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-4697f7a2f39461ef27bd8bddbe0bc4333d8c41e1a11a1a76f077c2ba30a1e36a | Two boats start in the same location and travel apart along straight lines which meet at an angle of \(\pi/3\). Boat A travels east at 9 miles per hour, and Boat B travels northeast at 18 miles per hour. How fast is the distance between the boats increasing after 4 hours? | [
"\\( 9\\sqrt{3} \\)"
] | [
"Two boats start in the same location and travel apart along straight lines which meet at an angle of \\(\\pi/3\\). Boat A travels east at 9 miles per hour, and Boat B travels northeast at 18 miles per hour. How fast is the distance between the boats increasing after 4 hours?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f983662787b8250881a008a59204d8fc1f9cf7bd32ba4ed3ff49627269c501e5 | Two boats start simultaneously from pier $A$ down the river to pier $B$. The first boat arrives at $B$ two hours earlier than the second. If the boats started from the piers simultaneously traveling towards each other (the first from $A$ and the second from $B$), they would meet 3 h later. The distance between the pier... | [
"$v_r=1$ km/h"
] | [
"Two boats start simultaneously from pier $A$ down the river to pier $B$. The first boat arrives at $B$ two hours earlier than the second. If the boats started from the piers simultaneously traveling towards each other (the first from $A$ and the second from $B$), they would meet 3 h later. The distance between the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-046c9ac2a34d6701da9fd76b2cd3faea4b2106c019232974c0e91b0df4e3b000 | Two boats, $A$ and $B$, start from opposite ends of a river segment, the Shire and Mordor respectively, and travel towards each other at constant speeds. They stop for 15 minutes at each end to drop off and pick up passengers. On their first meeting, they are 72 miles from the Shire, and on their second meeting, they a... | [
"13"
] | [
"Two boats, $A$ and $B$, start from opposite ends of a river segment, the Shire and Mordor respectively, and travel towards each other at constant speeds. They stop for 15 minutes at each end to drop off and pick up passengers. On their first meeting, they are 72 miles from the Shire, and on their second meeting, t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11ab4a59b344da061676e3c0e75bd2060e4a885e0b36a838cb909564cb174c54 | Two boats, A and B, leave a port C at the same time on different routes. Boat B travels on a bearing of 150 degrees, and boat A travels north of B. When boat A has traveled 8 km and boat B has traveled 10 km, the distance between them is 12 km. Calculate the bearing of A's route from C. | [
"97.181"
] | [
"Two boats, A and B, leave a port C at the same time on different routes. Boat B travels on a bearing of 150 degrees, and boat A travels north of B. When boat A has traveled 8 km and boat B has traveled 10 km, the distance between them is 12 km. Calculate the bearing of A's route from C."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a91f83602ceda64aebaf4f4bda29841c9bab0b8cfd08f06e10dfe10033c28cb1 | Two boats, each traveling at its own constant speed, leave at the same time from points A and B respectively and travel directly across a river. They first pass each other 200 feet from shore A and next pass each other 20 seconds later, 100 feet from shore B. Compute the speed of the faster boat in feet per second. | [
"30"
] | [
"Two boats, each traveling at its own constant speed, leave at the same time from points A and B respectively and travel directly across a river. They first pass each other 200 feet from shore A and next pass each other 20 seconds later, 100 feet from shore B. Compute the speed of the faster boat in feet per second... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-534ee5fe937a61b09f5ade46c06117e79034e96a9525d8ee771ed1dbdb73856d | Two bodies arc moving along the circumference of a circle. The first body makes the whole circle 5 seconds faster than the second. If they both move in one direction, they will come together every 100 seconds. What portion of the circumference (in degrees) does each body make in one second? | [
"$18^\\circ$; $14^\\circ 24'$"
] | [
"Two bodies arc moving along the circumference of a circle. The first body makes the whole circle 5 seconds faster than the second. If they both move in one direction, they will come together every 100 seconds. What portion of the circumference (in degrees) does each body make in one second?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df9a6c05040122d1caa38b309865a5511a5eee745b12c6361d8bbdc0a5eb26e3 | Two bodies are in uniform motion around a circle in the same direction, and one of them catches up with the other every 46 seconds. If these bodies were in motion in opposite directions, they would meet every 8 seconds. Determine the rates of the bodies if we know that the radius of the circle is equal to 184 cm. | [
"$10\\pi $ cm/sec and $27\\pi $ cm/sec"
] | [
"Two bodies are in uniform motion around a circle in the same direction, and one of them catches up with the other every 46 seconds. If these bodies were in motion in opposite directions, they would meet every 8 seconds. Determine the rates of the bodies if we know that the radius of the circle is equal to 184 cm."... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04cc613b92b887fc7c26cb125748ecbf5048de546dc57fcac80d45a672a817b2 | Two bodies begin moving simultaneously in the same direction from two points which are 20 m apart. One of the bodies, which is behind, moves with a uniform acceleration and covers 25 m in the first second and 1/3 m more than during the preceding second for every subsequent second; the other body, which moves with unifo... | [
"16 s"
] | [
"Two bodies begin moving simultaneously in the same direction from two points which are 20 m apart. One of the bodies, which is behind, moves with a uniform acceleration and covers 25 m in the first second and 1/3 m more than during the preceding second for every subsequent second; the other body, which moves with ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79645c692fa952a006c80404bd3eabb1fa4f087133bacc45104f0ab736dc7e3c | Two bodies move along two sides of a right angle towards its vertex. At the initial instant the body $A$ was 60 m away from the vertex, while the body $B$ -- 80 m. In 3 seconds, the distance between $A$ and $B$ became equal to 70 m, and in another 2 seconds to 50 m. Find the velocity of each body. | [
"6 m/s and 8 m/s"
] | [
"Two bodies move along two sides of a right angle towards its vertex. At the initial instant the body $A$ was 60 m away from the vertex, while the body $B$ -- 80 m. In 3 seconds, the distance between $A$ and $B$ became equal to 70 m, and in another 2 seconds to 50 m. Find the velocity of each body."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88e7ef3fa6e9f812e14abe7dbe05950221a8ca06cbfed259e4bf95b6bf45c638 | Two bodies move in a circle in opposite directions, one of them being in a uniform motion with linear speed $v$ and the other in a uniformly accelerated motion with linear acceleration $a$. At the initial moment of time the bodies are at the same point $A$, and the velocity of the second one is equal to zero. In what t... | [
"t_1=\\left(\\sqrt{5}-1\\right)\\frac{v}{a}"
] | [
"Two bodies move in a circle in opposite directions, one of them being in a uniform motion with linear speed $v$ and the other in a uniformly accelerated motion with linear acceleration $a$. At the initial moment of time the bodies are at the same point $A$, and the velocity of the second one is equal to zero. In w... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ef838ab7e818787c21da497cf194b575874d18bba32e6e125e5df489b000186 | Two bodies move in a circle uniformly in the same direction. The first body makes a circle 2 s quicker than the second and overtakes the second body every 12 s. How much time does it take each body to make a circle? | [
"4 s and 6 s"
] | [
"Two bodies move in a circle uniformly in the same direction. The first body makes a circle 2 s quicker than the second and overtakes the second body every 12 s. How much time does it take each body to make a circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f85fc8bf15938d227df433ff47809ae086f783f03a85c00bfe0db7195e23bf35 | Two bodies move towards each other from two points which are 390 m apart. The first body covers 6 m in the first second and 6 m more than during the preceding second for every subsequent second. The second body moves uniformly at 12 m/s, having started 5 s after the first body. How many seconds after the first start wi... | [
