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-1935cce1b24c2678918384cd9fde5155163c135c164a696780792b1e6675c0bd | Bob's favorite number is between $50$ and $100$. It is a multiple of $11$, but not a multiple of $2$. The sum of its digits is a multiple of $3$. What is Bob's favorite number? | [
"99"
] | [
"Bob's favorite number is between $50$ and $100$. It is a multiple of $11$, but not a multiple of $2$. The sum of its digits is a multiple of $3$. What is Bob's favorite number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aac0dd8ed0060a825bff58108f838850c3810cefc8114851c54fcdb35631f370 | Bob's house is located at (-2,3). He needs to travel to the negative x-axis, then to the positive y-axis, and finally to (-1,1). What is the shortest possible distance for this journey? Note that (0,0) can be considered as a point on both the negative x-axis and the positive y-axis. | [
"\\sqrt{13} + \\sqrt{2}"
] | [
"Bob's house is located at (-2,3). He needs to travel to the negative x-axis, then to the positive y-axis, and finally to (-1,1). What is the shortest possible distance for this journey? Note that (0,0) can be considered as a point on both the negative x-axis and the positive y-axis."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-df9d11570aed022fd2baba1755edd9fd1710a5c09506763079760f76be341893 | Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number? | [
"\\frac{9}{20}"
] | [
"Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62fa3af8f8c008d6f4e7e0698ffec1e9918543d11dfa928ac1c8461f0c86ecd4 | Bob, Alice, and Jack are playing a game where they take some number of stones from a group, with the number of stones taken being in the range $(0, m)$ where $m \ge 1$. The player who takes the last stone loses. If the number of stones in the group is $N$, find the smallest value of $N$ such that Bob and Alice can alwa... | [
"3"
] | [
"Bob, Alice, and Jack are playing a game where they take some number of stones from a group, with the number of stones taken being in the range $(0, m)$ where $m \\ge 1$. The player who takes the last stone loses. If the number of stones in the group is $N$, find the smallest value of $N$ such that Bob and Alice ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-bb51f8b1a8441c7ea9ff1acd2e4f961dce0aab99e62a80d08cbd07ea5820e4ef | Bob, Joe, Tom, and Sam each pick a prime number. They form groups of 2, 3, or 4. What is the probability that the sum of the prime numbers in any group equals 39? | [
"0"
] | [
"Bob, Joe, Tom, and Sam each pick a prime number. They form groups of 2, 3, or 4. What is the probability that the sum of the prime numbers in any group equals 39?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-718791256b0e35062e661c7deaef487b039f2af0d761346d86082d0dccce23ec | Bob, Martha, and George each have 2010 apples. They decide to share their apples with a group of their own friends. The total number of friends sharing the apples is 1273. What is the sum of the apples the three of them are left with after sharing? | [
"0"
] | [
"Bob, Martha, and George each have 2010 apples. They decide to share their apples with a group of their own friends. The total number of friends sharing the apples is 1273. What is the sum of the apples the three of them are left with after sharing?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fea51d99545d7050814d24ffa789c7d4f643324f2697cc91dc39c26ff6c2f5f | Bob, Mary, and Blubby are going to have 10 children together (they are toads, not people). Each child has an equal chance of being of gender A, B, or C (there are 3 toad genders). Mary is sure at least one of the children will be gender B for some reason. Assuming Mary is correct, what is the probability all 10 baby to... | [
"\\(\\frac{1}{19683}\\)"
] | [
"Bob, Mary, and Blubby are going to have 10 children together (they are toads, not people). Each child has an equal chance of being of gender A, B, or C (there are 3 toad genders). Mary is sure at least one of the children will be gender B for some reason. Assuming Mary is correct, what is the probability all 10 ba... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-029c9bdd2e963739cd79cb5927b7ca1487ebc451fa171cfb1497f9b439e4ae72 | Bob, Sam, and Wob start together from bus station $A$ to bus station $B$, which is 125 km away. Bob and Wob travel by bus at 25 km/h, while Sam walks at 5 km/h. When Bob and Wob stop at bus station $C$, Wob gets off and starts walking to bus station $B$ at 5 km/h. Bob goes back to bus station $A$, meets Sam at bus stat... | [
"\\dfrac{49}{5}"
] | [
"Bob, Sam, and Wob start together from bus station $A$ to bus station $B$, which is 125 km away. Bob and Wob travel by bus at 25 km/h, while Sam walks at 5 km/h. When Bob and Wob stop at bus station $C$, Wob gets off and starts walking to bus station $B$ at 5 km/h. Bob goes back to bus station $A$, meets Sam at bus... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e87ba785bbf4f9b888b3fe566c27e5bbf3a7d45741d9363a718b20aa5963d502 | Bob, Son, and Joe each randomly choose a positive number from 1 to 10. It turns out that among all three people, any two numbers differ by at most 2. Find the number of different ways this can occur. | [
"160"
] | [
"Bob, Son, and Joe each randomly choose a positive number from 1 to 10. It turns out that among all three people, any two numbers differ by at most 2. Find the number of different ways this can occur."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 8
} | 55 | 0 | 0 |
omr-ws-9cdf0e18f238115876210e5fe0901b83c2b25d3809ac0d4ebf6ad8bfb25244d1 | Bobbo starts swimming at 2 ft/s across a 100-foot wide river with a current of 5 ft/s. He realizes his predicament midway across the river, 50 feet from either bank. What is the minimum speed that Bobbo must increase to in order to make it to the other side of the river safely, given that there is a waterfall 175 feet ... | [
"3 ft/s"
] | [
"Bobbo starts swimming at 2 ft/s across a 100-foot wide river with a current of 5 ft/s. He realizes his predicament midway across the river, 50 feet from either bank. What is the minimum speed that Bobbo must increase to in order to make it to the other side of the river safely, given that there is a waterfall 175 ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6c0d7481cfe6ad47ea0b4438aba7217eb8820c7fb8e44dea3973009579c21cfe | Bobby Fisherman played a tournament against 2009 players, losing his first two games but winning 2002 games in total. After each game, he computed his winning percentage. Let \( w_1, w_2, \ldots, w_{2009} \) be his winning percentages after games 1, 2, ..., 2009 respectively. How many positive real numbers are necessar... | [
"1"
] | [
"Bobby Fisherman played a tournament against 2009 players, losing his first two games but winning 2002 games in total. After each game, he computed his winning percentage. Let \\( w_1, w_2, \\ldots, w_{2009} \\) be his winning percentages after games 1, 2, ..., 2009 respectively. How many positive real numbers are ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-b58d0d720c02ad567522ecef9df8f0ff989b71b324ec6a083f6ec2724e6422dd | Bobby and Randy are playing a coin game using an unfair coin that comes up heads with a probability of \( h \), where \( h > 0 \). They take turns, with Bobby going first and flipping the coin once. If the coin comes up heads, Bobby wins and the game is over. If the coin comes up tails, Randy flips the coin twice. If e... | [
"\\dfrac{3 - \\sqrt{5}}{2}"
] | [
"Bobby and Randy are playing a coin game using an unfair coin that comes up heads with a probability of \\( h \\), where \\( h > 0 \\). They take turns, with Bobby going first and flipping the coin once. If the coin comes up heads, Bobby wins and the game is over. If the coin comes up tails, Randy flips the coin tw... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30f6045572cb52ab35c98b57cbbeceea3145b3c32332dc75d97d3e0d6629a35c | Bobby continues shooting three moving targets until he knocks all of them down. The probability of hitting a target is $p$. Find the expected number of shots required to knock all of the targets down. | [
"\\dfrac{3}{p}"
] | [
