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-29345b0025672c592d5fae7f0ed9f34aad6706b10a3b19fd0862d3712c139786 | Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive? | [
"15"
] | [
"Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14675d180b3abf2471546fd8ac0e80a87c1edfda3c742b1824780b263fe8356e | Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of 2024 pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third, and they continue to make moves in the same order. In each move, the player making the move must choose a positive ... | [
"Charlie"
] | [
"Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of 2024 pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third, and they continue to make moves in the same order. In each move, the player making the move must choose a posi... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-8b1a7fed21a74b1f6eaf33ba1eb4a34da4195db19717bd14209257bc8971e9e6 | Three friends agree to meet between 3 p.m. and 4 p.m. Each friend will wait for a maximum of 30 minutes. What is the probability that all three friends will meet within this time frame? | [
"\\dfrac{1}{2}"
] | [
"Three friends agree to meet between 3 p.m. and 4 p.m. Each friend will wait for a maximum of 30 minutes. What is the probability that all three friends will meet within this time frame?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2d0a4ca3908eb69966fd4142f5d4c0e43292f8955154a2424cb1c49751ed8695 | Three friends agree to meet between noon and 1:00 PM. Each arrives at a random time in the interval and will wait up to 15 minutes for the others before leaving. What is the probability that they all meet? | [
"\\dfrac{5}{32}"
] | [
"Three friends agree to meet between noon and 1:00 PM. Each arrives at a random time in the interval and will wait up to 15 minutes for the others before leaving. What is the probability that they all meet?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 7
} | 38 | 0 | 0 |
omr-ws-5be751240cd06401abb371b437a6ffc09ec65d914f1eff40ac90c3761ccba3a5 | Three friends are driving to New York City and splitting the gas cost equally. At the last minute, 2 more friends joined the trip. The cost of gas was then evenly redistributed among all of the friends. The cost for each of the original 3 decreased by $\$$11.00. What was the total cost of the gas, in dollars? | [
"82.50"
] | [
"Three friends are driving to New York City and splitting the gas cost equally. At the last minute, 2 more friends joined the trip. The cost of gas was then evenly redistributed among all of the friends. The cost for each of the original 3 decreased by $\\$$11.00. What was the total cost of the gas, in dollars?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51926c6fde2264998ba0d8c408c9cb10a13c47a33611e049c9411cbf17e8dfaf | Three friends are located in Maykop and one friend is in Belorechensk. The distance between Maykop and Belorechensk is 24 km. Two friends in Maykop need to reach Belorechensk, and the friend in Belorechensk wants to reach Maykop. They have only one bike, which is initially in Maykop. Each person can walk at a maximum s... | [
"200"
] | [
"Three friends are located in Maykop and one friend is in Belorechensk. The distance between Maykop and Belorechensk is 24 km. Two friends in Maykop need to reach Belorechensk, and the friend in Belorechensk wants to reach Maykop. They have only one bike, which is initially in Maykop. Each person can walk at a maxi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-c5501e77ac9ae3218df7636da413d37a206d15bcc64e609b1c2aafc12409aa73 | Three friends are walking at 5 kph around a 1-km race track. One of them stops for 8 minutes and then resumes walking in the opposite direction. How many minutes later will he encounter his friends? | [
"2"
] | [
"Three friends are walking at 5 kph around a 1-km race track. One of them stops for 8 minutes and then resumes walking in the opposite direction. How many minutes later will he encounter his friends?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34d3c55bc4fbd665b70881d7c607e34f50068e8dac030ff854f66d99f6fa9097 | Three friends found a large box of nuts and decided to divide them the next morning. One friend woke up first, threw 3 nuts to squirrels, hid 1/4 of the remaining nuts, and went back to sleep. The second friend did the same, and the third friend did the same before dividing the remaining nuts into three bags, each with... | [
"445"
] | [
"Three friends found a large box of nuts and decided to divide them the next morning. One friend woke up first, threw 3 nuts to squirrels, hid 1/4 of the remaining nuts, and went back to sleep. The second friend did the same, and the third friend did the same before dividing the remaining nuts into three bags, each... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-552255d4945bf96dc011b16c55a392e4e0bc23ee35cf649d745de65be67eeb11 | Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen? | [
"10"
] | [
"Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0fbc3a4a2b315fa7750bf88c049b166679d3765af062e0b15ebd148e71e9a32 | Three friends need to travel between Maykop and Belorechensk, which are 24 km apart. One friend starts from Maykop with a bike, while two friends start from Belorechensk without a bike. The friend with the bike can ride at 18 km/h and the others can walk at 6 km/h. The bike can only be used by one person at a time, and... | [
"120"
] | [
"Three friends need to travel between Maykop and Belorechensk, which are 24 km apart. One friend starts from Maykop with a bike, while two friends start from Belorechensk without a bike. The friend with the bike can ride at 18 km/h and the others can walk at 6 km/h. The bike can only be used by one person at a time... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d227860350da24f1277cb6c8029edd0ccf4dd3661bbe699e62eaa37282f4ec2 | Three friends take turns saying 1 to 3 consecutive integers starting from 1 and going to 10. The friend who says 10 loses, while the other two win. Friend A starts, followed by friend B, then C, and then back to A. Is there an optimal strategy for friend A, B, or C that will allow them to win? | [
"A"
] | [
"Three friends take turns saying 1 to 3 consecutive integers starting from 1 and going to 10. The friend who says 10 loses, while the other two win. Friend A starts, followed by friend B, then C, and then back to A. Is there an optimal strategy for friend A, B, or C that will allow them to win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-b4ceef49ebe2f6b2745ca04d33d1b99e71424ff63543e9e0defc25416e359e96 | Three friends want to move from point $A$ to point $B$ on the external ring of a circular street layout with common center $O$. Dima goes along the external ring, Kostya goes from $A$ to $O$ and then to $B$. Sergey decides to take a path that involves traveling from $A$ to a point $P$ on the circumference, then to $O$,... | [
"2"
] | [
"Three friends want to move from point $A$ to point $B$ on the external ring of a circular street layout with common center $O$. Dima goes along the external ring, Kostya goes from $A$ to $O$ and then to $B$. Sergey decides to take a path that involves traveling from $A$ to a point $P$ on the circumference, then to... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d99f23affbd444296ed239354f7b2e26eaf867d79bd5cc5149b69c5b60fe4d99 | Three friends want to share five (identical) muffins so that each friend ends up with the same total amount of muffin. Nobody likes small pieces of muffin, so the friends cut up and distribute the muffins in such a way that they maximize the size of the smallest muffin piece. What is the size of this smallest piece? | [
"\\dfrac{1}{3}"
] | [
