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-daa33e437d91b7b24ea81b5f3d45ce87cfa095baef09b19d2a5c4b8e0742fe2a | A man is walking on a straight road and finds that the line joining two points on the same side of the road subtends an angle $\alpha$ at some point on the road. After walking a distance of $b$ along the road, he finds that the line joining the two points again subtends an angle $\alpha$. After walking a further distan... | [
"\\sqrt{a(a + b)}"
] | [
"A man is walking on a straight road and finds that the line joining two points on the same side of the road subtends an angle $\\alpha$ at some point on the road. After walking a distance of $b$ along the road, he finds that the line joining the two points again subtends an angle $\\alpha$. After walking a further... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-3fff79df8f89e2b4ec6c5be786187d2e98b209511ee5b6891f4cffb7182f39ef | A man leaves for work at a random time between 7:00 AM and 8:00 AM. His newspaper arrives at his house at a random time between 6:30 AM and 7:30 AM. What is the probability that he has his newspaper before he leaves for work? | [
"0.875"
] | [
"A man leaves for work at a random time between 7:00 AM and 8:00 AM. His newspaper arrives at his house at a random time between 6:30 AM and 7:30 AM. What is the probability that he has his newspaper before he leaves for work?"
] | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-1c337e45acd3d47a6ad04050a0c5c4b71b3074bde9089c9337dd373500daa8ad | A man leaves his money to his children in his will as follows: the first child receives $1000 plus 10% of what is left, the second child receives $2000 plus 10% of what is left, and so on, with each child receiving the same amount. How many children are there? | [
"9"
] | [
"A man leaves his money to his children in his will as follows: the first child receives $1000 plus 10% of what is left, the second child receives $2000 plus 10% of what is left, and so on, with each child receiving the same amount. How many children are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-759f32bafa05859462627ad000bcb2104b78a25e1934b7537651a8348dc09de2 | A man left his three sons to receive the \( n \)-th, \( m \)-th, and \( t \)-th parts of his \( N \) camels. It is known that \( N+1 \) is a common multiple of \( n \), \( m \), and \( t \). When Aldo added one camel to the set, the camels could be divided as desired, and Aldo took his camel back afterward. Find all so... | [
"\\( n = 2, \\, m = 3, \\, t = 6, \\, N = 5 \\) (and other solutions exist with different values of \\( n \\), \\( m \\), \\( t \\) and \\( N \\) satisfying the conditions)."
] | [
"A man left his three sons to receive the \\( n \\)-th, \\( m \\)-th, and \\( t \\)-th parts of his \\( N \\) camels. It is known that \\( N+1 \\) is a common multiple of \\( n \\), \\( m \\), and \\( t \\). When Aldo added one camel to the set, the camels could be divided as desired, and Aldo took his camel back a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 22
} | 41 | 0 | 0 |
omr-ws-a43de09b971ca24d61c129f7c58a9978805294b37e3a915735e3a0be6177c948 | A man lent two equal sums of money. One at 10% per annum simple interest for 5 years and the other at 15% for 3 years and 4 months. If the difference between the two sums of interest earned is $200, find both the sums of money lent out. | [
"20000"
] | [
"A man lent two equal sums of money. One at 10% per annum simple interest for 5 years and the other at 15% for 3 years and 4 months. If the difference between the two sums of interest earned is $200, find both the sums of money lent out."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-4e1848094b7229201daf547632911d1c287543c2a54bd42b10ae5407bd2bc1e0 | A man makes an annual deposit of 200 units in an account that pays 5% interest compounded annually. How much money should be in the account immediately after the 10th deposit? | [
"2515.58"
] | [
"A man makes an annual deposit of 200 units in an account that pays 5% interest compounded annually. How much money should be in the account immediately after the 10th deposit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-601e5206aeb1fbcd2b3c34a873d28ed8fdf674a0891d82e6d30d1e790b5f72f4 | A man may take a step forward, backward, left, or right with equal probability. Find the probability that after nine steps, he will be just one step away from his initial position. | [
"\\( \\frac{3969}{16384} \\)"
] | [
"A man may take a step forward, backward, left, or right with equal probability. Find the probability that after nine steps, he will be just one step away from his initial position."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 30
} | 47 | 0 | 0 |
omr-ws-085cd13fbcab395d6dc61b633ed48ad71faf289194dd3b031271db4bfd06a028 | A man meets 4 of his friends in a random order, 1 of which is infected. The probability of infecting someone after meeting them is 15% if you aren't wearing a mask. This time, however, everyone but himself are wearing masks. The probability of someone wearing a mask infecting someone who isn't is 5%. What is the expect... | [
"1.06125"
] | [
"A man meets 4 of his friends in a random order, 1 of which is infected. The probability of infecting someone after meeting them is 15% if you aren't wearing a mask. This time, however, everyone but himself are wearing masks. The probability of someone wearing a mask infecting someone who isn't is 5%. What is the e... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-1d0dead0e0c7bba3f6192f4ab495da34869b930f5444874a72f968d79276431c | A man meets 4 of his friends in a random order, 1 of which is infected. The probability of infecting someone after meeting them is 15%. What is the expected number of people that will be infected after he meets all 4 friends? (Note that the man could meet the infected friend and then infect other friends he meets after... | [
"1.18375"
] | [
"A man meets 4 of his friends in a random order, 1 of which is infected. The probability of infecting someone after meeting them is 15%. What is the expected number of people that will be infected after he meets all 4 friends? (Note that the man could meet the infected friend and then infect other friends he meets ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-9ae029efb5b15e0d669153cc141d4a32e96d34b5fbdc7e87cb2adf35f76536cb | A man meets his wife at the train station every day, and she drives him home. One day, he arrives an hour early and starts walking home. His wife eventually picks him up and they arrive home 20 minutes earlier than usual. How much time did the man spend walking? | [
"50 minutes"
] | [
"A man meets his wife at the train station every day, and she drives him home. One day, he arrives an hour early and starts walking home. His wife eventually picks him up and they arrive home 20 minutes earlier than usual. How much time did the man spend walking?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ec776c4833607bc6c1861ecd3df9fb7c57c1326a6937f292962b8744f6cef077 | A man named Juan has 3 rectangular solids, each with a volume of 128. The areas of two faces of the first solid are 4 and 32, the areas of two faces of the second solid are 64 and 16, and the areas of two faces of the third solid are 8 and 32. What is the minimum possible exposed surface area of the tallest tower Juan ... | [
"688"
] | [
"A man named Juan has 3 rectangular solids, each with a volume of 128. The areas of two faces of the first solid are 4 and 32, the areas of two faces of the second solid are 64 and 16, and the areas of two faces of the third solid are 8 and 32. What is the minimum possible exposed surface area of the tallest tower ... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-b51017ab9395bb7c8d34aa2efe72af5d1ab01e4f46b736409a72a3191fa4c1fb | A man on a dock is pulling in a boat by means of a rope attached to the bow of the boat 1 ft above the water and passing through a simple pulley located on the dock 8 ft above the water. If the boat is originally 45 ft from the dock, and he pulls in the rope at 2 ft/sec, what is the angle between the rope and the water... | [
"15.9^\\circ"
] | [
