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-af3e4d52411ee61b6d3bec46d26b04c5b5b5ce940024bdf5940b5dc70fa35469 | Three ants are placed at points (0,0), (2,0), and (0,3) in the plane. The ants move one at a time and are only allowed to move parallel to the line through the other two ants. Which points can be reached by the ants? | [
"\\text{All points in the plane}"
] | [
"Three ants are placed at points (0,0), (2,0), and (0,3) in the plane. The ants move one at a time and are only allowed to move parallel to the line through the other two ants. Which points can be reached by the ants?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-0bf7560a08841dd27dddd228128da2b25f88761622e96cf91e0d23a9e0b1c168 | Three ants sit at the centers of three adjacent faces of a unit cube (1x1x1). At time \( t = 0 \), each ant begins to walk towards the next ant in a cycle (1 -> 2 -> 3 -> 1) at a constant speed \( v = 1 \). How much time will it take for the ants to meet? | [
"1"
] | [
"Three ants sit at the centers of three adjacent faces of a unit cube (1x1x1). At time \\( t = 0 \\), each ant begins to walk towards the next ant in a cycle (1 -> 2 -> 3 -> 1) at a constant speed \\( v = 1 \\). How much time will it take for the ants to meet?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9303cad51c8f1bb728089032b242eddf0d96d671f2c4690ea226c8d4b6b06610 | Three apples, two bananas, and seven clementines cost 65p. Two apples and three clementines cost 5p more than two bananas. The price of each single fruit in pence is a positive integer. Find the cost of: (i) one apple and two clementines together. (ii) two bananas and one clementine together. | [
"23"
] | [
"Three apples, two bananas, and seven clementines cost 65p. Two apples and three clementines cost 5p more than two bananas. The price of each single fruit in pence is a positive integer. Find the cost of:\n(i) one apple and two clementines together.\n(ii) two bananas and one clementine together."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b42e8f4e80c634ad7362704efa55f8097645b657dbe84f56e0860b94d0f6b665 | Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the star... | [
"1.6"
] | [
"Three athletes ran at different constant speeds along a track of length $1$. They started moving at the same time at one end of the track. Having reached one of the ends of the track, the athlete immediately turned around and continued running in the opposite direction. After a while, all three athletes met at the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-972000e347bcb79f8c4af15adce84f71630fc3faf8725c192682ac7d122b5902 | Three automatic assembly lines turn out the same product, but they each have different outputs. The output of all the three assembly lines, operating simultaneously, is 1.5 times the output of the first and second lines operating jointly. A task assigned to the first line can be carried out by the second and third line... | [
"8 hours"
] | [
"Three automatic assembly lines turn out the same product, but they each have different outputs. The output of all the three assembly lines, operating simultaneously, is 1.5 times the output of the first and second lines operating jointly. A task assigned to the first line can be carried out by the second and third... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d8a0a9a37b94397f2c4ca524fb1efe7c5f6d71279a4cd40d4e24d6eed680f17 | Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are 50%, 25%, and 20%, respectively. All three bags of candy are dumped into one bowl. What is the closest ratio of yellow jelly beans to all beans in the bowl? | [
"31%"
] | [
"Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are 50%, 25%, and 20%, respectively. All three bags of candy are dumped into one bowl. What is the closest ratio of yellow jelly beans to all beans in the bowl?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-94d3a36a3832efa3acdca1123cc6c753620cade88ce027ed25c0e5f4a82674ba | Three balls are placed inside a hemispherical tent. The first two balls have a radius of $1$, touch each other, the roof, and the diameter of the tent base. The third ball of radius $r$ touches the other two, the roof, and the base of the tent. Find the radius $r$ of the third ball. | [
"\\(\\frac{\\sqrt{2}+4}{7}\\)"
] | [
"Three balls are placed inside a hemispherical tent. The first two balls have a radius of $1$, touch each other, the roof, and the diameter of the tent base. The third ball of radius $r$ touches the other two, the roof, and the base of the tent. Find the radius $r$ of the third ball."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 6,
"tir": 14
} | 41 | 0 | 0 |
omr-ws-265ccbe4ab8649459d9b82a52111a18cf5b30a48cec5e8754319c3b8f5ae6767 | Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\fra... | [
"55"
] | [
"Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \\ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4338f81778a8782c62dfe09ecd28eca41c72196962bf45d8aee8c7020d4148a1 | Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,...$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}... | [
"55"
] | [
"Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,...$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\\f... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b1680f5df316f6870cc074b375e3871442ebc035042f947761593cc2a3e9b33f | Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,\ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\fra... | [
"55"
] | [
"Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,\\ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-c5c23ae2ed1a3d5d99114a8c34518f9a8741446a55f222307522aa05d814abbf | Three balls are randomly chosen from a bag containing 3 identical red balls, 4 identical blue balls, and 5 identical green balls without replacement. What is the probability that all three balls have different colors? | [
"\\(\\frac{3}{11}\\)"
] | [
"Three balls are randomly chosen from a bag containing 3 identical red balls, 4 identical blue balls, and 5 identical green balls without replacement. What is the probability that all three balls have different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-11ba3f4a293d7f3aa9ea441ea6c856e3346029207bb75a94762f722217ca59d7 | Three balls marked 1, 2, and 3 are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated twice more. If the sum of the numbers recorded is 6, what is the probability that the ball numbered 2 was drawn all three times? | [
"\\(\\frac{1}{7}\\)"
] | [
"Three balls marked 1, 2, and 3 are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated twice more. If the sum of the numbers recorded is 6, what is the probability that the ball numbered 2 was drawn all three times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-572f6b4d1784ca7104673bacc62b5dc8080a486dffd8c8f64a3a4ac48acd2c7b | Three balls of radius $r$ are placed in a cone with base radius $2$ and generatrix $4$. They touch each other externally, the lateral surface of the cone, and the first two balls touch the base of the cone. Find the maximum value of $r$. | [
"\\( \\sqrt{3} - 1 \\)"
] | [
"Three balls of radius $r$ are placed in a cone with base radius $2$ and generatrix $4$. They touch each other externally, the lateral surface of the cone, and the first two balls touch the base of the cone. Find the maximum value of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 17
} | 54 | 0 | 0 |
omr-ws-84f8c3c6c9b6f7ebdc45ebffb02cb02ed75a78c0f9ae34eabdf0715d62da771c | Three balls of radius \( r \) lie on the lower base of a regular triangular prism, and each of them touches two other balls and two side faces of the prism. On these balls lies the fourth ball, which touches all the side faces and the upper base of the prism. Find the height of the prism. | [
"\\( r\\left( 2 + \\frac{\\sqrt{3}}{3}+\\sqrt{3+\\frac{4\\sqrt{3}}{3}}\\right) \\)"
] | [
"Three balls of radius \\( r \\) lie on the lower base of a regular triangular prism, and each of them touches two other balls and two side faces of the prism. On these balls lies the fourth ball, which touches all the side faces and the upper base of the prism. Find the height of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-f04df61f9917a017388cfd86aa85a5fc5daa33af576cdf49535f098c871b624c | Three balls of the same size fit snugly into a cylindrical container such that the balls touch the side, top, and bottom of the container. Determine the ratio of the volume of the balls to the volume of the space in the container around the balls. | [