"Bobby continues shooting three moving targets until he knocks all of them down. The probability of hitting a target is $p$. Find the expected number of shots required to knock all of the targets down."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-8a6f156456d06235e2ed36b19f0fa90aad2adcfd55d80e87060cca6f4e393f9b | Bobby has a square-based toolshed measuring 15 feet on each side. He tethers his dog to one vertex of the shed with a 10-foot leash. What is the area of the region where his dog can roam freely? Express your answer in terms of pi. | [
"75\\pi"
] | [
"Bobby has a square-based toolshed measuring 15 feet on each side. He tethers his dog to one vertex of the shed with a 10-foot leash. What is the area of the region where his dog can roam freely? Express your answer in terms of pi."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-9338b1df60bd20ee48175fec08d88e581cd24f3f06c04ee2c81d34a033f885e4 | Bobby is making a tablecloth for a circular table with a diameter of 3 feet. The tablecloth needs to hang down 5 inches over the edge of the table all the way around, and Bobby will fold under and sew down 1 inch of the material all around the edge. Bobby is using a single piece of rectangular fabric that is 60 inches ... | [
"48"
] | [
"Bobby is making a tablecloth for a circular table with a diameter of 3 feet. The tablecloth needs to hang down 5 inches over the edge of the table all the way around, and Bobby will fold under and sew down 1 inch of the material all around the edge. Bobby is using a single piece of rectangular fabric that is 60 in... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-1a9492f4d998cb8b6699267c171e44450fb797d3fb3edb030a89d06ed61b9e10 | Bobby is making a tablecloth for a circular table with a diameter of 3 feet. The tablecloth needs to hang down 5 inches over the edge of the table all the way around, and Bobby will fold under and sew down 1 inch of the material all around the edge. The fabric is 60 inches wide. What is the shortest length of fabric, i... | [
"48"
] | [
"Bobby is making a tablecloth for a circular table with a diameter of 3 feet. The tablecloth needs to hang down 5 inches over the edge of the table all the way around, and Bobby will fold under and sew down 1 inch of the material all around the edge. The fabric is 60 inches wide. What is the shortest length of fabr... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 4
} | 50 | 0 | 0 |
omr-ws-79fbbbf96a53f0fee67c977ba007d162e84c99876c377f3ed7c50c884d2e368b | Bobby puts his money in a bank in Britain that pays 6% simple interest per annum. He stays in Britain for 6 months. Calculate the interest he receives for the six months. Which equation would you use? | [
"3\\%"
] | [
"Bobby puts his money in a bank in Britain that pays 6% simple interest per annum. He stays in Britain for 6 months. Calculate the interest he receives for the six months. Which equation would you use?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-6c679fb74fbfde03c2cbf5da2686c5218748be68c7694f7ae68cea4274f5c291 | Bobby, Peter, Greg, Cindy, Jan, and Marcia line up for ice cream. In an acceptable lineup, Greg is ahead of Peter, Peter is ahead of Bobby, Marcia is ahead of Jan, and Jan is ahead of Cindy. Compute the number of acceptable lineups. | [
"20"
] | [
"Bobby, Peter, Greg, Cindy, Jan, and Marcia line up for ice cream. In an acceptable lineup, Greg is ahead of Peter, Peter is ahead of Bobby, Marcia is ahead of Jan, and Jan is ahead of Cindy. Compute the number of acceptable lineups."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-866961b9bf06b3d8d49d51be543bf6801b97c1c8dac4bf7486efcfdd732efd5b | Bobby’s booby-trapped safe requires a 3-digit code to unlock it. The probe responds "Fail" if no individual digit is correct and "Close" otherwise, including when all digits are correct. What is the smallest number of attempts needed to guarantee that Alex knows the correct code, following an optimal strategy? | [
"5"
] | [
"Bobby’s booby-trapped safe requires a 3-digit code to unlock it. The probe responds \"Fail\" if no individual digit is correct and \"Close\" otherwise, including when all digits are correct. What is the smallest number of attempts needed to guarantee that Alex knows the correct code, following an optimal strategy?... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-28dba67bef2b28363f9015290803b07d82703c16c7ca5a2a21cde82a8d1356a7 | Boberto #$1$, Boberto #$2$, Boberto #$3$, and Boberto #$4$ each have to collectively pay $1000$ dollars of library debt. Each Boberto can pay any non-negative amount, except Boberto #$n$ refuses to pay multiples of $n + 1$. Let $m$ be the number of ways they can exactly pay off their library debt. What is $m$ mod $1000... | [
"375"
] | [
"Boberto #$1$, Boberto #$2$, Boberto #$3$, and Boberto #$4$ each have to collectively pay $1000$ dollars of library debt. Each Boberto can pay any non-negative amount, except Boberto #$n$ refuses to pay multiples of $n + 1$. Let $m$ be the number of ways they can exactly pay off their library debt. What is $m$ mod ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a551a8548aaac8b5751405f8ac3774036eab79fada39aee0d015ad61a28e8b5 | Bobo the clown goes to the land of ice cream and says to Roy the clown, "How many ways can I make an ice cream cone if there are two scoops and 9 flavors which can be reused?" Roy doesn’t know. Help Roy. | [
"81"
] | [
"Bobo the clown goes to the land of ice cream and says to Roy the clown, \"How many ways can I make an ice cream cone if there are two scoops and 9 flavors which can be reused?\" Roy doesn’t know. Help Roy."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a2d59a670f9efaac8bef02c9425032771a8f1eb12aa329ef91a0ad7b45f9939c | Bobo visits a high school with 1,000 lockers all initially closed. He opens all the lockers, then toggles every second locker (opens if closed, closes if open), then every third locker, and so on, up to every 1,000th locker. After completing this process, how many lockers are left open and which ones are they? | [
"31"
] | [
"Bobo visits a high school with 1,000 lockers all initially closed. He opens all the lockers, then toggles every second locker (opens if closed, closes if open), then every third locker, and so on, up to every 1,000th locker. After completing this process, how many lockers are left open and which ones are they?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-87ac556820af94e31184d0f66d64e98d1a567d9bf3c0157962b441bb4ff2dbc6 | Bob’s piggy bank has $24.55 with exactly 200 coins consisting of n nickels, d dimes, and q quarters. Note that n, d, or q can be zero. Find the difference between the largest and smallest number of quarters possible in Bob’s piggy bank. | [
"41"
] | [
"Bob’s piggy bank has $24.55 with exactly 200 coins consisting of n nickels, d dimes, and q quarters. Note that n, d, or q can be zero. Find the difference between the largest and smallest number of quarters possible in Bob’s piggy bank."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-d80e35188c3376d12656bd1aebddd8d8733dacf8730ac58609013e9bb6b56ed3 | Bocchi rolls 21 fair six-sided dice numbered from 1 to 6 and records the number rolled on each one. She then computes the product of all 21 values and assigns that variable to the value \( N \). The expected value of the period of the repeating decimal \( \frac{1}{N} \) is \( \frac{p}{q} \) where \( p \) and \( q \) ar... | [
"88"
] | [
"Bocchi rolls 21 fair six-sided dice numbered from 1 to 6 and records the number rolled on each one. She then computes the product of all 21 values and assigns that variable to the value \\( N \\). The expected value of the period of the repeating decimal \\( \\frac{1}{N} \\) is \\( \\frac{p}{q} \\) where \\( p \\)... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-195738eb28fd85b0a7f74e7ee52657be7cbda1cd7f89a403affcae244fcb1673 | Bonham Garden Fertilizers produces Regular and Super-Gro formulations. There are 10 employees, or 400 worker-hours of production time each week. It takes 1/4 hour to produce and package each Regular and each Super-Gro. Bonham has $7000 to spend on raw materials. Raw materials for Regular cost $2 per package, and materi... | [
"\\left( \\dfrac{1000}{3}, \\dfrac{3800}{3} \\right)"
] | [