"Three friends want to share five (identical) muffins so that each friend ends up with the same total amount of muffin. Nobody likes small pieces of muffin, so the friends cut up and distribute the muffins in such a way that they maximize the size of the smallest muffin piece. What is the size of this smallest piec... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-51d479f6636a346d2671f5e8bcf5c2bb749449b92dc6116e88c8ec5b12fceeb5 | Three friends want to travel 32 miles from Derby to Kent. Andy travels by car at 8 mph, Bill by bike at 6 mph, and Charlie on foot at 4 mph. They can exchange modes of transportation only on the hour. What is the minimum time in hours it will take for all three friends to arrive at Kent simultaneously? | [
"6"
] | [
"Three friends want to travel 32 miles from Derby to Kent. Andy travels by car at 8 mph, Bill by bike at 6 mph, and Charlie on foot at 4 mph. They can exchange modes of transportation only on the hour. What is the minimum time in hours it will take for all three friends to arrive at Kent simultaneously?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-de9c986b678e1e0853838e3dff429e43e222095ec0983cbd655b9cd6e502f19e | Three friends, Anne, Bernard, and Casey, started the night with jellybeans in the ratio 7:6:5. At the end of the night, the ratio was 6:5:4. One of the friends won 1200 jellybeans. How many jellybeans did each of the friends bring with them on Saturday night? | [
"42000, 36000, 30000"
] | [
"Three friends, Anne, Bernard, and Casey, started the night with jellybeans in the ratio 7:6:5. At the end of the night, the ratio was 6:5:4. One of the friends won 1200 jellybeans. How many jellybeans did each of the friends bring with them on Saturday night?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-2d162a5f2c1e52ee54692c9e171860d212bf9db5d2d29975cadf3dd53a324c06 | Three friends, Xander, Yulia, and Zoe, have each planned to visit the same cafe one day. If each person arrives at the cafe at a random time between $2$ PM and $3$ PM and stays for $15$ minutes, what is the probability that all three friends will be there at the same time at some point? | [
"\\dfrac{5}{32}"
] | [
"Three friends, Xander, Yulia, and Zoe, have each planned to visit the same cafe one day. If each person arrives at the cafe at a random time between $2$ PM and $3$ PM and stays for $15$ minutes, what is the probability that all three friends will be there at the same time at some point?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ebd90c375a7e4e4bba2a5b7e8c5e7139d4e3f1c99ed452d8d7b3870dd877c49d | Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another such that the leapee is at the midpoint of the line segment whose endpoints are the starting and ending positions of the leaper. After 100 minutes, what is the sum of the coordinates of the position of the frog that started ... | [
"0"
] | [
"Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another such that the leapee is at the midpoint of the line segment whose endpoints are the starting and ending positions of the leaper. After 100 minutes, what is the sum of the coordinates of the position of the frog that sta... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3989758c6dfa9db7f64ffc0f3421369288a37759b4975c47332eac9e45f9b8a4 | Three gamblers start with sums of money in the ratio 7:6:5 and finish with sums of money in the ratio 6:5:4. One of the players won 120. How much did this player start with? | [
"4200"
] | [
"Three gamblers start with sums of money in the ratio 7:6:5 and finish with sums of money in the ratio 6:5:4. One of the players won 120. How much did this player start with?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4627050254cb9f40c0eeeec731e37b347030e8bc7db56304c322f1b7c7f73ff7 | Three gears have perimeters \(P_1\), \(P_2\), and \(P_3\) respectively. If the first gear is given a complete turn, then the second gear makes a complete turn, and finally, the third gear makes a complete turn (all clockwise), the gears return to their original positions. Determine the possible values of \((P_1, P_2, P... | [
"(P_1, P_2, P_3) \\text{ where } P_1 = P_2 = P_3"
] | [
"Three gears have perimeters \\(P_1\\), \\(P_2\\), and \\(P_3\\) respectively. If the first gear is given a complete turn, then the second gear makes a complete turn, and finally, the third gear makes a complete turn (all clockwise), the gears return to their original positions. Determine the possible values of \\(... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-cff0bfd1e00ce6dc41ecc00ac643bb7b20709e16969fd94ad096524a20b82744 | Three girls A, B, and C are standing on a circle of radius 5 meters. The distance between A and B is 6 meters, and the distance between B and C is 6 meters. Find the distance between A and C. | [
"9.6"
] | [
"Three girls A, B, and C are standing on a circle of radius 5 meters. The distance between A and B is 6 meters, and the distance between B and C is 6 meters. Find the distance between A and C."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-ec22ee56aa61d0963d65abc0846536c16aa4e4255813c7ed8db9d7cea3dfd202 | Three girls A, B, and C, and nine boys are to be lined up in a row. B must lie between A and C, and A and B must be separated by exactly 4 boys. Determine the value of $\left \lfloor \frac{n}{7!} \right \rfloor$, where $n$ is the number of ways to arrange them under these conditions. | [
"3024"
] | [
"Three girls A, B, and C, and nine boys are to be lined up in a row. B must lie between A and C, and A and B must be separated by exactly 4 boys. Determine the value of $\\left \\lfloor \\frac{n}{7!} \\right \\rfloor$, where $n$ is the number of ways to arrange them under these conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 11
} | 64 | 0 | 0 |
omr-ws-6ad7fb7edc4bdfd025ffef7e735e26a2fea3ce8c5aca59990a36e4b0bbe52432 | Three girls and two boys are on the Math Team. Two students wear white shirts, and three wear black shirts. Miken and Carry wear shirts of different colors. Bari and Jamie wear shirts of the same color. Pita and Carry are the same gender. Jamie and Miken are of different genders. The boy in the white shirt scored the m... | [
"Miken"
] | [
"Three girls and two boys are on the Math Team. Two students wear white shirts, and three wear black shirts. Miken and Carry wear shirts of different colors. Bari and Jamie wear shirts of the same color. Pita and Carry are the same gender. Jamie and Miken are of different genders. The boy in the white shirt scored ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-fcc7d4c5ab604fbfcba56178520aa8c01753736f4ed87a9fea1cfa035b237cb2 | Three girls' birth years are 1986, 1989, and 1992. Their last names are Kiss, Hunter, and Goldbach. Their first names are Olga, Maria, and Renata. In a certain year, each girl's age matched the number of letters in her full name (first name + last name). Determine the year this occurred and the full names of the girls. | [
"2000"
] | [
"Three girls' birth years are 1986, 1989, and 1992. Their last names are Kiss, Hunter, and Goldbach. Their first names are Olga, Maria, and Renata. In a certain year, each girl's age matched the number of letters in her full name (first name + last name). Determine the year this occurred and the full names of the g... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-b4bf433870df71d8df6b637abb61fb1846bbb8f52f2dde7e80d37ac6bfe053d6 | Three given circles have the same radius and pass through a common point \( P \). Their other points of pairwise intersections are \( A \), \( B \), and \( C \). We define triangle \( A'B'C' \), each of whose sides is tangent to two of the three circles. The three circles are contained in \( \triangle A'B'C' \). If the... | [
"4"
] | [
"Three given circles have the same radius and pass through a common point \\( P \\). Their other points of pairwise intersections are \\( A \\), \\( B \\), and \\( C \\). We define triangle \\( A'B'C' \\), each of whose sides is tangent to two of the three circles. The three circles are contained in \\( \\triangle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-e85c33d98fd59dd93d1d1266237ef73567bc42881c07add517f1305ef34b2c74 | Three grasshoppers \( A \), \( B \), and \( C \) are placed on a line, with \( B \) at the midpoint between \( A \) and \( C \). Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if \( X \) jumps over \( Y \) to point \( X' \), then \( XY = YX' \)). After sev... | [