"2:1"
] | [
"Three balls of the same size fit snugly into a cylindrical container such that the balls touch the side, top, and bottom of the container. Determine the ratio of the volume of the balls to the volume of the space in the container around the balls."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f1106ff365c04ac0b02d7f19ea1522325ed72162fcee38473598cb897c0f90c | Three beads, two white and one blue, are placed in a line in random order. What is the probability that no two neighboring beads are the same color? | [
"\\(\\frac{1}{3}\\)"
] | [
"Three beads, two white and one blue, are placed in a line in random order. What is the probability that no two neighboring beads are the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-20adfd977b1ca58ed0fc239002e436a6ca4395e51a444043cb45bc4ed9c28f69 | Three beetles start at the same point on a table and crawl to the vertices of a triangle with an inradius of $2$. Let $d_1$, $d_2$, and $d_3$ be the distances each beetle crawls to reach the vertices. Calculate the minimum possible value of $\max(d_1, d_2, d_3)$. | [
"4"
] | [
"Three beetles start at the same point on a table and crawl to the vertices of a triangle with an inradius of $2$. Let $d_1$, $d_2$, and $d_3$ be the distances each beetle crawls to reach the vertices. Calculate the minimum possible value of $\\max(d_1, d_2, d_3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed3a2e718021c2b7d5af37d4a3753727e5900726bcd27bd8050e7314c9ee4938 | Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, $\frac{4}{3}$ seconds, $\frac{5}{3}$ seconds, and $2$ seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in $1$ minute? (Include first and last.) | [
"85"
] | [
"Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, $\\frac{4}{3}$ seconds, $\\frac{5}{3}$ seconds, and $2$ seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in $1$ minute? (Include first and last.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-d5e42ba692559812b1194f14f75fbb056d8e7b860bb2cca9148b337bf8c5fdc0 | Three biased coins C1, C2, C3 lie on a table. Their respective probabilities of falling heads when tossed are 1/3, 2/3, and 1. A coin is picked at random, tossed, and observed to fall heads. Calculate the probability that it is Ck, for each k = 1, 2, 3. Given that a coin has been tossed once and observed to fall heads,... | [
"\\dfrac{6}{7}"
] | [
"Three biased coins C1, C2, C3 lie on a table. Their respective probabilities of falling heads when tossed are 1/3, 2/3, and 1. A coin is picked at random, tossed, and observed to fall heads. Calculate the probability that it is Ck, for each k = 1, 2, 3.\nGiven that a coin has been tossed once and observed to fall ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0af1d30c7dc0aabfc9b1773ae567bfe46d6487714e08b8ea598c23a7857cba14 | Three biased coins are tossed 1200 times simultaneously. The partial outcomes are given below: - No tails: 180 - One tail: 320 - Two tails: \( x \) The frequency of the occurrence of two tails is thrice that of no tails. Find \( x \). | [
"525"
] | [
"Three biased coins are tossed 1200 times simultaneously. The partial outcomes are given below:\n- No tails: 180\n- One tail: 320\n- Two tails: \\( x \\)\nThe frequency of the occurrence of two tails is thrice that of no tails. Find \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d34a7dd4bc3fd18b221aa21a29cd9a5bea955c27f7205a5b09eec497f10c7db | Three birds, four squirrels, and five bears decide to sit on a rotting log. The three birds choose their spots first, then the four squirrels, and finally the five bears. Each animal has an equal probability of choosing a spot. If any two bears sit next to each other, the log will break. What is the probability that th... | [
"\\(\\frac{7}{99}\\)"
] | [
"Three birds, four squirrels, and five bears decide to sit on a rotting log. The three birds choose their spots first, then the four squirrels, and finally the five bears. Each animal has an equal probability of choosing a spot. If any two bears sit next to each other, the log will break. What is the probability th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-ea08bec00e1c19570f38172b450c129e67ba6b53135f1387f41ec713a17881c2 | Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the distance traveled by each boat, \( D \). | [
"\\( D = 100 \\text{ km} \\)"
] | [
"Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the distance traveled by each boat, \\( D \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-656bb7af8f3d848f45183138fce88931b0c42ca1f9dd93994271d70087c2cb7a | Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the evolution of the triangle formed by the three boats. | [
"The triangle formed by the three boats will shrink and rotate, maintaining its equilateral shape until the boats meet at a single point."
] | [
"Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the evolution of the triangle formed by the three boats."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 12
} | 19 | 0 | 0 |
omr-ws-6d04a35c5c804af9ef0b8cdd2b9fa36fab8b242fa167b64c42eb4e39421ef9e3 | Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the time \( t \) at which they all meet. | [
"\\(\\frac{5}{3}\\) hours"
] | [
"Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the time \\( t \\) at which they all meet."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b03df911c0b0e25448dc873cbef26f423937a71dd4ca2996bdd2a85e1eda6200 | Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the velocity components of the three boats in the direction of the meeting point P, as a function ... | [
"15"
] | [
"Three boats start at the corners of an equilateral triangle with a side length of 50 km and each boat maintains a constant speed of 30 km/h. Each boat heads towards the neighboring boat in a clockwise direction. Determine the velocity components of the three boats in the direction of the meeting point P, as a func... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c7d8dcb2bcf58f5d89131ff6004b71bdc6fa283763f0c8d4fee9d997f95a3bbe | Three bodies move in the same straight line from point $A$ to point $B$. The second body began moving 5 seconds, and the third body 8 seconds later than the first one. The speed of the first body is less than that of the second body by 6 cm/s. The speed of the third body is equal to 30 cm/s. Find the distance $AB$ and ... | [
"Either 360 cm and 18 cm/s or 60 cm and 6 cm/s"
] | [
"Three bodies move in the same straight line from point $A$ to point $B$. The second body began moving 5 seconds, and the third body 8 seconds later than the first one. The speed of the first body is less than that of the second body by 6 cm/s. The speed of the third body is equal to 30 cm/s. Find the distance $AB$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da99d6be21872d89c5cbc0841968fa76d8e2c0cea63a122d93c24f9adeed1376 | Three bowmen, A, B, and C, are competing in a duel. The probabilities of hitting the target are as follows: A has a probability of 1/3, B has a probability of 2/3, and C always hits the target. They take turns shooting in the order A, B, C, and the cycle repeats. Each bowman uses the optimal strategy. Determine the pro... | [
"A = 19/63, B = 30/63, C = 14/63"
] | [
"Three bowmen, A, B, and C, are competing in a duel. The probabilities of hitting the target are as follows: A has a probability of 1/3, B has a probability of 2/3, and C always hits the target. They take turns shooting in the order A, B, C, and the cycle repeats. Each bowman uses the optimal strategy. Determine th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe2c41c82fa294fb16f4b14f3ace41a93867f3eb3db74f66a15ba6614eba21f7 | Three boxes A, B, and C contain 100, 50, and 80 marbles respectively, some of which are black. Box A has 15 black marbles. A box is selected at random, and then a marble is selected at random from that box. The probability of obtaining a black marble is 101/600. What is the greatest possible number of black marbles in ... | [
"22"
] | [