"10 s"
] | [
"Two bodies move towards each other from two points which are 390 m apart. The first body covers 6 m in the first second and 6 m more than during the preceding second for every subsequent second. The second body moves uniformly at 12 m/s, having started 5 s after the first body. How many seconds after the first sta... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eed16c1da525182f07d25b09e54e4b8c9d56eadc3627d51d473ccfdbbae2a319 | Two bodies moving along the circumference of a circle in the same direction come together every 56 minutes. If they were moving with the same speeds as before, but in opposite directions they would meet every 8 minutes. Also, when moving in opposite directions, the distance (along the circumference) between the approac... | [
"20 m/min; 15 m/min; 280 m"
] | [
"Two bodies moving along the circumference of a circle in the same direction come together every 56 minutes. If they were moving with the same speeds as before, but in opposite directions they would meet every 8 minutes. Also, when moving in opposite directions, the distance (along the circumference) between the ap... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33a416b6313c8c12edf060c5755e1beb326ae89c92dbe60e58e4aa1a90973dfd | Two bodies simultaneously emerge from points $A$ and $B$ towards each other. Each body reverses direction upon reaching point $A$ or $B$. The first body travels the path $AB$ in $101$ seconds, and the second body travels the same path $AB$ in $201$ seconds. After $2 \cdot 101 \cdot 201$ seconds, each body will be at it... | [
"100"
] | [
"Two bodies simultaneously emerge from points $A$ and $B$ towards each other. Each body reverses direction upon reaching point $A$ or $B$. The first body travels the path $AB$ in $101$ seconds, and the second body travels the same path $AB$ in $201$ seconds. After $2 \\cdot 101 \\cdot 201$ seconds, each body will b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-51bf730914dace55c8894321daf63325463ed9a90dabb829ea793c65cba3af35 | Two bodies simultaneously start out in the same direction from two points 20 meters apart. The one behind is in uniformly accelerated motion and covers 25 meters during the first second and $\frac{1}{3}$ of a meter more in the next second. The other body is in uniformly decelerated motion and passes 30 meters in the fi... | [
"16"
] | [
"Two bodies simultaneously start out in the same direction from two points 20 meters apart. The one behind is in uniformly accelerated motion and covers 25 meters during the first second and $\\frac{1}{3}$ of a meter more in the next second. The other body is in uniformly decelerated motion and passes 30 meters in ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73adf9fcfdbb691ecf37ec5fa8a06e8e7974deb15d72d2a29acc393bde4c33b5 | Two bodies spaced at $d$ meters are moving towards each other and meet in $a$ seconds. If they move at the same speed as before, but in one direction, they will meet in $b$ seconds. Determine the speed of each body. | [
"$v_1=\\frac{d}{2}\\left(\\frac{1}{a}+\\frac{1}{b}\\right)$; $v_2=\\frac{d}{2}\\left(\\frac{1}{a}-\\frac{1}{b}\\right)$. The problem is solvable only if $a<b$"
] | [
"Two bodies spaced at $d$ meters are moving towards each other and meet in $a$ seconds. If they move at the same speed as before, but in one direction, they will meet in $b$ seconds. Determine the speed of each body."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d0f99cd0c734d9e9442d3c240750be7c132c2ffd29105f41ca87294c26b06d9 | Two bodies, $M_1$ and $M_2$, are uniformly moving towards each other from $A$ and $B$ 60 meters apart. $M_1$ starts out from $A$ 15 seconds before $M_2$ starts out from $B$. At their respective terminals the two bodies turn around and immediately go back at the same speeds as before. Their first encounter takes place i... | [
"The speed of the body $M_1$ is equal to 2 m/s, and that of the body $M_2$, to 3 m/s"
] | [
"Two bodies, $M_1$ and $M_2$, are uniformly moving towards each other from $A$ and $B$ 60 meters apart. $M_1$ starts out from $A$ 15 seconds before $M_2$ starts out from $B$. At their respective terminals the two bodies turn around and immediately go back at the same speeds as before. Their first encounter takes pl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9e868d2775d41530d3488ca26f4e2cb914388e4f18630d4dc2a90f7b2c7a109 | Two bowls contain \( p \) and \( q \) coins respectively, where \( p, q \in \mathbb{N} \). Two players, Bugzpodder and Siuhochung, take turns making moves. A move consists of: 1. Taking one coin from one of the bowls. 2. Taking one coin from each bowl. 3. Moving one coin from one bowl to the other. Bugzpodder starts. T... | [
"Bugzpodder can force a victory if at least one of \\( p \\) or \\( q \\) is odd; Siuhochung can force a victory if both \\( p \\) and \\( q \\) are even."
] | [
"Two bowls contain \\( p \\) and \\( q \\) coins respectively, where \\( p, q \\in \\mathbb{N} \\). Two players, Bugzpodder and Siuhochung, take turns making moves. A move consists of:\n1. Taking one coin from one of the bowls.\n2. Taking one coin from each bowl.\n3. Moving one coin from one bowl to the other.\n\nB... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 16
} | 33 | 0 | 0 |
omr-ws-0e9a6220c5f4f94a66ede4304b07899370acf370c1f8419113c2acef5e1773ee | Two boxes $A$ and $B$ contain an equal number of toys. If one toy, valued at $\$5$, is transferred from $A$ to $B$, the average value of the toys in box $A$ decreases by $\$1$ while the average value of the toys in box $B$ increases by $\$1$. What is the total value of the toys in these two boxes? | [
"12"
] | [
"Two boxes $A$ and $B$ contain an equal number of toys. If one toy, valued at $\\$5$, is transferred from $A$ to $B$, the average value of the toys in box $A$ decreases by $\\$1$ while the average value of the toys in box $B$ increases by $\\$1$. What is the total value of the toys in these two boxes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-d9e910354989c7560591de088752acfbd495fac60b690a05398e0a3b481a3d4e | Two boxes A and B are given. Box A contains 6 balls labeled 1 through 6, and Box B is empty. A die is rolled 100,000 times, and each time a number \(i\) is rolled, the ball labeled \(i\) is moved from one box to the other. Find the probability that Box B is empty after 100,000 rolls. | [
"\\dfrac{1}{32}"
] | [
"Two boxes A and B are given. Box A contains 6 balls labeled 1 through 6, and Box B is empty. A die is rolled 100,000 times, and each time a number \\(i\\) is rolled, the ball labeled \\(i\\) is moved from one box to the other. Find the probability that Box B is empty after 100,000 rolls."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b97ac5c95573660b8cafb82854550837e07118a36bc593c0802ea77d4776d8a3 | Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red, or yellow. If you take any five balls of the same color, at least two of them will always be of the same size (radius). Let $N$ be the minimum number of balls that must be in one of the boxes to guarantee that there are ... | [
"33"
] | [
"Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red, or yellow. If you take any five balls of the same color, at least two of them will always be of the same size (radius). Let $N$ be the minimum number of balls that must be in one of the boxes to guarantee that there... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29c1d16bed39de0ff2a0db1a46d5682f6f615a0ed96a2ae514feab839354cc28 | Two boxes each contain 4 cupcakes. One box has 3 chocolate and 1 vanilla, and the other box has 2 chocolate and 2 vanilla. A box is randomly selected, opened, and a cupcake is randomly selected. This first cupcake is vanilla. If one more cupcake is randomly selected from the same box, what is the probability that it wi... | [
"\\(\\frac{1}{6}\\)"
] | [
"Two boxes each contain 4 cupcakes. One box has 3 chocolate and 1 vanilla, and the other box has 2 chocolate and 2 vanilla. A box is randomly selected, opened, and a cupcake is randomly selected. This first cupcake is vanilla. If one more cupcake is randomly selected from the same box, what is the probability that ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d45ebf4a2d80ec8c973b04f853a5194d9f49770999c2502eac6a7c397fdfcfc | Two boys $A$ and $B$ start at the same time to ride from Port Jervis to Poughkeepsie, which is 60 miles away. $A$ travels 4 miles per hour slower than $B$. $B$ reaches Poughkeepsie and immediately turns back, meeting $A$ 12 miles from Poughkeepsie. What was the rate of $A$ in miles per hour? | [