"0"
] | [
"Three grasshoppers \\( A \\), \\( B \\), and \\( C \\) are placed on a line, with \\( B \\) at the midpoint between \\( A \\) and \\( C \\). Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if \\( X \\) jumps over \\( Y \\) to point \\( X' \\), then \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-dc02c05589558a0ee81411129b69c39d70ceea999821a959805ddf2bf179312a | Three grasshoppers jump at different speeds along the path along the contour of the square. Each of the grasshoppers makes one jump in one second. The fastest grasshopper overcomes the side of the square in $5$ jumps, the second in $15$ jumps, and the third in $N$ jumps ($N$ is natural). Find $N$ if grasshoppers meet o... | [
"20"
] | [
"Three grasshoppers jump at different speeds along the path along the contour of the square. Each of the grasshoppers makes one jump in one second. The fastest grasshopper overcomes the side of the square in $5$ jumps, the second in $15$ jumps, and the third in $N$ jumps ($N$ is natural). Find $N$ if grasshoppers m... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a263b6694afff37340593f2939d66359ddf3f433a2a9d412350f88660a38aa81 | Three grasshoppers sit on a plane at points \((0,1)\), \((1,1)\), and \((1,0)\). One of the grasshoppers hops symmetrically over one of the others. No grasshopper jumps twice in a row. Determine the number of distinct points \((a, b)\) with \(0 \leq a, b \leq 10\) and \(a\) and \(b\) relatively prime, to which one of t... | [
"65"
] | [
"Three grasshoppers sit on a plane at points \\((0,1)\\), \\((1,1)\\), and \\((1,0)\\). One of the grasshoppers hops symmetrically over one of the others. No grasshopper jumps twice in a row. Determine the number of distinct points \\((a, b)\\) with \\(0 \\leq a, b \\leq 10\\) and \\(a\\) and \\(b\\) relatively pri... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-455bc12f872ffacc2ba90b7e9fa24b22683b464fefff83f30d50850b1337abd6 | Three groups of lines divide a plane into \( D \) regions. Every pair of lines in the same group are parallel. Let \( x, y \), and \( z \) respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines in groups 1 and 2, then the least number of lines requ... | [
"102"
] | [
"Three groups of lines divide a plane into \\( D \\) regions. Every pair of lines in the same group are parallel. Let \\( x, y \\), and \\( z \\) respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines in groups 1 and 2, then the least number of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-866e9434a4f65fad9703944d8dffcc8e3c684800344b0b9c6465843d0fdc5c35 | Three heads of the Serpent Gorynych ate two identical cakes in $15$ minutes, with each head eating as much time as it would take the other two to eat the same cake together. In how many minutes would the three heads eat one cake together? | [
"\\(\\frac{10}{3}\\)"
] | [
"Three heads of the Serpent Gorynych ate two identical cakes in $15$ minutes, with each head eating as much time as it would take the other two to eat the same cake together. In how many minutes would the three heads eat one cake together?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66459b06616411c0db080595c62a80f3e763070d19c467fc8a9ecd06acca9aac | Three horsemen are riding in the same direction along a circular road with constant and pairwise distinct speeds. They can only surpass one another at a single designated point on the road. Let $T$ be the minimum time after which all possible surpassing sequences among the three horsemen have occurred at least once. As... | [
"1"
] | [
"Three horsemen are riding in the same direction along a circular road with constant and pairwise distinct speeds. They can only surpass one another at a single designated point on the road. Let $T$ be the minimum time after which all possible surpassing sequences among the three horsemen have occurred at least onc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-a2370dfa17a4226cedade341148cbc3365661c81e3a2ec97aabae2e3bae275b4 | Three hundred students participated in a mathematical olympiad. Any two students either know or don't know each other, and there aren't three students who all know each other. Each student knows at most $n$ other students, and for every $m$ (with $1 \le m \le n$), there is at least one student who knows exactly $m$ oth... | [
"150"
] | [
"Three hundred students participated in a mathematical olympiad. Any two students either know or don't know each other, and there aren't three students who all know each other. Each student knows at most $n$ other students, and for every $m$ (with $1 \\le m \\le n$), there is at least one student who knows exactly ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-0015b1a919720dba7ae82deff6bc3c7ab61d9535e25eb423f28475a6d24c9c47 | Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at least one p... | [
"\\(\\frac{3}{4}\\)"
] | [
"Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at least ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-36a686919c11caa940c16e181346cad1ff61e70422cd59eb4f85f291253d61d2 | Three identical balls of radius $1$ are located on a plane, each touching the other two and the plane. Find the radius of the fourth ball that touches each of these three balls and the plane. | [
"\\(\\frac{1}{3}\\)"
] | [
"Three identical balls of radius $1$ are located on a plane, each touching the other two and the plane. Find the radius of the fourth ball that touches each of these three balls and the plane."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-c2ab90282dde6ba5f29f65aef8af8811ce2e794b2e7feef6cda77dbb77f8f116 | Three identical balls, each with a radius of 3 cm, are placed side by side so they are just touching each other. An identical fourth ball is placed on top of the other three balls so that it just touches the other three balls. What is the height from the top of the fourth ball to the ground surface on which the balls h... | [
"\\(2\\sqrt{6} + 6\\)"
] | [
"Three identical balls, each with a radius of 3 cm, are placed side by side so they are just touching each other. An identical fourth ball is placed on top of the other three balls so that it just touches the other three balls. What is the height from the top of the fourth ball to the ground surface on which the ba... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 58
} | 122 | 0 | 0 |
omr-ws-d946c6cd48b45cb6f55506ce5f517af83d8c99537e123b105ed3d38a180727aa | Three identical circles are touching each other. Three equilateral triangles are drawn: the first by joining the centers of the circles, the second by joining points on the circumferences where the circles touch, and the third by joining the points of tangency of the circles. What is the ratio of the perimeters of thes... | [
"1:1+2\\sqrt3:2+\\sqrt3"
] | [
"Three identical circles are touching each other. Three equilateral triangles are drawn: the first by joining the centers of the circles, the second by joining points on the circumferences where the circles touch, and the third by joining the points of tangency of the circles. What is the ratio of the perimeters of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a678cace5228b3e22ab73f1bfe38008f7586cfeccc3bc8cb183d240c13168f1e | Three identical cones with apex \( A \) touch each other externally. Each of them internally touches a fourth cone with apex at point \( A \) and an apex angle \( \frac{2\pi}{3} \). Find the apex angle for the identical cones. (The angle at the apex of the cone is the angle between its generatrices in the axial section... | [
"\\( 2 \\text{ arccot} \\frac{4+\\sqrt{3}}{3} \\)"
] | [