"Bonham Garden Fertilizers produces Regular and Super-Gro formulations. There are 10 employees, or 400 worker-hours of production time each week. It takes 1/4 hour to produce and package each Regular and each Super-Gro. Bonham has $7000 to spend on raw materials. Raw materials for Regular cost $2 per package, and m... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ebb3cf95718095d5fe5659d124ea7ed7e278fcc8f387e81f60b0f5e925de780c | Bonnie makes the frame of a cube out of 12 pieces of wire that are each six inches long. Meanwhile Roark uses 1-inch-long pieces of wire to make a collection of unit cube frames that are not connected to each other. The total volume of Roark's cubes is the same as the volume of Bonnie's cube. What is the ratio of the t... | [
"\\dfrac{1}{36}"
] | [
"Bonnie makes the frame of a cube out of 12 pieces of wire that are each six inches long. Meanwhile Roark uses 1-inch-long pieces of wire to make a collection of unit cube frames that are not connected to each other. The total volume of Roark's cubes is the same as the volume of Bonnie's cube. What is the ratio of ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-658dad250ed186e4311e90aa84a661837652e9abf348a514c5c4fbe26471b80e | Boris and Eugene are playing a game on a circle where they mark points in turns. Boris marks the first point and paints it white, and Eugene paints his points black. Once each has marked $n$ points, any unmarked point on the circle is colored with the color of the nearest marked point. The sum of the arc lengths colore... | [
"\\dfrac{1}{5}"
] | [
"Boris and Eugene are playing a game on a circle where they mark points in turns. Boris marks the first point and paints it white, and Eugene paints his points black. Once each has marked $n$ points, any unmarked point on the circle is colored with the color of the nearest marked point. The sum of the arc lengths c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e5cfce4203101529bcc2214fba6c8a095490e2315185a15c6e8eb35123ab35e | Boris and Michael are jogging around a lake. It takes Boris 6 minutes to complete a lap and then he rests for 1 minute. Michael takes 8 minutes to complete a lap and then he rests for 2 minutes. If they started jogging together, how long, in minutes, does it take until they are next together? | [
"20 minutes",
"70"
] | [
"Boris and Michael are jogging around a lake. It takes Boris 6 minutes to complete a lap and then he rests for 1 minute. Michael takes 8 minutes to complete a lap and then he rests for 2 minutes. If they started jogging together, how long, in minutes, does it take until they are next together?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-08f92b46dead9887e1bbc92aa703c05d9c61a6a92e35ddff045437402feda58b | Boris flips a fair coin until he has flipped $5$ heads in a row. What is the expected number of tails he has flipped? | [
"31"
] | [
"Boris flips a fair coin until he has flipped $5$ heads in a row. What is the expected number of tails he has flipped?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81073c21ce94175f3f81f441a0cd616269a46e398a70b1f299f49ef2514f5686 | Boris plays a game on a 5x5 table of distinct integers by repeatedly choosing the minimal number in the table, adding it to his sum, and then erasing its row and column. Let $S_B$ be Boris's total sum after 5 turns. Let $S_M$ be the maximum possible sum of 5 numbers chosen from the table such that no two numbers are in... | [
"20"
] | [
"Boris plays a game on a 5x5 table of distinct integers by repeatedly choosing the minimal number in the table, adding it to his sum, and then erasing its row and column. Let $S_B$ be Boris's total sum after 5 turns. Let $S_M$ be the maximum possible sum of 5 numbers chosen from the table such that no two numbers a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50a9a0ef069f5136b177505ae58963010fbb516c892b255dd296e260e005bb48 | Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 399, what is the sum of $a$ and $b$? | [
"21"
] | [
"Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 399, what is the sum of $a$ and $b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-642a000426480f62dd31b6df8b461d59e6c3b8bffbbeeac928b0cf6b76bbda75 | Both beginning on the starting line, Sally and Sam run in opposite directions on a 450 meter-long circular track. Sally runs at 4 m/sec, Sam also runs at 4 m/sec, that is, until he passes Sally. Wanting to impress her, he speeds up by 1 m/sec each time he passes her and maintains that pace until their next meeting. How... | [
"960.8 meters"
] | [
"Both beginning on the starting line, Sally and Sam run in opposite directions on a 450 meter-long circular track. Sally runs at 4 m/sec, Sam also runs at 4 m/sec, that is, until he passes Sally. Wanting to impress her, he speeds up by 1 m/sec each time he passes her and maintains that pace until their next meeting... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8d5a646956eaa2709350900580e077825d39ea2147508a1f34d678d64411fe4 | Both me and my brother separately pick three not necessarily distinct digits (0, 1, 2, ..., 9) and whisper them to our sister. Call a three-digit number formed from the permutation of three digits a "permute". Our sister tells us that all of my permutes are relatively prime to all of my brother's permutes. She also tel... | [
"100"
] | [
"Both me and my brother separately pick three not necessarily distinct digits (0, 1, 2, ..., 9) and whisper them to our sister. Call a three-digit number formed from the permutation of three digits a \"permute\". Our sister tells us that all of my permutes are relatively prime to all of my brother's permutes. She a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-74dc2c7db67c6693cb4df3c17b844e38befb51f2edc63b11f68722b503d70021 | Both points $(3, 1)$ and $(5, 2)$ are the same distance from the line $y=mx$, but one point lies above the line, while the other lies below the line. Determine the equation of the line. | [
"\\( y = \\frac{3}{8}x \\)"
] | [
"Both points $(3, 1)$ and $(5, 2)$ are the same distance from the line $y=mx$, but one point lies above the line, while the other lies below the line. Determine the equation of the line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2f916e2987f159461035a7ef16fb81ad78ebc6ffb0116b105dd2d89cdd3fc670 | Both roots of the quadratic equation $x^2 - 63 x + k = 0$ are prime numbers. How many possible values of $k$ are there? | [
"1"
] | [
"Both roots of the quadratic equation $x^2 - 63 x + k = 0$ are prime numbers. How many possible values of $k$ are there?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa4d1ebd1731baafbf4104e6031a40d59b3630a3ef299547e048a5f818ab3444 | Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k.$ | [
"1"
] | [
"Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ab1715e4255054a3a5af0cd2dbd6bda14ccace329a4ed495655126cebdde658 | Both the Hawking and the Gabe follow a linear model for fuel efficiency with respect to speed. The Hawking has a fuel efficiency of 20 mpg at 30 mph and 40 mpg at 60 mph. The Gabe has a fuel efficiency of 25 mpg at 30 mph and 35 mpg at 60 mph. Find equations for fuel efficiency, \( E \), in terms of speed, \( S \), for... | [
"\\( \\left| \\frac{1}{3}S - 15 \\right| \\)"
] | [
"Both the Hawking and the Gabe follow a linear model for fuel efficiency with respect to speed. The Hawking has a fuel efficiency of 20 mpg at 30 mph and 40 mpg at 60 mph. The Gabe has a fuel efficiency of 25 mpg at 30 mph and 35 mpg at 60 mph. Find equations for fuel efficiency, \\( E \\), in terms of speed, \\( S... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc26caeaa3ebcb907f2d1e54422b6787076b43a724dc5f4da8482c3dd26666b6 | Bottleguy has 400 out of 500 different types of bottles. If he buys a 6-pack of bottles where no two bottles are the same, what is the probability that he will get at least one new bottle type? | [
"74%"
] | [
"Bottleguy has 400 out of 500 different types of bottles. If he buys a 6-pack of bottles where no two bottles are the same, what is the probability that he will get at least one new bottle type?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-c3d6b2d36c7f5ed805ba691b777bbf0a49e94c55f9d40456a8cb6fb9accd6dfa | Bound $x$ given the inequality $\sin x > (\cos x)^2$. | [
"\\( x \\in \\left( \\arcsin \\left( \\frac{-1 + \\sqrt{5}}{2} \\right) + 2k\\pi, \\pi - \\arcsin \\left( \\frac{-1 + \\sqrt{5}}{2} \\right) + 2k\\pi \\right) \\) for integer \\( k \\)."