"A man on a dock is pulling in a boat by means of a rope attached to the bow of the boat 1 ft above the water and passing through a simple pulley located on the dock 8 ft above the water. If the boat is originally 45 ft from the dock, and he pulls in the rope at 2 ft/sec, what is the angle between the rope and the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-6b1b462db71cdc3f4b012a65609490b03c3b9b5f9eecc68798eb3c8b76b941d4 | A man ordered a specific number of duck eggs and received exactly that number. He needs to go up to the 23rd floor using the stairs because the elevator is out of order. At each floor, a guard takes half of the eggs he is carrying and returns one egg. How many eggs must he initially have to ensure he has at least one e... | [
"2"
] | [
"A man ordered a specific number of duck eggs and received exactly that number. He needs to go up to the 23rd floor using the stairs because the elevator is out of order. At each floor, a guard takes half of the eggs he is carrying and returns one egg. How many eggs must he initially have to ensure he has at least ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-4a65ebea379c3011f87b6ac55be6895eadf4a21477f7eebbd8d55310c7474468 | A man picks a number \( x \) uniformly in the interval \([0, 1]\) and writes it down on a piece of paper. On another piece of paper, he writes \( x^2 \). You enter the room and choose one of the two pieces of paper. You have the choice between: 1. Keeping the chosen piece of paper and receiving the amount written on it... | [
"Keep if \\( Y > 0.31944846 \\), Switch if \\( Y < 0.31944846 \\)"
] | [
"A man picks a number \\( x \\) uniformly in the interval \\([0, 1]\\) and writes it down on a piece of paper. On another piece of paper, he writes \\( x^2 \\). You enter the room and choose one of the two pieces of paper. You have the choice between:\n1. Keeping the chosen piece of paper and receiving the amount w... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-79b7aa728633b0536433bf59d420348ec4b8e815e6c04db0447f5e607e191f21 | A man puts money in a savings bank and one year later earns an interest of 15 roubles. Having added another 85 roubles, he deposits the money for another year. After the expiry of this period the sum-total of the principal and its interest is 420 roubles. What sum of money was originally deposited and what interest doe... | [
"300 roub., 5\\%"
] | [
"A man puts money in a savings bank and one year later earns an interest of 15 roubles. Having added another 85 roubles, he deposits the money for another year. After the expiry of this period the sum-total of the principal and its interest is 420 roubles. What sum of money was originally deposited and what interes... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93e65658ffec7a0a902f116a52754b451c3d75121480192de130b379336a2c8f | A man realizes that the product of his two sons' ages is his age and that the product of his daughters' ages is his wife's age. He also realizes that the sum of his kids' ages will surpass the sum of him and his wife's ages in 25 years. Additionally, he realizes that $n$ years ago, the product of his kids' ages was les... | [
"3"
] | [
"A man realizes that the product of his two sons' ages is his age and that the product of his daughters' ages is his wife's age. He also realizes that the sum of his kids' ages will surpass the sum of him and his wife's ages in 25 years. Additionally, he realizes that $n$ years ago, the product of his kids' ages wa... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-fbf39d2d43578050dff446276afeb1bacd0421ac42f5fb06e74306bb1c063d07 | A man rows a boat upstream under the first bridge and his hat falls into the river exactly under the second bridge, which is 1 km away. After 15 minutes, he realizes his hat is missing and starts rowing back downstream. He catches his hat under the first bridge. Determine the speed of the river and the speed of the boa... | [
"U = 2 \\text{ km/h}"
] | [
"A man rows a boat upstream under the first bridge and his hat falls into the river exactly under the second bridge, which is 1 km away. After 15 minutes, he realizes his hat is missing and starts rowing back downstream. He catches his hat under the first bridge. Determine the speed of the river and the speed of th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-de871c6b1aad1257d33c2781c30f68fa4574554663dd185a4359e693a813cdb3 | A man rows his boat against the stream in a river from point $M$. After traveling 2 kilometers, he drops his hat but doesn't realize it. The hat floats downstream for 10 minutes before he notices and turns back to row at the same speed. He catches his hat at point $M$. What is the speed of the stream? | [
"\\( 6 \\) km/h"
] | [
"A man rows his boat against the stream in a river from point $M$. After traveling 2 kilometers, he drops his hat but doesn't realize it. The hat floats downstream for 10 minutes before he notices and turns back to row at the same speed. He catches his hat at point $M$. What is the speed of the stream?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-78a5c6cef62de6df3d5c8cf528c4cdf01140db02e7295500409444e1d081cf2f | A man rows upstream 1 mile and then drops a bottle into the river. He continues rowing upstream for ten more minutes before turning around and rowing downstream. He and the bottle arrive back at the dock at the same time. How fast is the river flowing? | [
"3"
] | [
"A man rows upstream 1 mile and then drops a bottle into the river. He continues rowing upstream for ten more minutes before turning around and rowing downstream. He and the bottle arrive back at the dock at the same time. How fast is the river flowing?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52eb42fec112abed40b9a6a5c8e35883a75314fcac9f8f4d9ed35e0716362fb4 | A man speaks the truth $3$ out of $5$ times. He throws a dice and reports that it is $1$. Find the probability that the dice actually landed on $1$. | [
"\\(\\frac{3}{13}\\)"
] | [
"A man speaks the truth $3$ out of $5$ times. He throws a dice and reports that it is $1$. Find the probability that the dice actually landed on $1$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-bacfb1c07983563694e0a49719928be2a46b9474556eb356ddc4fdea2037cb6c | A man speaks the truth 3 out of 4 times. He throws a dice and reports that a 6 occurred. Find the probability that a 6 actually occurred. | [
"\\(\\frac{3}{4}\\)"
] | [
"A man speaks the truth 3 out of 4 times. He throws a dice and reports that a 6 occurred. Find the probability that a 6 actually occurred."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-d1548d33920d2d8dd8a5ce8ea5107add50dddc6b34e233221178bc8058dcd747 | A man speaks the truth 3 out of 4 times. He throws a die and reports that it is a six. What is the probability that it is actually a six? | [
"\\(\\frac{3}{8}\\)"
] | [
"A man speaks the truth 3 out of 4 times. He throws a die and reports that it is a six. What is the probability that it is actually a six?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 4
} | 8 | 0 | 0 |
omr-ws-fedbb7786a1e4f6abf2b6e6513cbf9977ece0c90105e1d1d61f419a6a7b2c80d | A man standing $x$ metres to the north of a tower finds the angle of elevation of its top to be $30^{\circ}$. He then walks $\frac{x}{2}$ metres towards the tower, turns east and walks $\frac{x}{2}$ metres, and then turns south and walks $\frac{x}{2}$ metres. Find the angle of elevation of the top of the tower from his... | [
"\\arctan\\left( \\dfrac{2\\sqrt{3}}{3} \\right)"
] | [
"A man standing $x$ metres to the north of a tower finds the angle of elevation of its top to be $30^{\\circ}$. He then walks $\\frac{x}{2}$ metres towards the tower, turns east and walks $\\frac{x}{2}$ metres, and then turns south and walks $\\frac{x}{2}$ metres. Find the angle of elevation of the top of the tower... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed5668e00737be5a17b8f349c707304d0332fd53e75944d80dcfcc7f0548360e | A man standing on the deck of a ship, which is 16 meters above the water level, observes the angle of elevation of the top of a cliff as 6 degrees and the angle of depression of the base of the cliff as 30 degrees. Calculate the distance of the cliff from the ship and the height of the cliff. | [
"Distance of the cliff from the ship: \\( 16\\sqrt{3} \\) meters, Height of the cliff: 80 meters."