"Three identical cones with apex \\( A \\) touch each other externally. Each of them internally touches a fourth cone with apex at point \\( A \\) and an apex angle \\( \\frac{2\\pi}{3} \\). Find the apex angle for the identical cones. (The angle at the apex of the cone is the angle between its generatrices in the ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5676cb96dc04c3fa79ab88a07306976e08fe9dce3541f452510ee475e05e9950 | Three identical cones, each 120 cm tall with a base diameter of 100 cm, are placed on a table. A sphere with a radius of 50 cm is placed in the middle of the third cone. Determine the height of the sphere's center from the surface of the table. | [
"50"
] | [
"Three identical cones, each 120 cm tall with a base diameter of 100 cm, are placed on a table. A sphere with a radius of 50 cm is placed in the middle of the third cone. Determine the height of the sphere's center from the surface of the table."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 30,
"genselect": 5
} | 35 | 0 | 0 |
omr-ws-df028b2232156e2d049f1cb7badb57d4f2010c49937f242bef8879c8d1f617a0 | Three identical rectangles are combined to form rectangle ABCD. Given that the shorter side of each rectangle is 5 feet, what is the area of rectangle ABCD in square feet? | [
"150"
] | [
"Three identical rectangles are combined to form rectangle ABCD. Given that the shorter side of each rectangle is 5 feet, what is the area of rectangle ABCD in square feet?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4aac8195282374e70dd2f97c5bf402c45c99f3e6aadbbbdf792b3a65c80411b8 | Three identical rectangles are put together to form rectangle \(ABCD\). Given that the length of the shorter side of each of the rectangles is 5 feet, what is the area in square feet of rectangle \(ABCD\)? | [
"150"
] | [
"Three identical rectangles are put together to form rectangle \\(ABCD\\). Given that the length of the shorter side of each of the rectangles is 5 feet, what is the area in square feet of rectangle \\(ABCD\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-1dc9921850b144552670bb9f4dfa44249378a202e196f43c54d88819d907da4b | Three identical rectangles are put together to form rectangle \(ABCD\). Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle \(ABCD\)? | [
"150"
] | [
"Three identical rectangles are put together to form rectangle \\(ABCD\\). Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle \\(ABCD\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 7,
"tir": 1
} | 54 | 0 | 0 |
omr-ws-8dab8ec76ba57bbe65a44e494d8d7eec521d63ec39f4ca92ff3f2cd3ef311b36 | Three identical regular triangular pyramids, each with a volume of $36\sqrt{2}$, are placed such that they share a common side edge and a common top. Find the maximum side length of the base of the pyramids. | [
"12"
] | [
"Three identical regular triangular pyramids, each with a volume of $36\\sqrt{2}$, are placed such that they share a common side edge and a common top. Find the maximum side length of the base of the pyramids."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-10ce3e92ce43bf52620a508778dcece1e80116937b120155ea5d2bce39cd3903 | Three identical right-angled triangles are placed inside an equilateral triangle, one in each corner. The total area covered twice (dark grey) is equal to the area left uncovered (white). What fraction of the area of the equilateral triangle does one right-angled triangle cover? | [
"\\(\\frac{1}{3}\\)"
] | [
"Three identical right-angled triangles are placed inside an equilateral triangle, one in each corner. The total area covered twice (dark grey) is equal to the area left uncovered (white). What fraction of the area of the equilateral triangle does one right-angled triangle cover?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-f21ab6243792698ee9df89ee2a4a4a13959fe91e88ecdbeebdaa8f0c23a619d8 | Three identical snowplows start plowing down a long straight road at noon, 1:00 p.m., and 2:00 p.m., respectively. The snow started falling at a constant rate before noon. All three plows collide with each other at the same time. Assuming that the plowing rates are inversely proportional to the snow depth, determine wh... | [
"\\frac{\\sqrt{5} - 1}{2}"
] | [
"Three identical snowplows start plowing down a long straight road at noon, 1:00 p.m., and 2:00 p.m., respectively. The snow started falling at a constant rate before noon. All three plows collide with each other at the same time. Assuming that the plowing rates are inversely proportional to the snow depth, determi... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a82f5f3630dd7347600364cbf8db4d71eeead209933bf0bafeb052ca9b9575dc | Three identical spheres are in the interior of a regular triangular pyramid with the length of the edge of the base equal to $a$ and the base dihedral angle equal to $60^\circ$. Each sphere touches the other two spheres, the plane of the base and two lateral faces. Find the radii of the spheres. | [
"$\\frac{a}{8}$"
] | [
"Three identical spheres are in the interior of a regular triangular pyramid with the length of the edge of the base equal to $a$ and the base dihedral angle equal to $60^\\circ$. Each sphere touches the other two spheres, the plane of the base and two lateral faces. Find the radii of the spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e171b1ed2c1f6ce0deb3cb5309f21f2ed249ca53bde2ead1e9254ad3d05d2067 | Three identical spheres are inscribed into a right circular cylinder with the radius of the base $r = 1$ and the altitude $H=12/\left(3+2\sqrt 3\right)$ so that the spheres touch the upper base of the cylinder, its lateral surface and pairwise touch one another. Find the volume of a right circular cone whose base coinc... | [
"$\\frac{4\\pi }{9}$"
] | [
"Three identical spheres are inscribed into a right circular cylinder with the radius of the base $r = 1$ and the altitude $H=12/\\left(3+2\\sqrt 3\\right)$ so that the spheres touch the upper base of the cylinder, its lateral surface and pairwise touch one another. Find the volume of a right circular cone whose ba... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11704636a3ecbf13cfdb0d65b2947f3abd2a0b6e9cf166e2399a000bd40cb192 | Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has a volume of $108$ cm$^3$. What is the side length of the bottom of the glass? | [
"\\(\\sqrt[3]{4}\\)"
] | [
"Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has a volume of $108$ cm$^3$. What is the side length of the bottom of the glass?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-474081a9076e7ed1c81e48b363ad4a52d42f69056b3e8bed691e2c0d5e09dcfb | Three identical spheres of radius $r$ are in the interior of a right circular cone, with the vertex angle of $60^\circ$ in the axial section of the cone, so that each sphere touches the other two spheres, the lateral surface of the cone and the plane of the base. Find the radius of the base of the cone. | [
"$\\frac{5}{\\sqrt 3}r$"
] | [
"Three identical spheres of radius $r$ are in the interior of a right circular cone, with the vertex angle of $60^\\circ$ in the axial section of the cone, so that each sphere touches the other two spheres, the lateral surface of the cone and the plane of the base. Find the radius of the base of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e37155b000fa3ec38539f4b1867b5ef39c1d405960dbc5061a45fa828e13aa9 | Three identical spheres with radii 4 cm, 5 cm, and 6 cm are tangent to each other. Determine the distances from each center of the spheres to the center of gravity of the three spheres. | [
"\\dfrac{\\sqrt{1689}}{9}"
] | [
"Three identical spheres with radii 4 cm, 5 cm, and 6 cm are tangent to each other. Determine the distances from each center of the spheres to the center of gravity of the three spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d6496dbaa4c3ff14aeaa8fad9cd788341e650e3b407b53b48934bec195fc2c94 | Three identical spheres, each with radius \( a \), are inserted into a cylinder such that: 1. Each sphere is tangent to the lateral surface of the cylinder. 2. Each sphere is tangent to the other two spheres. 3. Two spheres touch the lower base while the third one touches the upper base. 4. The height of the cylinder i... | [
