take turns moving their pieces to adjacent vertices. The first cow to move its piece to the vertex occupied by the opponent's piece wins. If the cows move their pieces randomly, what is the probability that the first cow t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a4aca74b0680337e9061b8c82d9e12f56c95ddd13924978c5c1e1920398c632 | Two cows play a game where each has one playing piece, they begin by having the two pieces on opposite vertices of an octahedron, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex occupied by its opponent’s piece. Because cows are not ... | [
"\\dfrac{2}{5}"
] | [
"Two cows play a game where each has one playing piece, they begin by having the two pieces on opposite vertices of an octahedron, and the two cows take turns moving their piece to an adjacent vertex. The winner is the first player who moves its piece to the vertex occupied by its opponent’s piece. Because cows are... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-20d30db2fe4dd25c6e0d6ba5455f0492141f21720883e6d642f5bd4e635d0c20 | Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $216\sqrt{3}$ square feet and $486\sqrt{3}$ square feet. The two planes are $8$ feet apart. How far from the apex of the pyramid is the larger cross section... | [
"24"
] | [
"Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $216\\sqrt{3}$ square feet and $486\\sqrt{3}$ square feet. The two planes are $8$ feet apart. How far from the apex of the pyramid is the larger cross ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2ecc320a0276e39d94828c6ab3d5055393debedfba7fae11b7cf74e98a88e4c | Two cubes are inscribed in a sphere of radius \( R \). Calculate the sum of the squares of all segments connecting the vertices of one cube with the vertices of the other cube. | [
"\\( 256R^2 \\)"
] | [
"Two cubes are inscribed in a sphere of radius \\( R \\). Calculate the sum of the squares of all segments connecting the vertices of one cube with the vertices of the other cube."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b6239fa6745ecc35c60b33aa6446dd71a16b8d5b9a6120cf38d7ca843a0f351 | Two cubes have their faces painted either red or blue. The first cube has five red faces and one blue face. When the two cubes are rolled, the probability that they show the same color is 1/2. How many red faces are there on the second cube? | [
"3"
] | [
"Two cubes have their faces painted either red or blue. The first cube has five red faces and one blue face. When the two cubes are rolled, the probability that they show the same color is 1/2. How many red faces are there on the second cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6516db5a590820d39be53aa13400dd19c9f8a44bd2d4a95b7036347281329489 | Two cubes with edge length 1 share the same center. What is the smallest possible volume of their intersection? | [
"\\(\\frac{1}{6}\\)"
] | [
"Two cubes with edge length 1 share the same center. What is the smallest possible volume of their intersection?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e05b8a8bf26980f1f8022c06b15b0bc13b189396b2b19c46b03e190e622ab5aa | Two cubes with the faces numbered 1 through 6 are tossed and the numbers shown on the top faces are added. What is the probability that the sum is even? Express your answer as a common fraction. | [
"\\frac12"
] | [
"Two cubes with the faces numbered 1 through 6 are tossed and the numbers shown on the top faces are added. What is the probability that the sum is even? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a30c4aeb18381bbb42f19c3c07b4f8d3bbf21a80c3c11063ee6a91daea9e6b0 | Two cubes, each numbered with the integers 4 through 9, are tossed. What is the probability that the product of the two numbers rolled is greater than 29 and less than 61? Express your answer as a common fraction. | [
"\\(\\frac{11}{18}\\)"
] | [
"Two cubes, each numbered with the integers 4 through 9, are tossed. What is the probability that the product of the two numbers rolled is greater than 29 and less than 61? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-12f9822ef8a93ef819e1ab43ce2b9caecfef4212827fc1a480d4d9547bfd1900 | Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probabil... | [
"\\(\\frac{2}{11}\\)"
] | [
"Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the pro... | [
"aops_c5_contests_amp_programs"
] | [
"genselect",
"tir"
] | {
"genselect": 7,
"tir": 8
} | 15 | 0 | 0 |
omr-ws-62e1ac1371b451f76e02ab172679bb40b6826169aae7d761a65dbc61f014f5a2 | Two curves in polar coordinates are given by $r = f(\theta)$ and $r = g(\theta)$. Suppose these curves intersect at a point where they meet at right angles. If $f(\theta) = 2\theta$ and $g(\theta) = \frac{1}{\theta}$, find the value of $\theta$ where this condition is satisfied. | [
"no_valid_answer_found"
] | [
"Two curves in polar coordinates are given by $r = f(\\theta)$ and $r = g(\\theta)$. Suppose these curves intersect at a point where they meet at right angles. If $f(\\theta) = 2\\theta$ and $g(\\theta) = \\frac{1}{\\theta}$, find the value of $\\theta$ where this condition is satisfied."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3bba2c4fe169d93c86e7e8221672ef05b212a2ec0aeacdf13a9e80f045f53dd | Two cyclists and a pedestrian started from point $A$ to point $B$ at the same time. More than an hour after his start, the first cyclist had an accident with his bicycle and continued on foot, traveling 4.5 times slower than he traveled on the bicycle. The second cyclist overtook him 5/8 h after the accident and the pe... | [
"In 4 h"
] | [
"Two cyclists and a pedestrian started from point $A$ to point $B$ at the same time. More than an hour after his start, the first cyclist had an accident with his bicycle and continued on foot, traveling 4.5 times slower than he traveled on the bicycle. The second cyclist overtook him 5/8 h after the accident and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f81a8df4c7226241d3505aaea1cdf5b0ea44a621e37fd490e257e6bca02324dc | Two cyclists leave simultaneously from point $P$ on a circular runway with constant velocities $v_1$ and $v_2$ ($v_1 > v_2$) in the same direction. A pedestrian also leaves $P$ at the same time, moving with velocity $v_3 = \frac{v_1 + v_2}{12}$ in the opposite direction. The pedestrian meets the second cyclist 91 secon... | [
"the opposite of P"
] | [
"Two cyclists leave simultaneously from point $P$ on a circular runway with constant velocities $v_1$ and $v_2$ ($v_1 > v_2$) in the same direction. A pedestrian also leaves $P$ at the same time, moving with velocity $v_3 = \\frac{v_1 + v_2}{12}$ in the opposite direction. The pedestrian meets the second cyclist 91... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 8
} | 31 | 0 | 0 |
omr-ws-20b4b7ceb1993dea35605ab6309d95e9201525c3e206e46c1dcb956f894556c9 | Two cyclists left point $A$ at the same time and went in the same direction. The first cyclist traveled at a speed of 7 km/h and the second at a speed of 10 km/h. Thirty minutes later a third cyclist left point $A$ and went in the same direction. He overtook the first cyclist, and 1.5 h later he overtook the second cyc... | [