] | [
"A man standing on the deck of a ship, which is 16 meters above the water level, observes the angle of elevation of the top of a cliff as 6 degrees and the angle of depression of the base of the cliff as 30 degrees. Calculate the distance of the cliff from the ship and the height of the cliff."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-47129b22f35b14d04cbc4987b7da020f43ae397d4617dc8ce739c3e3329cf1db | A man stands $a$ metres due east of a tower and finds the angle of elevation of the top of the tower to be $\theta$. He then walks $x$ metres north west and finds the angle of elevation to be $\theta$ again. Find the value of $x$. | [
"\\( x = a\\sqrt{2} \\)"
] | [
"A man stands $a$ metres due east of a tower and finds the angle of elevation of the top of the tower to be $\\theta$. He then walks $x$ metres north west and finds the angle of elevation to be $\\theta$ again. Find the value of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-35785465d7d5d1fe892259ccf19db92c3ffa800afd7e91564d524fc3bf6294ca | A man stands 30 feet from the base of a tree and holds a ruler 6 inches in front of his eye. The bottom of the ruler aligns with the base of the tree, and the top of the tree aligns with a point 8 inches high on the ruler. Using this information, determine the height of the tree to confirm if it is over 45 feet tall. | [
"40"
] | [
"A man stands 30 feet from the base of a tree and holds a ruler 6 inches in front of his eye. The bottom of the ruler aligns with the base of the tree, and the top of the tree aligns with a point 8 inches high on the ruler. Using this information, determine the height of the tree to confirm if it is over 45 feet ta... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-0624df9cd7a8eb1dceaaccf69f15bac0b57f31e386083246731edfa8cbb0beb0 | A man stands 4 feet behind a cliff and rolls a fair 6-sided die. If the die lands on any number from 1 to 4, he takes a step forward (1 foot); if it lands on 5 or 6, he steps one foot backwards. Four feet behind him is his family, who will grab him immediately and save him, allowing him to survive. If he continues the ... | [
"\\dfrac{1}{17}"
] | [
"A man stands 4 feet behind a cliff and rolls a fair 6-sided die. If the die lands on any number from 1 to 4, he takes a step forward (1 foot); if it lands on 5 or 6, he steps one foot backwards. Four feet behind him is his family, who will grab him immediately and save him, allowing him to survive. If he continues... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24ae69fcd41d7f45001ac95f3d4e1276612a1cc1dbe2a33d8321d37c2600bb7c | A man stands at the edge of a cliff (facing it). On every "move", he either takes two steps backwards with probability $\frac{1}{2}$ or one step forward also with probability $\frac{1}{2}$. What is the probability that after some number of moves, the man falls off the cliff? | [
"\\(\\frac{\\sqrt{5} - 1}{2}\\)"
] | [
"A man stands at the edge of a cliff (facing it). On every \"move\", he either takes two steps backwards with probability $\\frac{1}{2}$ or one step forward also with probability $\\frac{1}{2}$. What is the probability that after some number of moves, the man falls off the cliff?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-bf130e79f8c333bd922b5cacb3c0f5b26665680447994762507a0fe6fcbd08ba | A man stands on the number line at $0$, and each second, he walks left with a probability of $25$ percent, and right with a probability of $75$ percent. Let $P_n$ be the probability that the man is at position $0$ after $n$ seconds. Calculate the value of $P_{100}$ and express your answer as a fraction in lowest terms.... | [
"\\binom{100}{50} \\cdot 3^{50} + 2^{200}"
] | [
"A man stands on the number line at $0$, and each second, he walks left with a probability of $25$ percent, and right with a probability of $75$ percent. Let $P_n$ be the probability that the man is at position $0$ after $n$ seconds. Calculate the value of $P_{100}$ and express your answer as a fraction in lowest t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-904ddb9c5c409602ca2ca71258d1dde8d4639b1bbc3b681edfce0df90748d289 | A man starts walking from the origin of a rectangular coordinate system. His first step is of unit length along the positive $x$-axis. He then spins on his heels through a counterclockwise angle of $\theta$ and walks a second step of distance one-half unit. He continues this process, spinning through the angle $\theta$... | [
"\\(\\sqrt{\\left( \\frac{\\ln 2}{2} \\right)^2 + \\left( \\frac{\\pi}{4} \\right)^2}\\)"
] | [
"A man starts walking from the origin of a rectangular coordinate system. His first step is of unit length along the positive $x$-axis. He then spins on his heels through a counterclockwise angle of $\\theta$ and walks a second step of distance one-half unit. He continues this process, spinning through the angle $\... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-954c23f39fbba9722d09b6cf47dad5157a0b0bb2687bce638ca665c8563a96fe | A man starts walking north at $2$ ft/s from a point P. Five minutes later a woman starts walking south at $7$ ft/s from a point $500$ ft due east of P. At what rate (in ft/s) are the people moving apart $15$ minutes after the woman starts walking? (Round your answer to two decimal places.) | [
"8.97"
] | [
"A man starts walking north at $2$ ft/s from a point P. Five minutes later a woman starts walking south at $7$ ft/s from a point $500$ ft due east of P. At what rate (in ft/s) are the people moving apart $15$ minutes after the woman starts walking? (Round your answer to two decimal places.)"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4e32208d030d10455004293375b0359f1ded5686018616560f4ba18a06f47097 | A man starts walking north at 4 ft/s from a point P. Five minutes later, a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking? | [
"539.3 ft/s"
] | [
"A man starts walking north at 4 ft/s from a point P. Five minutes later, a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-160fa6b80cd95db60681dacde558226543c3fb7f2876e44c0903f57ed0a9a941 | A man starts walking north at 4 ft/sec from a point P. Five minutes later, a woman starts walking south at 5 ft/sec from a point 100 feet due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking? Given: - The man has traveled for 20 minutes (1200 seconds) at 4 ft/sec, covering 4... | [
"6.26 ft/sec"
] | [
"A man starts walking north at 4 ft/sec from a point P. Five minutes later, a woman starts walking south at 5 ft/sec from a point 100 feet due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking?\n\nGiven:\n- The man has traveled for 20 minutes (1200 seconds) at 4 ft/sec, c... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d39b0174619c2dd66c69761eb9c7187d50965a81b110d5e4b644305ea0defb6 | A man starts walking north at 5 ft/s from a point \( P \). Five minutes later, a woman starts walking south at 4 ft/s from a point 1300 ft due east of \( P \). At what rate are the people moving apart 20 minutes after the woman starts walking? | [
"\\( 6.4 \\, \\frac{\\text{ft.}}{\\text{s}} \\)"
] | [
"A man starts walking north at 5 ft/s from a point \\( P \\). Five minutes later, a woman starts walking south at 4 ft/s from a point 1300 ft due east of \\( P \\). At what rate are the people moving apart 20 minutes after the woman starts walking?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-995b81317c8de6bf1038afd2814b9918b32f0838f12945d41c708ef28d329cf7 | A man starts with 100 Joules of energy and walks on an infinite square grid, expending 1 Joule of energy for each step north, south, east, or west. He stops when he has 0 Joules of energy. How many different places could he have ended up? | [
"10201"
] | [
"A man starts with 100 Joules of energy and walks on an infinite square grid, expending 1 Joule of energy for each step north, south, east, or west. He stops when he has 0 Joules of energy. How many different places could he have ended up?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-979bbac0c661662a2ab6fff2c585ac21937885b1f82a292b024a07e61e7afe8a | A man travels from A to D via B and C, where there are 4 roads from A to B, 2 roads from B to C, and 3 roads from C to D. If he returns from D to A by a different route, how many ways can he complete this journey? | [
"552"
] | [
"A man travels from A to D via B and C, where there are 4 roads from A to B, 2 roads from B to C, and 3 roads from C to D. If he returns from D to A by a different route, how many ways can he complete this journey?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-26bbf977614f09f95d0e5d324d7ae8e812fd73f5ce87a39bd103f4622e2ec60e | A man travels from city A to city B at 6 am every morning with a probability of 1/2 and returns from city B to city A at 6 pm every evening with a probability of 2/3. If he takes an Airbus A320 Air India flight from city A to city B on July 1, 2024, at 6 am, what is the probability that he will be in city B at 5:30 am ... | [