"\\( a \\left(\\frac{3 + 2\\sqrt{3}}{3}\\right) \\)"
] | [
"Three identical spheres, each with radius \\( a \\), are inserted into a cylinder such that:\n1. Each sphere is tangent to the lateral surface of the cylinder.\n2. Each sphere is tangent to the other two spheres.\n3. Two spheres touch the lower base while the third one touches the upper base.\n4. The height of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-60b31d5fed35d006e926d4dc7b3ee968a2edda14b823d5b36400cc39b28d2619 | Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in a $24$-sided polygon. The area of this polygon can be expressed in the for... | [
"147"
] | [
"Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in a $24$-sided polygon. The area of this polygon can be expressed in ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-31226d8c232086b5faaf2d98d14f9f7dc4442edd6003df278c3082f996653e4c | Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30° about its center and the top sheet is rotated clockwise 60° about its center, resulting in a 24-sided polygon. The area of this polygon can be expressed in the form \(a - b\sqrt{c}\... | [
"147"
] | [
"Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30° about its center and the top sheet is rotated clockwise 60° about its center, resulting in a 24-sided polygon. The area of this polygon can be expressed in the form \\(a - b\\s... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-0cfd9f3da19d66f8f0345306b78de465273868ab1e760dd94e0c13cdf155f4b4 | Three identical squares with side length $4$ units are spaced in a straight line, $1$ unit apart. The rightmost square is translated upwards $2$ units. The middle square is adjusted so that each of its $4$ corners are $a$ units away from its closest corner, other than itself. Find $a^2$. | [
"2"
] | [
"Three identical squares with side length $4$ units are spaced in a straight line, $1$ unit apart. The rightmost square is translated upwards $2$ units. The middle square is adjusted so that each of its $4$ corners are $a$ units away from its closest corner, other than itself. Find $a^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-367ac9135d67e861e5c9ff3ea2f6d1cd589cf3764657dd031a74aa339785f712 | Three identical test tubes are half filled with alcohol solutions. When the content of the third tube was divided in half and each half was poured into each of the first two tubes, the volume concentration of alcohol in the first tube became smaller by 20\% of its value and that in the second tube increased by 10\% of ... | [
"13/4 times"
] | [
"Three identical test tubes are half filled with alcohol solutions. When the content of the third tube was divided in half and each half was poured into each of the first two tubes, the volume concentration of alcohol in the first tube became smaller by 20\\% of its value and that in the second tube increased by 10... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db3fd880197f642739f8259af9d689db2f58f68d845fd1c522d3073d7fb4dc39 | Three independent meteorologists, each of whom is wrong 1/4 of the time, all predict that it will be sunny tomorrow. What is the probability that it will be sunny tomorrow? | [
"\\(\\frac{27}{28}\\)"
] | [
"Three independent meteorologists, each of whom is wrong 1/4 of the time, all predict that it will be sunny tomorrow. What is the probability that it will be sunny tomorrow?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c65562ef74833362294f6e265765c56761372d3f54caba721b27b0657fac0794 | Three integers (not necessarily distinct) form a geometric progression with a sum of 741, and the common ratio is an integer. If the terms (not necessarily in order) of the progression are represented by \( x \), \( y \), and \( z \) where \( x \leq y \leq z \), find the sum of all possible different values of \( z \). | [
"1397"
] | [
"Three integers (not necessarily distinct) form a geometric progression with a sum of 741, and the common ratio is an integer. If the terms (not necessarily in order) of the progression are represented by \\( x \\), \\( y \\), and \\( z \\) where \\( x \\leq y \\leq z \\), find the sum of all possible different val... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-5eae8510f4fbc3171af05e99be16194284a855776d9e1e6d5c620dc273155df9 | Three integers \( a, b \), and \( c \) have a product of 12012. Further, \( \frac{a}{4} = b + 2 = c - 8 \). What is \( a + b + c \)? | [
"84"
] | [
"Three integers \\( a, b \\), and \\( c \\) have a product of 12012. Further, \\( \\frac{a}{4} = b + 2 = c - 8 \\). What is \\( a + b + c \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-2d5d281e9f5ac462b8f19f237012c35ea43cd5129e25ae4359e6d1c95c72fae9 | Three integers \(a\), \(b\), and \(c\) are three consecutive terms of an arithmetic sequence, and \(a \cdot b \cdot c\) is a positive prime number. Find \((a, b, c)\). | [
"(-3, -1, 1)"
] | [
"Three integers \\(a\\), \\(b\\), and \\(c\\) are three consecutive terms of an arithmetic sequence, and \\(a \\cdot b \\cdot c\\) is a positive prime number. Find \\((a, b, c)\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 3
} | 16 | 0 | 0 |
omr-ws-6eef272e46dd899a0cffec2c192808a990bcbbbcd34452d767b149440ecd7d56 | Three integers \(a\), \(b\), and \(c\) have a product of \(9!\) and satisfy \(a \le b \le c\). Find the smallest possible value of \(c - a\). | [
"2"
] | [
"Three integers \\(a\\), \\(b\\), and \\(c\\) have a product of \\(9!\\) and satisfy \\(a \\le b \\le c\\). Find the smallest possible value of \\(c - a\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-5fbe1bb03e18e9b3a96b65ded2bde93be308fd5f19986a86ae396f5e7eee4d96 | Three integers are chosen at random from the range $[1, 5]$. Find the probability that their product has exactly $4$ factors. | [
"\\(\\frac{51}{125}\\)"
] | [
"Three integers are chosen at random from the range $[1, 5]$. Find the probability that their product has exactly $4$ factors."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3f522f1794eba9e8410c00a67ed2761585b1fbb8f75607055c17551f17dca9d | Three integers are selected simultaneously from the set $\{1, 2, 3, \ldots, 25\}$. Find the probability that the product of the selected numbers is divisible by 4. | [
"\\dfrac{773}{1150}"
] | [
"Three integers are selected simultaneously from the set $\\{1, 2, 3, \\ldots, 25\\}$. Find the probability that the product of the selected numbers is divisible by 4."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32a9085a080dec8564e72d1f9ce06baa0e161d808e9d8329d67f5891b1ab65ed | Three integers form a geometric progression. Their sum is $21$ and the sum of their reciprocals is $\frac{7}{12}$. Find the largest integer. | [
"12"
] | [
"Three integers form a geometric progression. Their sum is $21$ and the sum of their reciprocals is $\\frac{7}{12}$. Find the largest integer."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f963556166429d4ed08d36ceee50d3c0f9688490eacc3b1eac8e515f2e948d04 | Three isometric circles that pass through a common point also meet at the points M, N, and P. Let M'N'P' be the triangle containing the three circles and whose sides are each tangent to two of the circles. If the area of triangle MNP is 4, find the minimum possible area of triangle M'N'P'. | [
"\\dfrac{100}{9}"
] | [
"Three isometric circles that pass through a common point also meet at the points M, N, and P. Let M'N'P' be the triangle containing the three circles and whose sides are each tangent to two of the circles. If the area of triangle MNP is 4, find the minimum possible area of triangle M'N'P'."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7fac068172b4a3d3c1646fcde51ce5c7654bbfa8f5c4ba7a9747316063524a7 | Three isometrical circles intersect at points $A$, $B$, and $C$. Let $A'B'C'$ be the triangle containing all three circles, with each side tangent to two of the circles. If the area of triangle $A'B'C'$ is $16$, find the maximum possible area of triangle $ABC$. | [