"Three boxes A, B, and C contain 100, 50, and 80 marbles respectively, some of which are black. Box A has 15 black marbles. A box is selected at random, and then a marble is selected at random from that box. The probability of obtaining a black marble is 101/600. What is the greatest possible number of black marble... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 15
} | 62 | 0 | 0 |
omr-ws-a6ec5f2fb1ebee0d053bc9a8299ab0914ddc2755a2fa8fbc49f454a9407b86d8 | Three boxes are set out on a table. One contains two black balls, one contains two white balls, and one contains a black and a white ball. Matt removes a ball at random from box A, and it is black. If he then removes a ball from box B, compute the probability it is white. | [
"2/3"
] | [
"Three boxes are set out on a table. One contains two black balls, one contains two white balls, and one contains a black and a white ball. Matt removes a ball at random from box A, and it is black. If he then removes a ball from box B, compute the probability it is white."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-9c3883bc6710a2862a5bd5521c04bc0531810bc5a07aa3a0ca86cbf911540ed1 | Three boxes contain 600 balls each: one box with red balls, one with white balls, and one with blue balls. In how many ways can 900 balls be chosen from these boxes? | [
"270901"
] | [
"Three boxes contain 600 balls each: one box with red balls, one with white balls, and one with blue balls. In how many ways can 900 balls be chosen from these boxes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 63
} | 95 | 0 | 0 |
omr-ws-284e083896a8089180eff09aee71ddaafe7129ac2c60f4928df5ec2abbb0abb3 | Three boxes contain 600 balls each: the first box has 600 identical red balls, the second has 600 identical blue balls, and the third has 600 identical white balls. In how many ways can a total of 900 balls be chosen from these three boxes? | [
"270901"
] | [
"Three boxes contain 600 balls each: the first box has 600 identical red balls, the second has 600 identical blue balls, and the third has 600 identical white balls. In how many ways can a total of 900 balls be chosen from these three boxes?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-981e5b6ad9467bd2bf4afc7ffc8d675b89d0191af1716fcd3df99e5745873d7a | Three boxes contain 64.2 kg of sugar. The second box contains $\frac{4}{5}$ of the contents of the first, and the third contains $42\frac{1}{2}\%$ of what there is in the second box. How much sugar is there in each box? | [
"30 kg, 24 kg, 10.2 kg"
] | [
"Three boxes contain 64.2 kg of sugar. The second box contains $\\frac{4}{5}$ of the contents of the first, and the third contains $42\\frac{1}{2}\\%$ of what there is in the second box. How much sugar is there in each box?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44cb29b78674cf1a3f5df34d3a5ac2075bda7f2c6f259e4022a76b762d50481b | Three boys A, B, and C sit around a round table in that order. A has a ball in his hand. Starting from A, the boy with the ball passes it to either of the other two boys. After 5 passes, the ball goes back to A. How many different ways can the ball be passed? | [
"14"
] | [
"Three boys A, B, and C sit around a round table in that order. A has a ball in his hand. Starting from A, the boy with the ball passes it to either of the other two boys. After 5 passes, the ball goes back to A. How many different ways can the ball be passed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-324d14365b6342f0066769e99367ca5c5c99b57f55a5c473d12434cc6443e68a | Three boys and two girls stand in a queue. What is the probability that the number of boys ahead of every girl is at least one more than the number of girls ahead of her? | [
"\\dfrac{1}{2}"
] | [
"Three boys and two girls stand in a queue. What is the probability that the number of boys ahead of every girl is at least one more than the number of girls ahead of her?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-209a78427e12b90ffe01888465d939fe1b9408921c9a75d17cc80993b7714dfe | Three boys decide to trade marbles, jacks, and checkers. They agree that m marbles can be traded for J jacks and j jacks can be traded for c checkers. The number of marbles needed for C checkers is...? | [
"\\(\\frac{mC}{j}\\)\n\nExplanation:\nThe problem states that \\(m\\) marbles can be traded for \\(J\\) jacks and \\(j\\) jacks can be traded for \\(c\\) checkers. To find the number of marbles needed for \\(C\\) checkers, we need to establish the conversion rate between marbles and checkers through jacks.\n\nGiven... | [
"Three boys decide to trade marbles, jacks, and checkers. They agree that m marbles can be traded for J jacks and j jacks can be traded for c checkers. The number of marbles needed for C checkers is...?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-fa97132c235fd0b6862b4a2ba37bf552c2863f09dfcc0914ec5c419f41af5923 | Three boys have the same birthday but were born in different years. On one such birthday, their mean age is 6 years. At least how much older must the eldest boy be than the youngest boy in years? | [
"2"
] | [
"Three boys have the same birthday but were born in different years. On one such birthday, their mean age is 6 years. At least how much older must the eldest boy be than the youngest boy in years?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f0449583a580f430b13093fa429ec5fb4d6867e30157959e6093767d41b26683 | Three boys, A, B, and C, are sitting at a round table, and A has a ball. If there are a total of 5 passes, how many ways are there such that the final pass is to A? | [
"10"
] | [
"Three boys, A, B, and C, are sitting at a round table, and A has a ball. If there are a total of 5 passes, how many ways are there such that the final pass is to A?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-22ff155fcdf573db3b6dfe2c2e9849214091a1d0cf61d939a3ab6d24a2fa09dc | Three brothers and their wives each made one visit to a friend in the hospital, resulting in six visits in total. Some visits overlapped such that each brother met the wives of the other two brothers during a visit. Let $x$ be the number of visits in which a brother met his own wife. Calculate the minimum possible valu... | [
"1"
] | [
"Three brothers and their wives each made one visit to a friend in the hospital, resulting in six visits in total. Some visits overlapped such that each brother met the wives of the other two brothers during a visit. Let $x$ be the number of visits in which a brother met his own wife. Calculate the minimum possible... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-3dbac8a8829072941b05dcfc8e64cf383347b979fcc5880c59688b31aeb6b8fb | Three brothers share some money in proportion to their age. The numbers expressing their age form a geometric progression. If they shared this money in proportion to their age in three years, then the youngest would get 105 roubles more and the middle brother 15 roubles more than now. How old is each brother if it is k... | [
"12; 18; 27"
] | [
"Three brothers share some money in proportion to their age. The numbers expressing their age form a geometric progression. If they shared this money in proportion to their age in three years, then the youngest would get 105 roubles more and the middle brother 15 roubles more than now. How old is each brother if it... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba442eabfd948e5e8224ffb3d2985a822ba8169205facdb5bec1365c0879363f | Three brothers whose ages form a geometric progression divide a certain sum of money in proportion to their ages. If they do that three years later, when the youngest is half the age of the oldest, then he will receive 105 roubles, and the middle brother will get 15 roubles more than he gets now. Give the ages of the b... | [
"27, 18 and 12 years old"
] | [
"Three brothers whose ages form a geometric progression divide a certain sum of money in proportion to their ages. If they do that three years later, when the youngest is half the age of the oldest, then he will receive 105 roubles, and the middle brother will get 15 roubles more than he gets now. Give the ages of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac9c30bbf15196cc3863dbdcb4835851267b0fe6f1da18b0a6f3a653a80a7546 | Three brothers, Thing One, Thing Two, and Thing Three, are painting the walls of a room. Thing Two can paint twice as fast as Thing One, and Thing Three can paint three times as fast as Thing Two. Thing One starts alone and paints until one third of the walls are painted. Then Thing Two joins and they work together unt... | [