"\\dfrac{11}{54}"
] | [
"A man travels from city A to city B at 6 am every morning with a probability of 1/2 and returns from city B to city A at 6 pm every evening with a probability of 2/3. If he takes an Airbus A320 Air India flight from city A to city B on July 1, 2024, at 6 am, what is the probability that he will be in city B at 5:3... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28d9dca813ca4298494be1b3ab0bc74c17edd4e07d4aaa3954ff1d258dede4e5 | A man walked a certain distance at a constant rate. If he had gone $\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\frac{1}{2}$ mile per hour slower, he would have been $2\frac{1}{2}$ hours longer on the road. The distance in miles he walked was. | [
"15"
] | [
"A man walked a certain distance at a constant rate. If he had gone $\\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\\frac{1}{2}$ mile per hour slower, he would have been $2\\frac{1}{2}$ hours longer on the road. The distance in miles he walked was.... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b7aac90f9ca09ab4e8202eeb49315760b54b7cc042fe112afd38fe5706ae5610 | A man walked a total of 5 hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks 4 miles per hour on the level, 3 miles per hour uphill, and \(r\) miles per hour downhill. For what values of \(r\) will this information unique... | [
"6"
] | [
"A man walked a total of 5 hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks 4 miles per hour on the level, 3 miles per hour uphill, and \\(r\\) miles per hour downhill. For what values of \\(r\\) will this informati... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-741ce074080356d3e5e521698c042548e5781860d541a73b8795ede84594926b | A man walks 1 km east and then 1 km northwest. How far is he from the starting point? | [
"\\(\\sqrt{2-\\sqrt{2}}\\) km"
] | [
"A man walks 1 km east and then 1 km northwest. How far is he from the starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2214675d1ff9ffaa6eb20f934f252a27217a50e13f849b8b895fea21fc25c1aa | A man walks 300m East along a river until he reaches a mill. Then he walks 200m in the direction S15E to a post on the edge of a road. Finally, he follows the road back to his starting spot. What is the total area of the quadrilateral formed by his path? | [
"7764.5"
] | [
"A man walks 300m East along a river until he reaches a mill. Then he walks 200m in the direction S15E to a post on the edge of a road. Finally, he follows the road back to his starting spot. What is the total area of the quadrilateral formed by his path?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-7d4f1dfc852cebca2775e18164fa0ea156d851bd2447e24c798b33da7b8ff8c3 | A man walks a distance of 3 units from the origin towards the north-east direction. From there, he walks a distance of 4 units towards the north-east direction to reach point P. What is the position of point P in the Argand plane? | [
"\\((3+4i)e^{\\frac{i\\pi}{4}}\\)"
] | [
"A man walks a distance of 3 units from the origin towards the north-east direction. From there, he walks a distance of 4 units towards the north-east direction to reach point P. What is the position of point P in the Argand plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-e0844bd54ccbfeb930f697a4141c24d91e3e077f218b6739b8001bfc23f3fb3f | A man walks around a circle of length 1 with an irrational step length 'a' (measured along the circumference). The circle has a ditch of width 'e' > 0. Let $N(e)$ be the smallest number of steps such that the man steps into the ditch, regardless of his starting position. If $e = \frac{1}{100}$, find the smallest intege... | [
"100"
] | [
"A man walks around a circle of length 1 with an irrational step length 'a' (measured along the circumference). The circle has a ditch of width 'e' > 0. Let $N(e)$ be the smallest number of steps such that the man steps into the ditch, regardless of his starting position. If $e = \\frac{1}{100}$, find the smallest ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-5f1ab330f1518f748979917a3073c844cb032b79b344719074de85edbd69576a | A man walks in a circle with circumference 1, taking steps of length $\alpha$ where $\alpha$ is irrational. There is a ditch of length $\epsilon = \frac{1}{100}$ on the circle. Calculate the minimum number of steps $N$ such that the man is guaranteed to fall into the ditch at least once during his walk. | [
"100"
] | [
"A man walks in a circle with circumference 1, taking steps of length $\\alpha$ where $\\alpha$ is irrational. There is a ditch of length $\\epsilon = \\frac{1}{100}$ on the circle. Calculate the minimum number of steps $N$ such that the man is guaranteed to fall into the ditch at least once during his walk."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-5baff38e30c1459eddd81d4b74e78af9538e28ca3ce512f860b6148057f29dc2 | A man walks on the real line starting from 3. He moves to \( n+1 \) from \( n \) with probability \( \frac{2}{3} \) and to \( n-1 \) with probability \( \frac{1}{3} \). What is the probability that he reaches 5 before reaching 0? | [
"\\(\\frac{28}{31}\\)"
] | [
"A man walks on the real line starting from 3. He moves to \\( n+1 \\) from \\( n \\) with probability \\( \\frac{2}{3} \\) and to \\( n-1 \\) with probability \\( \\frac{1}{3} \\). What is the probability that he reaches 5 before reaching 0?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-31ac88512f4cdcdff11b1ff9eafbdfa05771264981e41faa8a0040c2d744ca6d | A man wants to measure the width of a river from a point A on one bank to a point B on the opposite bank. He marks a point C on a perpendicular to AB at a distance of \(a\) meters from A. He measures the angle \(\angle BCA\) to be \(\alpha\) degrees and finds that \(\sin \alpha = c\). If the distance from A to the near... | [
"60"
] | [
"A man wants to measure the width of a river from a point A on one bank to a point B on the opposite bank. He marks a point C on a perpendicular to AB at a distance of \\(a\\) meters from A. He measures the angle \\(\\angle BCA\\) to be \\(\\alpha\\) degrees and finds that \\(\\sin \\alpha = c\\). If the distance f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-b0fa5cba309cc5c869c4091e433a53dd26dfdf1268e026c0ce26023df5c5d880 | A man was born on April 1st, 20 BCE and died on April 1st, 24 CE. How many years did he live? | [
"43"
] | [
"A man was born on April 1st, 20 BCE and died on April 1st, 24 CE. How many years did he live?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a339c8ba0a4f6a0e9ce8d955b1f2d6fd5fd06d75cbd01bfdc7ede0636d72ea4 | A man wearing a circular sombrero of radius 3 feet walks in a circle of radius 5 feet. What is the area of dead grass that results from the sombrero blocking sunlight? | [
"\\(60\\pi\\)"
] | [
"A man wearing a circular sombrero of radius 3 feet walks in a circle of radius 5 feet. What is the area of dead grass that results from the sombrero blocking sunlight?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-5397dacf13852299179d737b3122a5cb3253b4bac7bf1c0623af1476487528bb | A man weighing 70 kg lies in a hammock whose ropes make angles of 30 degrees and 45 degrees with the horizontal. What is the tension in each rope? | [
"615 \\, \\text{N}"
] | [
"A man weighing 70 kg lies in a hammock whose ropes make angles of 30 degrees and 45 degrees with the horizontal. What is the tension in each rope?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-feefa10d3c33d96e79636a5d7bc5109a27002e883e574e6a26774653e2522dc1 | A man went to cash a check. The teller misread the check, giving the man cents for dollars and dollars for cents. The man gave 5 cents to a beggar and then found out he had twice the amount of the check in his pocket. Before going to the bank, the man had no money. What is the amount of the check? | [
"$31.63"
] | [
"A man went to cash a check. The teller misread the check, giving the man cents for dollars and dollars for cents. The man gave 5 cents to a beggar and then found out he had twice the amount of the check in his pocket. Before going to the bank, the man had no money. What is the amount of the check?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-29035992d030e3e6b481658436729554b7ffb8a0086d32a1dfb82d64420b67ee | A man who has walked three-fifths of the distance across a bridge sees a train coming towards him. The speed of the train is 60 miles per hour. By running, the man can reach either end of the bridge just as the train reaches that end. What is the man's running speed, in miles per hour? | [