"8"
] | [
"Three isometrical circles intersect at points $A$, $B$, and $C$. Let $A'B'C'$ be the triangle containing all three circles, with each side tangent to two of the circles. If the area of triangle $A'B'C'$ is $16$, find the maximum possible area of triangle $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-88faeb82432263bd2bce2e6bd41fd4432df4a0dff99e125984ee54508733353f | Three isosceles right triangles are erected from the larger side of a rectangle into the interior of the rectangle, where M is the midpoint of that side. Five circles are inscribed tangent to some of the sides and to one another, with one of the circles touching the vertex of the largest triangle. Find the ratios among... | [
"1 : \\sqrt{2} - 1 : (\\sqrt{2} - 1)^2 : (\\sqrt{2} - 1)^3 : (\\sqrt{2} - 1)^4"
] | [
"Three isosceles right triangles are erected from the larger side of a rectangle into the interior of the rectangle, where M is the midpoint of that side. Five circles are inscribed tangent to some of the sides and to one another, with one of the circles touching the vertex of the largest triangle. Find the ratios ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-6f4a4414d92fe91d4ca2e57d4630e97cff0e1329a266d302913c173d3498ac6e | Three large cubes with sides 1m, 3m, and 5m are glued together at their faces. What is the least possible surface area of the resulting solid figure? | [
"188"
] | [
"Three large cubes with sides 1m, 3m, and 5m are glued together at their faces. What is the least possible surface area of the resulting solid figure?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-319d93755576db942486e1b61aaf0eac3af1d7b30c63ecec89adc09e4df7c5ff | Three large squares of tin, each with edges 1m long, have four small, equal squares cut from their corners. All twelve resulting small squares are to be of the same size. The three large cross-shaped pieces are then folded and welded to make boxes with no tops, and the twelve small squares are used to make two small cu... | [
"\\( 0.25 \\, \\text{m}^3 \\)"
] | [
"Three large squares of tin, each with edges 1m long, have four small, equal squares cut from their corners. All twelve resulting small squares are to be of the same size. The three large cross-shaped pieces are then folded and welded to make boxes with no tops, and the twelve small squares are used to make two sma... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-26811b2a3a98561894b9f433628b23e8fc9c53d546c3ac7750bedb3de9df6fed | Three lattice points (points with integer coordinates) are chosen at random with replacement in the interior of the square defined by $-99 \le x \le 100$ and $-99 \le y \le 100$. The probability that the area of the triangle thus formed (which may be degenerate) is an integer can be expressed in the form $m/n$, where $... | [
"13"
] | [
"Three lattice points (points with integer coordinates) are chosen at random with replacement in the interior of the square defined by $-99 \\le x \\le 100$ and $-99 \\le y \\le 100$. The probability that the area of the triangle thus formed (which may be degenerate) is an integer can be expressed in the form $m/n$... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-e99cd6bd76682de677f0a532685a8a3385d552bff668064bca191ccdea2a6eb4 | Three lattice points (points with integer coordinates) are chosen uniformly at random from a large section of the plane to form a triangle. What is the probability that the area of the triangle is an integer? | [
"\\dfrac{5}{8}"
] | [
"Three lattice points (points with integer coordinates) are chosen uniformly at random from a large section of the plane to form a triangle. What is the probability that the area of the triangle is an integer?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b17dc7a7cff21c9db8de573a8d546cffb0e3ecb355c821b8560fc9a0c2f2bfa | Three letters are randomly picked from the word PRECAL. What are the odds that the set of 3 letters chosen consists of at least 1 vowel? | [
"4:1"
] | [
"Three letters are randomly picked from the word PRECAL. What are the odds that the set of 3 letters chosen consists of at least 1 vowel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a08b5a39df3a71fc8e17f6d51df721dd0e5562d45f6888c2029fd00a8b3411fb | Three line segments, each of length 1, form a connected figure on the plane. Any point that is common to two of these line segments is an endpoint of both segments. Find the maximum area of the convex hull of the figure. | [
"\\(\\frac{\\sqrt{3}}{4}\\)"
] | [
"Three line segments, each of length 1, form a connected figure on the plane. Any point that is common to two of these line segments is an endpoint of both segments. Find the maximum area of the convex hull of the figure."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a681577c57c516baf76bc8ee18d02d214c25aab6e4aee1b6d53a1eb1453d43cf | Three lines $l, m, n$ in space pass through point $S$. A plane perpendicular to $m$ intersects $l, m, n$ at $A, B, C$ respectively. Suppose that $\angle ASB = \angle BSC = 45^\circ$ and $\angle ABC = 90^\circ$. Compute $\angle ASC$. | [
"60^\\circ"
] | [
"Three lines $l, m, n$ in space pass through point $S$. A plane perpendicular to $m$ intersects $l, m, n$ at $A, B, C$ respectively. Suppose that $\\angle ASB = \\angle BSC = 45^\\circ$ and $\\angle ABC = 90^\\circ$. Compute $\\angle ASC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-a28085e1495730349e24d5026595e2d256a8d8bd0ccabaa35e103282f84ad198 | Three lines \( k, l, m \) are drawn through a point \( P \) inside a triangle \( ABC \) such that \( k \) meets \( AB \) at \( A_1 \) and \( AC \) at \( A_2 \ne A_1 \) with \( PA_1 = PA_2 \), \( l \) meets \( BC \) at \( B_1 \) and \( BA \) at \( B_2 \ne B_1 \) with \( PB_1 = PB_2 \), and \( m \) meets \( CA \) at \( C... | [
"14"
] | [
"Three lines \\( k, l, m \\) are drawn through a point \\( P \\) inside a triangle \\( ABC \\) such that \\( k \\) meets \\( AB \\) at \\( A_1 \\) and \\( AC \\) at \\( A_2 \\ne A_1 \\) with \\( PA_1 = PA_2 \\), \\( l \\) meets \\( BC \\) at \\( B_1 \\) and \\( BA \\) at \\( B_2 \\ne B_1 \\) with \\( PB_1 = PB_2 \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94e9f065ad6d285e9e8d30fec51f946b06512d40602865e0d81ea3eb38635812 | Three lines \( m \), \( n \), and \( l \) lie in a plane such that no two are parallel. Lines \( m \) and \( n \) meet at an acute angle of 14 degrees, and lines \( m \) and \( l \) meet at an acute angle of 20 degrees. Find the sum of all possible acute angles formed by lines \( n \) and \( l \). | [
"40"
] | [
"Three lines \\( m \\), \\( n \\), and \\( l \\) lie in a plane such that no two are parallel. Lines \\( m \\) and \\( n \\) meet at an acute angle of 14 degrees, and lines \\( m \\) and \\( l \\) meet at an acute angle of 20 degrees. Find the sum of all possible acute angles formed by lines \\( n \\) and \\( l \\)... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-cc17de6193797c1c79e6ae837827cfa03ec4c7df904ee857e7198eb8a1657ed9 | Three lines \(X\), \(Y\), and \(Z\) are incident with a point \(P\) and are orthogonal to the sides \(c\), \(a\), and \(b\) of a triangle \(ABC\), respectively. These lines \(X\), \(Y\), and \(Z\) are reflected at the midpoints of \(c\), \(a\), and \(b\) to lines \(u\), \(v\), and \(w\), respectively. If the coordinate... | [
"\\left(2, \\dfrac{1}{3}\\right)"
] | [
"Three lines \\(X\\), \\(Y\\), and \\(Z\\) are incident with a point \\(P\\) and are orthogonal to the sides \\(c\\), \\(a\\), and \\(b\\) of a triangle \\(ABC\\), respectively. These lines \\(X\\), \\(Y\\), and \\(Z\\) are reflected at the midpoints of \\(c\\), \\(a\\), and \\(b\\) to lines \\(u\\), \\(v\\), and \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12c6ff3199b3411db2519a762c78643f90ec81b3148e94c7f5297113cee81080 | Three lines are drawn through a point in a triangle parallel to its sides. The segments intercepted on these lines by the triangle have the same length. Given the triangle's side lengths $a, b$, and $c$, find the length of the segments. | [