"4 mph"
] | [
"Two boys $A$ and $B$ start at the same time to ride from Port Jervis to Poughkeepsie, which is 60 miles away. $A$ travels 4 miles per hour slower than $B$. $B$ reaches Poughkeepsie and immediately turns back, meeting $A$ 12 miles from Poughkeepsie. What was the rate of $A$ in miles per hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-1635ef9eb4da156182857aff2eaa10970e1af3772a5059b6e49f81fa6fea1d33 | Two boys are given a bag of potatoes, each bag containing 150 tubers. They take turns transferring the potatoes, where in each turn they transfer a non-zero number of tubers from their bag to the other boy's bag. Their moves must satisfy the following condition: In each move, a boy must move more tubers than he had in ... | [
"32"
] | [
"Two boys are given a bag of potatoes, each bag containing 150 tubers. They take turns transferring the potatoes, where in each turn they transfer a non-zero number of tubers from their bag to the other boy's bag. Their moves must satisfy the following condition: In each move, a boy must move more tubers than he ha... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bc249b1b099271eba1aab09f4b5ad19890a5a1bb66b618ff750c76889a39849 | Two boys play a game where they take turns writing a digit, either at the right or left of the current number, starting with the number 1. The first boy wants to avoid forming a perfect square, while the second boy wants to form a perfect square. If the second boy follows an optimal strategy, what is the smallest numbe... | [
"7"
] | [
"Two boys play a game where they take turns writing a digit, either at the right or left of the current number, starting with the number 1. The first boy wants to avoid forming a perfect square, while the second boy wants to form a perfect square. If the second boy follows an optimal strategy, what is the smallest ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-dacd121816454105dc31f99db754dccbb98e99b2ce3dba04d443c360027d40d9 | Two boys were 3/8 of the way across a railroad bridge when they saw a train approaching at 60 km/hr. They ran at the same speed towards opposite ends of the bridge, each arriving at an end of the bridge just as the train arrived at that end. How fast did the boys run, in km/hr? | [
"15 km/hr"
] | [
"Two boys were 3/8 of the way across a railroad bridge when they saw a train approaching at 60 km/hr. They ran at the same speed towards opposite ends of the bridge, each arriving at an end of the bridge just as the train arrived at that end. How fast did the boys run, in km/hr?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-8e70e14c3b7faaef2c487fb530e396b747be2df76f0890217d5badb0253815bd | Two boys were walking through a narrow railroad tunnel which was exactly one mile long. The tunnel had distance markers every tenth of a mile, and just as they got to the 0.6 mile marker, they heard a train whistle from the direction they were walking. Both boys immediately started running in opposite directions. One b... | [
"50 miles per hour."
] | [
"Two boys were walking through a narrow railroad tunnel which was exactly one mile long. The tunnel had distance markers every tenth of a mile, and just as they got to the 0.6 mile marker, they heard a train whistle from the direction they were walking. Both boys immediately started running in opposite directions. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-081d41e64e9c4f37bef5a6147697dd5876d7f2902098b9246ac3fdf3a1ce5ce5 | Two boys, $A$ and $B$, start at the same time to ride from Irvine to San Diego, which is 60 miles away. $A$ travels 4 miles per hour slower than $B$. Once $B$ arrives at San Diego, he turns back and meets $A$ 12 miles from San Diego. What is the rate of $A$? | [
"8 miles per hour"
] | [
"Two boys, $A$ and $B$, start at the same time to ride from Irvine to San Diego, which is 60 miles away. $A$ travels 4 miles per hour slower than $B$. Once $B$ arrives at San Diego, he turns back and meets $A$ 12 miles from San Diego. What is the rate of $A$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-17d025afb242cf6cfb3f6fc877ff131079c438acbda6ea86a625076a33dfff21 | Two brothers are walking to the next bus stop. After covering $1/3$ of the distance, they see a bus approaching the stop. One brother runs backwards towards the starting point, and the other runs forward towards the bus stop, both at the same speed. They both reach their respective stops exactly when the bus arrives. I... | [
"10 km/h"
] | [
"Two brothers are walking to the next bus stop. After covering $1/3$ of the distance, they see a bus approaching the stop. One brother runs backwards towards the starting point, and the other runs forward towards the bus stop, both at the same speed. They both reach their respective stops exactly when the bus arriv... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-55700af7ed2cfb8bf18cc54c82df1a784281ad4cc22508857808613154605489 | Two brothers had a flock of sheep. They sold the flock and got as many rubles for every sheep as was the number of the sheep in the flock. The money was shared in the following way: first the elder brother took ten rubles from the cash, then the younger brother took ten rubles, after which the elder brother took ten ru... | [
"Two rubles"
] | [
"Two brothers had a flock of sheep. They sold the flock and got as many rubles for every sheep as was the number of the sheep in the flock. The money was shared in the following way: first the elder brother took ten rubles from the cash, then the younger brother took ten rubles, after which the elder brother took t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89fbbab9cffc79100a4c386f70d771fa424da1ce0afed461d2912ed2f0222434 | Two brothers sold a herd of sheep, receiving as many dollars per sheep as the number of sheep in the herd. They divided the money by alternately taking $10 each, until the younger brother found less than $10 remaining on his turn. To make the division fair, the older brother gave the younger brother his penknife. How m... | [
"4"
] | [
"Two brothers sold a herd of sheep, receiving as many dollars per sheep as the number of sheep in the herd. They divided the money by alternately taking $10 each, until the younger brother found less than $10 remaining on his turn. To make the division fair, the older brother gave the younger brother his penknife. ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-616ac7865318827dd74d10c5ea499afe0e36812d1c264a5107d790e29c8498c0 | Two brothers sold their flock of sheep for as many pieces of gold per sheep as the number of sheep in the original flock. They shared the money by giving the first 10 pieces to the elder brother, the next 10 pieces to the younger brother, and so on. At the last turn, the younger brother’s share was less than 10 pieces ... | [
"2"
] | [
"Two brothers sold their flock of sheep for as many pieces of gold per sheep as the number of sheep in the original flock. They shared the money by giving the first 10 pieces to the elder brother, the next 10 pieces to the younger brother, and so on. At the last turn, the younger brother’s share was less than 10 pi... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-683e0e29188fdf3df54278e5cb68dc6f002cd5ebf4738ed80aa2f60aa1c7146d | Two bugs are crawling on a cube, starting from the same vertex $A$. Each minute, each bug moves to one of the three adjacent vertices. Find the probability that both bugs reach the farthest vertex $B$ from $A$ within 2 minutes of each other. | [
"\\(\\frac{1}{27}\\)"
] | [
"Two bugs are crawling on a cube, starting from the same vertex $A$. Each minute, each bug moves to one of the three adjacent vertices. Find the probability that both bugs reach the farthest vertex $B$ from $A$ within 2 minutes of each other."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f32a20642fd0d916e5708a22554ff65c568561f4c004e9f72a4feb40862b39ec | Two bugs are placed at opposite vertices of a regular octahedron. Each day, each bug randomly selects with equal probability one of the adjacent vertices and walks along the edge connecting the two vertices. If $\mathbb{E}$ is the expected number of days it will take for the two bugs to meet each other for the first ti... | [
"46"
] | [