"$9+\\sqrt {11}$ km/h"
] | [
"Two cyclists left point $A$ at the same time and went in the same direction. The first cyclist traveled at a speed of 7 km/h and the second at a speed of 10 km/h. Thirty minutes later a third cyclist left point $A$ and went in the same direction. He overtook the first cyclist, and 1.5 h later he overtook the sec... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7c8b5d72a180827774201837c9b5268ff956db77fa3e1ca4a258a924e8023c7 | Two cyclists left point $A$ for point $B$ at the same time. The first of them stopped 42 minutes later, when he was within 1 km of $B$, and the second cyclist stopped in 52 minutes, when 2 fern remained to $B$. If the first cyclist covered as many km as the second and the second as many as the first, then the first of ... | [
"15 km"
] | [
"Two cyclists left point $A$ for point $B$ at the same time. The first of them stopped 42 minutes later, when he was within 1 km of $B$, and the second cyclist stopped in 52 minutes, when 2 fern remained to $B$. If the first cyclist covered as many km as the second and the second as many as the first, then the fir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0fc63a4121334dade9bfaed49bdc220b56d5b8283ef5fda8f74a221418faeb12 | Two cyclists live in cities A and B, which are 140 km apart. They plan to meet in city C, located halfway between A and B. The cyclist from B leaves 1 hour earlier than the cyclist from A because their speed is 9 km/h less than the cyclist from A. The cyclist from A experiences a half-hour breakdown and increases their... | [
"30"
] | [
"Two cyclists live in cities A and B, which are 140 km apart. They plan to meet in city C, located halfway between A and B. The cyclist from B leaves 1 hour earlier than the cyclist from A because their speed is 9 km/h less than the cyclist from A. The cyclist from A experiences a half-hour breakdown and increases ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-8287ef387cf4d55059e8262437709146f704d6d68f2fffec0378c35606330bb5 | Two cyclists move at constant speeds along their own straight paths. At 11:00, the distance between them was 4 km. At 11:08, the distance between them was still 4 km. At 11:17, the distance between them was 4.5 km. Determine the relative speed of one cyclist relative to the other in km/h. | [
"\\(\\frac{10}{3}\\) km/h"
] | [
"Two cyclists move at constant speeds along their own straight paths. At 11:00, the distance between them was 4 km. At 11:08, the distance between them was still 4 km. At 11:17, the distance between them was 4.5 km. Determine the relative speed of one cyclist relative to the other in km/h."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ee32f1a627ededfc057195d9d76b9920e1c959ab78e041ba1436d534f5815729 | Two cyclists ride on two intersecting circles at constant speeds, starting from one of the points of intersection. Let $C_1$ and $C_2$ be the centers of the two circles, and let $r_1$ and $r_2$ be their respective radii. If the cyclists complete one lap and meet again at the starting point, find the distance between $C... | [
"5"
] | [
"Two cyclists ride on two intersecting circles at constant speeds, starting from one of the points of intersection. Let $C_1$ and $C_2$ be the centers of the two circles, and let $r_1$ and $r_2$ be their respective radii. If the cyclists complete one lap and meet again at the starting point, find the distance betwe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-327d07dff7b154a8b9ff6eaad251531701223ed1dd56186700234bfb48d82d0f | Two cyclists start at the same time from opposite ends of a 5-mile course. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they began will they meet? | [
"\\(\\frac{1}{6}\\) hours or 12 minutes"
] | [
"Two cyclists start at the same time from opposite ends of a 5-mile course. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they began will they meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-926583fd2222bfc7bef56d504cf7a80803dcbfa758e195ca5122a5277072e82d | Two cyclists start simultaneously from points $A$ and $B$ towards each other and, traveling at constant speeds, meet $2\frac{2}{5}$ h later. If the first cyclist increased his speed by 50\% and the second by 20\%, then it would take the first cyclist 2/3 h more to cover the distance between $A$ and $B$ than the second.... | [
"6 h and 4 h"
] | [
"Two cyclists start simultaneously from points $A$ and $B$ towards each other and, traveling at constant speeds, meet $2\\frac{2}{5}$ h later. If the first cyclist increased his speed by 50\\% and the second by 20\\%, then it would take the first cyclist 2/3 h more to cover the distance between $A$ and $B$ than t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4bd5da3ab5122300182bc5acee4f502827a53b6f6238d2f75a1fe77ec3104aa | Two cyclists start simultaneously from two points 270 km apart and travel towards each other. The second cyclist covers 1.5 km less per hour than the first cyclist and meets him in as many hours as the first cyclist covers in an hour. Find the speed of each cyclist. | [
"12 km/h, 10.5 km/h"
] | [
"Two cyclists start simultaneously from two points 270 km apart and travel towards each other. The second cyclist covers 1.5 km less per hour than the first cyclist and meets him in as many hours as the first cyclist covers in an hour. Find the speed of each cyclist."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efd4f47d6c72db3970626f85beb39abefd27ab7a870c503be84dc0e65b1a5264 | Two cyclists start simultaneously towards each other from points $A$ and $B$ which are 120 km apart and meet more than 5 hours later. The next day they start simultaneously in the same direction from points $C$ and $D$ which are 36 km apart. The cyclist, who is leading, pedals at a speed which is 6 km/h higher than it ... | [
"It will be enough"
] | [
"Two cyclists start simultaneously towards each other from points $A$ and $B$ which are 120 km apart and meet more than 5 hours later. The next day they start simultaneously in the same direction from points $C$ and $D$ which are 36 km apart. The cyclist, who is leading, pedals at a speed which is 6 km/h higher th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64deada43163fa5a541a4f33abfb0dd6048af792af0a87c9324aaa4d43f355d2 | Two cyclists started simultaneously from points $A$ and $B$ and met each other 2.4 hours later. Had the first cyclist traveled 50\% faster and the second 20\% faster, then it would have taken the first cyclist $\frac{2}{3}$ hour more than the second cyclist to ride from $A$ to $B$. How much time does it take each cycli... | [
"6 hours and 4 hours"
] | [