"364 minutes"
] | [
"Three brothers, Thing One, Thing Two, and Thing Three, are painting the walls of a room. Thing Two can paint twice as fast as Thing One, and Thing Three can paint three times as fast as Thing Two. Thing One starts alone and paints until one third of the walls are painted. Then Thing Two joins and they work togethe... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aec09acc645f98200370070ce9f9c1ab543162fe6cc462cc429f48138873d4ef | Three brothers, whose ages form a geometric progression, divide among themselves a certain sum of money in direct proportion to the age of each. If this were done in three years time, when the youngest becomes one-half the age of the oldest, then the youngest would receive 105 roubles, and the middle one, 15 roubles mo... | [
"27, 18 and 12 years old"
] | [
"Three brothers, whose ages form a geometric progression, divide among themselves a certain sum of money in direct proportion to the age of each. If this were done in three years time, when the youngest becomes one-half the age of the oldest, then the youngest would receive 105 roubles, and the middle one, 15 roubl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab1e87e5b4e1555e115eac99797f3e6405cb0dd4718f154efe1cc8a0473e1830 | Three builders are scheduled to build a house in $60$ days. However, they do nothing for the first $50$ days. To complete the house on time, they must hire more workers and work twice as fast as originally planned. If the new workers also work at the doubled rate, how many new workers will they need to hire? | [
"6"
] | [
"Three builders are scheduled to build a house in $60$ days. However, they do nothing for the first $50$ days. To complete the house on time, they must hire more workers and work twice as fast as originally planned. If the new workers also work at the doubled rate, how many new workers will they need to hire?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d6ff71c7feff5f240854dfa43636e672e3401eede9d573bfded295031c73107 | Three candies and three marbles are grouped into two boxes such that each box contains three items. At a time, we move one item from one box to the other and vice versa. Calculate the probability that the first box contains exactly one candy after a move. | [
"\\(\\frac{7}{9}\\)"
] | [
"Three candies and three marbles are grouped into two boxes such that each box contains three items. At a time, we move one item from one box to the other and vice versa. Calculate the probability that the first box contains exactly one candy after a move."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33d5c7d7c7311dbca1afe5d0ff8ef2e271acce4be887af6af90a115f52940638 | Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen? | [
"\\(\\frac{467}{425}\\)"
] | [
"Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-368b1e081f2c64cf4f1d16583e3811dbca5a83fcaa0517ab69adc01d83c44067 | Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2? | [
"\\frac{1}{663}"
] | [
"Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a 4, the second card is a $\\clubsuit$, and the third card is a 2?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb0a48f0dae9f7a327ce11169cfa75acee79aa624f5e78965fad9725b9ba6231 | Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Jack, the second card is a Queen, and the third card is a King? Assuming the dealing is done without replacement. | [
"\\frac{8}{16575}"
] | [
"Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Jack, the second card is a Queen, and the third card is a King? Assuming the dealing is done without replacement."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55fca3931bec23bf2f311c611d04fad1361dd66dd328ace79b359e3f79e360dd | Three cards labeled \(a\), \(b\), and \(c\) (with \(a > b > c\)) are drawn from a set of 10 cards labeled from 1 to 10. Find the probability that \(\int_0^a (x^2 - 2bx + 3c) \, dx = 0\). | [
"\\(\\frac{1}{40}\\)"
] | [
"Three cards labeled \\(a\\), \\(b\\), and \\(c\\) (with \\(a > b > c\\)) are drawn from a set of 10 cards labeled from 1 to 10. Find the probability that \\(\\int_0^a (x^2 - 2bx + 3c) \\, dx = 0\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-f022c4c71ff1770aa9c965545ca9c075b2dd26211ac8b56b208b5e50d69562bc | Three cards, each with a positive integer written on it, are lying face-down on a table. The numbers are all different, they sum to 13, and they are in increasing order from left to right. Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then... | [
"4"
] | [
"Three cards, each with a positive integer written on it, are lying face-down on a table. The numbers are all different, they sum to 13, and they are in increasing order from left to right. Casey looks at the number on the leftmost card and says, \"I don't have enough information to determine the other two numbers.... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 43
} | 75 | 0 | 0 |
omr-ws-0e4bff69f81f5bab11e52bfa2531110c415ea3e197a7e331a1bc8150407aeef4 | Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two f... | [
"1/3"
] | [
"Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-5557c82c0e1b2d4ed994ae16582488dccd7f5e6d5f1712f136c179999590e5e3 | Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two f... | [
"\\(\\frac{2}{3}\\)"
] | [
"Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-d396b6ee949d75d8cf693c1e858464b3646f9de823a5b0da3ccc4c15ef02c990 | Three carpets have a combined area of $200 m^2$. By overlapping the three carpets to cover a total floor area of $140 m^2$, the area which is covered by exactly two layers of carpet is $24 m^2$. What area of floor is covered by exactly three layers of carpet? | [
"18"
] | [
"Three carpets have a combined area of $200 m^2$. By overlapping the three carpets to cover a total floor area of $140 m^2$, the area which is covered by exactly two layers of carpet is $24 m^2$. What area of floor is covered by exactly three layers of carpet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-38c21c1dc16cd86d0a8a8ee772d9be197fbb3770e4d0b0d00eef090301f57b81 | Three cars $a$, $b$, and $c$ leave at 6 am from three different towns $A$, $B$, and $C$ and travel straight paths $AD$, $BE$, and $CF$ that intersect in pairs. Each car travels at a constant speed such that each pair of cars reaches the intersection of their respective roads at the same time. The cars $a$ and $b$ inter... | [
"44"
] | [
"Three cars $a$, $b$, and $c$ leave at 6 am from three different towns $A$, $B$, and $C$ and travel straight paths $AD$, $BE$, and $CF$ that intersect in pairs. Each car travels at a constant speed such that each pair of cars reaches the intersection of their respective roads at the same time. The cars $a$ and $b$ ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d52cbf6ee14beddb5b79c2d83990c59bc4ce50c441317f325fce4353bc02182d | Three cars move uniformly along three different straight roads. At time $t=0$, the positions of the cars are $(x_1(0), y_1(0))$, $(x_2(0), y_2(0))$, and $(x_3(0), y_3(0))$ respectively. It is known that at $t=0$, the three cars are collinear. If the velocities of the cars are $(v_{x1}, v_{y1})$, $(v_{x2}, v_{y2})$, and... | [
"0"
] | [
"Three cars move uniformly along three different straight roads. At time $t=0$, the positions of the cars are $(x_1(0), y_1(0))$, $(x_2(0), y_2(0))$, and $(x_3(0), y_3(0))$ respectively. It is known that at $t=0$, the three cars are collinear. If the velocities of the cars are $(v_{x1}, v_{y1})$, $(v_{x2}, v_{y2})$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21d6274d04c0b68b9e7c2990f3c6737e6a2b651cdefc6598793131ec465581c0 | Three cevians are drawn in a triangle that do not intersect at one point, forming 4 triangles and 3 quadrilaterals. If the area of each of the four triangles is 8, find the sum of the areas of the three quadrilaterals. | [
"24"
] | [