"Two bugs are placed at opposite vertices of a regular octahedron. Each day, each bug randomly selects with equal probability one of the adjacent vertices and walks along the edge connecting the two vertices. If $\\mathbb{E}$ is the expected number of days it will take for the two bugs to meet each other for the fi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df4404164fb5fd5c06dd46ffd097ec8d1c3bf1491732c456f5c595da7de9a784 | Two bugs start at a vertex of an equilateral triangle with side length 3 and travel a distance of 1 towards a second vertex. Bug A continues by turning \(60^\circ\) towards the third vertex and traveling \(\frac{2}{3}\) of the distance it just traveled, repeating this process. Bug B travels \(\frac{2}{3}\) of its previ... | [
"B"
] | [
"Two bugs start at a vertex of an equilateral triangle with side length 3 and travel a distance of 1 towards a second vertex. Bug A continues by turning \\(60^\\circ\\) towards the third vertex and traveling \\(\\frac{2}{3}\\) of the distance it just traveled, repeating this process. Bug B travels \\(\\frac{2}{3}\\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5ffd3ec5c00288b9e8c06e7dd8a0dfd73636832e1645a7a0e8d375333b95d9cb | Two buildings are 12 feet and 24 feet above the ground. A rope is connected from the top of each building to the bottom of the other building. These ropes touch at a point $N$ inches from the ground. Find $N$. | [
"96 inches"
] | [
"Two buildings are 12 feet and 24 feet above the ground. A rope is connected from the top of each building to the bottom of the other building. These ropes touch at a point $N$ inches from the ground. Find $N$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ef493d5d756cd619ce1bc2bcf5d7112378a558fe917a523f2ce8c53fad7a292 | Two buildings are connected by a rope that is 20 meters long. The rope droops 10 meters below the roofs of the buildings. How far apart are the buildings from each other? | [
"0"
] | [
"Two buildings are connected by a rope that is 20 meters long. The rope droops 10 meters below the roofs of the buildings. How far apart are the buildings from each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2661cfb8c4066c19adb87f4c08965a858740b77571a99a413c42fca8ac02e4f0 | Two bullet trains are each traveling towards each other at 180 kph. A passenger notices that it takes 5 seconds for the other train to pass him. How long is the other train? | [
"500 m"
] | [
"Two bullet trains are each traveling towards each other at 180 kph. A passenger notices that it takes 5 seconds for the other train to pass him. How long is the other train?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-346e382e5089f8e15a06f0a52c10dc6f7d342dd7962173d4e32c07e507756fa0 | Two bus routes stop at a certain bus stop. The A bus comes every one hour, and the B bus comes at regular intervals of a different length. When grandma rests on the bench, one A bus and two B buses come by. Later, when grandpa rests on the same bench, eight A buses come by. What is the minimum number of B buses that mu... | [
"more than 9."
] | [
"Two bus routes stop at a certain bus stop. The A bus comes every one hour, and the B bus comes at regular intervals of a different length. When grandma rests on the bench, one A bus and two B buses come by. Later, when grandpa rests on the same bench, eight A buses come by. What is the minimum number of B buses th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1b3a19cb75d117e63fd620ffd7546f90f1f7b0892fded9574b32e6ce491b189 | Two buses P and Q start simultaneously from cities A and B respectively towards city C. By the time Q reaches C, P is exactly halfway to C. Immediately after Q reaches C, it starts travelling towards A and it crosses P at a point 165 km from A. The ratio of the speeds of P and Q is 3 : 5. Assume that the roads joining ... | [
"48"
] | [
"Two buses P and Q start simultaneously from cities A and B respectively towards city C. By the time Q reaches C, P is exactly halfway to C. Immediately after Q reaches C, it starts travelling towards A and it crosses P at a point 165 km from A. The ratio of the speeds of P and Q is 3 : 5. Assume that the roads joi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-dedfd011a99de9bf2f54f6cf98c804c0da4ff2196d72f13b3164179f34b3cb9a | Two buses start at the same time from points A and B to B and A respectively, each with constant velocity. They reach their destinations after 1 hour and 4 hours, respectively, of passing each other. What is the ratio of the velocities of the two buses? | [
"2"
] | [
"Two buses start at the same time from points A and B to B and A respectively, each with constant velocity. They reach their destinations after 1 hour and 4 hours, respectively, of passing each other. What is the ratio of the velocities of the two buses?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5435c273405e490d2fb902f9cd23ebf3b18534855a988db26b19337b52888771 | Two candles - a green one and a cyan one - are lit at the same time. The green candle is twice as long as the cyan candle, but it takes the cyan candle twice as long to burn out completely as the green one. When the green candle becomes one-half as long as the cyan candle, what fraction of the green candle's original l... | [
"\\(\\frac{1}{7}\\)"
] | [
"Two candles - a green one and a cyan one - are lit at the same time. The green candle is twice as long as the cyan candle, but it takes the cyan candle twice as long to burn out completely as the green one. When the green candle becomes one-half as long as the cyan candle, what fraction of the green candle's origi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e0300e5e50cbedb927700d34147097e15d734471824c4f3ea6c5b8a4168b36a2 | Two candles of equal length start burning at the same time. One candle burns down in 3 hours and the other takes 5 hours to burn down. Find the exact number of hours of burning before one candle is three times the length of the other. | [
"\\(\\frac{5}{2}\\)"
] | [
"Two candles of equal length start burning at the same time. One candle burns down in 3 hours and the other takes 5 hours to burn down. Find the exact number of hours of burning before one candle is three times the length of the other."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-21c346a085e205a59642057661c094d2ce82198eee2603db48262f87ebd59ca1 | Two candles of equal length start burning at the same time. One candle burns down in 4 hours, and the other in 5 hours. How many hours will they burn before one candle is 3 times the length of the other? | [
"\\( \\frac{40}{11} \\)"
] | [
"Two candles of equal length start burning at the same time. One candle burns down in 4 hours, and the other in 5 hours. How many hours will they burn before one candle is 3 times the length of the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c7f461db3c12a99271d68b84b9d8c88db9ceac5085d6406dffc19835f8fc8093 | Two candles of equal length start burning at the same time. One of the candles will burn in $4$ hours, and the other in $5$ hours. How long in hours will they have to burn before one candle is $3$ times the length of the other? | [
"\\dfrac{40}{11}"
] | [
"Two candles of equal length start burning at the same time. One of the candles will burn in $4$ hours, and the other in $5$ hours. How long in hours will they have to burn before one candle is $3$ times the length of the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f6d5bae4580ab2e34a6cd4b89e4d19ed0256fafbdd990e02a718cb1228930a9 | Two candles of equal length start burning at the same time. One will last 4 hours and the other 6 hours. How many hours will they burn before one candle is twice the length of the other? | [
"12"
] | [
"Two candles of equal length start burning at the same time. One will last 4 hours and the other 6 hours. How many hours will they burn before one candle is twice the length of the other?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b0d0bb66d259a996a9cd997afb5cbe855e961ff51303c95a7706488c4a5cc76 | Two candles of the same height are lit at the same time. One candle burns completely in 4 hours, and the other in 5 hours. At what time will one candle be twice the height of the other candle? | [
"200 minutes"
] | [
"Two candles of the same height are lit at the same time. One candle burns completely in 4 hours, and the other in 5 hours. At what time will one candle be twice the height of the other candle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3bb8ebafe8815153ac080c3863608a36b79a9e7375428c9b5cb0e44ccb36bab6 | Two candles of the same initial length are made of different materials so that one burns out completely in 3 hours and the other in 4 hours. At what time p.m. should the candles be lit so that, at 4 p.m., one stub is twice the length of the other? | [
"1:36 p.m."