"Two cyclists started simultaneously from points $A$ and $B$ and met each other 2.4 hours later. Had the first cyclist traveled 50\\% faster and the second 20\\% faster, then it would have taken the first cyclist $\\frac{2}{3}$ hour more than the second cyclist to ride from $A$ to $B$. How much time does it take ea... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b57ae84e49addb5ff8e9c7b9f6d16f391dd842781cab488ac0d0444bd07158c | Two cyclists started simultaneously from points $A$ and $B$ to meet each other. The one driving from $A$ reached $B$ in four hours, and the other driving from $B$ reached $A$ in nine hours alter they had met. How much time does it take each cyclist to cover the distance? | [
"15 hours and 10 hours"
] | [
"Two cyclists started simultaneously from points $A$ and $B$ to meet each other. The one driving from $A$ reached $B$ in four hours, and the other driving from $B$ reached $A$ in nine hours alter they had met. How much time does it take each cyclist to cover the distance?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aad9ec4f85acc197de098d27776f4eda52c4fc9bcbb1e98c9c0986c226b6a52c | Two cyclists started simultaneously towards each other from points $A$ and $B$ and met 12 km from point $B$. After the meeting, they arrived at points $B$ and $A$ respectively, started back immediately and met again 6 km from point $A$. Find the speeds of the cyclists and the distance $AB$ if the second cyclist is know... | [
"30 km/h, 20 km/h and 30 km"
] | [
"Two cyclists started simultaneously towards each other from points $A$ and $B$ and met 12 km from point $B$. After the meeting, they arrived at points $B$ and $A$ respectively, started back immediately and met again 6 km from point $A$. Find the speeds of the cyclists and the distance $AB$ if the second cyclist ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37e223019a84e082e58f8dbfe99f487a8729782472a1b97f31b37897614fcf78 | Two cyclists started, the second 2 minutes after the first. The second cyclist overtook the first 1 km from the start. If the second cyclist, having covered another 5 km, returned back, then he would meet the first cyclist 20 minutes after the first cyclist had started cycling. Find the speed of the second cyclist. | [
"20 km/h"
] | [
"Two cyclists started, the second 2 minutes after the first. The second cyclist overtook the first 1 km from the start. If the second cyclist, having covered another 5 km, returned back, then he would meet the first cyclist 20 minutes after the first cyclist had started cycling. Find the speed of the second cyclist... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-714c7ab2684cf919681e3b3fa4f146f66bda160841fc6a247ccf0af4ee4913e1 | Two cyclists, Dobi and Bodi, start from two cities, Dobruisk and Bodruisk, respectively, which are 40 km apart. Dobi travels at 23 km/h and Bodi travels at 17 km/h towards each other. A fly starts from Dobi's nose and flies towards Bodi at 40 km/h. Upon meeting Bodi, the fly turns back and flies towards Dobi at 30 km/h... | [
"70 km"
] | [
"Two cyclists, Dobi and Bodi, start from two cities, Dobruisk and Bodruisk, respectively, which are 40 km apart. Dobi travels at 23 km/h and Bodi travels at 17 km/h towards each other. A fly starts from Dobi's nose and flies towards Bodi at 40 km/h. Upon meeting Bodi, the fly turns back and flies towards Dobi at 30... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d1ac28118ecc29d1151e44876ffbb435ffe07a5730f7bb2ad6587d95284e902 | Two cylinders are made from 8.5x11 sized pieces of paper. One cylinder has a height of 11 and the other has a height of 8.5. Compare the lateral areas, total areas, and volumes of the two cylinders. | [
"93.5"
] | [
"Two cylinders are made from 8.5x11 sized pieces of paper. One cylinder has a height of 11 and the other has a height of 8.5. Compare the lateral areas, total areas, and volumes of the two cylinders."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-ca89ddd3226e360b2cee54cdbbe97ff5aea503863fc059ac670f2d2eb1e0d415 | Two cylindrical cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form. | [
"12\\sqrt{3}"
] | [
"Two cylindrical cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7eb31bcb100690ccc59baa5a7d39ee81501172ab3a3a570c221acbc35391d5f3 | Two cylindrical cans, made of the same material with the same thickness and shape, are full of coffee. The first can weighs 2 pounds and is 12 inches tall, while the second can weighs 1 kg and is 9.5 inches tall. Considering the negligible thickness of the cans, what is the approximate net weight of the coffee in each ... | [
"2 pounds and 2.20462 pounds"
] | [
"Two cylindrical cans, made of the same material with the same thickness and shape, are full of coffee. The first can weighs 2 pounds and is 12 inches tall, while the second can weighs 1 kg and is 9.5 inches tall. Considering the negligible thickness of the cans, what is the approximate net weight of the coffee in ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 11
} | 13 | 0 | 0 |
omr-ws-003e6263a957243b6789201b319f75f6d77b3c613d745c7a501a7d60cfbc6427 | Two darts are thrown at a circular dartboard of radius $1$ lightyear. Given that the darts always land on the dartboard, what is the expected value of the distance between the two points on the dartboard on which the two darts landed? Express your answer in lightyears. | [
"\\dfrac{128}{45\\pi}"
] | [
"Two darts are thrown at a circular dartboard of radius $1$ lightyear. Given that the darts always land on the dartboard, what is the expected value of the distance between the two points on the dartboard on which the two darts landed? Express your answer in lightyears."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d91c5af7cc7313f2d353822045e258b1834954a8b9eb9c486f80624fa18a86ae | Two decks each consisting of $n$ cards are pooled and shuffled together. The combined deck is then split into two parts, each containing $n$ cards. The cards from each part are matched randomly. Find the probability of at least one match. | [
"1 - \\dfrac{2^n}{\\dbinom{2n}{n}}"
] | [
"Two decks each consisting of $n$ cards are pooled and shuffled together. The combined deck is then split into two parts, each containing $n$ cards. The cards from each part are matched randomly. Find the probability of at least one match."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-028a38f5b5c59cbbebac458c7bb71207482d53c11d26db1474b21de251b76921 | Two decks of cards are arranged such that the larger Joker is on top, followed by the smaller Joker, then the four Aces, four 2's, four 3's, ..., four Queens, and four Kings, with each type of card arranged in the order Spades, Hearts, Diamonds, Clubs. One deck is placed on top of the other. The dealer removes cards by... | [
"8 of Clubs"
] | [
"Two decks of cards are arranged such that the larger Joker is on top, followed by the smaller Joker, then the four Aces, four 2's, four 3's, ..., four Queens, and four Kings, with each type of card arranged in the order Spades, Hearts, Diamonds, Clubs. One deck is placed on top of the other. The dealer removes car... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 15
} | 26 | 0 | 0 |