"Three cevians are drawn in a triangle that do not intersect at one point, forming 4 triangles and 3 quadrilaterals. If the area of each of the four triangles is 8, find the sum of the areas of the three quadrilaterals."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0b7ac2911518f694212b82382a877a0edce78660cb1b656a2c59ca4efc7257a9 | Three cevians concur at a point inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form a geometric progression with a common ratio of $r$. If the lengths of the cevians are $a$, $ar$, and $ar^2$, find the value of $r$ given that the shortest segment has a... | [
"\\dfrac{1 + \\sqrt{5}}{2}"
] | [
"Three cevians concur at a point inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form a geometric progression with a common ratio of $r$. If the lengths of the cevians are $a$, $ar$, and $ar^2$, find the value of $r$ given that the shortest segment ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6584f1099ed98c505132155d47b971ccf18cd8846320180c1bcd3ad9661230c9 | Three chemical stations are located at coordinates (4000,0), (4000,1000), and (-4000,0). The second station picks up a signal 1 second after the first station, and the third station picks up the signal 5 seconds after the first station. Given that the speed of sound is 337.6 m/s, determine the coordinates of the source... | [
"\\( (855.247, -640.448) \\)"
] | [
"Three chemical stations are located at coordinates (4000,0), (4000,1000), and (-4000,0). The second station picks up a signal 1 second after the first station, and the third station picks up the signal 5 seconds after the first station. Given that the speed of sound is 337.6 m/s, determine the coordinates of the s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 6
} | 11 | 0 | 0 |
omr-ws-8722084e7827db5270aa9f8b0bee9645346d0d72ba1758bd89b336f2e393ee93 | Three children and two adults want to cross a river using a rowing boat. The boat can carry no more than a single adult or, in case no adult is in the boat, a maximum of two children. Find the least number of times the boat needs to cross the river to transport all five people. | [
"11"
] | [
"Three children and two adults want to cross a river using a rowing boat. The boat can carry no more than a single adult or, in case no adult is in the boat, a maximum of two children. Find the least number of times the boat needs to cross the river to transport all five people."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 3,
"tir": 10
} | 40 | 0 | 0 |
omr-ws-2f561169c1d7387300c33ea55eed5d884450cc45947f1d568d7581243c6dbda6 | Three children need to travel 84 km to visit their grandmother. They have two bikes and can walk at 5 km/h or cycle at 20 km/h. They cannot have two children on one bike at the same time or ride one bike while wheeling the other. What is the shortest time in which they can complete the journey, and how is this achieved... | [
"8.4 hours"
] | [
"Three children need to travel 84 km to visit their grandmother. They have two bikes and can walk at 5 km/h or cycle at 20 km/h. They cannot have two children on one bike at the same time or ride one bike while wheeling the other. What is the shortest time in which they can complete the journey, and how is this ach... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-26ed74f287afa23a85123381cf382777c33d85a7b1e208b5e147d48a2c3561ce | Three children, Alice, Brandy, and Christine, play a game of laser tag. Proceeding in alphabetical order, they take turns choosing one of the other two players and then shooting at them with a harmless laser beam. Alice has an 80% chance of hitting her target, Brandy a 60% chance, and Christine a 40% chance. Each playe... | [
"\\( \\frac{61}{119} \\)"
] | [
"Three children, Alice, Brandy, and Christine, play a game of laser tag. Proceeding in alphabetical order, they take turns choosing one of the other two players and then shooting at them with a harmless laser beam. Alice has an 80% chance of hitting her target, Brandy a 60% chance, and Christine a 40% chance. Each ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-fef68ea76b01ff7049950dca066d3b484d7336974565e2db068803322e49eecd | Three chips are inserted into a box with side length 1. Let \(a\), \(b\), and \(c\) denote the distance from the center of each chip to the closest side (not including the bottom). What is the probability that \(a + b + c > 1\)? | [
"\\(\\frac{1}{90}\\)"
] | [
"Three chips are inserted into a box with side length 1. Let \\(a\\), \\(b\\), and \\(c\\) denote the distance from the center of each chip to the closest side (not including the bottom). What is the probability that \\(a + b + c > 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d30f30097d4ce21e0199e2c1a521a6cb77071706dd3aedb0b544d0c2dc73409 | Three chords \(AA_1\), \(BB_1\), and \(CC_1\) of a circle intersect at a common point \(K\) and form six angles of \(60^\circ\). If \(KA = 3\), \(KB = 4\), and \(KC = 5\), calculate the value of \(KA_1 + KB_1 + KC_1\). | [
"\\dfrac{188}{17}"
] | [
"Three chords \\(AA_1\\), \\(BB_1\\), and \\(CC_1\\) of a circle intersect at a common point \\(K\\) and form six angles of \\(60^\\circ\\). If \\(KA = 3\\), \\(KB = 4\\), and \\(KC = 5\\), calculate the value of \\(KA_1 + KB_1 + KC_1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5c005bd3e6deb7cc421a14719dac6684ff2f2a3f41555e51bcbb89935a35170 | Three chords are drawn from point M on a circle. MN = 1, MP = 6, MQ = 2, and angles NMP and PMQ are congruent. Find the radius of the circle. | [
"\\(\\frac{2\\sqrt{34}}{\\sqrt{15}}\\)"
] | [
"Three chords are drawn from point M on a circle. MN = 1, MP = 6, MQ = 2, and angles NMP and PMQ are congruent. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 12
} | 25 | 0 | 0 |
omr-ws-0a7c653de03d0c54742f0f9a18e2ae67c638cc94563c8ac3fd30dc90aeeb2d2a | Three chords are drawn in a circle to create seven regions. The numbers from 1 to 7 are to be placed, one in each region, so that, for each chord, the total numbers in the circle on one side of the chord is equal to the total of the numbers on the other side. How many possible values are there for the number \( x \) in... | [
"3"
] | [
"Three chords are drawn in a circle to create seven regions. The numbers from 1 to 7 are to be placed, one in each region, so that, for each chord, the total numbers in the circle on one side of the chord is equal to the total of the numbers on the other side. How many possible values are there for the number \\( x... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3029acb6868487a3529cca02634dd2d780d3bd39562400467b04749c0668219c | Three chords in a circle have lengths \(a\), \(b\), and \(c\), where \(c = a + b\). If the chord of length \(a\) subtends an arc of \(30^\circ\) and the chord of length \(b\) subtends an arc of \(90^\circ\), what is the number of degrees in the smaller arc subtended by the chord of length \(c\)? | [
"\\( 150^\\circ \\)"
] | [
"Three chords in a circle have lengths \\(a\\), \\(b\\), and \\(c\\), where \\(c = a + b\\). If the chord of length \\(a\\) subtends an arc of \\(30^\\circ\\) and the chord of length \\(b\\) subtends an arc of \\(90^\\circ\\), what is the number of degrees in the smaller arc subtended by the chord of length \\(c\\)... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-3bee9f0895aeffe6096f27fe91b20c6597c13405fb7a80c4d78eb1843ad34e10 | Three chords of a sphere intersect at a point $X$ inside the sphere but are not coplanar. A sphere passing through one endpoint of each chord and $X$ touches the sphere passing through the other endpoints and $X$. If the radius of the original sphere is $R$ and the distance from the center of the original sphere to $X$... | [
"2\\sqrt{R^2 - d^2}"
] | [
"Three chords of a sphere intersect at a point $X$ inside the sphere but are not coplanar. A sphere passing through one endpoint of each chord and $X$ touches the sphere passing through the other endpoints and $X$. If the radius of the original sphere is $R$ and the distance from the center of the original sphere t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-1974bf1e5f165dfce75901d14f218d099042d9a3b2f5f6c46bc872a5d768c24e | Three chords of a sphere, each having lengths 5, 6, and 7, intersect at a single point inside the sphere and are pairwise perpendicular. For \( R \) the maximum possible radius of this sphere, find \( R^2 \). | [
"15"
] | [