"12"
] | [
"A man who has walked three-fifths of the distance across a bridge sees a train coming towards him. The speed of the train is 60 miles per hour. By running, the man can reach either end of the bridge just as the train reaches that end. What is the man's running speed, in miles per hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-57ad94713d8888976864909d7b67dd92cf394cdecf6d8a8a0141e9375f2045e5 | A man with his camel is at point A and needs to carry 3000 bananas to point B, which is 1000 km away. The camel can carry at most 1000 bananas at a time and must eat one banana to travel 1 km. What is the maximum number of bananas that can be delivered to point B? (Fractions of bananas or kilometers are allowed.) | [
"533"
] | [
"A man with his camel is at point A and needs to carry 3000 bananas to point B, which is 1000 km away. The camel can carry at most 1000 bananas at a time and must eat one banana to travel 1 km. What is the maximum number of bananas that can be delivered to point B? (Fractions of bananas or kilometers are allowed.)"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-0cfd0ef2e7eeb127c5e0a3b2c9788f5f18febf853e9d34c79c197bd6f36459a9 | A man with irrational step lengths $\alpha$ walks along the circumference of a circle of length $1$. There exists a ditch of length $\epsilon > 0$ along the circumference of the circle. Calculate the smallest positive integer $N$ such that the man will have stepped into the ditch at least once after taking $N$ steps, g... | [
"100"
] | [
"A man with irrational step lengths $\\alpha$ walks along the circumference of a circle of length $1$. There exists a ditch of length $\\epsilon > 0$ along the circumference of the circle. Calculate the smallest positive integer $N$ such that the man will have stepped into the ditch at least once after taking $N$ s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-52f133ae255a788f37f82c86fba004b3fd72ae0bc896a9565b11d0dd08d3fdb2 | A man's fortune increases at a rate proportional to the square of his present wealth. If he had 1 million dollars one year ago and has 2 million today, how much will he have at the end of six months? At the end of two years? Comment on the plausibility of this model. | [
"-2"
] | [
"A man's fortune increases at a rate proportional to the square of his present wealth. If he had 1 million dollars one year ago and has 2 million today, how much will he have at the end of six months? At the end of two years? Comment on the plausibility of this model."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b35cd67f36e27a2ba9ae2f11e79e55163290aa71f2c2b787d52f83fc63ddf74c | A manager of a company has 8 workers. Each meeting lasts 1 hour, no break between two meetings. Three workers attend each meeting. Any two workers have attended at least one common meeting. Any worker cannot leave until he finishes all his meetings. How long does the worker who works the longest work at least? | [
"3"
] | [
"A manager of a company has 8 workers. Each meeting lasts 1 hour, no break between two meetings. Three workers attend each meeting. Any two workers have attended at least one common meeting. Any worker cannot leave until he finishes all his meetings. How long does the worker who works the longest work at least?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1d687c8613b354359e42dad6d7b875ff6160c330857f607114671eb6e1a030f | A manufacturer collects a sample of 9 parts and measures their weight in kilograms. The average weight in the sample is found to be 2.75 kilograms. Assume the population standard deviation is 0.45 kilograms. Assume the distribution of weights is normal. a) Construct a 95% confidence interval for the population mean wei... | [
"(2.456, 3.044) for the 95% confidence interval and (2.3636, 3.1364) for the 99% confidence interval."
] | [
"A manufacturer collects a sample of 9 parts and measures their weight in kilograms. The average weight in the sample is found to be 2.75 kilograms. Assume the population standard deviation is 0.45 kilograms. Assume the distribution of weights is normal.\na) Construct a 95% confidence interval for the population me... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-a70bdef7442e8226f18f32cc10639677b13e494568399eb8e7d1eeab939c2f2e | A manufacturer has 99% of its products fine. A test gives a 90% probability that a product is fine when it is indeed fine, but also gives a 13% probability that a product is fine when it is not fine. If the test says a randomly chosen product is not fine, what is the probability that the product is actually not fine? | [
"\\( 0.0808 \\)"
] | [
"A manufacturer has 99% of its products fine. A test gives a 90% probability that a product is fine when it is indeed fine, but also gives a 13% probability that a product is fine when it is not fine. If the test says a randomly chosen product is not fine, what is the probability that the product is actually not fi... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1f7d22fbfb95524900320eca5675b2245331ee1b7e27828c4deaa0888bb75bae | A manufacturer of HDTVs must ship at least 100 TVs to two West Coast warehouses. Each warehouse can hold a maximum of 100 TVs. The northern warehouse already has 25 TVs, and the southern warehouse has 20 TVs. It costs $12 to ship a TV to the northern warehouse and $10 to ship a TV to the southern warehouse. Shipping a ... | [
"50 TVs to each warehouse, minimum cost is $1100."
] | [
"A manufacturer of HDTVs must ship at least 100 TVs to two West Coast warehouses. Each warehouse can hold a maximum of 100 TVs. The northern warehouse already has 25 TVs, and the southern warehouse has 20 TVs. It costs $12 to ship a TV to the northern warehouse and $10 to ship a TV to the southern warehouse. Shippi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 28
} | 30 | 0 | 0 |
omr-ws-a19da2d2f3386032c5a2b726bc5f8b43122798bc5260e5aa068660d62d3ac771 | A manufacturer of cotton pins knows that 5% of the product he is manufacturing is defective. If he sells the pins in boxes of 100 and guarantees that no more than 4 pins will be defective, what is the approximate probability that the box will fail to meet the guaranteed quality? | [
"0.5595"
] | [
"A manufacturer of cotton pins knows that 5% of the product he is manufacturing is defective. If he sells the pins in boxes of 100 and guarantees that no more than 4 pins will be defective, what is the approximate probability that the box will fail to meet the guaranteed quality?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c28a6d0a7b251a959890f9ff022a42bac67fa5495c39ca9950801af2f1f16ca1 | A manufacturing company has 100 square feet of tin available to make cylindrical cans. The cans can have any radius and height, but the total surface area of all cans must be less than or equal to 100 square feet. What is the optimal number of cans and their dimensions to maximize the volume of food stored? | [
"Optimal number of cans \\( k = 1 \\), radius \\( r = \\sqrt{\\frac{50}{3\\pi}} \\), height \\( h = \\sqrt{\\frac{200}{3\\pi}} \\)"
] | [
"A manufacturing company has 100 square feet of tin available to make cylindrical cans. The cans can have any radius and height, but the total surface area of all cans must be less than or equal to 100 square feet. What is the optimal number of cans and their dimensions to maximize the volume of food stored?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 31
} | 40 | 0 | 0 |
omr-ws-cd5935423e684b2df503accad67e769f9bf8ace8228349ba37d9d894cbcf0a5c | A map \( T \) takes a point \( (x, y) \) in the first quadrant and transforms it into \( (\sqrt{x}, \sqrt{y}) \). Given a quadrilateral \( Q \) with vertices \( (900, 300) \), \( (1800, 600) \), \( (300, 900) \), and \( (600, 1800) \), find the greatest integer not exceeding the area of \( T(Q) \). | [
"600"
] | [
"A map \\( T \\) takes a point \\( (x, y) \\) in the first quadrant and transforms it into \\( (\\sqrt{x}, \\sqrt{y}) \\). Given a quadrilateral \\( Q \\) with vertices \\( (900, 300) \\), \\( (1800, 600) \\), \\( (300, 900) \\), and \\( (600, 1800) \\), find the greatest integer not exceeding the area of \\( T(Q) ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-186108a605095efeef665904812296d22f04521d2e1421fc76bb9f4c187071fb | A map \( T \) takes a point \((x, y)\) in the first quadrant to the point \((\sqrt{x}, \sqrt{y})\). \( Q \) is the quadrilateral with vertices \((900, 300)\), \((1800, 600)\), \((600, 1800)\), \((300, 900)\). Find the greatest integer not exceeding the area of \( T(Q) \). | [
"300"
] | [