"\\(\\frac{2abc}{ab + bc + ca}\\)"
] | [
"Three lines are drawn through a point in a triangle parallel to its sides. The segments intercepted on these lines by the triangle have the same length. Given the triangle's side lengths $a, b$, and $c$, find the length of the segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 43,
"tir": 13
} | 56 | 0 | 0 |
omr-ws-694e8e7cbe6a6ccc6969101f8b3c515cdc6473201716af3f17a60b8e08ee8f0e | Three lines are given by $\vec{r} = \lambda \vec{i}, \lambda \in \mathbb{R}$, $\vec{r} = \mu (\vec{i} + \vec{j}), \mu \in \mathbb{R}$, and $\vec{r} = \upsilon(\vec{i} + \vec{j} + \vec{k}), \upsilon \in \mathbb{R}$. These lines intersect the plane $x + y + z = 1$ at points $A$, $B$, and $C$ respectively. If the area of ... | [
"\\(\\frac{3}{4}\\)"
] | [
"Three lines are given by $\\vec{r} = \\lambda \\vec{i}, \\lambda \\in \\mathbb{R}$, $\\vec{r} = \\mu (\\vec{i} + \\vec{j}), \\mu \\in \\mathbb{R}$, and $\\vec{r} = \\upsilon(\\vec{i} + \\vec{j} + \\vec{k}), \\upsilon \\in \\mathbb{R}$. These lines intersect the plane $x + y + z = 1$ at points $A$, $B$, and $C$ res... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f8e4f089690eb99a0a848c588faef7f2a31c74250dceab8f8b983bede84f97c1 | Three lines passing through point \( O \) form equal angles by pairs. Points \( A_1 \) and \( A_2 \) are on the first line, and points \( B_1 \) and \( B_2 \) are on the second line such that the common point \( C_1 \) of \( A_1B_1 \) and \( A_2B_2 \) lies on the third line. Let \( C_2 \) be the common point of \( A_1B... | [
"\\dfrac{8}{9}"
] | [
"Three lines passing through point \\( O \\) form equal angles by pairs. Points \\( A_1 \\) and \\( A_2 \\) are on the first line, and points \\( B_1 \\) and \\( B_2 \\) are on the second line such that the common point \\( C_1 \\) of \\( A_1B_1 \\) and \\( A_2B_2 \\) lies on the third line. Let \\( C_2 \\) be the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c1fbdf416369f87f09ee456bf13d61e2d7dd0896aa99a6941cd850487a3d2bc | Three lines were drawn through the point $X$ in space, crossing a sphere at six points. The distances from point $X$ to five of these points are $2$ cm, $3$ cm, $4$ cm, $5$ cm, and $6$ cm. What can be the distance from point $X$ to the sixth point? | [
"\\(\\frac{12}{5}\\)"
] | [
"Three lines were drawn through the point $X$ in space, crossing a sphere at six points. The distances from point $X$ to five of these points are $2$ cm, $3$ cm, $4$ cm, $5$ cm, and $6$ cm. What can be the distance from point $X$ to the sixth point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 21,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-b10a258ff9b654d4f11b03b048051b3cbba43e6cb5f070a4a1392468217bc106 | Three lines with slopes \(a\), \(b\), and \(c\) intersect at lattice points on the coordinate plane. What is the minimum possible area of the triangle formed by their points of intersection? | [
"\\(\\frac{1}{2}\\)"
] | [
"Three lines with slopes \\(a\\), \\(b\\), and \\(c\\) intersect at lattice points on the coordinate plane. What is the minimum possible area of the triangle formed by their points of intersection?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-767f94a7d0056cffdfa3abcc7c4610f8d6118af1f069e1fe6f936b8192ebda88 | Three listening stations located at $(3300,0)$, $(3300,1100)$, and $(-3300,0)$ monitor an explosion. The last two stations detect the explosion $1$ second and $4$ seconds after the first, respectively. Determine the coordinates of the explosion. Assume that the coordinate system is measured in feet and that sound trave... | [
"\\((3300, -2750)\\)"
] | [
"Three listening stations located at $(3300,0)$, $(3300,1100)$, and $(-3300,0)$ monitor an explosion. The last two stations detect the explosion $1$ second and $4$ seconds after the first, respectively. Determine the coordinates of the explosion. Assume that the coordinate system is measured in feet and that sound ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-646c075e11bc591ebd932bbd0314ee7b4b4a141ad3c578c3f5e242427f7cfcec | Three logicians, Carroll, Kurt, and Kleene, are each placed in a cell with a unique number of coins ranging from 1 to 9. Carroll asks if the total number of coins is even, and Moriarty answers "No." Kurt asks if the total number of coins is a prime number, and Moriarty answers "No." If Kleene has 5 coins in his cell, w... | [
"Is the total number of coins 15?"
] | [
"Three logicians, Carroll, Kurt, and Kleene, are each placed in a cell with a unique number of coins ranging from 1 to 9. Carroll asks if the total number of coins is even, and Moriarty answers \"No.\" Kurt asks if the total number of coins is a prime number, and Moriarty answers \"No.\" If Kleene has 5 coins in hi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 9
} | 10 | 0 | 0 |
omr-ws-709bf8720bcadd9953b2633816a83c9e467ffdb0f348b79f8206649bcfd11539 | Three lorries (one red, one yellow, and one green) are filled with bags of cement. Initially, the ratio of the numbers of bags in each lorry is R : Y : G = 7 : 6 : 5. After moving bags between lorries, the new ratio is R : Y : G = 6 : 5 : 4. Determine which lorry lost the most bags. | [
"green lorry"
] | [
"Three lorries (one red, one yellow, and one green) are filled with bags of cement. Initially, the ratio of the numbers of bags in each lorry is R : Y : G = 7 : 6 : 5. After moving bags between lorries, the new ratio is R : Y : G = 6 : 5 : 4. Determine which lorry lost the most bags."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-ff4d3161f384ed5c8a5d5b5fb770b2c4c2608091202321afbd28ba7baade7763 | Three lorries (one red, one yellow, and one green) are filled with bags of cement. Initially, the ratio of the numbers of bags in each lorry is R : Y : G = 7 : 6 : 5. After moving bags between lorries, the new ratio is R : Y : G = 6 : 5 : 4. The lorry that gained the most bags gained 12 bags. Determine the total number... | [
"1080"
] | [
"Three lorries (one red, one yellow, and one green) are filled with bags of cement. Initially, the ratio of the numbers of bags in each lorry is R : Y : G = 7 : 6 : 5. After moving bags between lorries, the new ratio is R : Y : G = 6 : 5 : 4. The lorry that gained the most bags gained 12 bags. Determine the total n... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a76325889d71719ea00a23e481a119fe166980184f27354efd2ebf6109c3cdec | Three machines P, Q, and R, working together, can complete a job in \( x \) hours. When working alone, P needs an additional 6 hours to complete the job, Q needs an additional 1 hour, and R needs an additional \( x \) hours. Determine the value of \( x \). | [
"\\(\\frac{2}{3}\\)"
] | [
"Three machines P, Q, and R, working together, can complete a job in \\( x \\) hours. When working alone, P needs an additional 6 hours to complete the job, Q needs an additional 1 hour, and R needs an additional \\( x \\) hours. Determine the value of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-882015fb9c7dc05a5e5023ea99e6a5f584cef3104487530b32e38c9ee1df1bad | Three marbles are randomly selected, without replacement, from a bag containing two red, two blue and two green marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction. | [
"\\frac{2}{5}"
] | [
"Three marbles are randomly selected, without replacement, from a bag containing two red, two blue and two green marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b66ddcc7d3e11390ccb770ed381479ae5c43a292fcff37aa45aee129c39e10e0 | Three married couples are sitting around a round table. If they randomly choose their seats, what is the probability that no one sits next to his/her spouse? Express the probability as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, and find $m + n$. | [
"6"
] | [