"Three chords of a sphere, each having lengths 5, 6, and 7, intersect at a single point inside the sphere and are pairwise perpendicular. For \\( R \\) the maximum possible radius of this sphere, find \\( R^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b936e6732c4dccf106d9bf4598ecd822acd0f7b4d4721088fe1e6fe5ed831289 | Three circles $(c_1), (c_2), (c_3)$ pass through two points $A$ and $B$. A line $(l)$ passing through point $A$ intersects $(c_1), (c_2), (c_3)$ at points $D, E, F$ respectively. Let $T_1, T_2, T_3$ be the points where the tangents to $(c_1), (c_2), (c_3)$ at $D, E, F$ intersect the circumcircle of triangle $DEF$. If t... | [
"\\dfrac{9}{25}"
] | [
"Three circles $(c_1), (c_2), (c_3)$ pass through two points $A$ and $B$. A line $(l)$ passing through point $A$ intersects $(c_1), (c_2), (c_3)$ at points $D, E, F$ respectively. Let $T_1, T_2, T_3$ be the points where the tangents to $(c_1), (c_2), (c_3)$ at $D, E, F$ intersect the circumcircle of triangle $DEF$.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-be6f28ad07a208ea70275a04d591e8088303798225e8169fc0309a380e78656c | Three circles $(c_1), (c_2), (c_3)$ with centers $O_1, O_2, O_3$ pass through two points $A$ and $B$. If a line $(l)$ passing through point $A$ intersects circles $(c_1), (c_2), (c_3)$ at points $D, E, F$ respectively, calculate the value of $\angle O_1BO_2 + \angle O_2BO_3 + \angle O_3BO_1$ in degrees. | [
"180"
] | [
"Three circles $(c_1), (c_2), (c_3)$ with centers $O_1, O_2, O_3$ pass through two points $A$ and $B$. If a line $(l)$ passing through point $A$ intersects circles $(c_1), (c_2), (c_3)$ at points $D, E, F$ respectively, calculate the value of $\\angle O_1BO_2 + \\angle O_2BO_3 + \\angle O_3BO_1$ in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6c4127ec40aeba3f1138da0577bf342cccb2f387f39350c9028daae4465b214 | Three circles $A$, $B$, and $C$ are given. Circles $A$ and $B$ have the same radius and are colored to distinguish them from each other. The radius of circle $C$ is smaller. What is the number of arrangements (identified by the number of their common points) of the three circles on a plane if the larger circles have no... | [
"5"
] | [
"Three circles $A$, $B$, and $C$ are given. Circles $A$ and $B$ have the same radius and are colored to distinguish them from each other. The radius of circle $C$ is smaller. What is the number of arrangements (identified by the number of their common points) of the three circles on a plane if the larger circles ha... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-29468eefb69e88bcc70d5453a6d44529ee7c0244d22d4595fb559c5cd536f6ed | Three circles $C_A, C_B, C_C$ of radius 1 are centered at points $A, B, C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap. | [
"\\(\\frac{\\pi - \\sqrt{3}}{2}\\)"
] | [
"Three circles $C_A, C_B, C_C$ of radius 1 are centered at points $A, B, C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-a500434ae388140601002d1aa5902bdc038bb00b4fc030646991bbb2ffe36007 | Three circles $\Gamma, \Gamma_1, \Gamma_2$ are concurrent at $A$. The centers of $\Gamma_1$ and $\Gamma_2$, denoted as $B$ and $C$ respectively, are on $\Gamma$. $\Gamma_1$ and $\Gamma_2$ intersect $\Gamma$ again at $D$ and $E$, respectively. Let $X$ be the second point of intersection of $\Gamma_1$ and $\Gamma_2$. If ... | [
"(0, 5 - \\sqrt{21})"
] | [
"Three circles $\\Gamma, \\Gamma_1, \\Gamma_2$ are concurrent at $A$. The centers of $\\Gamma_1$ and $\\Gamma_2$, denoted as $B$ and $C$ respectively, are on $\\Gamma$. $\\Gamma_1$ and $\\Gamma_2$ intersect $\\Gamma$ again at $D$ and $E$, respectively. Let $X$ be the second point of intersection of $\\Gamma_1$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-089a0e524e6786b70ac7bd1f93cb86d6651ee856e197f7277ae02b3ccd0034a8 | Three circles $\Gamma_1, \Gamma_2, \Gamma_3$ are inscribed into an angle such that $\Gamma_1$ has the smallest radius and $\Gamma_3$ has the largest radius. Circle $\Gamma_2$ touches $\Gamma_1$ at point $A$ and $\Gamma_3$ at point $B$. Let $\ell$ be a tangent to $\Gamma_1$ at point $A$. Consider circles $\omega$ that t... | [
"$\\Gamma_2 \\cap AB \\backslash \\{A, B\\}$"
] | [
"Three circles $\\Gamma_1, \\Gamma_2, \\Gamma_3$ are inscribed into an angle such that $\\Gamma_1$ has the smallest radius and $\\Gamma_3$ has the largest radius. Circle $\\Gamma_2$ touches $\\Gamma_1$ at point $A$ and $\\Gamma_3$ at point $B$. Let $\\ell$ be a tangent to $\\Gamma_1$ at point $A$. Consider circles ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db3436197d4c17fcf0a0b407caf4dc1a8ca53d7ce869e765e6cb20eb20a5c5b6 | Three circles $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ at the point $X \ne O$. Simila... | [
"1"
] | [
"Three circles $\\Gamma_A$, $\\Gamma_B$, and $\\Gamma_C$ share a common point of intersection $O$. The other common point of $\\Gamma_A$ and $\\Gamma_B$ is $C$, that of $\\Gamma_A$ and $\\Gamma_C$ is $B$, and that of $\\Gamma_C$ and $\\Gamma_B$ is $A$. The line $AO$ intersects the circle $\\Gamma_A$ at the point $X... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6f6df36eb7328ae575d457994b9e3887ef8c76643e5b9aa9b8260d55ff54c69 | Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$, and $\mathcal C_3(O_3)$ share a common point $P$ and meet again pairwise at the points $A$, $B$, and $C$. If the points $A$, $B$, and $C$ are collinear, and the radius of $\mathcal C_1$ is $r_1$, the radius of $\mathcal C_2$ is $r_2$, and the radius of $\mathcal C... | [
"1"
] | [
"Three circles $\\mathcal C_1(O_1)$, $\\mathcal C_2(O_2)$, and $\\mathcal C_3(O_3)$ share a common point $P$ and meet again pairwise at the points $A$, $B$, and $C$. If the points $A$, $B$, and $C$ are collinear, and the radius of $\\mathcal C_1$ is $r_1$, the radius of $\\mathcal C_2$ is $r_2$, and the radius of $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-928b1db721f89eb5b5beef63e7710684fbfac60cdf09a4c974d6c66b413148ee | Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ with radii $R_1$, $R_2$, $R_3$ respectively, pass through the point $O$ and intersect two by two at points $A$, $B$, $C$. The point $O$ lies inside the triangle $ABC$. Let $A_1$, $B_1$, $C_1$ be the intersection points of the lines $AO$, $BO$, $CO$ with th... | [
"no_valid_answer_found"
] | [
"Three circles $\\mathcal{K}_1$, $\\mathcal{K}_2$, $\\mathcal{K}_3$ with radii $R_1$, $R_2$, $R_3$ respectively, pass through the point $O$ and intersect two by two at points $A$, $B$, $C$. The point $O$ lies inside the triangle $ABC$. Let $A_1$, $B_1$, $C_1$ be the intersection points of the lines $AO$, $BO$, $CO$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c76535c3ff04c2c4545dcbf7c2204da2c9b82bc6437f69a2bb2ef99769c916ad | Three circles $\omega_1$, $\omega_2$, and $\omega_3$ are externally tangent to each other at points $A$, $B$, and $C$ respectively. Points $D$ and $E$ are chosen on circles $\omega_1$ and $\omega_2$ such that $DB$ is a common tangent to $\omega_1$ and $\omega_2$, and $EC$ is a common tangent to $\omega_3$ and $\omega_1... | [
"\\sqrt{5}"
] | [
"Three circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ are externally tangent to each other at points $A$, $B$, and $C$ respectively. Points $D$ and $E$ are chosen on circles $\\omega_1$ and $\\omega_2$ such that $DB$ is a common tangent to $\\omega_1$ and $\\omega_2$, and $EC$ is a common tangent to $\\omega_3$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-774fc321240dcd3643dcd2341eb17135869d81bbc1051469d84dea3da825e725 | Three circles $\omega_1$, $\omega_2$, and $\omega_3$ pass through a common point $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$, and $P_{3,2}$ are similarly defined. Let $O_1$, $O_2$, and $O_3$ be th... | [
"1"
] | [
"Three circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ pass through a common point $P$. The tangent line to $\\omega_1$ at $P$ intersects $\\omega_2$ and $\\omega_3$ at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$, and $P_{3,2}$ are similarly defined. Let $O_1$, $O_2$, and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-c6cbe9ecb73061696e99fcabcf592abe2634bb6c1acf17ea3806ecc5ef765060 | Three circles $\omega_1, \omega_2, \omega_3$ are tangent to line $l$ at points $A, B, C$ respectively, with $B$ lying between $A$ and $C$. Circle $\omega_2$ is externally tangent to the other two circles. Let $X, Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1$ and $\omeg... | [