"A map \\( T \\) takes a point \\((x, y)\\) in the first quadrant to the point \\((\\sqrt{x}, \\sqrt{y})\\). \\( Q \\) is the quadrilateral with vertices \\((900, 300)\\), \\((1800, 600)\\), \\((600, 1800)\\), \\((300, 900)\\). Find the greatest integer not exceeding the area of \\( T(Q) \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-b6c0ac7e54327001f6c819e9eb69f5771ad99df1b230d879e2f0146504e8d21d | A map is drawn using a scale of 1 cm to 8 km. a) Find the representative fraction (R.F.) of the map. b) Find the distance on the map, in cm, between two places that are 72 km apart. c) Find the area on the map, in cm², of a forest that has an area of 496 km². | [
"7.75"
] | [
"A map is drawn using a scale of 1 cm to 8 km. \na) Find the representative fraction (R.F.) of the map.\nb) Find the distance on the map, in cm, between two places that are 72 km apart.\nc) Find the area on the map, in cm², of a forest that has an area of 496 km²."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1322982e658f872c726d0457efd24c9583df092e823420674c21a9ec96f57537 | A map of the town that Annie, Barbara, and Charlie live in can be represented by the Cartesian plane. Annie is located at $(6,-20)$ and Barbara is located at $(1, 14)$. They agree to meet at the closest point that is equidistant from their current locations and walk upwards together to get to Charlie's location at $\le... | [
"5"
] | [
"A map of the town that Annie, Barbara, and Charlie live in can be represented by the Cartesian plane. Annie is located at $(6,-20)$ and Barbara is located at $(1, 14)$. They agree to meet at the closest point that is equidistant from their current locations and walk upwards together to get to Charlie's location at... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bf56bffc7b3e581624b32355d93c058a29495f694cdcfa6c03001074a023ef46 | A marathon is $26$ miles and $385$ yards. One mile equals $1760$ yards. Leila has run ten marathons in her life. If the total distance Leila covered in these marathons is $m$ miles and $y$ yards, where $0\le y<1760$, what is the value of $y$? | [
"330"
] | [
"A marathon is $26$ miles and $385$ yards. One mile equals $1760$ yards.\n\nLeila has run ten marathons in her life. If the total distance Leila covered in these marathons is $m$ miles and $y$ yards, where $0\\le y<1760$, what is the value of $y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e03adc23591040868cd1a4c7323c54fb63611111bf0d5d1aa2626d107731891c | A marathon race followed by four SMUs: Peacock, Pigeon, Sparrow, and Swallow. Each high school sent five runners. Runners who enter the 1st, 2nd, 3rd, 4th, 5th, 6th finishes obtain consecutive values of 7, 5, 4, 3, 2, 1. The value of each SMU is the sum of the values of the fifth runner. The SMU with the highest value ... | [
"15"
] | [
"A marathon race followed by four SMUs: Peacock, Pigeon, Sparrow, and Swallow. Each high school sent five runners. Runners who enter the 1st, 2nd, 3rd, 4th, 5th, 6th finishes obtain consecutive values of 7, 5, 4, 3, 2, 1. The value of each SMU is the sum of the values of the fifth runner. The SMU with the highest v... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-fd063761e994905207c2f13208ae7b3080d36aa604ed14541fc567fbb2bc320b | A marathon runner has a peculiar way of training for a marathon. On the first day of week \( 1 \), the runner runs a distance equivalent to the first prime number. On the second day, the runner runs a distance equal to the second prime number, continuing this pattern until the \( 7 \)th day of the week. Each successive... | [
"19"
] | [
"A marathon runner has a peculiar way of training for a marathon. On the first day of week \\( 1 \\), the runner runs a distance equivalent to the first prime number. On the second day, the runner runs a distance equal to the second prime number, continuing this pattern until the \\( 7 \\)th day of the week. Each s... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c2f6289099ebfa58986e41cddd4ed19ee04f29e5a9b58da637492e0096eb05f | A marble is placed on each of the 33 unit squares of a $10 \times 10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board? | [
"438"
] | [
"A marble is placed on each of the 33 unit squares of a $10 \\times 10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 45
} | 45 | 0 | 0 |
omr-ws-34a57d6bfa905d870e17ccfb7807141e0095382c416b9b72f377705a92d61e2a | A marble is taken from Box I, which contains 5 red marbles and 6 black marbles, and placed into Box II, which initially contains 10 red marbles and 5 black marbles. Afterward, two marbles are taken from Box II without replacement. Calculate the expected number of red marbles obtained from Box II. | [
"\\dfrac{115}{88}"
] | [
"A marble is taken from Box I, which contains 5 red marbles and 6 black marbles, and placed into Box II, which initially contains 10 red marbles and 5 black marbles. Afterward, two marbles are taken from Box II without replacement. Calculate the expected number of red marbles obtained from Box II."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5dd37c4115d5042797541fc8a142543853121f4487895f09957c1776013c982 | A mark is placed on the perimeter of a wheel where it touches the road. The wheel is rolled along the road without skidding through less than half a rotation. The mark has advanced horizontally from its starting position by \( x \). Given \( x \) and the radius of the wheel \( r \), what is the height \( y \) of the ma... | [
"r \\left(1 - \\cos\\left( \\frac{x}{r} \\right) \\right)"
] | [
"A mark is placed on the perimeter of a wheel where it touches the road. The wheel is rolled along the road without skidding through less than half a rotation. The mark has advanced horizontally from its starting position by \\( x \\). Given \\( x \\) and the radius of the wheel \\( r \\), what is the height \\( y ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-65722f06d4b5aadc572303d3d193afe4ad3b9b679bfe5c8d8cfd24b01e7fc4b7 | A marker is placed at point $A$ on a circle with points $A$, $B$, and $C$ arranged clockwise. On each move, a six-sided die is thrown: - If the die shows $1$ or $2$, the marker stays at its current position. - If the die shows $3$ or $4$, the marker moves clockwise. - If the die shows $5$ or $6$, the marker moves count... | [
"\\(\\frac{1}{3}\\)"
] | [
"A marker is placed at point $A$ on a circle with points $A$, $B$, and $C$ arranged clockwise. On each move, a six-sided die is thrown:\n- If the die shows $1$ or $2$, the marker stays at its current position.\n- If the die shows $3$ or $4$, the marker moves clockwise.\n- If the die shows $5$ or $6$, the marker mov... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ce514a03495e051f8d82875b30ff46f9a04d6b1704ad80a91d85d5a46068a677 | A marker is placed on each of the 4 vertices of a shape. Each marker is red on one side and black on the reverse side, with all red sides facing up initially. On each move, you swap any two markers and flip them both over. The goal is to return each marker to its starting vertex with all markers black side up. Can this... | [
"6"
] | [
"A marker is placed on each of the 4 vertices of a shape. Each marker is red on one side and black on the reverse side, with all red sides facing up initially. On each move, you swap any two markers and flip them both over. The goal is to return each marker to its starting vertex with all markers black side up. Can... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-d29246e9d45974e1f26e540dac9357b94a124dee0458cfde839da4d4af9ae246 | A married couple has $50,000$ dollars in their joint account. In anticipation of their upcoming anniversary, they agreed to split-up evenly and buy each one a gift. However, they agreed that they must have enough money left such that when combined, a minimum bank balance of $5,000$ dollars is maintained. If the couple ... | [
"\\dfrac{49}{50}"
] | [
"A married couple has $50,000$ dollars in their joint account. In anticipation of their upcoming anniversary, they agreed to split-up evenly and buy each one a gift. However, they agreed that they must have enough money left such that when combined, a minimum bank balance of $5,000$ dollars is maintained. If the co... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4aa29e0b1ce9b075ce75f608367fb99ffc7394287a29d57b4114d96e6f1dd43f | A married couple has PhP 50,000 in their joint account. They agreed to split the money evenly to buy gifts for each other, with the condition that they must maintain a minimum combined bank balance of PhP 5,000 after purchasing the gifts. If each gift is randomly priced between PhP 0 and PhP 25,000, what is the probabi... | [
"\\(\\frac{49}{50}\\)"
] | [