] | [
"Bound $x$ given the inequality $\\sin x > (\\cos x)^2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-8cb94a674a9aa9b91aaf66433ea80acb5b2241fd8f4941361d4135955fe4950b | Bound the minimum number of circles \( k \) that can cover all integer points \( (x,y) \in \{0,1,...,n\} \times \{0,1,...,n\} \) in the plane except \( (0,0) \) (every point must be on the circumference of some circle). | [
"2n - 1"
] | [
"Bound the minimum number of circles \\( k \\) that can cover all integer points \\( (x,y) \\in \\{0,1,...,n\\} \\times \\{0,1,...,n\\} \\) in the plane except \\( (0,0) \\) (every point must be on the circumference of some circle)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-e0fe40ee43c0cb6a455ad6da6c27434078c6a64973ea0e1f6781bbaad21ea806 | Bound the sequence defined by $a_0=1$ and $a_{n+1} = a_n + \frac{1}{a_n^{2005}}$ using exponential functions of $n$. | [
"2^n"
] | [
"Bound the sequence defined by $a_0=1$ and $a_{n+1} = a_n + \\frac{1}{a_n^{2005}}$ using exponential functions of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-9c29deba99484b83a3919bdcf80ebf2702eee57af43eb649d0c5a24c9b8cee02 | Bowl A has 6 red balls and 4 blue balls. We randomly choose 5 balls and move them to bowl B, which was empty. Given that a ball chosen from bowl B is blue, what is the probability that we moved 2 red balls and 3 blue balls? | [
"\\(\\frac{5}{14}\\)"
] | [
"Bowl A has 6 red balls and 4 blue balls. We randomly choose 5 balls and move them to bowl B, which was empty. Given that a ball chosen from bowl B is blue, what is the probability that we moved 2 red balls and 3 blue balls?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-400a43c1e67433c398cb5740f6cf538af0d55fd2f2c8a540e3c32b458fdcd2d7 | Box $A$ contains 1 red ball and 3 white balls, and box $B$ contains 1 red ball and 3 white balls. At each trial, one ball is chosen at random from each box and exchanged. After $n$ repetitions of this trial, find the probability $p_n$ that box $A$ contains 1 red ball and 3 white balls. | [
"\\( p_n = \\frac{3}{7 \\cdot 8^n} + \\frac{4}{7} \\)"
] | [
"Box $A$ contains 1 red ball and 3 white balls, and box $B$ contains 1 red ball and 3 white balls. At each trial, one ball is chosen at random from each box and exchanged. After $n$ repetitions of this trial, find the probability $p_n$ that box $A$ contains 1 red ball and 3 white balls."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-e089833bf2cedd5b78044e4323e4e3863fa493da3ea4d554fb967eda4025fce7 | Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$. Then take the same number of balls from $B$ and place them in $A$. Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$? | [
"equal to"
] | [
"Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$. Then take the same number of balls from $B$ and place them in $A$. Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-9b6eb80c2a89f9d444f35c81e1ea4f8832db4315ba437cefe0f2786fccf7408d | Box 1 contains 1 ball of color 1, 2 balls of color 2, 3 balls of color 3, up to n balls of color n. Box 2 contains n balls of color 1, n-1 balls of color 2, n-2 balls of color 3, down to 1 ball of color n. A ball is randomly chosen from Box 1 and placed into Box 2. A ball is then randomly chosen from Box 2. Which color... | [
"1"
] | [
"Box 1 contains 1 ball of color 1, 2 balls of color 2, 3 balls of color 3, up to n balls of color n. Box 2 contains n balls of color 1, n-1 balls of color 2, n-2 balls of color 3, down to 1 ball of color n. A ball is randomly chosen from Box 1 and placed into Box 2. A ball is then randomly chosen from Box 2. Which ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-551015b2081ec1d1b60fac8d6617a16c7b2af431e6594476393d81bfed842678 | Box A contains 5 blue balls and 2 red balls. Box B contains 3 blue balls and 4 red balls. A ball is randomly selected from Box A and placed in Box B. If a blue ball is selected from Box B, what is the probability that a red ball was transferred from Box A to Box B? | [
"\\(\\frac{3}{13}\\)"
] | [
"Box A contains 5 blue balls and 2 red balls. Box B contains 3 blue balls and 4 red balls. A ball is randomly selected from Box A and placed in Box B. If a blue ball is selected from Box B, what is the probability that a red ball was transferred from Box A to Box B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-abc5c75f01b196724412982b612440c00a9812fc030bcd46453bbfb392308fe9 | Box A contains black balls and box B contains white balls. Take a certain number of balls from A and place them in B. Then take the same number of balls from B and place them in A. Is the number of white balls in A then greater, equal to, or less than the number of black balls in B? | [
"equal to"
] | [
"Box A contains black balls and box B contains white balls. Take a certain number of balls from A and place them in B. Then take the same number of balls from B and place them in A. Is the number of white balls in A then greater, equal to, or less than the number of black balls in B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6261725ce2343a88d0893331383f52e87e4f577cf5bf8edae55256971892908a | Box A contains marbles numbered 1-8 and Box B contains marbles numbered 1-11. If two marbles are drawn without replacement from Box A and placed into Box B, and then two marbles are drawn with replacement from Box B, find the probability that the product of the two numbers drawn from Box B is odd. | [
"\\(\\frac{2}{7}\\)"
] | [
"Box A contains marbles numbered 1-8 and Box B contains marbles numbered 1-11. If two marbles are drawn without replacement from Box A and placed into Box B, and then two marbles are drawn with replacement from Box B, find the probability that the product of the two numbers drawn from Box B is odd."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa4fddeb37f2469f70c97f6eb472ab3cd0bd8a4e416b5f284f16a656b05b405e | Box W and Box V each contain several blue sticks and red sticks, and all of the red sticks have the same length. The length of each red stick is 19 inches less than the average length of the sticks in Box W and 6 inches greater than the average length of the sticks in Box V. What is the average (arithmetic mean) length... | [
"25"
] | [
"Box W and Box V each contain several blue sticks and red sticks, and all of the red sticks have the same length. The length of each red stick is 19 inches less than the average length of the sticks in Box W and 6 inches greater than the average length of the sticks in Box V. What is the average (arithmetic mean) l... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9acfa4c47b59d5dd1ef9770f07f864207862491cacd16afa6a91aad7741d541 | Box is thinking of a number whose digits are all "1". When he squares the number, the sum of the digits of the squared number is 85. How many digits does Box's number have? | [
"11"
] | [
"Box is thinking of a number whose digits are all \"1\". When he squares the number, the sum of the digits of the squared number is 85. How many digits does Box's number have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e6587e165382e7a644782e793026aa2fc569cef40c3208eea2b4cbbee61fb4f3 | Boxes are arranged in a 20-layer tower. The top layer has three boxes, and each subsequent layer has one more than twice the number of boxes in the layer above it. If $2^n$ is the largest power of two that is a factor of the total number of boxes, what is the value of $n$? | [