] | [
"Two candles of the same initial length are made of different materials so that one burns out completely in 3 hours and the other in 4 hours. At what time p.m. should the candles be lit so that, at 4 p.m., one stub is twice the length of the other?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 30
} | 47 | 0 | 0 |
omr-ws-e3648e7ed50407561524911eaf60af1c3136cb927ce7feec881c168e18ca64d6 | Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in 3 hours and the other in 4 hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? | [
"1:36"
] | [
"Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in 3 hours and the other in 4 hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a04bc8e9be45f80095e76f27fba0eeaef57b962d49ba7a1306ffc4cd7a6d2132 | Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in 3 hours and the other in 4 hours. At what time PM should the candles be lit so that, at 4 PM, one stub is twice the length of the other? | [
"3:35 PM"
] | [
"Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in 3 hours and the other in 4 hours. At what time PM should the candles be lit so that, at 4 PM, one stub is twice the length of the other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fdb4702596f860b6d8f75b9890edf41af86f45e8fa04f93cd3f3d353e19dd71 | Two cans containing coffee have the same shape and are made out of the same tin. One can weighs 2 kg and is 12 cm high; the other weighs 1 kg and is 9.5 cm high. Find the net weight of the coffee in both cans. | [
"2.904"
] | [
"Two cans containing coffee have the same shape and are made out of the same tin. One can weighs 2 kg and is 12 cm high; the other weighs 1 kg and is 9.5 cm high. Find the net weight of the coffee in both cans."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-b9e2b9751ba599c358bb88577922a58b4b1a780f6b4b10c65cc52d830ba1b106 | Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12? | [
"\\frac{35}{663}"
] | [
"Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9abb31c826284f019f2b5a180b2dd83c66408be2dfef66fa2196b68c0870894 | Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the second card is a 10? | [
"\\frac{1}{52}"
] | [
"Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the second card is a 10?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b64bb5067380ecb788a0fdae35f58ab1b6d2a08ba0a77b9a82f1dd270ad55cf | Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 6 and the second card is a Queen? | [
"\\dfrac{4}{663}"
] | [
"Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 6 and the second card is a Queen?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72e5a41f18b648ee0b1354b9a9c0381182c6499a69b82f0a1972041204b428f4 | Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\diamondsuit$ and the second card is an ace? | [
"\\dfrac{1}{52}"
] | [
"Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\\diamondsuit$ and the second card is an ace?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc46423e0e0a8be0ba507bd3c0ac6e559a9a8c5807c74428cda0489abf1e4813 | Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\heartsuit$? | [
"\\dfrac{1}{52}"
] | [
"Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\\heartsuit$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bc275a9e9d9e76910ada04b5cc3ba3425ce317973a7de21ca6ce53263cdd59a | Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\diamondsuit$ and the second card dealt is a $\spadesuit$? | [
" \\frac{13}{204}"
] | [
"Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\diamondsuit$ and the second card dealt is a $\\spadesuit$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3c55e79e8d3e9406a54c9536144470a19b4d68baeb9d62ebec8b9b92f88ab0c | Two cards are drawn at random from a deck of 52 cards, then returned and the deck is shuffled. Two more cards are drawn at random. What is the probability that exactly one card is common between the two draws? | [
"\\(\\frac{100}{(26 \\cdot 51)^2}\\)"
] | [
"Two cards are drawn at random from a deck of 52 cards, then returned and the deck is shuffled. Two more cards are drawn at random. What is the probability that exactly one card is common between the two draws?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab3688e8404e8e4b166ee52c74c3321b3c88c9bcf8a952e40b1935413e5af9fc | Two cards are drawn from a regular deck of 52 cards without replacement. Let \( x \) be the value of the first card and \( y \) be the value of the second card, where \( A=1 \), \( J=11 \), \( Q=12 \), and \( K=13 \). What is the probability that \( x \) and \( y \) satisfy the equation \( \log_3 (x+y) - \log_3 x - \lo... | [
"\\(\\frac{11}{663}\\)"
] | [
"Two cards are drawn from a regular deck of 52 cards without replacement. Let \\( x \\) be the value of the first card and \\( y \\) be the value of the second card, where \\( A=1 \\), \\( J=11 \\), \\( Q=12 \\), and \\( K=13 \\). What is the probability that \\( x \\) and \\( y \\) satisfy the equation \\( \\log_3... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 30
} | 42 | 0 | 0 |
omr-ws-7df58437fcce90429943e5f614aed966b5186c62f5c7bd95f825e68166c94000 | Two cards are drawn from a well-shuffled pack of 52 cards. Find the probability that one of them is a red card and the other is a queen. | [
"\\(\\frac{101}{1326}\\)"
] | [
"Two cards are drawn from a well-shuffled pack of 52 cards. Find the probability that one of them is a red card and the other is a queen."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-f16c112530beae4358fde2010805a79bdac68f4a97b2e3f33d634ef540eb68cf | Two cards are drawn without replacement from a shuffled deck of 52 cards. Determine the probability that the first card is a heart and the second card is the queen of hearts. | [
"\\(\\frac{1}{221}\\)"
] | [
"Two cards are drawn without replacement from a shuffled deck of 52 cards. Determine the probability that the first card is a heart and the second card is the queen of hearts."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-eccfb2027f52272dc5da6b43e10fd027a1dab82d2b9983fc115c1613a56ea3c8 | Two cards are drawn without replacement from a shuffled deck of 52 cards. Determine the probability that the first card is the queen of hearts and the second card is a heart. | [
"\\(\\frac{1}{221}\\)"
] | [
"Two cards are drawn without replacement from a shuffled deck of 52 cards. Determine the probability that the first card is the queen of hearts and the second card is a heart."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-f5c5f90c4471f1c705e5966faf5f1e549e1a25090d89e6b76fb77d369c944a00 | Two cards are drawn without replacement from a standard deck of cards. If it is known that neither card is a spade, what is the probability that both cards are diamonds? | [
"\\(\\frac{2}{19}\\)"
] | [
"Two cards are drawn without replacement from a standard deck of cards. If it is known that neither card is a spade, what is the probability that both cards are diamonds?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-9a226c3dc5069955eee44d3493eff46c963d726eba61e212a882142c79770a08 | Two cards are randomly drawn without replacement from a standard deck. Let B be the event that both cards are aces. Let A_s be the event that the ace of spades is chosen. Let A be the event that at least one card is an ace. Find: (a) P(B|A_s) (b) P(B|A) | [
"\\( P(B|A_s) = \\frac{1}{17} \\) and \\( P(B|A) = \\frac{1}{33} \\)"
] | [
"Two cards are randomly drawn without replacement from a standard deck. Let B be the event that both cards are aces. Let A_s be the event that the ace of spades is chosen. Let A be the event that at least one card is an ace. Find:\n(a) P(B|A_s)\n(b) P(B|A)"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-0324d375a133e20b18c6409ba3f266ee3ace0090c23eeecc45858b1dc602151d | Two cards are taken out from a pack of 52 cards. What is the probability that a card drawn from the remaining 50 cards is an Ace? | [
"\\(\\frac{4}{52}\\) or simplified \\(\\frac{1}{13}\\)."