"A married couple has PhP 50,000 in their joint account. They agreed to split the money evenly to buy gifts for each other, with the condition that they must maintain a minimum combined bank balance of PhP 5,000 after purchasing the gifts. If each gift is randomly priced between PhP 0 and PhP 25,000, what is the pr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5bde47bdc678e17b226a8fb79420e61c543578f88abc081a76364ca9a3426f56 | A mason has bricks with dimensions $2 \times 5 \times 8$ and other bricks with dimensions $2 \times 3 \times 7$. She also has a box with dimensions $10 \times 11 \times 14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bric... | [
"24"
] | [
"A mason has bricks with dimensions $2 \\times 5 \\times 8$ and other bricks with dimensions $2 \\times 3 \\times 7$. She also has a box with dimensions $10 \\times 11 \\times 14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-9d18f59590ead76ee80943bce66a24838f3a38a8f2895e9f73f8ecca684f791b | A math class consists of five pairs of best friends who need to form five groups of two for an upcoming project. However, to promote synergy in the classroom, the teacher forbids any student to work with his best friend. In how many ways can the students pair up? | [
"544"
] | [
"A math class consists of five pairs of best friends who need to form five groups of two for an upcoming project. However, to promote synergy in the classroom, the teacher forbids any student to work with his best friend. In how many ways can the students pair up?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-bcfb3fc1be8b2bb400e6dd72d78617d6a5c928188e53b2ebab11a8e20974ea4c | A math class has fewer than $40$ students. When the students try to sit in rows of $7$, $3$ students sit in the last row. When the students try to sit in rows of $5$, $1$ student is left in the last row. How many students are in this class? | [
"31"
] | [
"A math class has fewer than $40$ students. When the students try to sit in rows of $7$, $3$ students sit in the last row. When the students try to sit in rows of $5$, $1$ student is left in the last row. How many students are in this class?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31bf5360d41e26afa98ef26bfe03bd8305dd8c2ca536f534698fd01fb7a10756 | A math conference is presenting a lecture series with six different lecturers. If Dr. Smith's lecture depends on Dr. Jones's lecture, so that Dr. Smith must be scheduled at some time after Dr. Jones, in how many orders can the six lecturers be scheduled? | [
"360"
] | [
"A math conference is presenting a lecture series with six different lecturers. If Dr. Smith's lecture depends on Dr. Jones's lecture, so that Dr. Smith must be scheduled at some time after Dr. Jones, in how many orders can the six lecturers be scheduled?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-587216e7737f7b85464864a214ec4c278ee656940f21b83abc72e2be30cc1d7d | A math contest consists of 9 objective type questions and 6 fill in the blanks questions. All students attempted exactly 14 out of 15 questions. Let \( O_1, O_2, \dots, O_9 \) be the nine objective questions and \( F_1, F_2, \dots, F_6 \) be the six fill in the blanks questions. Let \( a_{ij} \) be the number of studen... | [
"21"
] | [
"A math contest consists of 9 objective type questions and 6 fill in the blanks questions. All students attempted exactly 14 out of 15 questions. Let \\( O_1, O_2, \\dots, O_9 \\) be the nine objective questions and \\( F_1, F_2, \\dots, F_6 \\) be the six fill in the blanks questions. Let \\( a_{ij} \\) be the num... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
omr-ws-98419537a98d59929d7226f1ec686bb5fb027d383614fa5fd2da2aaf13758a2c | A math exam consists of $n$ problems. For each problem, the student's answers are either right or wrong, with no partial credits. We are told that for every 2 distinct problems $a$ and $b$, there is at least one student who got $a$ right and $b$ wrong, and one student who got $a$ wrong but $b$ right. What is the minimu... | [
"n"
] | [
"A math exam consists of $n$ problems. For each problem, the student's answers are either right or wrong, with no partial credits. We are told that for every 2 distinct problems $a$ and $b$, there is at least one student who got $a$ right and $b$ wrong, and one student who got $a$ wrong but $b$ right. What is the m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-c5c383bb4d48efe479ee776d7cb5de98c9648463bdc2f95155ff9cb45b16e253 | A math field day competition is held in a room with many tables, and there are 6 stools at each table. Each stool has 3 legs, and each table has 4 legs. If there is a total of 484 legs on all the tables and stools in the room, how many tables are in the room? | [
"22"
] | [
"A math field day competition is held in a room with many tables, and there are 6 stools at each table. Each stool has 3 legs, and each table has 4 legs. If there is a total of 484 legs on all the tables and stools in the room, how many tables are in the room?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0ebd3a4933d18734221eecc3fa9f28ecd6a5ad8d5f49118d6aff7ddd243fe7f | A math professor has a pack of nuts to distribute among 11 math, 13 biology, and 17 political science students. When distributing the nuts equally among all students, some nuts are left over. Similarly, when distributing among only two groups (math/biology, math/political science, or biology/political science), some nu... | [
"1607"
] | [
"A math professor has a pack of nuts to distribute among 11 math, 13 biology, and 17 political science students. When distributing the nuts equally among all students, some nuts are left over. Similarly, when distributing among only two groups (math/biology, math/political science, or biology/political science), so... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a17ee3a504d321ccc6ad807a9b1846956f91385ef474aeb0f5c0cece2230b57 | A math professor wants to distribute nuts among 11 math, 13 biology, and 17 political science students such that each student receives the same maximal number of nuts, with some nuts left over. When distributing among only two groups (math/biology, math/political science, or biology/political science), there are also n... | [
"2179"
] | [
"A math professor wants to distribute nuts among 11 math, 13 biology, and 17 political science students such that each student receives the same maximal number of nuts, with some nuts left over. When distributing among only two groups (math/biology, math/political science, or biology/political science), there are a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-59d50b172b3f639270357bd828afaec4eba1db4deb90ed7727e2de2063b41d32 | A math professor will select three out of 8 theorems to put on an exam. What is the least number of theorems the student should learn in order to be certain of being able to prove a theorem on the exam? | [
"6"
] | [
"A math professor will select three out of 8 theorems to put on an exam. What is the least number of theorems the student should learn in order to be certain of being able to prove a theorem on the exam?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1a2117c77b0850ff79c6c5f25e4dbfceb2388777192d3a269a54e071ccb31ab | A math teacher has a bag containing \( n > 3 \) identically sized and shaped weights weighing 1 kilogram to \( n \) kilograms. You are to weigh them one at a time on a digital scale. If the successive weighings of any two weights differed by one kilogram, the teacher will pay you a dollar. (a) What is the minimum and m... | [
"Yes"
] | [
"A math teacher has a bag containing \\( n > 3 \\) identically sized and shaped weights weighing 1 kilogram to \\( n \\) kilograms. You are to weigh them one at a time on a digital scale. If the successive weighings of any two weights differed by one kilogram, the teacher will pay you a dollar.\n(a) What is the min... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75b4c2beaa5f1158561efaef537940de9c593c68156d78b4f8af5a043aa12c79 | A math teacher requires Noelle to do one homework assignment for each of the first five homework points she wants to earn; for each of the next five homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\text{th}}$ homework point, she has to do $n\div5$ (rounded up) homework assi... | [
"75"
] | [
"A math teacher requires Noelle to do one homework assignment for each of the first five homework points she wants to earn; for each of the next five homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\\text{th}}$ homework point, she has to do $n\\div5$ (rounded up) homewo... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d33056ab8892df3bdd4c55dee22f22b057ab487f6d96657117732484fe165b7d | A math tournament has a test which contains 10 questions, each of which comes from one of three different subjects. The subject of each question is chosen uniformly at random from the three subjects, and independently of the subjects of all the other questions. The test is unfair if any one subject appears at least 5 t... | [