"3"
] | [
"Boxes are arranged in a 20-layer tower. The top layer has three boxes, and each subsequent layer has one more than twice the number of boxes in the layer above it. If $2^n$ is the largest power of two that is a factor of the total number of boxes, what is the value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-216e24d7f859a05cea59cb2b9b877f81586557dce5aecef2eaafe9592151b67c | Boy A and 2016 flags are on a circumference of length 1. Boy A wants to collect all flags by moving a distance \( l \) on the circumference. Determine the minimum value of \( l \) such that Boy A can collect all flags regardless of their initial positions. Note that Boy A does not need to return to the starting point a... | [
"\\dfrac{2015}{2016}"
] | [
"Boy A and 2016 flags are on a circumference of length 1. Boy A wants to collect all flags by moving a distance \\( l \\) on the circumference. Determine the minimum value of \\( l \\) such that Boy A can collect all flags regardless of their initial positions. Note that Boy A does not need to return to the startin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-5973c25be801afb9fe8e9656cdb206a1837f76e61225b04a20d80e83f1406db3 | Boynton’s sheet cake measures 18 × 24 inches and has a height of 4 inches. However, these measurements include a 3/4-inch thick layer of frosting on the top and sides. What is the volume of Boynton’s cake excluding the frosting? Express your answer to the nearest whole number. | [
"928"
] | [
"Boynton’s sheet cake measures 18 × 24 inches and has a height of 4 inches. However, these measurements include a 3/4-inch thick layer of frosting on the top and sides. What is the volume of Boynton’s cake excluding the frosting? Express your answer to the nearest whole number."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-c31b42ed41ebb4458c0a48fa382c47f36fde8730445c0986597f81e7f6f572e6 | Bozo the Clown wants to buy lunch at a food truck. The truck has 2 entrees available: beef for $8 and chicken for $5. The truck also has 3 sides available: chips for $1, salad for $30, and a cookie for $5. Drinks are $2. How many ways can Bozo pay for a lunch including an entree, side, and drink, using dollars of the v... | [
"77"
] | [
"Bozo the Clown wants to buy lunch at a food truck. The truck has 2 entrees available: beef for $8 and chicken for $5. The truck also has 3 sides available: chips for $1, salad for $30, and a cookie for $5. Drinks are $2. How many ways can Bozo pay for a lunch including an entree, side, and drink, using dollars of ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 27
} | 47 | 0 | 0 |
omr-ws-7011d874b443809f93af81c8fdafe86c9def6e3ca5dd3339d80732170a4b5743 | Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel? | [
"3"
] | [
"Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09151644565e5f41f50472b9c27dfa3edcaf710a66cc252ee2fedd926fada5f8 | Bradley is standing at meter 50 of a 100 meter long bridge. He repeatedly flips a coin. If heads, he steps forward 1 meter, and if tails, he steps back 1 meter. What is the expected value of the number of steps he must take to get off the bridge (reach meter 0 or 100)? | [
"2500"
] | [
"Bradley is standing at meter 50 of a 100 meter long bridge. He repeatedly flips a coin. If heads, he steps forward 1 meter, and if tails, he steps back 1 meter. What is the expected value of the number of steps he must take to get off the bridge (reach meter 0 or 100)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-852b47bcf95a236ed983d52470cc34a66b46d8adc8649a92d24312fd106d1016 | Brain weights of Swedish men are normally distributed with a mean of 1.40 kg and a standard deviation of 0.11 kg. Determine the percentage of all samples of three Swedish men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.4 kg. Interpret your answer in terms of sampling error. | [
"94.06%"
] | [
"Brain weights of Swedish men are normally distributed with a mean of 1.40 kg and a standard deviation of 0.11 kg. Determine the percentage of all samples of three Swedish men that have mean brain weights within 0.1 kg of the population mean brain weight of 1.4 kg. Interpret your answer in terms of sampling error."... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f0870e8198e50ab6fea308b08b9a17d7ce0f20f76ba6684fedd012c8a10697d | Brand X soda advertises, ``We will give you 20$\%$ more soda than Brand Y for a total price that is 10$\%$ less than Brand Y's price!'' What is the ratio of the unit price of Brand X soda to the unit price of Brand Y soda? Express your answer as a common fraction. | [
"\\frac{3}{4}"
] | [
"Brand X soda advertises, ``We will give you 20$\\%$ more soda than Brand Y for a total price that is 10$\\%$ less than Brand Y's price!'' What is the ratio of the unit price of Brand X soda to the unit price of Brand Y soda? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0107b05028701a608b33ac5812e048904e118ded408c3721df23f082ede0848c | Brandon is bored in calculus class, so he writes $5$ digits $N = d_1d_2d_3d_4d_5$ on his desk (Note that the leading digit can be $0$. For instance, if $d_1 = d_2 = 0$ and $d_3 = 3, d_4 = 5, d_5 = 5$, then $N = 355.$) He then notices something peculiar about the number he wrote: $\frac{d_{1+k}d_{2+k}d_{3+k}d_{4+k}d_{5+... | [
"18"
] | [
"Brandon is bored in calculus class, so he writes $5$ digits $N = d_1d_2d_3d_4d_5$ on his desk (Note that the leading digit can be $0$. For instance, if $d_1 = d_2 = 0$ and $d_3 = 3, d_4 = 5, d_5 = 5$, then $N = 355.$) He then notices something peculiar about the number he wrote: $\\frac{d_{1+k}d_{2+k}d_{3+k}d_{4+k... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-e4eb60073f395e7812c74c7758330c1670e2f8d42095558b938b1451783cd2f1 | Brandon writes a 5-digit number \( N = d_1d_2d_3d_4d_5 \) on his desk, where the leading digit can be 0. He notices that for all positive integers \( 1 \leq k \leq 4 \), the expression \(\frac{d_{1+k}d_{2+k}d_{3+k}d_{4+k}d_{5+k}}{N}\) is an integer, where \( d_{n} = d_{5+n} \). How many distinct positive integers \( N ... | [
"18"
] | [
"Brandon writes a 5-digit number \\( N = d_1d_2d_3d_4d_5 \\) on his desk, where the leading digit can be 0. He notices that for all positive integers \\( 1 \\leq k \\leq 4 \\), the expression \\(\\frac{d_{1+k}d_{2+k}d_{3+k}d_{4+k}d_{5+k}}{N}\\) is an integer, where \\( d_{n} = d_{5+n} \\). How many distinct positiv... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-84816d32d821056955e006612f9800665ebdd05ec049bf3fc1283c8ae437fdcb | Braving blizzard conditions on the planet Hoth, Luke Skywalker sets out at top speed in his snow speeder for a rebel base 4800 miles away. He travels into a steady headwind and makes the trip in 3 hours. Returning, he finds that the trip back, still at top speed but now with a tailwind, takes only 2 hours. Find the top... | [
"\\( x = 2000 \\) miles per hour, \\( w = 400 \\) miles per hour."