] | [
"Two cards are taken out from a pack of 52 cards. What is the probability that a card drawn from the remaining 50 cards is an Ace?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4a8ca05d9cb80138e11aac4b2a2aef789c3bf9d9e4056f87295d99e99f053ea7 | Two cars are moving towards each other at constant speeds of 40 and 50 km/h respectively along two intersecting streets. Assuming that the streets intersect at right angles and knowing that at a certain moment in time the cars are respectively 2 and 3 km away from the intersection, when will the two cars be closest? | [
"$\\frac{23}{410}$ h"
] | [
"Two cars are moving towards each other at constant speeds of 40 and 50 km/h respectively along two intersecting streets. Assuming that the streets intersect at right angles and knowing that at a certain moment in time the cars are respectively 2 and 3 km away from the intersection, when will the two cars be closes... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9655fd36e6528417f9ee5488cefe070cb3a6cc7ef4c17bbfd41bdf0a2a3e362 | Two cars drive along a highway one after the other at a distance of 20 m from each other at 24 m/s. The drivers notice an obstacle, apply the brakes, and change to a uniformly decelerated motion with the accelerations $a_1$ and $a_2$ ($a_1<0$ and $a_2<0$) until they come to a complete stop. The driver of the front car ... | [
"The second car was the first to stop. $a_2=-8\\mbox{ m/s}^2$"
] | [
"Two cars drive along a highway one after the other at a distance of 20 m from each other at 24 m/s. The drivers notice an obstacle, apply the brakes, and change to a uniformly decelerated motion with the accelerations $a_1$ and $a_2$ ($a_1<0$ and $a_2<0$) until they come to a complete stop. The driver of the fron... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c447f5e760b5df555d838e528144817c176b018cc7aa036f964fc565c354a294 | Two cars enter the Florida Turnpike at Commercial BLVD at 8am, each heading for Wildwood. One car's average speed is 10 miles per hour more than the other's. The faster car arrives at Wildwood at 11am, 1/2 hour before the other car. What is the average speed of each car? How far did each car travel? | [
"The average speed of the faster car is 70 mph, and the average speed of the slower car is 60 mph. Each car traveled 210 miles."
] | [
"Two cars enter the Florida Turnpike at Commercial BLVD at 8am, each heading for Wildwood. One car's average speed is 10 miles per hour more than the other's. The faster car arrives at Wildwood at 11am, 1/2 hour before the other car. What is the average speed of each car? How far did each car travel?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-e00b8094f3541c5fbc14f28973be79a2fdbdd4358f1e38434d6a957aeec07687 | Two cars leave a restaurant at the same time and travel along a straight highway in opposite directions. After three hours, they are 300 miles apart. If one car travels 20 mph faster than the other, find the speed of each car. | [
"40 mph and 60 mph"
] | [
"Two cars leave a restaurant at the same time and travel along a straight highway in opposite directions. After three hours, they are 300 miles apart. If one car travels 20 mph faster than the other, find the speed of each car."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 29
} | 31 | 0 | 0 |
omr-ws-3d8ef229bb4a0dec63192f8fa32ab243b0f8dab1205a8b7223398628e2fdf1ca | Two cars leave an intersection at the same time, one headed northwest at 35 mph, the other headed east at 40 mph. To the nearest 1 mph, how fast are they moving apart? | [
"69 mph"
] | [
"Two cars leave an intersection at the same time, one headed northwest at 35 mph, the other headed east at 40 mph. To the nearest 1 mph, how fast are they moving apart?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-ef92e24324c0c2cb4f75c8785612a2803a6bba1473f81cba9fa710ae2867d30f | Two cars leave from Bobtown and AvinashCity towards each other and meet at a place 60 km from Bobtown. After reaching their destinations, they turn around and meet again at a place 50 km from AvinashCity. Find the distance between AvinashCity and Bobtown. | [
"115 km"
] | [
"Two cars leave from Bobtown and AvinashCity towards each other and meet at a place 60 km from Bobtown. After reaching their destinations, they turn around and meet again at a place 50 km from AvinashCity. Find the distance between AvinashCity and Bobtown."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7f096ded3d50cce18118953bbd43b19b97828403a867269c494c297ccba94e77 | Two cars leave the same point simultaneously and start in the same direction. The speed of the first car is 40 km/h, the speed of the other car is 125\% that of the first. Thirty minutes later, a third car starts from the same point in the same direction. It overtakes the first car and 1.5 h later it overtakes the seco... | [
"60 km/h"
] | [
"Two cars leave the same point simultaneously and start in the same direction. The speed of the first car is 40 km/h, the speed of the other car is 125\\% that of the first. Thirty minutes later, a third car starts from the same point in the same direction. It overtakes the first car and 1.5 h later it overtakes th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4f894a6b8a6b075fc7409cb23ac7c03e6fcd61a3750091013a59b048222b135 | Two cars leave the towns of Knossos and Phaistos at the same time and travel towards each other, meeting at Gortyns. The car from Knossos travels 20 km/h faster than the car from Phaistos. The distance from Knossos to Gortyns is 10 km more than the distance from Phaistos to Gortyns. The car from Phaistos takes 35 minut... | [
"130km"
] | [
"Two cars leave the towns of Knossos and Phaistos at the same time and travel towards each other, meeting at Gortyns. The car from Knossos travels 20 km/h faster than the car from Phaistos. The distance from Knossos to Gortyns is 10 km more than the distance from Phaistos to Gortyns. The car from Phaistos takes 35 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-7b702a6dfe6b7fbf6e25f69ffcb010591f4da11262d2974776ba5c6499a47f70 | Two cars leave the towns of Knossos and Phaistos at the same time and travel towards each other, passing each other at Gortyns. The car from Knossos travels 20 km/h faster than the car from Phaistos. The distance from Knossos to Gortyns is 10 km more than the distance from Phaistos to Gortyns. The car from Phaistos tak... | [
"130 km"
] | [
"Two cars leave the towns of Knossos and Phaistos at the same time and travel towards each other, passing each other at Gortyns. The car from Knossos travels 20 km/h faster than the car from Phaistos. The distance from Knossos to Gortyns is 10 km more than the distance from Phaistos to Gortyns. The car from Phaisto... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-d1bb687339e885b265cea42dbc9d15856df324192a4d59709f0b902ff8d7e74d | Two cars lie on the Cartesian plane, one at the origin and one at $(1, 0)$. The car at $(1, 0)$ starts moving in the positive $y$ direction, and the car at the origin starts moving at the same speed, always directly toward the other car. What is the path traced by the car starting at the origin? | [
"y = \\frac{x^2}{4} - \\frac{x}{2} - \\frac{1}{2} \\ln(1 - x)"
] | [