"5r^2"
] | [
"Three circles $\\omega_1, \\omega_2, \\omega_3$ are tangent to line $l$ at points $A, B, C$ respectively, with $B$ lying between $A$ and $C$. Circle $\\omega_2$ is externally tangent to the other two circles. Let $X, Y$ be the intersection points of $\\omega_2$ with the other common external tangent of $\\omega_1$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-770bbd92172fdce561508923def37e6e4eae96c9f99448b798db55e66e8d2d28 | Three circles $\omega_1, \omega_2, \omega_3$ are tangent to line $l$ at points $A, B, C$ respectively. Circle $\omega_2$ is tangent to both $\omega_1$ and $\omega_3$. The common tangent to $\omega_1$ and $\omega_3$ other than $l$ meets $\omega_2$ at points $X$ and $Y$. Point $Z$ lies on $\omega_2$ such that $ZB \perp A... | [
"2r"
] | [
"Three circles $\\omega_1, \\omega_2, \\omega_3$ are tangent to line $l$ at points $A, B, C$ respectively. Circle $\\omega_2$ is tangent to both $\\omega_1$ and $\\omega_3$. The common tangent to $\\omega_1$ and $\\omega_3$ other than $l$ meets $\\omega_2$ at points $X$ and $Y$. Point $Z$ lies on $\\omega_2$ such t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-0969ac43e5fe0953593dd4af037ea75cf12d7bf9c330d130115d308408a12578 | Three circles $\omega_1, \omega_2, \omega_3$ of increasing radii are inscribed in one angle. Let $A_1, A_2, A_3$ be the touch points on one side of the angle, and $B_1, B_2, B_3$ lie on the other side. Let the circles $\omega_1$ and $\omega_2$ meet at the points $M$ and $N$, and the circles $\omega_2$ and $\omega_3$ at... | [
"\\(1:2:3\\)"
] | [
"Three circles $\\omega_1, \\omega_2, \\omega_3$ of increasing radii are inscribed in one angle. Let $A_1, A_2, A_3$ be the touch points on one side of the angle, and $B_1, B_2, B_3$ lie on the other side. Let the circles $\\omega_1$ and $\\omega_2$ meet at the points $M$ and $N$, and the circles $\\omega_2$ and $\... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4687151c08da51db8dd2c4dcd0eaf8deb55383e23c7871768d9c6f3a7d24bdda | Three circles $\omega_1, \omega_2,$ and $\omega_3$ pass through a common point $D$. Let $A$ be the intersection of $\omega_1$ and $\omega_3$, $E$ be the intersection of $\omega_3$ and $\omega_2$, and $F$ be the intersection of $\omega_2$ and $\omega_1$. It is known that $\omega_3$ passes through the center $B$ of the c... | [
"0"
] | [
"Three circles $\\omega_1, \\omega_2,$ and $\\omega_3$ pass through a common point $D$. Let $A$ be the intersection of $\\omega_1$ and $\\omega_3$, $E$ be the intersection of $\\omega_3$ and $\\omega_2$, and $F$ be the intersection of $\\omega_2$ and $\\omega_1$. It is known that $\\omega_3$ passes through the cent... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-13ed9295828889b260b5414d0313f126bb0ac1a0d5d3dcd34497303326dffd5a | Three circles (not necessarily equal) are tangent to each other at points \( T_a \), \( T_b \), and \( T_c \) and are also tangent to the sides of triangle \( ABC \). Let \( P \) be the point of concurrency of the lines \( AT_a \), \( BT_b \), and \( CT_c \). If the area of triangle \( ABC \) is 60 and the radius of th... | [
"4"
] | [
"Three circles (not necessarily equal) are tangent to each other at points \\( T_a \\), \\( T_b \\), and \\( T_c \\) and are also tangent to the sides of triangle \\( ABC \\). Let \\( P \\) be the point of concurrency of the lines \\( AT_a \\), \\( BT_b \\), and \\( CT_c \\). If the area of triangle \\( ABC \\) is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4cf4f547632cc76c9fcb962d251f7294a59020f1cae9c9dc18a5ed9a1ac68cf1 | Three circles C, D, and E with radii 1, 2, and 3 respectively touch each other externally. The centers of circles C and D lie on the x-axis, and circle E touches them from the top. Find the y-coordinate (ordinate) of the center of a circle that lies in the region enclosed by circles C, D, and E and touches all three ci... | [
"\\dfrac{20}{23}"
] | [
"Three circles C, D, and E with radii 1, 2, and 3 respectively touch each other externally. The centers of circles C and D lie on the x-axis, and circle E touches them from the top. Find the y-coordinate (ordinate) of the center of a circle that lies in the region enclosed by circles C, D, and E and touches all thr... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-9ca3fd8a51b2e76e8a444dbdc10f3e03dbf5aad33001a24d04cdedde26973cfc | Three circles \( C_1(O_1) \), \( C_2(O_2) \), and \( C_3(O_3) \) share a common point \( Q \) and meet again pairwisely at points \( A \), \( B \), and \( C \). If points \( A \), \( B \), and \( C \) are collinear, and the radii of \( C_1 \), \( C_2 \), and \( C_3 \) are 5, 12, and 13 respectively, calculate the radiu... | [
"\\dfrac{13}{2}"
] | [
"Three circles \\( C_1(O_1) \\), \\( C_2(O_2) \\), and \\( C_3(O_3) \\) share a common point \\( Q \\) and meet again pairwisely at points \\( A \\), \\( B \\), and \\( C \\). If points \\( A \\), \\( B \\), and \\( C \\) are collinear, and the radii of \\( C_1 \\), \\( C_2 \\), and \\( C_3 \\) are 5, 12, and 13 re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80fb5de2a1d995a9cd07d910078013590c07c699f15393698d8ca5c81fcbc3b4 | Three circles \( K, L, M \) are tangent externally to each other at points \( A, B, C \). Circles \( K \) and \( L \) are tangent at \( A \), and circles \( L \) and \( M \) are tangent at \( B \). If the line \( AC \) meets circle \( M \) at \( D \), find the measure of angle \( \angle ABD \) in degrees. | [
"90"
] | [
"Three circles \\( K, L, M \\) are tangent externally to each other at points \\( A, B, C \\). Circles \\( K \\) and \\( L \\) are tangent at \\( A \\), and circles \\( L \\) and \\( M \\) are tangent at \\( B \\). If the line \\( AC \\) meets circle \\( M \\) at \\( D \\), find the measure of angle \\( \\angle ABD... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b31d103fde668153c5ccd9414353cfae4c8689e24a1bcab002b87f70c4da372 | Three circles \( P, P_1, P_2 \) are given; \( P_1 \) and \( P_2 \) touch \( P \) internally at points \( A \) and \( B \) respectively. It is known that the radius of \( P \) is equal to the sum of the radii of \( P_1 \) and \( P_2 \). If the radii of \( P_1 \) and \( P_2 \) are \( r_1 \) and \( r_2 \) respectively, an... | [
"r_1 + r_2"
] | [
"Three circles \\( P, P_1, P_2 \\) are given; \\( P_1 \\) and \\( P_2 \\) touch \\( P \\) internally at points \\( A \\) and \\( B \\) respectively. It is known that the radius of \\( P \\) is equal to the sum of the radii of \\( P_1 \\) and \\( P_2 \\). If the radii of \\( P_1 \\) and \\( P_2 \\) are \\( r_1 \\) a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-b9e14fbd1d28b27296624cfa4b88a30fec13725fdbac81b33ed26708c27ee09f | Three circles \( P, P_1, P_2 \) are given; \( P_1 \) and \( P_2 \) touch \( P \) internally at points \( A \) and \( B \) respectively. It is known that the radius of \( P \) is equal to the sum of the radii of \( P_1 \) and \( P_2 \). Let \( C \) be the other intersection point of \( P_1 \) and \( P_2 \). If \( O \) i... | [
"90^\\circ"
] | [
"Three circles \\( P, P_1, P_2 \\) are given; \\( P_1 \\) and \\( P_2 \\) touch \\( P \\) internally at points \\( A \\) and \\( B \\) respectively. It is known that the radius of \\( P \\) is equal to the sum of the radii of \\( P_1 \\) and \\( P_2 \\). Let \\( C \\) be the other intersection point of \\( P_1 \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-7e34060684545ce2e2481c45473cc3b7c9b3b44a2972b830965d224fa07b2f38 | Three circles \( S_1 \), \( S_2 \), and \( S_3 \) pass through the point \( M \). Let the lines \( \ell_1 \) and \( \ell_2 \) passing through \( M \) intersect these circles for the second time at the points \( A_1 \), \( A_2 \), \( A_3 \) and \( B_1 \), \( B_2 \), \( B_3 \), respectively, with \( A_1A_2 = A_2A_3 \) an... | [
"1"
] | [
"Three circles \\( S_1 \\), \\( S_2 \\), and \\( S_3 \\) pass through the point \\( M \\). Let the lines \\( \\ell_1 \\) and \\( \\ell_2 \\) passing through \\( M \\) intersect these circles for the second time at the points \\( A_1 \\), \\( A_2 \\), \\( A_3 \\) and \\( B_1 \\), \\( B_2 \\), \\( B_3 \\), respective... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90481a3eefeddc4a6247eecdd54617f2b876fb204e274f11841c33ef2b7bf352 | Three circles \( S_A, S_B \), and \( S_C \) in the plane with centers \( A, B \), and \( C \), respectively, are mutually tangential on the outside. The touchpoint between \( S_A \) and \( S_B \) is \( C' \), the touchpoint between \( S_A \) and \( S_C \) is \( B' \), and the touchpoint between \( S_B \) and \( S_C \) ... | [