omr-ws-08a34d3490524915d88d74d95573491879ca97fbb9d5b3b6f6dec8341001406d | Two decks of playing cards are shuffled and placed side by side. The top cards of both decks are flipped one by one. What is the probability that there will be at least one matching pair? | [
"\\(\\sum_{k=1}^{52}(-1)^{k+1}\\frac{1}{k!}\\)"
] | [
"Two decks of playing cards are shuffled and placed side by side. The top cards of both decks are flipped one by one. What is the probability that there will be at least one matching pair?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 12
} | 15 | 0 | 0 |
omr-ws-32db4f02c1dc255431b578799f0ea91c37e0eacb6d2d1bff06a814b6533998b1 | Two diagonals are selected at random from a regular n-sided polygon such that they do not share any endpoints. What is the probability that they intersect? | [
"\\(\\frac{\\binom{n}{4}}{\\binom{\\binom{n}{2} - n}{2}}\\)"
] | [
"Two diagonals are selected at random from a regular n-sided polygon such that they do not share any endpoints. What is the probability that they intersect?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-0aef725a91578bf1ea3547ea9a02a61fe45911c2f62069b985f6c7fbf56933b2 | Two diagonals of a circumscribed quadrilateral \(ABCD\) intersect at \(O\). Let \(E\), \(F\), \(G\), and \(H\) be the centers of the inscribed circles of triangles \(AOB\), \(BOC\), \(COD\), and \(DOA\), respectively. If the radius of the circle circumscribing quadrilateral \(EFGH\) is \(R\), and the distance from \(O\... | [
"6 - 4\\sqrt{2}"
] | [
"Two diagonals of a circumscribed quadrilateral \\(ABCD\\) intersect at \\(O\\). Let \\(E\\), \\(F\\), \\(G\\), and \\(H\\) be the centers of the inscribed circles of triangles \\(AOB\\), \\(BOC\\), \\(COD\\), and \\(DOA\\), respectively. If the radius of the circle circumscribing quadrilateral \\(EFGH\\) is \\(R\\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d6dbc28a52501ff71dcc35e9e89dc91ce096b589d550725e5e502a4e92272d9 | Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Fin... | [
"071"
] | [
"Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integer... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-87455f9ea2e43a3dcf3a949237d0ff5a41bd3daece65c3ee64ee5cbce74ffa28 | Two dice are given as A and B, where A has 3 red and 3 white faces, and B has 4 red and 2 white faces. You randomly choose one of the two dice and roll it. If you get a red face, you keep the same die. If you get a white face, you switch to the other die and repeat the experiment n times. Let \( P_n \) be the probabili... | [
"\\dfrac{3}{5} - \\dfrac{1}{10} \\left( \\dfrac{1}{6} \\right)^n"
] | [
"Two dice are given as A and B, where A has 3 red and 3 white faces, and B has 4 red and 2 white faces. You randomly choose one of the two dice and roll it. If you get a red face, you keep the same die. If you get a white face, you switch to the other die and repeat the experiment n times. Let \\( P_n \\) be the pr... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-5d7ef06cabc4cc83da1a7963520b0259ed4dcd53b761f1faab2378b05b1a6226 | Two dice are rolled until the sum of points in their upper faces is 5 or 6. Find the probability that the sum 5 appears before the sum 6. | [
"\\(\\frac{4}{9}\\)"
] | [
"Two dice are rolled until the sum of points in their upper faces is 5 or 6. Find the probability that the sum 5 appears before the sum 6."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-6dfe0e30f95faeefe0c7997a602d77c4f53052938354b4b3e2e03f7b8058876b | Two dice are rolled. Let \( X \) be the smaller number of points and \( Y \) the larger number of points. If both dice show the same number, say \( z \), then \( X = Y = z \). Calculate the value of \( P(X=2 \mid Y=4) \). | [
"\\dfrac{2}{7}"
] | [
"Two dice are rolled. Let \\( X \\) be the smaller number of points and \\( Y \\) the larger number of points. If both dice show the same number, say \\( z \\), then \\( X = Y = z \\). Calculate the value of \\( P(X=2 \\mid Y=4) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ac847edb2053d3ebeed194b40b95d26edae0e17b4d3c3d50794c7d3ebe464a0 | Two dice are rolled. Let \( X \) be the smaller number of points and \( Y \) the larger number of points. If both dice show the same number, say \( z \), then \( X = Y = z \). a) Find the joint probability density function (pdf) of \( (X, Y) \). | [
"\\[ p(X=a, Y=b) = \\begin{cases} \n0 & \\text{if } a > b, \\\\\n\\frac{1}{36} & \\text{if } a = b, \\\\\n\\frac{1}{18} & \\text{if } a < b.\n\\end{cases} \\]"
] | [
"Two dice are rolled. Let \\( X \\) be the smaller number of points and \\( Y \\) the larger number of points. If both dice show the same number, say \\( z \\), then \\( X = Y = z \\).\na) Find the joint probability density function (pdf) of \\( (X, Y) \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-87bac8ae6804cf76e9de9fa9c3fc79b55563344ca22a5ca96170f5daf47a836f | Two dice are rolled. Let \( X \) be the smaller number of points and \( Y \) the larger number of points. If both dice show the same number, say \( z \), then \( X = Y = z \). c) If the smaller number shown is 2, what is the probability that the larger one will be 5? | [
"\\(\\frac{2}{9}\\)"
] | [
"Two dice are rolled. Let \\( X \\) be the smaller number of points and \\( Y \\) the larger number of points. If both dice show the same number, say \\( z \\), then \\( X = Y = z \\).\nc) If the smaller number shown is 2, what is the probability that the larger one will be 5?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-8944cad6903e01cd87e4b2ad3852ad5de2a205527b329949b3376edd89e4ed92 | Two dice are rolled. What is the probability that the number on the first die is greater than the number on the second die and the number on the first die is greater than 4? | [
"\\( \\frac{5}{9} \\)"
] | [
"Two dice are rolled. What is the probability that the number on the first die is greater than the number on the second die and the number on the first die is greater than 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6adaecbc3bb4cced1a2ee0fd70814b516b9959e1d0235878379b47efd061addd | Two dice are thrown $n$ times in succession. Compute the probability that consecutive double $6$s appear at least once. How large should $n$ be to make this probability at least $\frac{1}{2}$? | [
"25"
] | [
"Two dice are thrown $n$ times in succession. Compute the probability that consecutive double $6$s appear at least once. How large should $n$ be to make this probability at least $\\frac{1}{2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-270ce5dc22af7f442652afbb8b3f1ae8ab543ba9855a548a085c8fb1fe0b5702 | Two dice are thrown once simultaneously. Let \( E \) be the event "Sum of numbers appearing on the dice." What are the members of \( E \)? Can you load these dice (not necessarily in the same way) such that all members of \( E \) are equally likely? Provide justification. | [
"\\( E = \\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\\} \\) and it is not possible to load the dice such that all members of \\( E \\) are equally likely."