"Two cars lie on the Cartesian plane, one at the origin and one at $(1, 0)$. The car at $(1, 0)$ starts moving in the positive $y$ direction, and the car at the origin starts moving at the same speed, always directly toward the other car. What is the path traced by the car starting at the origin?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-47c8541b6ee31efff79d340294e5177322204a230c880c7975db36645a22695b | Two cars pass exit $A$ simultaneously and head towards exit $B$, which is 245 miles away. Three hours later, the distance of the slower car from $A$ is nine times the distance of the faster car from $B$. If the difference in their speeds is a prime number, how fast (in mph) is the faster car traveling? | [
"74 mph"
] | [
"Two cars pass exit $A$ simultaneously and head towards exit $B$, which is 245 miles away. Three hours later, the distance of the slower car from $A$ is nine times the distance of the faster car from $B$. If the difference in their speeds is a prime number, how fast (in mph) is the faster car traveling?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-8dbf9ae5f7cfd1db82fdd4325af09d8ea4fa6671741086dfd9cfee8c7a48f595 | Two cars simultaneously start out from a point and proceed in the same direction, one of them going at a speed of 50 km/h and the other at 40 km/h. In half an hour a third car starts out from the same point and overtakes the first car 1.5 hours after catching up with the second car. Determine the speed of the third car... | [
"60 km/h"
] | [
"Two cars simultaneously start out from a point and proceed in the same direction, one of them going at a speed of 50 km/h and the other at 40 km/h. In half an hour a third car starts out from the same point and overtakes the first car 1.5 hours after catching up with the second car. Determine the speed of the thir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12bb9bbe3b14e421eb9e097961f261c2ef9b1291219d3ec9f54348d07944c8d3 | Two cars start moving from the same point. One travels south at 30 km/h and the other travels west at 72 km/h. At what rate is the distance between the cars increasing two hours later? | [
"78"
] | [
"Two cars start moving from the same point. One travels south at 30 km/h and the other travels west at 72 km/h. At what rate is the distance between the cars increasing two hours later?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16e90584049314c577c72d7460794494ea71f37df598c71cf5c29e4c7130a716 | Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later? | [
"65 mi/h"
] | [
"Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3a91e6436c32e697a9b1c6e30389c08439caf91a80cb500627fe11a433887b69 | Two cars started simultaneously from point $A$ and drove along the highway and an hour later a third car started after them. An hour later the distance between the third and the first car decreased by a factor of 1.5 and that between the third and the second car decreased two times. The speed of which car, the first or... | [
"The speed of the first car is 9/8 times as high as that of the second"
] | [
"Two cars started simultaneously from point $A$ and drove along the highway and an hour later a third car started after them. An hour later the distance between the third and the first car decreased by a factor of 1.5 and that between the third and the second car decreased two times. The speed of which car, the fi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34170eeaaef3c21bac13fc185693a3f8072559b54cfc1b1f8f26d0ac9739ae35 | Two cars travel around a track at equal and constant speeds, each completing a lap every hour. The first car starts at time \( t = 0 \) and the second car starts at an arbitrary later time \( t = T > 0 \). Let \( N \) be the total number of hours during the motion in which the first car has completed exactly twice as m... | [
"1"
] | [
"Two cars travel around a track at equal and constant speeds, each completing a lap every hour. The first car starts at time \\( t = 0 \\) and the second car starts at an arbitrary later time \\( t = T > 0 \\). Let \\( N \\) be the total number of hours during the motion in which the first car has completed exactly... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-678a6a2518df1419d928fa9570938adf478cb835a08a4b198e323ebf0bf1b743 | Two cars travel on a straight road at a speed of 60 km/h in the same direction. There are traffic lights along the road, each with a cycle of 1 minute green and 1 minute red. Let the distance between the two cars be \(d\) km. Determine the maximum value of \(d\) such that the two cars can still potentially meet at some... | [
"1"
] | [
"Two cars travel on a straight road at a speed of 60 km/h in the same direction. There are traffic lights along the road, each with a cycle of 1 minute green and 1 minute red. Let the distance between the two cars be \\(d\\) km. Determine the maximum value of \\(d\\) such that the two cars can still potentially mee... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-55e4da5832df6b28c755bb0cad5c1b1457fd43d0d224c28cbd04f15047fd5364 | Two cats, Bill and Tom, play a game where they take turns eating fishes from a heap of 50 fishes. On each turn, a player can eat exactly 1, 4, or 7 fishes. The player who eats the last fish wins. If both cats play optimally, who wins the game? | [
"Tom wins."
] | [
"Two cats, Bill and Tom, play a game where they take turns eating fishes from a heap of 50 fishes. On each turn, a player can eat exactly 1, 4, or 7 fishes. The player who eats the last fish wins. If both cats play optimally, who wins the game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-654779e655910f64215426b9400ecff8dc420c4754088b68e863d80880fe78a0 | Two ceiling fans hang one above the other. The top fan has three blades and makes $1.4$ clockwise revolutions per second, while the bottom fan has four blades and makes $0.75$ clockwise revolutions per second. Assuming that no blades are initially aligned, how many times in one minute does one of the top blades overtak... | [
"468"
] | [
"Two ceiling fans hang one above the other. The top fan has three blades and makes $1.4$ clockwise revolutions per second, while the bottom fan has four blades and makes $0.75$ clockwise revolutions per second. Assuming that no blades are initially aligned, how many times in one minute does one of the top blades ov... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-58290cd17e937747795c1f785c2234ae139735179e9f0a48fa6adf2c4e25dd05 | Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to C. Initially, there are 40 grams of A and 50 grams of B, and for each gram of B, 2 grams of A are used. It is observed that 10 grams of C are formed i... | [
"29 grams (at 20 minutes), 60 grams (limiting amount)"
] | [
"Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to C. Initially, there are 40 grams of A and 50 grams of B, and for each gram of B, 2 grams of A are used. It is observed that 10 grams of C are for... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d02a974675d5d6d6088dcd53952bd9e549a89fce2efb0a182583eb2343fd66e2 | Two children play a game with 27 oranges, where each child can eat between 1 and 4 oranges per turn. The winner is the child who eats an even number of oranges in total. How should the first child play to ensure they always win? | [
"The first child should eat 3 oranges on their first turn and then adjust their number of oranges eaten in each subsequent turn to ensure the total number of oranges eaten by both children in each round is 5."