"\\dfrac{4}{3}"
] | [
"Three circles \\( S_A, S_B \\), and \\( S_C \\) in the plane with centers \\( A, B \\), and \\( C \\), respectively, are mutually tangential on the outside. The touchpoint between \\( S_A \\) and \\( S_B \\) is \\( C' \\), the touchpoint between \\( S_A \\) and \\( S_C \\) is \\( B' \\), and the touchpoint between... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-7e8e38bf5ae5641f1e1bb6f2a04061d310c1094ca8c8c81b983578438886a6fa | Three circles \( c, c_1, c_2 \) are tangent to each other and are all tangent to the line \( \ell \). The tangent points of \( c_1c_2, c_1c, c_2c \) are \( A, B, C \) respectively. The perpendicular bisectors of \( AB \) and \( AC \) intersect the line \( \ell \) at \( U \) and \( V \) respectively. Let \( O_1 \) and \... | [
""
] | [
"Three circles \\( c, c_1, c_2 \\) are tangent to each other and are all tangent to the line \\( \\ell \\). The tangent points of \\( c_1c_2, c_1c, c_2c \\) are \\( A, B, C \\) respectively. The perpendicular bisectors of \\( AB \\) and \\( AC \\) intersect the line \\( \\ell \\) at \\( U \\) and \\( V \\) respecti... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-0cb55ee44adfcd4d70bfeb642d7ea62406707f1aaededeb98c92ec95bc4bb08e | Three circles \( k_1 \), \( k_2 \), and \( k_3 \) pass through points \( A \) and \( B \). A secant passes through point \( A \), intersecting circles \( k_1 \), \( k_2 \), and \( k_3 \) at points \( C \), \( D \), and \( E \) respectively. If the radii of \( k_1 \), \( k_2 \), and \( k_3 \) are \( 5 \), \( 10 \), and ... | [
"1:1"
] | [
"Three circles \\( k_1 \\), \\( k_2 \\), and \\( k_3 \\) pass through points \\( A \\) and \\( B \\). A secant passes through point \\( A \\), intersecting circles \\( k_1 \\), \\( k_2 \\), and \\( k_3 \\) at points \\( C \\), \\( D \\), and \\( E \\) respectively. If the radii of \\( k_1 \\), \\( k_2 \\), and \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-0134588d4626c3d0d845a8c4da00c44cefd5ad1cb518204d24f4417f51f0213e | Three circles \( k_1, k_2, \) and \( k_3 \) pass through points \( A \) and \( B \). A secant through \( A \) intersects the circles \( k_1, k_2, \) and \( k_3 \) again at points \( C, D, \) and \( E \) respectively. If the radii of \( k_1, k_2, \) and \( k_3 \) are 5, 7, and 9 respectively, and the distance \( AB = 8 ... | [
"\\dfrac{\\sqrt{33} - 3}{\\sqrt{65} - \\sqrt{33}}"
] | [
"Three circles \\( k_1, k_2, \\) and \\( k_3 \\) pass through points \\( A \\) and \\( B \\). A secant through \\( A \\) intersects the circles \\( k_1, k_2, \\) and \\( k_3 \\) again at points \\( C, D, \\) and \\( E \\) respectively. If the radii of \\( k_1, k_2, \\) and \\( k_3 \\) are 5, 7, and 9 respectively, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e8532437f8062b1447c845df2fe27ac9627be831f5b70a2ba06bee9e6f4747d | Three circles \( w1 \), \( w2 \), and \( w3 \) are mutually tangent at points \( A \), \( B \), and \( C \). The three external common tangents of each pair of circles intersect at points \( D \), \( E \), and \( F \). If the coordinates of \( A \), \( B \), and \( C \) are \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \) ... | [
"0"
] | [
"Three circles \\( w1 \\), \\( w2 \\), and \\( w3 \\) are mutually tangent at points \\( A \\), \\( B \\), and \\( C \\). The three external common tangents of each pair of circles intersect at points \\( D \\), \\( E \\), and \\( F \\). If the coordinates of \\( A \\), \\( B \\), and \\( C \\) are \\( (1, 1) \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-cf7d77ae7ffc97c3b03c048c2aeafb1c465dd3d1a337a89ca3abbf3f8ef019de | Three circles \( w_1 \), \( w_2 \), and \( \Gamma \) are given on the same side of a line \( l \) such that \( w_1 \) and \( w_2 \) are tangent to \( l \) at points \( K \) and \( L \), respectively, and tangent to \( \Gamma \) at points \( M \) and \( N \), respectively. The circles \( w_1 \) and \( w_2 \) do not inte... | [
"20"
] | [
"Three circles \\( w_1 \\), \\( w_2 \\), and \\( \\Gamma \\) are given on the same side of a line \\( l \\) such that \\( w_1 \\) and \\( w_2 \\) are tangent to \\( l \\) at points \\( K \\) and \\( L \\), respectively, and tangent to \\( \\Gamma \\) at points \\( M \\) and \\( N \\), respectively. The circles \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-237576bd2ff985eddd3a3ea12daf8933e17e4f0c8c50754a0e616584381ff82a | Three circles \((O; R)\), \((O'; R')\), and \((I; r)\) are all tangent to a line \(d\) and to each other. If \(R = 16\) and \(R' = 9\), find the value of \(r\). | [
"\\dfrac{144}{49}"
] | [
"Three circles \\((O; R)\\), \\((O'; R')\\), and \\((I; r)\\) are all tangent to a line \\(d\\) and to each other. If \\(R = 16\\) and \\(R' = 9\\), find the value of \\(r\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63ce4c3b6a293711888d3af50f936c919008aaa0f68a1ab4e8f56c3555dd577c | Three circles \((O_1), (O_2), (O_3)\) intersect at a common point \(O\). Points \(A_1, A_2, A_3\) lie on circles \((O_1), (O_2), (O_3)\) respectively such that \(OA_1, OA_2, OA_3\) are parallel to \(O_2O_3, O_3O_1, O_1O_2\) respectively. If the radii of the circles \((O_1), (O_2), (O_3)\) are 5, 7, and 9 respectively, ... | [
"\\dfrac{\\sqrt{155}}{2}"
] | [
"Three circles \\((O_1), (O_2), (O_3)\\) intersect at a common point \\(O\\). Points \\(A_1, A_2, A_3\\) lie on circles \\((O_1), (O_2), (O_3)\\) respectively such that \\(OA_1, OA_2, OA_3\\) are parallel to \\(O_2O_3, O_3O_1, O_1O_2\\) respectively. If the radii of the circles \\((O_1), (O_2), (O_3)\\) are 5, 7, a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b2b08b95c3a914b677936f70ab7c6a5ebba85c51a9b2157ed46a870eb61cd4d3 | Three circles \((O_1)\), \((O_2)\), and \((O_3)\) are concurrent at point \(O\). A line \(\Delta\) passes through \(O\) and intersects \((O_1)\), \((O_2)\), and \((O_3)\) at points \(A_1\), \(A_2\), and \(A_3\) respectively. Lines \(\Delta_1\), \(\Delta_2\), and \(\Delta_3\) pass through \(A_1\), \(A_2\), and \(A_3\) a... | [
"(1, 1)"
] | [
"Three circles \\((O_1)\\), \\((O_2)\\), and \\((O_3)\\) are concurrent at point \\(O\\). A line \\(\\Delta\\) passes through \\(O\\) and intersects \\((O_1)\\), \\((O_2)\\), and \\((O_3)\\) at points \\(A_1\\), \\(A_2\\), and \\(A_3\\) respectively. Lines \\(\\Delta_1\\), \\(\\Delta_2\\), and \\(\\Delta_3\\) pass ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50d45d7402e64b377dfe584c3779e1f75c0fe638acc741db0cd496692e7fed30 | Three circles \((O_1, R)\), \((O_2, R)\), and \((O_3, R)\) intersect at a common point \(O\). Circles \((O_1)\) and \((O_2)\) intersect at \(O\) and \(P\). The third circle \((O_3)\) intersects \((O_1)\) and \((O_2)\) at points \(E\) and \(F\) respectively, but does not pass through \(P\). If \(R = 1\), calculate the l... | [
"1"
] | [
"Three circles \\((O_1, R)\\), \\((O_2, R)\\), and \\((O_3, R)\\) intersect at a common point \\(O\\). Circles \\((O_1)\\) and \\((O_2)\\) intersect at \\(O\\) and \\(P\\). The third circle \\((O_3)\\) intersects \\((O_1)\\) and \\((O_2)\\) at points \\(E\\) and \\(F\\) respectively, but does not pass through \\(P\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-9733a36fce7baa1f25377646a371dc3898d3be03674fa153f6ddadebc91ea9d7 | Three circles \((O_{1}; R_{1})\), \((O_{2}; R_{2})\), and \((O_{3}; R_{3})\) are tangent to each other at points \(A\), \(B\), and \(C\). Let \(r\) be the radius of the inscribed circle of triangle \(ABC\). If \(R_{1} = 6\), \(R_{2} = 8\), and \(R_{3} = 10\), find the value of \(6\sqrt{3}r\). | [
"24"
] | [
"Three circles \\((O_{1}; R_{1})\\), \\((O_{2}; R_{2})\\), and \\((O_{3}; R_{3})\\) are tangent to each other at points \\(A\\), \\(B\\), and \\(C\\). Let \\(r\\) be the radius of the inscribed circle of triangle \\(ABC\\). If \\(R_{1} = 6\\), \\(R_{2} = 8\\), and \\(R_{3} = 10\\), find the value of \\(6\\sqrt{3}r\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
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