] | [
"Two dice are thrown once simultaneously. Let \\( E \\) be the event \"Sum of numbers appearing on the dice.\" What are the members of \\( E \\)? Can you load these dice (not necessarily in the same way) such that all members of \\( E \\) are equally likely? Provide justification."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 21
} | 32 | 0 | 0 |
omr-ws-f3a09c3d381380432f89468e471283dc29ca3de7af9d6df86c08aaec6b67b9fc | Two dice are thrown simultaneously. The probability that the sum of the two numbers that appear is prime is ... | [
"\\dfrac{5}{12}"
] | [
"Two dice are thrown simultaneously. The probability that the sum of the two numbers that appear is prime is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0dc3cf747ddb4a3a5f7109c993dc916dbcd1e6245f5ad1ff1ea29a8fdd9bda34 | Two dice are thrown successively, and the difference between the points of the higher and lower dice is calculated. If the difference is 0, 1, or 2, Juan earns a point. If the difference is 3, 4, or 5, Pedro earns a point. Determine the probabilities of Juan and Pedro winning a point on a single turn. | [
"\\( \\frac{2}{3} \\) and \\( \\frac{1}{3} \\)"
] | [
"Two dice are thrown successively, and the difference between the points of the higher and lower dice is calculated. If the difference is 0, 1, or 2, Juan earns a point. If the difference is 3, 4, or 5, Pedro earns a point. Determine the probabilities of Juan and Pedro winning a point on a single turn."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 32
} | 36 | 0 | 0 |
omr-ws-b9048b5c04eec4c2138f087a5d9eb8699168bda39ae85c90e80b9c92a0105bbc | Two dice are tossed. What is the probability that the sum is greater than three? | [
"\\frac{11}{12}"
] | [
"Two dice are tossed. What is the probability that the sum is greater than three?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-256cb2b9defdecaa5500e5d1064ee71efedff60c4c3b0608ae53bd6c9ee2841a | Two dice have been thrown, giving a total of at least 10. What is the probability that the throw of a third die will bring the total of the three numbers shown to 15 or higher? | [
"\\(\\frac{4}{9}\\)"
] | [
"Two dice have been thrown, giving a total of at least 10. What is the probability that the throw of a third die will bring the total of the three numbers shown to 15 or higher?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-dd721cb23f6c460affe061c150575f85827828b3a5777a0562f8235d989478ea | Two different 2-digit numbers are randomly chosen and multiplied together. What is the probability that the resulting product is even? | [
"$\\frac{67}{89}$"
] | [
"Two different 2-digit numbers are randomly chosen and multiplied together. What is the probability that the resulting product is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-273b2f7f9c7fa6027658755f3cab3e9f9b57e4f9c6df3265ad1d7d793f4e39e0 | Two different 3-digit numbers are picked. For each number, the sum of all 5 numbers formed by permuting its digits is calculated. Let $S_1$ and $S_2$ be the sums for the two different 3-digit numbers. Find the smallest possible positive difference between $S_1$ and $S_2$. | [
"9"
] | [
"Two different 3-digit numbers are picked. For each number, the sum of all 5 numbers formed by permuting its digits is calculated. Let $S_1$ and $S_2$ be the sums for the two different 3-digit numbers. Find the smallest possible positive difference between $S_1$ and $S_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-47ec364199a9ce9617fb04f28e85f50843dd534aa91ed0a87c47e9daf59997a9 | Two different base 120 units digits are chosen randomly and their product is written in base 120. What is the probability that the units digit of the base 120 product is 1? | [
"\\dfrac{2}{1785}"
] | [
"Two different base 120 units digits are chosen randomly and their product is written in base 120. What is the probability that the units digit of the base 120 product is 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc6b014dc0fde5af70940092b00f9216c5abe7ea19f4efa33412b36ee06de385 | Two different circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Let $C$ be a point on $\omega_2$ different from $A$ and $B$. Lines $CA$ and $CB$ intersect $\omega_1$ again at points $F$ and $G$ respectively. The tangents to $\omega_1$ through $F$ and $G$ intersect at $D$. The lines $AG$ and $BF$ inters... | [
"-\\dfrac{1}{2}"
] | [
"Two different circles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. Let $C$ be a point on $\\omega_2$ different from $A$ and $B$. Lines $CA$ and $CB$ intersect $\\omega_1$ again at points $F$ and $G$ respectively. The tangents to $\\omega_1$ through $F$ and $G$ intersect at $D$. The lines $AG$ and $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7983f9fdc06cc7347d5b9315651d05c21469426687eee5c71e3a7bd4adfea78d | Two different circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect at points $A$ and $B$. The line $O_1B$ intersects $\omega_2$ at point $F \ne B$, and the line $O_2B$ intersects $\omega_1$ at point $E \ne B$. A line through point $B$ parallel to $EF$ intersects $\omega_1$ at point $M \... | [
"1"
] | [
"Two different circles $\\omega_1$ and $\\omega_2$ with centers $O_1$ and $O_2$ respectively intersect at points $A$ and $B$. The line $O_1B$ intersects $\\omega_2$ at point $F \\ne B$, and the line $O_2B$ intersects $\\omega_1$ at point $E \\ne B$. A line through point $B$ parallel to $EF$ intersects $\\omega_1$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c30cd827fdbadf0d2a4ec036dda5dd9c0871eba82f091c0d86c15dad18c0eaed | Two different circles are inscribed in the angle \(PCQ\). Let \(A\) be the touchpoint of the first circle with side \(CP\), and \(B\) be the touchpoint of the second circle with side \(CQ\). The circle circumscribed around triangle \(ABC\) intersects the first circle again at point \(L\) and the second circle again at ... | [
"1"
] | [
"Two different circles are inscribed in the angle \\(PCQ\\). Let \\(A\\) be the touchpoint of the first circle with side \\(CP\\), and \\(B\\) be the touchpoint of the second circle with side \\(CQ\\). The circle circumscribed around triangle \\(ABC\\) intersects the first circle again at point \\(L\\) and the seco... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-53461b8b2e01f2fc7ac2fbcd825aafee0a6224370afdb065cc10422965814d40 | Two different cubes of the same size are painted with each face being either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? | [