] | [
"Two children play a game with 27 oranges, where each child can eat between 1 and 4 oranges per turn. The winner is the child who eats an even number of oranges in total. How should the first child play to ensure they always win?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af999c0404a766bc9acbf8bb062e59387dcfb964feb287428894b664cd965dfa | Two children, Alina and Bogdan, play a game with natural numbers from 1 to 100 written in ascending order. They take turns filling the 99 spaces between consecutive numbers with either "+" or "*". Alina starts. If the final result is an odd number, Alina wins; otherwise, Bogdan wins. Determine which child has a winning... | [
"Alina"
] | [
"Two children, Alina and Bogdan, play a game with natural numbers from 1 to 100 written in ascending order. They take turns filling the 99 spaces between consecutive numbers with either \"+\" or \"*\". Alina starts. If the final result is an odd number, Alina wins; otherwise, Bogdan wins. Determine which child has ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-56afa34c2fd59f1d3ccd0951724eac9a85fda512130fac07e9d496e49203b7b2 | Two chords $AB$ and $CD$ intersect at point $P$ inside a circle. If $AP = 5$, $PB = x$, $CP = 4$, and $PD = y$, find the value of $xy$. | [
"20"
] | [
"Two chords $AB$ and $CD$ intersect at point $P$ inside a circle. If $AP = 5$, $PB = x$, $CP = 4$, and $PD = y$, find the value of $xy$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3972782c49560615b644d86791142107831ea341abd5c853eb33ce3a31f743b8 | Two chords $AB$ and $CD$ of a circle intersect at point $K$. Point $A$ divides the arc $CAD$ into two equal parts. Given that $AK = 2$ and $KB = 6$, determine the length of the chord $AD$. | [
"4"
] | [
"Two chords $AB$ and $CD$ of a circle intersect at point $K$. Point $A$ divides the arc $CAD$ into two equal parts. Given that $AK = 2$ and $KB = 6$, determine the length of the chord $AD$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c270a253e8d1343b3da80bf834c896703e6915dd3229307aaa88e9db1a1c79a | Two chords $AB$ and $CD$ of a circumference form an angle of $60^\circ$, and the distance from the center of the circumference to each of these chords is equal to $\sqrt[4]{3}$. If the area of the shaded surface inside the circle is $16$, find the unshaded surface area. | [
"17"
] | [
"Two chords $AB$ and $CD$ of a circumference form an angle of $60^\\circ$, and the distance from the center of the circumference to each of these chords is equal to $\\sqrt[4]{3}$. If the area of the shaded surface inside the circle is $16$, find the unshaded surface area."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1a9b5c2d3443058d1ea6a371839ac711f87a55b336a44ec4a982f24e9f18937 | Two chords $AC$ and $BD$ of a circle with center $M$ intersect at $P$. The circumcenters of the triangles $PBC$ and $PDA$ are $E$ and $F$, respectively, and the circumcircles intersect for a second time at $Q$. If $MF = 10$, find the length of $QE$. | [
"10"
] | [
"Two chords $AC$ and $BD$ of a circle with center $M$ intersect at $P$. The circumcenters of the triangles $PBC$ and $PDA$ are $E$ and $F$, respectively, and the circumcircles intersect for a second time at $Q$. If $MF = 10$, find the length of $QE$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d585eb76f015f3243f309c05162532e306dc67f46def9bae61a82f1c6eec0ebf | Two chords AB and CD of a circle intersect at a point E inside the circle. Let M be a point on the line segment EB, distinct from E and B. A circle passes through the points D, E, and M. The tangent to this circle at E intersects the lines BC and AC at points F and G, respectively. If AM/AB = t, determine the value of ... | [
"\\dfrac{t}{1 - t}"
] | [
"Two chords AB and CD of a circle intersect at a point E inside the circle. Let M be a point on the line segment EB, distinct from E and B. A circle passes through the points D, E, and M. The tangent to this circle at E intersects the lines BC and AC at points F and G, respectively. If AM/AB = t, determine the valu... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-1285ee75be50572513c88b936860d77d35ecefb2efc7b87e27a22fce709adda0 | Two chords \(AC\) and \(BD\) in a circle with center \(O\) intersect at \(E\). The circumcircles of triangles \(EAB\) and \(ECD\) intersect again at \(F\). If the radius of the circle is 10 units, calculate the length of \(OF\) given that \(EF = 6\) units. | [
"8"
] | [
"Two chords \\(AC\\) and \\(BD\\) in a circle with center \\(O\\) intersect at \\(E\\). The circumcircles of triangles \\(EAB\\) and \\(ECD\\) intersect again at \\(F\\). If the radius of the circle is 10 units, calculate the length of \\(OF\\) given that \\(EF = 6\\) units."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-308b92dc9487d8c60bfe3eb9a73c8393dd7e4fe5c42ba9f7e56d88d8f5b025a3 | Two chords in a circle share a common endpoint and divide the circle into three parts of equal area. Find the angle between the chords. | [
"\\( \\theta \\approx 30.725834588294227^\\circ \\)"
] | [
"Two chords in a circle share a common endpoint and divide the circle into three parts of equal area. Find the angle between the chords."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 7,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-3c5f61b3141e4b4d78dc2301548e23c98d3a09a49f1c9eda9adae856e86de406 | Two chords of a circle \( AB \) and \( CD \) intersect at \( E \), with \( AE = 8 \) and \( BE = 24.5 \). The length of \( CE \) and the length of \( DE \) are integers. What is the difference between the smallest and largest possible lengths of \( CD \)? | [
"169"
] | [
"Two chords of a circle \\( AB \\) and \\( CD \\) intersect at \\( E \\), with \\( AE = 8 \\) and \\( BE = 24.5 \\). The length of \\( CE \\) and the length of \\( DE \\) are integers. What is the difference between the smallest and largest possible lengths of \\( CD \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47e8332fa3aa798bdd6ad310789ab1812e3622dbc5f87f6a822f5fc9b800a6bc | Two chords of a circle are perpendicular to each other and intersect at a point. The segments of one chord are 3 and 4, while the segments of the other chord are 2 and 6. Find the diameter of the circle. | [
"\\sqrt{65}"
] | [
"Two chords of a circle are perpendicular to each other and intersect at a point. The segments of one chord are 3 and 4, while the segments of the other chord are 2 and 6. Find the diameter of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b523c6d03289fddb782d356b36231a8ee0b391cfee561b0a1e83b3fd6c8182e1 | Two chords of a circle intersect at right angles, dividing one chord into lengths 2 and 6. One length from the intersection to the circle along the other chord is 1. What is the distance from the center of the circle to the intersection of the two chords? | [
"\\(\\frac{\\sqrt{137}}{2}\\)"
] | [
"Two chords of a circle intersect at right angles, dividing one chord into lengths 2 and 6. One length from the intersection to the circle along the other chord is 1. What is the distance from the center of the circle to the intersection of the two chords?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-bc8c20835448a743d8ecae8ce5f3c9b5b238ceaa25af0143e55626c129b8343c | Two chords of a circle of radius $R$ intersect at right angles. The portions of one chord are as $4 : 5$, those of the other as $5 : 16$. Find the lengths of the four arcs into which the endpoints of the chords divide the circle. | [
"$R[(\\pi /2)+\\alpha -\\beta ]$, $R[(\\pi /2)-\\alpha +\\beta ]$, $R[-(\\pi /2)+\\alpha +\\beta ]$, $R[(3\\pi /2)-\\alpha -\\beta ]$, where $\\alpha =\\arcsin \\left(18/\\sqrt {445}\\right)$, $\\beta =\\arcsin \\left(21/\\sqrt {445}\\right)$"
] | [
"Two chords of a circle of radius $R$ intersect at right angles. The portions of one chord are as $4 : 5$, those of the other as $5 : 16$. Find the lengths of the four arcs into which the endpoints of the chords divide the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c548482810d04733c5d52658372e63cd070d58ae25d0d139f5ceeabe13cc7c8d | Two chords of lengths $a$ and $b$ are drawn from the same point of the circle. Find the radius of the circle if the distance between the midpoints of the given chords is $c/2$. | [
"$\\frac{abc}{\\sqrt {(a+b+c)(a+b-c)(a+c-b)(b+c-a)}}$"
] | [
"Two chords of lengths $a$ and $b$ are drawn from the same point of the circle. Find the radius of the circle if the distance between the midpoints of the given chords is $c/2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-19177530daf2df6de110cc71baaa4ba03c057f1021b41ceb795c8ea428e1c3c6 | Two chords, $AB$ and $CD,$ meet inside a circle at $P.$ If $AP = 3$ and $CP = 8,$ then what is $\frac{BP}{DP}$? | [
"\\frac{8}{3}"
] | [
"Two chords, $AB$ and $CD,$ meet inside a circle at $P.$ If $AP = 3$ and $CP = 8,$ then what is $\\frac{BP}{DP}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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