] | [
"Braving blizzard conditions on the planet Hoth, Luke Skywalker sets out at top speed in his snow speeder for a rebel base 4800 miles away. He travels into a steady headwind and makes the trip in 3 hours. Returning, he finds that the trip back, still at top speed but now with a tailwind, takes only 2 hours. Find th... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 31
} | 31 | 0 | 0 |
omr-ws-1777ca32584d5621ece16b352e6b6593333ed9a83a5a00ffaa0add5b26a7339d | Break down the transformation $z \mapsto 2e^{i\pi/4}(z-1+i)$ into three steps and describe each step: - Step 1: $z \mapsto (z-1+i)$ (translation) - Step 2: $z \mapsto e^{i\pi/4}(z-1+i)$ (rotation) - Step 3: $z \mapsto 2e^{i\pi/4}(z-1+i)$ (magnification) | [
"z \\mapsto 2e^{i\\pi/4}(z-1+i)"
] | [
"Break down the transformation $z \\mapsto 2e^{i\\pi/4}(z-1+i)$ into three steps and describe each step:\n - Step 1: $z \\mapsto (z-1+i)$ (translation)\n - Step 2: $z \\mapsto e^{i\\pi/4}(z-1+i)$ (rotation)\n - Step 3: $z \\mapsto 2e^{i\\pi/4}(z-1+i)$ (magnification)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-af2cbe1c4691e9dc32ce6b81505691f13c4861cb6d386d3f7cf52b9899f10226 | Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | [
"350"
] | [
"Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?"
... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 96,
"tir": 19
} | 115 | 0 | 0 |
omr-ws-93ab12edc41a426d108c05cf43f1cf70c68711d1d8153f21e6490bc74c0224e4 | Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel? | [
"2\\sqrt{41}"
] | [
"Brenda is going from $(-4,5)$ to $(5,-4)$, but she needs to stop by the origin on the way. How far does she have to travel?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81db125e9318a28e3c0f20abb390baa71028f05e62587234a48413bc529f7364 | Brenda is located at the origin of the Cartesian plane. A circle with radius \( R \) is centered at every integer coordinate point. Brenda cannot see any circle with a coordinate point of magnitude greater than 2. If the smallest possible radius \( R \) of the circle can be written as \( \frac{m}{\sqrt{n}} \), where \(... | [
"6"
] | [
"Brenda is located at the origin of the Cartesian plane. A circle with radius \\( R \\) is centered at every integer coordinate point. Brenda cannot see any circle with a coordinate point of magnitude greater than 2. If the smallest possible radius \\( R \\) of the circle can be written as \\( \\frac{m}{\\sqrt{n}} ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-952a5311938223a1d12e5ec93302964aefe299bd59ac4e7cb7fc2b5ec34a472a | Brenda left the airport and drove north at an average speed of 60 km/h. Chelsea left one hour later and drove in the same direction but with an average speed of 75 km/h. Find the number of hours Brenda drove before Chelsea caught up. | [
"5"
] | [
"Brenda left the airport and drove north at an average speed of 60 km/h. Chelsea left one hour later and drove in the same direction but with an average speed of 75 km/h. Find the number of hours Brenda drove before Chelsea caught up."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2940ba2549a95f46278be4532d58b101d1e1d87908d0a67f69fdb76f988347aa | Brenda starts with no money and collects either a quarter or a dime from each relative with equal probability. Compute the probability that at some point Brenda will have exactly $1.00. | [
"\\( \\frac{233}{1024} \\)"
] | [
"Brenda starts with no money and collects either a quarter or a dime from each relative with equal probability. Compute the probability that at some point Brenda will have exactly $1.00."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-0bfdca214704cccd0d8905a5a36109daaefae825bea0c8655f5c884b81b29b0a | Brennan chooses a set \( A = \{a, b, c, d, e\} \) of five real numbers with \( a \leq b \leq c \leq d \leq e \). Delaney determines the subsets of \( A \) containing three numbers and adds up the numbers in these subsets. She obtains the sums \( 0, 3, 4, 8, 9, 10, 11, 12, 14, 19 \). What are the five numbers in Brennan... | [
"\\{-3, -1, 4, 7, 8\\}"
] | [
"Brennan chooses a set \\( A = \\{a, b, c, d, e\\} \\) of five real numbers with \\( a \\leq b \\leq c \\leq d \\leq e \\). Delaney determines the subsets of \\( A \\) containing three numbers and adds up the numbers in these subsets. She obtains the sums \\( 0, 3, 4, 8, 9, 10, 11, 12, 14, 19 \\). What are the five... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-4923b3bd676b5d30ae22ab81d87dd95b674ef124595bf581d1ba6add04155568 | Brent rolls a fair dodecahedral die with numbers $1,2,3,...,12$ on its faces. What is the expected number of digits in the number he obtains? Express your answer in a decimal. | [
"1.25"
] | [
"Brent rolls a fair dodecahedral die with numbers $1,2,3,...,12$ on its faces. What is the expected number of digits in the number he obtains? Express your answer in a decimal."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd967295fb56ebb3cc1fb17f36800412a6c65dbbaed0bc317bac14a06db62cb2 | Brett has an $n \times n \times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $1 \times 1 \times 1$ cubes, and notices that the number of unpainted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of B... | [
"4"
] | [
"Brett has an $n \\times n \\times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $1 \\times 1 \\times 1$ cubes, and notices that the number of unpainted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side le... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73f50d72ae5a4245e1d19e2869599079ab0a3d8618d0649f84fde6bb76ddc676 | Brian and Stephen play a game where Brian writes "MPTF" on a chalkboard 100 times. Each letter has a point value: 'M' is worth 3 points, 'P' is worth 6 points, 'T' is worth 9 points, and 'F' is worth 12 points. On each turn, a player erases one character. Brian's goal is to erase the entire string "MPTF" all by himself... | [
"1800"
] | [
"Brian and Stephen play a game where Brian writes \"MPTF\" on a chalkboard 100 times. Each letter has a point value: 'M' is worth 3 points, 'P' is worth 6 points, 'T' is worth 9 points, and 'F' is worth 12 points. On each turn, a player erases one character. Brian's goal is to erase the entire string \"MPTF\" all b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-f1173a4198ec4083dbd91cbe152043132cd8513f4654109fdfce7bf8f2180070 | Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2N$ ho... | [
"\\dfrac{N(k - 1)}{k}"
] | [
"Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2b4d85866a8d7e458fb079fd5afc40d1f87299061e2b1c166abdb2f93976c8a | Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in 45 minutes. How long will it take to do the job if Brian worked with only three of his friends? | [
"\\dfrac{12}{11}"
] | [
"Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in 45 minutes. How long will it take to do the job if Brian worked with only three of his friends?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e5a70e54879da010c5458b5cc720267e81d68099fb9592af7b6004221489ad4 | Brian has a segment of length $6$. He divides it into three equal pieces and forms a triangle using the three pieces. Then, he divides it into six equal pieces and forms a hexagon in a similar fashion. The ratio of the area between the hexagon and triangle can be expressed as $\frac{a}{b}$. Find $a + b$. | [
"5"
] | [
"Brian has a segment of length $6$. He divides it into three equal pieces and forms a triangle using the three pieces. Then, he divides it into six equal pieces and forms a hexagon in a similar fashion. The ratio of the area between the hexagon and triangle can be expressed as $\\frac{a}{b}$. Find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-83e54f8bc9ada5b51ceab09686a75794de3697051574de5317c03c9f1fea9f9a | Brian is at the upper-left corner of a 4x4 grid. Every second, he moves to an adjacent square (no diagonal movements allowed) with equal probability. Immediately after moving, there is a 1/3 chance that a trap door opens beneath him. What is the probability that the trap door he ultimately falls into is one of the cent... | [