"\\dfrac{15}{256}"
] | [
"Two different cubes of the same size are painted with each face being either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7be0cfa0ce7eb0bb763a164c306dfb6af8cf0c252c55c96a71108f1797d43611 | Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? | [
"\\(\\frac{147}{1024}\\)"
] | [
"Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-6994267e051960f9a4f12268d05fa1fca266274dbbd1b110c015374c5f0957f0 | Two different geometric progressions both have sum 1 and the same second term. One has a third term of \( \frac{1}{8} \). Find its second term. | [
"\\(\\frac{1}{2 + 2\\sqrt{5}}\\)"
] | [
"Two different geometric progressions both have sum 1 and the same second term. One has a third term of \\( \\frac{1}{8} \\). Find its second term."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 30
} | 76 | 0 | 0 |
omr-ws-854f5cf23e012a45374928f9123eeeb7fba1d1631e2e731b5270e2fd533340d4 | Two different integers are randomly chosen from the set $$\{ -5, -8, 7, 4, -2 \}.$$ What is the probability that their product is negative? Express your answer as a common fraction. | [
"\\frac{3}{5}"
] | [
"Two different integers are randomly chosen from the set $$\\{ -5, -8, 7, 4, -2 \\}.$$ What is the probability that their product is negative? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3808fa629672ad88e9c43191472f4c2271d3f0c2313736a3c654daa101c15a31 | Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction. | [
"\\dfrac{14}{95}"
] | [
"Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e72b432b76aea0c59ab8dcfa1d3fb8fe34f32608e79a11a2521bd08f14e2c3e0 | Two different lines intersect the interior of a regular pentagon and thereby divide it into polygons. Let $S$ denote the sum of the measures of the interior angles of all these polygons. Find all possible values of $S$. | [
"540°"
] | [
"Two different lines intersect the interior of a regular pentagon and thereby divide it into polygons. Let $S$ denote the sum of the measures of the interior angles of all these polygons. Find all possible values of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-b84bdccd8545d9524ecf7410160bb4f9584078f7516b824678dac64675a21a97 | Two different natural numbers are selected from the set $\ \allowbreak \{1, 2, 3, \ldots, 6\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | [
" \\frac{11}{15}"
] | [
"Two different natural numbers are selected from the set $\\ \\allowbreak \\{1, 2, 3, \\ldots, 6\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26f8422ec20f5f56e907b5c3404ca80893c0296d0ee957509f6c4d0c1a24cb3d | Two different numbers are randomly selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. The probability that their sum is 12 would be greater if the number $n$ had first been removed from set $S$. What is the value of $n$? | [
"6"
] | [
"Two different numbers are randomly selected from the set $S = \\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$. The probability that their sum is 12 would be greater if the number $n$ had first been removed from set $S$. What is the value of $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4dab238b1e79c2f9388d814dcc3566d4db77c532b4fd2b511303d783cd0b45b | Two different numbers are randomly selected from the set $\{-2, -1, 0, 3, 4, 5\}$ and multiplied together. What is the probability that the product is $0$? | [
"\\(\\frac{1}{3}\\)"
] | [
"Two different numbers are randomly selected from the set $\\{-2, -1, 0, 3, 4, 5\\}$ and multiplied together. What is the probability that the product is $0$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-5c7beda51ae312dc75806607f0e768e9e26b077b9ecb82e393e753bdde93c92f | Two different numbers are selected at random from the set $\{2,3,5,7,11,13,17,19,23,29\}$. What is the probability that their sum is 24? | [
"\\(\\frac{1}{15}\\)"
] | [
"Two different numbers are selected at random from the set $\\{2,3,5,7,11,13,17,19,23,29\\}$. What is the probability that their sum is 24?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-d3eedc9e154e43db65ffacc6087f53305bb9c3431ea0b962c0506b143aabcd02 | Two different numbers are selected at random from the set \(\{1, 2, 3, 4, 5\}\) and multiplied together. What is the probability that the product is even? | [
"0.7"
] | [
"Two different numbers are selected at random from the set \\(\\{1, 2, 3, 4, 5\\}\\) and multiplied together. What is the probability that the product is even?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-02d0ba1f6a9ab62ec61f5ce373a7077b57dc633835a0f7c4396f00728778c556 | Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7\}$. What is the probability that the positive difference between the two numbers is $2$ or greater? Express your answer as a common fraction. | [
"\\frac{5}{7}"
] | [
"Two different numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5, 6, 7\\}$. What is the probability that the positive difference between the two numbers is $2$ or greater? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fba1a5abda8a450d25bce4b7cde8e28c291b1d3496792ad08e29e6f1a2495573 | Two different points $A$ and $B$ are marked on the line $\ell$. Consider all possible pairs of circles touching each other and the line $\ell$ at points $A$ and $B$ respectively. For each pair, let $M$ be the midpoint of the segment tangent to these circles, not lying on $\ell$. Find the locus of points $M$. | [
"\\( x^2 - bx + y^2 = \\frac{3}{4}b^2 \\)"
] | [
"Two different points $A$ and $B$ are marked on the line $\\ell$. Consider all possible pairs of circles touching each other and the line $\\ell$ at points $A$ and $B$ respectively. For each pair, let $M$ be the midpoint of the segment tangent to these circles, not lying on $\\ell$. Find the locus of points $M$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f7390b579bc428265c04a5258a81744702e38f731a0dfe6bc2bc53fe676f542 | Two different points $A$ and $B$ have been marked on the circle $\omega$. For all points $X$ on the circle $\omega$ (different from $A$ and $B$), let $Y$ be the midpoint of the chord $AX$ and $Z$ be the projection of point $A$ onto the line $BX$. Let $P$ be the fixed point through which all straight lines $YZ$ pass. If... | [
"(1,0)"
] | [