"Three married couples are sitting around a round table. If they randomly choose their seats, what is the probability that no one sits next to his/her spouse? Express the probability as $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, and find $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06542566fb1f0aa9c0f8da9a06031a9d34b103d3e0eca1905caa0dca26db751e | Three married couples are to be seated in a row with six seats in a cinema hall. In how many ways can they be seated if all the ladies sit together? | [
"144"
] | [
"Three married couples are to be seated in a row with six seats in a cinema hall. In how many ways can they be seated if all the ladies sit together?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a29f7ef6f7f3a45a22fece9fb54acbcbf223e18e0f1966c7be8a13e6e5d82e6 | Three married couples arrange a party. They arrive one at a time, and each person shakes the hand of everyone already there except their own spouse. When everyone has arrived, someone asks all the others how many hands they shook upon arriving, and gets five different answers. How many hands did he himself shake upon a... | [
"2"
] | [
"Three married couples arrange a party. They arrive one at a time, and each person shakes the hand of everyone already there except their own spouse. When everyone has arrived, someone asks all the others how many hands they shook upon arriving, and gets five different answers. How many hands did he himself shake u... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-62115a42a5062c7d31f857a7a364b44c7af159e4cd8501bea99307c1fde5a02a | Three married couples sit at a rectangular table with twelve chairs, six chairs on one side and six chairs on the other side. What is the probability that every man will sit next to his wife? | [
"\\(\\frac{41}{6930}\\)"
] | [
"Three married couples sit at a rectangular table with twelve chairs, six chairs on one side and six chairs on the other side. What is the probability that every man will sit next to his wife?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe75fd4877086df843bd4e7ef3dbeafde360648eaed8e9810d22c5a620b1e735 | Three married couples sit at a square table with twelve chairs, three chairs on each side. What is the probability that every man will sit next to his wife? | [
"0.0023088023"
] | [
"Three married couples sit at a square table with twelve chairs, three chairs on each side. What is the probability that every man will sit next to his wife?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-f777922bc4716a3f6822b508c6a913cdbbee1a3171a8d11facca9ea1d230348b | Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives? | [
"\\dfrac{1}{3}"
] | [
"Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d5f20727bd6263b4418c62b12df0a402c954d77765ace1e33fd7f363586d12b | Three married couples sit in a row. Find the probability that no husband sits next to his wife. | [
"\\(\\frac{1}{3}\\)"
] | [
"Three married couples sit in a row. Find the probability that no husband sits next to his wife."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-058cc9a23f76f340829764eb49f996c5827c6a009a1b1d3845d1b86e2bfda26c | Three married couples were invited for a dinner. Each person who arrives gives a present to all those who arrived before them, except their spouse. When everyone arrived, John asked how many times each person gave a present and received 5 different answers. At what position did John arrive? | [
"4"
] | [
"Three married couples were invited for a dinner. Each person who arrives gives a present to all those who arrived before them, except their spouse. When everyone arrived, John asked how many times each person gave a present and received 5 different answers. At what position did John arrive?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-33de3939e071406ced735c70fb068137f0a6886f30f083f9382d36ef288108fb | Three married couples, the Alvarezes, the Barros, and the Castros, went to buy Christmas gifts. Each woman bought a quantity \(p\) of gifts in \(1000p^2\) pesos and each woman (Miriam, Francisca, and Gladys) spent \(147,000\) pesos more than her respective husband. Miriam is known to have bought 9 fewer gifts than Mr. ... | [
"Castro"
] | [
"Three married couples, the Alvarezes, the Barros, and the Castros, went to buy Christmas gifts. Each woman bought a quantity \\(p\\) of gifts in \\(1000p^2\\) pesos and each woman (Miriam, Francisca, and Gladys) spent \\(147,000\\) pesos more than her respective husband. Miriam is known to have bought 9 fewer gift... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a64ab47513300bf42bc13098e4d0596204bd9664deb34f3adc83827c16cf6e51 | Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs? | [
"\\dfrac{1}{4}"
] | [
"Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c8fd479eeadf31d3e6402f4d29cec4a89bcc23fae05d288d3527fdb3c15a146 | Three mathematicians and five physicists are sitting at a round table. Because mathematicians strongly dislike each other, they refuse to sit next to each other. How many distinct ways can they be seated? | [
"32"
] | [
"Three mathematicians and five physicists are sitting at a round table. Because mathematicians strongly dislike each other, they refuse to sit next to each other. How many distinct ways can they be seated?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-555d57b69035b80d4650a59e9c51922887f4d9982d6e71619e8177cad821e606 | Three mathematicians are given two numbers, both less than 100 and neither equal to 1. Mr. P is given the product of the numbers, and Mr. S is given the sum of the numbers. The following conversation occurs: - Mr. P: "I don't know what the numbers are." - Mr. S: "I knew you wouldn't know." - Mr. P: "I know the numbers ... | [
"\\(4\\) and \\(13\\)"
] | [
"Three mathematicians are given two numbers, both less than 100 and neither equal to 1. Mr. P is given the product of the numbers, and Mr. S is given the sum of the numbers. The following conversation occurs:\n\n- Mr. P: \"I don't know what the numbers are.\"\n- Mr. S: \"I knew you wouldn't know.\"\n- Mr. P: \"I kn... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 20
} | 27 | 0 | 0 |
omr-ws-2d769dec3c96ef7130a5a828c3fb6fa2bcc1291ae274f7b7059469c16b7be536 | Three mathematicians go to a coffee shop some time between 1 and 2. Each one stays exactly 10 minutes and then leaves. What is the probability that they are all there at some point? | [
"\\dfrac{2}{27}"
] | [
"Three mathematicians go to a coffee shop some time between 1 and 2. Each one stays exactly 10 minutes and then leaves. What is the probability that they are all there at some point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-9da34705e61a7179294684e641c977c19ce57c6d0cdbb31732d5d431d4229128 | Three mathematicians take a lunch break separately between 12:00 and 1:00 and stay for $\frac{m}{n}$ minutes. If the probability that two of them meet but the third doesn't meet with either of them is 0.4, compute $m+n$. | [
"no_valid_answer_found"
] | [
"Three mathematicians take a lunch break separately between 12:00 and 1:00 and stay for $\\frac{m}{n}$ minutes. If the probability that two of them meet but the third doesn't meet with either of them is 0.4, compute $m+n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe80af84c6a1add80447db5911944af690de2aadd48dc05128a688dea62a0919 | Three members of the Euclid Middle School girls' softball team have uniform numbers that are 2-digit primes. Bethany's birthday date is the sum of Ashley's and Caitlin's uniform numbers, Caitlin's birthday date is the sum of Ashley's and Bethany's uniform numbers, and today's date is the sum of Bethany's and Caitlin's ... | [
"11"
] | [
"Three members of the Euclid Middle School girls' softball team have uniform numbers that are 2-digit primes. Bethany's birthday date is the sum of Ashley's and Caitlin's uniform numbers, Caitlin's birthday date is the sum of Ashley's and Bethany's uniform numbers, and today's date is the sum of Bethany's and Caitl... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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