"3/14"
] | [
"Brian is at the upper-left corner of a 4x4 grid. Every second, he moves to an adjacent square (no diagonal movements allowed) with equal probability. Immediately after moving, there is a 1/3 chance that a trap door opens beneath him. What is the probability that the trap door he ultimately falls into is one of the... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-45684ffde00d76f8d78754013710f672f02d233055d85847346eaa3dcf93c3e2 | Brian starts at the point \((1,0)\) in the plane. Every second, he can move from \((a,b)\) to \((a-b,a+b)\) or from \((a,b)\) to \((2a-b,a+2b)\). How many different paths can he take to end up at \((28,-96)\)? | [
"70"
] | [
"Brian starts at the point \\((1,0)\\) in the plane. Every second, he can move from \\((a,b)\\) to \\((a-b,a+b)\\) or from \\((a,b)\\) to \\((2a-b,a+2b)\\). How many different paths can he take to end up at \\((28,-96)\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-075170620529810386f749a3333d7ec42c9301ba9499bd1100f72270b8f6dc62 | Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$. What is the sum of the possible values for $w$? | [
"31"
] | [
"Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$. What is the sum of the possible values for $w$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52aad5c810937af1d840296bc4cfdb1e42e7a494029dd2c3f46efe9b734bbdb0 | Brian writes down four integers \( w > x > y > z \) whose sum is 44. The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9. What is the sum of the possible values for \( w \)? | [
"31"
] | [
"Brian writes down four integers \\( w > x > y > z \\) whose sum is 44. The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9. What is the sum of the possible values for \\( w \\)?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-841427d34151c08f245348e92b9d3d89012f49df94b5d4f0869d1712511f116e | Brian writes down four integers \( w > x > y > z \) whose sum is 44. The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9. What is the sum of the possible values of \( w \)? | [
"31"
] | [
"Brian writes down four integers \\( w > x > y > z \\) whose sum is 44. The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9. What is the sum of the possible values of \\( w \\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 9
} | 48 | 0 | 0 |
omr-ws-da931609dfd86dbc7017370da45803cb39ac2a2ceaf10d33be8631b31d11027c | Brian, Chris, and Paul's scores on a math test are in an arithmetic progression, and their scores on a history test are in a geometric progression. The sum of Brian's scores is $168$, the sum of Chris' scores is $169$, and the sum of Paul's scores is $172$. The sum of the scores from their math tests is $255$. If Paul ... | [
"30"
] | [
"Brian, Chris, and Paul's scores on a math test are in an arithmetic progression, and their scores on a history test are in a geometric progression. The sum of Brian's scores is $168$, the sum of Chris' scores is $169$, and the sum of Paul's scores is $172$. The sum of the scores from their math tests is $255$. If ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-4f85793bc6dc004162f69d96bf1b7f308affb6da614731c00e58425b71e80a7a | Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? | [
"\\(\\frac{2}{5}\\)"
] | [
"Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-70cb87a400a82c349ae7e8e1cf9d36ebe6271c285244687942a6e90923abda56 | Brian’s lawn is in the shape of a circle with radius 10 meters. Brian can throw a frisbee up to 50 meters from where he stands. What is the area of the region (in square meters) in which the frisbee can land, if Brian can stand anywhere on his lawn? | [
"\\(3600\\pi\\)"
] | [
"Brian’s lawn is in the shape of a circle with radius 10 meters. Brian can throw a frisbee up to 50 meters from where he stands. What is the area of the region (in square meters) in which the frisbee can land, if Brian can stand anywhere on his lawn?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-84bc8300e7016bb8d978fb75db3608a7d862700b6dbf0337d09302bb6c2bc25f | Bridget places a box that is $4 \times 6 \times 8$ on the floor and then places a $2 \times 3 \times 5$ box on top of it, forming a two-box tower. She will paint all the surfaces she can paint without moving either of the boxes, aiming to minimize the painted area. What is the total area she has to paint? | [
"192"
] | [
"Bridget places a box that is $4 \\times 6 \\times 8$ on the floor and then places a $2 \\times 3 \\times 5$ box on top of it, forming a two-box tower. She will paint all the surfaces she can paint without moving either of the boxes, aiming to minimize the painted area. What is the total area she has to paint?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-8c90cf6e6b85734565ed873edd00c0ccf30f4b0a0bc5e8de8bd0eb9c8cc51641 | Bristy starts with the set \( A = \{0, 42\} \). At any step, she can add an integer \( x \) to the set \( A \) if it is a root of a polynomial whose coefficients are the already existing integers in \( A \). She continues this process until she cannot add any more numbers. How many numbers will be in \( A \) after she ... | [
"17"
] | [
"Bristy starts with the set \\( A = \\{0, 42\\} \\). At any step, she can add an integer \\( x \\) to the set \\( A \\) if it is a root of a polynomial whose coefficients are the already existing integers in \\( A \\). She continues this process until she cannot add any more numbers. How many numbers will be in \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-53b629e1cdc6ee32346b0e07f0aa2747fe9ea2e97e32154a1a9430e9e9e51a60 | Bristy starts with the set \( A = \{0, 42\} \). At any step, she can add an integer \( x \) to the set \( A \) if it is a root of a polynomial whose coefficients are the integers already in \( A \). She continues this process until no more numbers can be added. How many numbers will be in \( A \) after this process ter... | [
"17"
] | [
"Bristy starts with the set \\( A = \\{0, 42\\} \\). At any step, she can add an integer \\( x \\) to the set \\( A \\) if it is a root of a polynomial whose coefficients are the integers already in \\( A \\). She continues this process until no more numbers can be added. How many numbers will be in \\( A \\) after... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-42589fea735b57f7c910068e277b4f543ae6d83618bc2c7810502ccd42d6aeb3 | Broady The Boar is playing a game on a circle with 2021 points labeled $0, 1, 2, \ldots, 2020$ in clockwise order. Broady starts at point $0$ and rolls a 2020-sided die, moving clockwise by the number of points rolled. If Broady is at point $2020$ and rolls a number $n$, they move to point $n-1$. Broady continues rolli... | [
"2021"
] | [
"Broady The Boar is playing a game on a circle with 2021 points labeled $0, 1, 2, \\ldots, 2020$ in clockwise order. Broady starts at point $0$ and rolls a 2020-sided die, moving clockwise by the number of points rolled. If Broady is at point $2020$ and rolls a number $n$, they move to point $n-1$. Broady continues... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e7a472020b85756eb491c78035b5a4fe830cced00cb00d448593f620189ed482 | Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of 10 miles per hour per minute until she reaches the speed limit of 25 miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of 20 miles per hour ... | [
"35"
] | [
"Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of 10 miles per hour per minute until she reaches the speed limit of 25 miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of 20 miles per ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-edc657c3675c11cedd34ec24a9cfd72950e7c7743f0aa46eeee30e83653955de | Brothers found a treasure of gold and silver and split it so that each brother received 100 kg. The oldest brother got 1/5 of all the gold and 1/7 of all the silver. The youngest brother got 1/7 of all the gold. What part of all the silver did the youngest brother get? | [