"\\dfrac{4111}{6561}"
] | [
"A math tournament has a test which contains 10 questions, each of which comes from one of three different subjects. The subject of each question is chosen uniformly at random from the three subjects, and independently of the subjects of all the other questions. The test is unfair if any one subject appears at leas... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48d5b02b9b6a6ecddc1c46022a0a40a0a8af8d55be3129e35e16da5d28153d21 | A mathematical competition is attended by $120$ participants from several teams. At the closing ceremony, each participant gives a souvenir to every other participant from the same team, and also one souvenir to one participant from every other team. Given that there were $3840$ souvenirs that changed hands, what is th... | [
"30"
] | [
"A mathematical competition is attended by $120$ participants from several teams. At the closing ceremony, each participant gives a souvenir to every other participant from the same team, and also one souvenir to one participant from every other team. Given that there were $3840$ souvenirs that changed hands, what ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 9
} | 56 | 0 | 0 |
omr-ws-0b21e706f71bb5928cd8848b797903d632a5f2c12cf34337959e930b1aaff67d | A mathematical competition is attended by 120 participants from several teams. At the closing ceremony, each participant gives a souvenir to every other participant from the same team, and also one souvenir to one participant from every other team. Given that there were 3840 souvenirs that changed hands, what is the ma... | [
"30"
] | [
"A mathematical competition is attended by 120 participants from several teams. At the closing ceremony, each participant gives a souvenir to every other participant from the same team, and also one souvenir to one participant from every other team. Given that there were 3840 souvenirs that changed hands, what is t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 14
} | 60 | 0 | 0 |
omr-ws-ffc9902d1095d9051ca16895f2ebc6b331a69d622453b12e0574ac38a47598c3 | A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer \( n \), then it can jump either to \( n+1 \) or to \( n + 2^{m_n+1} \) where \( 2^{m_n+1} \) is the largest power of 2 that is a factor of \( n \). Let \( J(n) \) denote the minim... | [
"7"
] | [
"A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer \\( n \\), then it can jump either to \\( n+1 \\) or to \\( n + 2^{m_n+1} \\) where \\( 2^{m_n+1} \\) is the largest power of 2 that is a factor of \\( n \\). Let \\( J(n) \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9a50bd56a0ef99072533cbcadd293cc365af80c86823f5a951c4e8133ba6662 | A mathematical moron is given the values \( b \), \( c \), and \( A \) for a triangle \( ABC \) and is required to find \( a \). He uses the cosine rule \( a^2 = b^2 + c^2 - 2bc \cos A \) but misapplies the logarithm to get \( \log a^2 = \log b^2 + \log c^2 - \log(2bc \cos A) \). Despite this mistake, he evaluates the ... | [
"\\( b = a \\) or \\( c = a \\) or \\( a = b = c \\)"
] | [
"A mathematical moron is given the values \\( b \\), \\( c \\), and \\( A \\) for a triangle \\( ABC \\) and is required to find \\( a \\). He uses the cosine rule \\( a^2 = b^2 + c^2 - 2bc \\cos A \\) but misapplies the logarithm to get \\( \\log a^2 = \\log b^2 + \\log c^2 - \\log(2bc \\cos A) \\). Despite this m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 7,
"tir": 14
} | 63 | 0 | 0 |
omr-ws-015183a02b22bb5ba1732ec1328c1a3634ebcf5604b8629f4130c60e0f5808ce | A mathematical moron is given the values \( b, c, A \) for a triangle \( ABC \) and is required to find \( a \). He uses the cosine rule \( a^2 = b^2 + c^2 - 2bc \cos(A) \) but misapplies the logarithm to get \( \log a^2 = \log b^2 + \log c^2 + \log(2bc \cos A) \). Despite this mistake, he evaluates the right-hand side... | [
"\\dfrac{13}{33240}"
] | [
"A mathematical moron is given the values \\( b, c, A \\) for a triangle \\( ABC \\) and is required to find \\( a \\). He uses the cosine rule \\( a^2 = b^2 + c^2 - 2bc \\cos(A) \\) but misapplies the logarithm to get \\( \\log a^2 = \\log b^2 + \\log c^2 + \\log(2bc \\cos A) \\). Despite this mistake, he evaluate... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4828cc8b1446f2806efd998c471d870b66ca114bf84b79b87a4cf22d570c5e70 | A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in $2007$. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in $200... | [
"372"
] | [
"A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in $2007$. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-0a57f159914c22d03cafe5bc04ce621276ba3373250e1868bb8c8b82b5ffef10 | A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. ... | [
"372"
] | [
"A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5af477a503d4e3fa9eeb349cbbe7503ecf905bed1d1336155c80b2bc6f40af3a | A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. ... | [
"372"
] | [
"A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 15
} | 36 | 0 | 0 |
omr-ws-b0b84270497a847302403b55c90723c86f7689ddd6c7e455d01d53fb126652da | A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. ... | [
"372"
] | [
"A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-c225306288a60a91b2cad14ca8026d833edb4fbad1d5269bc975a02ceff6c1c1 | A mathematician tells his friend that the product of his grandchildren's ages is $36$ and the sum of their ages is equal to the number of floors in a building. The friend cannot determine the ages. The mathematician then adds that his older grandson is named Sandro. What are the ages of the grandchildren? | [
"2, 2, 9"
] | [
"A mathematician tells his friend that the product of his grandchildren's ages is $36$ and the sum of their ages is equal to the number of floors in a building. The friend cannot determine the ages. The mathematician then adds that his older grandson is named Sandro. What are the ages of the grandchildren?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-bf6ad57b53af94875351ea217ef7447ea34574f586d0ae10cf239a383ee147dc | A mathematician used to boast that he was $x$ years old in the year $x^2$. Another mathematician claimed he was $a^2 + b^2$ years old in the year $a^4 + b^4$, $2m$ years old in the year $2m^2$, and $3n$ years old in the year $3n^4$. Determine the birth years of both mathematicians and find the integers $a, b, m, n$. | [
"5"
] | [
"A mathematician used to boast that he was $x$ years old in the year $x^2$. Another mathematician claimed he was $a^2 + b^2$ years old in the year $a^4 + b^4$, $2m$ years old in the year $2m^2$, and $3n$ years old in the year $3n^4$. Determine the birth years of both mathematicians and find the integers $a, b, m, n... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-a7b33bf769d246ead424f104c6098746dc434b59a700ebe9e1031137e6ae154e | A mathematics teacher reveals the number of positive divisors of a 2-digit natural number \( n \) to student S and the sum of the digits of \( n \) to student P. The following conversation occurs: - P: I cannot determine \( n \). - S: I can't either, but I know whether \( n \) is even or not. - P: Now I know what \( n ... | [
"56"
] | [
"A mathematics teacher reveals the number of positive divisors of a 2-digit natural number \\( n \\) to student S and the sum of the digits of \\( n \\) to student P. The following conversation occurs:\n- P: I cannot determine \\( n \\).\n- S: I can't either, but I know whether \\( n \\) is even or not.\n- P: Now I... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-69977ae22ca4d345aa05629b8bd9546f84d6e90ebbf9d06e21f7515283bb2d2c | A matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is said to be singular if $ad - bc = 0$. If the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is created at random by choosing integer values $a, b, c, d$ at random from the interval $[-3, 3]$, what is the probability that the matrix will be singular? | [
"\\dfrac{289}{2401}"
] | [
"A matrix $\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ is said to be singular if $ad - bc = 0$. If the matrix $\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ is created at random by choosing integer values $a, b, c, d$ at random from the interval $[-3, 3]$, what is the probability that the matrix will be s... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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