"Two different points $A$ and $B$ have been marked on the circle $\\omega$. For all points $X$ on the circle $\\omega$ (different from $A$ and $B$), let $Y$ be the midpoint of the chord $AX$ and $Z$ be the projection of point $A$ onto the line $BX$. Let $P$ be the fixed point through which all straight lines $YZ$ p... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b17e9550a25ecba7817abb41ac5c75031a5f64dff6d4a86a3a382cc7c39c21ea | Two different points $X$ and $Y$ are marked on the side $AB$ of a triangle $ABC$ such that $\frac{AX \cdot BX}{CX^2} = \frac{AY \cdot BY}{CY^2}$. If $AC = 10$ and $BC = 15$, find the value of $\frac{AC}{BC} \cdot \frac{\sin(\angle ACX)}{\sin(\angle BCY)}$. | [
"\\dfrac{2}{3}"
] | [
"Two different points $X$ and $Y$ are marked on the side $AB$ of a triangle $ABC$ such that $\\frac{AX \\cdot BX}{CX^2} = \\frac{AY \\cdot BY}{CY^2}$. If $AC = 10$ and $BC = 15$, find the value of $\\frac{AC}{BC} \\cdot \\frac{\\sin(\\angle ACX)}{\\sin(\\angle BCY)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-69a68bd22c74f9bc858ec238ee28cdbe8efaee9a636bb01e83e0a3a420e9bc1f | Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"059",
"59"
] | [
"Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 1
} | 64 | 0 | 0 |
omr-ws-a9dd0f323b30475673a2f2b2c473e69fce0f04a857ab5d4ac0cc0f051c8023c7 | Two different regular 1997-gons with a common vertex are drawn on the plane. Using one ruler, construct the intersection points of their circumscribed circles. | [
"B"
] | [
"Two different regular 1997-gons with a common vertex are drawn on the plane. Using one ruler, construct the intersection points of their circumscribed circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-597fa308703031bdb4c30fa8f831d5c67d79e201dfa2d20e4fbc8ecbaa259841 | Two different tangents are drawn to a circle at points $A$ and $B$. The distance from a point $M$ on the circle to these tangents is $p$ and $q$, respectively. Find the distance from point $M$ to the line $AB$. | [
"\\(\\sqrt{pq}\\)"
] | [
"Two different tangents are drawn to a circle at points $A$ and $B$. The distance from a point $M$ on the circle to these tangents is $p$ and $q$, respectively. Find the distance from point $M$ to the line $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-8cc9133d10062d414a538e2fc01704958110e3b55fcf4452adf137f77cb08176 | Two different teams with four students each were training for an international math contest. The students were given an assessment exam and their scores were ranked. What is the probability that the four best scores come from only one of the teams, assuming that no two students got the same score? | [
"\\(\\frac{1}{35}\\)"
] | [
"Two different teams with four students each were training for an international math contest. The students were given an assessment exam and their scores were ranked. What is the probability that the four best scores come from only one of the teams, assuming that no two students got the same score?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-883b4fdb927e232c199499aec93914e8dc591532b5a2e4f006ce5a3c3fbd8239 | Two different terms are randomly chosen from the expansion of $(1 + a + b^3)^8 - (2 + a^2 + b)^8$. If their coefficients are $x$ and $y$, and the expected value of $x + y$ can be expressed as $\frac{x}{y}$, where $y$ is positive, find $x + y$. | [
"-58936"
] | [
"Two different terms are randomly chosen from the expansion of $(1 + a + b^3)^8 - (2 + a^2 + b)^8$. If their coefficients are $x$ and $y$, and the expected value of $x + y$ can be expressed as $\\frac{x}{y}$, where $y$ is positive, find $x + y$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-24cc9cf3b970c86b632bb1254fb7aa6bc42e13a0efccbca1e6c2bde207089a65 | Two different tractors working together, ploughed a field in 8 days. If at first one tractor had ploughed half the field and then both tractors together had ploughed the other half, the whole job would have been done in 10 days. How many days would it take each tractor to plough the field individually? | [
"It would take the first tractor 12 days, and the second, 24 days to plough the field"
] | [
"Two different tractors working together, ploughed a field in 8 days. If at first one tractor had ploughed half the field and then both tractors together had ploughed the other half, the whole job would have been done in 10 days. How many days would it take each tractor to plough the field individually?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0bf1506251ee95d5706021e559059bb8b241176e80a642c181b9b8e1933a1b1b | Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy \[ \frac{f'(x)}{g'(x)} = e^{f(x) - g(x)} \] for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f \) and \( g \). | [
"\\( 1 - \\ln 2 \\)"
] | [
"Two differentiable real functions \\( f(x) \\) and \\( g(x) \\) satisfy\n\\[ \\frac{f'(x)}{g'(x)} = e^{f(x) - g(x)} \\]\nfor all \\( x \\), and \\( f(0) = g(2003) = 1 \\). Find the largest constant \\( c \\) such that \\( f(2003) > c \\) for all such functions \\( f \\) and \\( g \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-bf7f3f9d89525eaf6b6ea9a17c8938f4cce58437ed18b1afcf3116b2e13cd802 | Two digits are written at the ends of a segment. In the middle, their sum is written, which is the number 2. Then, in the middle between each two neighboring numbers, their sum is written again, and this process is repeated 1973 times. How many times will the number 1973 be written? | [
"0"
] | [
"Two digits are written at the ends of a segment. In the middle, their sum is written, which is the number 2. Then, in the middle between each two neighboring numbers, their sum is written again, and this process is repeated 1973 times. How many times will the number 1973 be written?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-14da7dacdb8dba5d59e74392d68c2fe1be8fa24abaa030bce017989a8521c38d | Two directly homothetic triangles are such that the incircle of one of them is the circumcircle of the other. If the ratio of the areas of these two triangles is 4, find the value of $k$ where $k$ is the ratio of the side lengths of the larger triangle to the smaller triangle. | [
"2"
] | [