"\\dfrac{9}{49}"
] | [
"Brothers found a treasure of gold and silver and split it so that each brother received 100 kg. The oldest brother got 1/5 of all the gold and 1/7 of all the silver. The youngest brother got 1/7 of all the gold. What part of all the silver did the youngest brother get?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d7da45d8997c2396ec6fb765a1fb405a1331d08b990b1edafc3a214a30f9fd6c | Bruno Mars is at $1$ degree Fahrenheit right now (time $t = 0$), and his heat $h(t)$ is given by $h(t) = 2 \cdot h(t - 1) + t$. If he will explode when he is over $1,000,000$ degrees Fahrenheit, at what time $t$ will Bruno Mars explode? | [
"19"
] | [
"Bruno Mars is at $1$ degree Fahrenheit right now (time $t = 0$), and his heat $h(t)$ is given by $h(t) = 2 \\cdot h(t - 1) + t$. If he will explode when he is over $1,000,000$ degrees Fahrenheit, at what time $t$ will Bruno Mars explode?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cbc40193d2af938b62697d38d3b823992e633ee68a0265f50cfd151c552692b | Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: \(\textbf{1 4 1 2 0 7 0 8}\). In each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of `$+$', `$-$', or `$\times$'. If he picks a random placement out of the \(3^7\) possible placements, what is the expected value... | [
"0"
] | [
"Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: \\(\\textbf{1 4 1 2 0 7 0 8}\\). In each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of `$+$', `$-$', or `$\\times$'. If he picks a random placement out of the \\(3^7\\) possible placements, what is the exp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2564e2d8724f05dfde33836a2a57d5ee9eb140bfd69cadb3d43faa1cefc32718 | Bryan has some 3 cent stamps and some 4 cent stamps. What is the least number of stamps he can combine so the value of the stamps is 33 cents? | [
"9"
] | [
"Bryan has some 3 cent stamps and some 4 cent stamps. What is the least number of stamps he can combine so the value of the stamps is 33 cents?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-696615551ad364f0b3cf912150662e3a0de381cca820e8edfd578b8c07a0525f | Bryce starts with nine tokens and flips a coin nine times. If the coin lands on heads, he gains two tokens, and if it lands on tails, he loses one token. If after nine flips, Bryce ends up with nine tokens, what is the probability that Bryce always had at least nine tokens during the process? | [
"\\(\\frac{5}{84}\\)"
] | [
"Bryce starts with nine tokens and flips a coin nine times. If the coin lands on heads, he gains two tokens, and if it lands on tails, he loses one token. If after nine flips, Bryce ends up with nine tokens, what is the probability that Bryce always had at least nine tokens during the process?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a074afef539ac323de9d06fd54808c323a36fe64a6a0933c350be5b89046cdb | Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Bubble Boy wants to reach Bubble Girl as quickly as possible, but he needs to pick up a gift from the $x$-axis first. Find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach ... | [
"\\sqrt{929} - 2"
] | [
"Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Bubble Boy wants to reach Bubble Girl as quickly as possible, but he needs to pick up a gift from the $x$-axis first. Find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then r... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-b485a69470dbd600bc1962f9049998f5753884c97be5e96c6b0c3ddb5f13bb3c | Bubble Boy and Bubble Girl live in bubbles of unit radii centered at (20, 13) and (0, 10) respectively. Bubble Boy wants to reach Bubble Girl as quickly as possible, but he needs to bring a gift from the x-axis first. Find the length of the shortest path Bubble Boy can take to visit the x-axis and then reach Bubble Gir... | [
"\\(\\sqrt{929}\\)"
] | [
"Bubble Boy and Bubble Girl live in bubbles of unit radii centered at (20, 13) and (0, 10) respectively. Bubble Boy wants to reach Bubble Girl as quickly as possible, but he needs to bring a gift from the x-axis first. Find the length of the shortest path Bubble Boy can take to visit the x-axis and then reach Bubbl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-812d3e663d1b900f5bc40c3177384e45ccd4c0a7f23daf930eada18f8ba715ae | Bucket A contains 3 liters of syrup. Bucket B contains \(n\) liters of water. Bucket C is empty. The following operations can be performed: - Pour away the entire amount in bucket X, - Pour the entire amount in bucket X into bucket Y, - Pour from bucket X into bucket Y until buckets Y and Z contain the same amount. (a)... | [
"n \\geq 7"
] | [
"Bucket A contains 3 liters of syrup. Bucket B contains \\(n\\) liters of water. Bucket C is empty. The following operations can be performed:\n- Pour away the entire amount in bucket X,\n- Pour the entire amount in bucket X into bucket Y,\n- Pour from bucket X into bucket Y until buckets Y and Z contain the same a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-aedcafe6cfe229d5f99e2a8eeae69680341385de3d788ad05a0d3264a1bc78a2 | Budi arranges fourteen balls, each with a radius of 10 cm. The first nine balls are placed on the table to form a square and touch each other. The next four balls are placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the four ... | [
"10 + 40\\sqrt{2}"
] | [
"Budi arranges fourteen balls, each with a radius of 10 cm. The first nine balls are placed on the table to form a square and touch each other. The next four balls are placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d814d12eefd2f52f0d8a4c152bd283fdb486ad5c03f9290e1adbe245e7dd9efc | Bugs Bunny has a carrot. In one bite, he eats some random amount of it (from no carrot to the entire carrot). He continues taking bites, with each bite eating the same amount of carrot as the first bite, until there is no longer enough carrot to take a full bite. What is the expected fraction of the carrot eaten by Bug... | [
"\\dfrac{\\pi^2}{12}"
] | [
"Bugs Bunny has a carrot. In one bite, he eats some random amount of it (from no carrot to the entire carrot). He continues taking bites, with each bite eating the same amount of carrot as the first bite, until there is no longer enough carrot to take a full bite. What is the expected fraction of the carrot eaten b... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-083c182a44c500e1f254f56fc1039ad442da6fd91fbc0dd9a4fce0e9f59af8c1 | Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). Suppose Bugs Bunny's first two hops we... | [
"10"
] | [
"Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \\geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \\geq 3$). Suppose Bugs Bunny's first two ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c8b1f2c9844f17ef8511156a1d805f0ca3d0dc467374ec98f1589a9bd8033d4 | Build a bijective function \( f: [0,1] \to [0,1] \) such that \( f \) is not monotonic on any specific interval \( J \subseteq [0,1] \). | [
"\\( f(x) = \\begin{cases} \nx & \\text{if } x \\text{ is irrational} \\\\\n1-x & \\text{if } x \\text{ is rational}\n\\end{cases} \\)"
] | [
"Build a bijective function \\( f: [0,1] \\to [0,1] \\) such that \\( f \\) is not monotonic on any specific interval \\( J \\subseteq [0,1] \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-0cd91a963668d1b6f33e2fc81fa127b98c4027ef508d6b760251f809d8c69954 | Build a bijective function \( f: [0,1] \to \mathbb{R} \). | [
"\\( f(x) = \\begin{cases} \n\\frac{1}{x - \\frac{1}{2}} - 2 & \\text{if } x > \\frac{1}{2} \\\\\n\\frac{1}{x - \\frac{1}{2}} + 2 & \\text{if } x \\in \\left[ 0, \\frac{1}{2} \\right) \\setminus \\left\\{ 0, a_{1}, a_{2}, a_{3}, \\ldots \\right\\} \\\\\n-1 & \\text{if } x = \\frac{1}{2} \\\\\n-\\frac{1}{2} & \\text... | [
"Build a bijective function \\( f: [0,1] \\to \\mathbb{R} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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