"Two directly homothetic triangles are such that the incircle of one of them is the circumcircle of the other. If the ratio of the areas of these two triangles is 4, find the value of $k$ where $k$ is the ratio of the side lengths of the larger triangle to the smaller triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-873b660e3a523533c3189c5bba6cb89ebdd1b4e327f87122ce00afbbf118e1e2 | Two discs, each of radius \( a \), are placed one on top of the other. The top disc begins to move with a velocity \( v \) such that \( v \propto \frac{1}{A'} \), where \( A' \) is the area of the stationary disc underneath that is exposed. Find the time taken by the moving disc to completely expose the stationary disc... | [
"\\dfrac{2a^3(3\\pi - 4)}{3k}"
] | [
"Two discs, each of radius \\( a \\), are placed one on top of the other. The top disc begins to move with a velocity \\( v \\) such that \\( v \\propto \\frac{1}{A'} \\), where \\( A' \\) is the area of the stationary disc underneath that is exposed. Find the time taken by the moving disc to completely expose the ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-7448ccfb1a796aaa1dcfd6372cf7a04ffc52d573e53fdbaebc5ad6e383937445 | Two discs, each of radius \( a \), are placed one over the other. The top disc begins to move with a velocity \( v \) such that \( v \) is inversely proportional to the area of the stationary disc underneath that is exposed. Find the time taken by the moving disc to completely expose the disc underneath. | [
"\\dfrac{2a^3}{k} \\left( \\pi - \\dfrac{4}{3} \\right)"
] | [
"Two discs, each of radius \\( a \\), are placed one over the other. The top disc begins to move with a velocity \\( v \\) such that \\( v \\) is inversely proportional to the area of the stationary disc underneath that is exposed. Find the time taken by the moving disc to completely expose the disc underneath."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-146bf32512297af45604fc2c3ca2cc7f97ac720d8e578ab99d113fb845fb7116 | Two discs, each of radius \( r \) cm, model an eclipse where the area of intersection represents the hidden area of the sun's disc. Calculate the distance between the centers of the two discs when 10% of the sun's disc is hidden. | [
"\\( 1.1855 \\cdot r \\)"
] | [
"Two discs, each of radius \\( r \\) cm, model an eclipse where the area of intersection represents the hidden area of the sun's disc. Calculate the distance between the centers of the two discs when 10% of the sun's disc is hidden."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76c6b4e259ab8d2f3abf4391c2cf9755fef687bbf2af46545438e3fd5e972b63 | Two discs, each of radius \( r \) cm, model an eclipse where the area of intersection represents the hidden area of the sun's disc. Calculate the distance between the centers of the two discs when 80% of the sun's disc is hidden. | [
"\\( 1.97946 \\cdot r \\)"
] | [
"Two discs, each of radius \\( r \\) cm, model an eclipse where the area of intersection represents the hidden area of the sun's disc. Calculate the distance between the centers of the two discs when 80% of the sun's disc is hidden."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60489efd27c35a172c62f1745589d007338cda9369cbcfe6afde9d0893e3fe15 | Two disjoint circles $\omega_1$ and $\omega_2$ are inscribed into an angle. Consider all pairs of parallel lines $l_1$ and $l_2$ such that $l_1$ touches $\omega_1$ and $l_2$ touches $\omega_2$ ($\omega_1$, $\omega_2$ lie between $l_1$ and $l_2$). Let $C$ be the center of the circle that is tangent to all the medial lin... | [
"2\\sqrt{r_1 r_2}"
] | [
"Two disjoint circles $\\omega_1$ and $\\omega_2$ are inscribed into an angle. Consider all pairs of parallel lines $l_1$ and $l_2$ such that $l_1$ touches $\\omega_1$ and $l_2$ touches $\\omega_2$ ($\\omega_1$, $\\omega_2$ lie between $l_1$ and $l_2$). Let $C$ be the center of the circle that is tangent to all the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80d2152bacaa7a611ccdbf599f4e55d696b51be7bb50cbb35facc11df6eb5c18 | Two disjoint circles $\omega_1$ and $\omega_2$ have points $A$ and $B$ on them such that the segment $AB$ has the greatest possible length. Let $AR$ and $AS$ be tangents drawn from point $A$ to circle $\omega_2$, and $BP$ and $BQ$ be tangents from point $B$ to the circle $\omega_1$. The circle $\gamma_1$ touches the ci... | [
"\\dfrac{2}{3} r_2"
] | [
"Two disjoint circles $\\omega_1$ and $\\omega_2$ have points $A$ and $B$ on them such that the segment $AB$ has the greatest possible length. Let $AR$ and $AS$ be tangents drawn from point $A$ to circle $\\omega_2$, and $BP$ and $BQ$ be tangents from point $B$ to the circle $\\omega_1$. The circle $\\gamma_1$ touc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-aca8d0eae0b9c8970d2f0ae10f7b818f13f2e5f5c10bf36f80fd3ae584182000 | Two disjoint circles \( k_1 \) and \( k_2 \) with centers \( O_1 \) and \( O_2 \) respectively lie on the same side of a line \( p \) and touch the line at \( A_1 \) and \( A_2 \) respectively. The segment \( O_1O_2 \) intersects \( k_1 \) at \( B_1 \) and \( k_2 \) at \( B_2 \). If \( O_1A_1 = 4 \) and \( O_2A_2 = 9 \... | [
"144"
] | [
"Two disjoint circles \\( k_1 \\) and \\( k_2 \\) with centers \\( O_1 \\) and \\( O_2 \\) respectively lie on the same side of a line \\( p \\) and touch the line at \\( A_1 \\) and \\( A_2 \\) respectively. The segment \\( O_1O_2 \\) intersects \\( k_1 \\) at \\( B_1 \\) and \\( k_2 \\) at \\( B_2 \\). If \\( O_1... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5afa3c04356c65d1536f8f24f915a569a6acafaf4518ec6b45081b7b15601969 | Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about triangl... | [
"\\dfrac{d}{4 \\sin \\frac{a}{2}}"
] | [
"Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about tr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-fb704fe103f9d5f32254dd177e58fd8a02bda091ad0da6852bd120038d7822da | Two disjoint circles are inscribed in an angle. They touch one side of the corner at points $K$ and $L$, the other at points $M$ and $N$. $C$ is the midpoint of the segment $KL$, and $A$ and $B$ are the points of intersection of the segments $CM$ and $CN$ with the circles. If the radius of the circle touching at $K$ an... | [
"\\dfrac{2}{5}"
] | [
"Two disjoint circles are inscribed in an angle. They touch one side of the corner at points $K$ and $L$, the other at points $M$ and $N$. $C$ is the midpoint of the segment $KL$, and $A$ and $B$ are the points of intersection of the segments $CM$ and $CN$ with the circles. If the radius of the circle touching at $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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