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-a372103b4eea01aabcb797a6a6fb95407c6a6008c88b4d1363a8d67d6466cfc5 | A bunny is 80 feet away from its rabbit hole and can only hop either 5 feet or 9 feet at a time. What is the least number of hops it needs to get back to the rabbit hole? | [
"12"
] | [
"A bunny is 80 feet away from its rabbit hole and can only hop either 5 feet or 9 feet at a time. What is the least number of hops it needs to get back to the rabbit hole?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-00075adfaab5fdacc42fbf258d1f7c0153d84c3754b86d2b3c54afb64edae161 | A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler? | [
"75\\%"
] | [
"A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f14f64b7da74cbfdbcba9a5baa02b58bd3e172d48219c013ecda2bf61a3b9e9 | A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cyclist... | [
"10:50 a.m."
] | [
"A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cy... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 7
} | 27 | 0 | 0 |
omr-ws-7df80784f8a81f0a948321544e28148cbdb3ed645b7ec9f836fcec90c26f450c | A bus arrives at a bus stop every 15 minutes starting at 7am. If a passenger arrives at the bus stop at a random time between 7 and 7:30, what is the probability that the passenger will have to wait at least 12 minutes? | [
"\\(\\frac{1}{5}\\)"
] | [
"A bus arrives at a bus stop every 15 minutes starting at 7am. If a passenger arrives at the bus stop at a random time between 7 and 7:30, what is the probability that the passenger will have to wait at least 12 minutes?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-1e9216930e0f7849346b45f50fd4cf6ec948cdf9d65589e14074162a473c94ed | A bus arrives at a bus stop every 15 minutes starting at 7am. If a passenger arrives at the bus stop at a random time between 7 and 7:30, what is the probability that the passenger will have to wait less than 10 minutes? | [
"\\dfrac{2}{3}"
] | [
"A bus arrives at a bus stop every 15 minutes starting at 7am. If a passenger arrives at the bus stop at a random time between 7 and 7:30, what is the probability that the passenger will have to wait less than 10 minutes?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-916f48f1f78e2c04db334f47b948061906aac85c7ea3c50039e1dcb09022391d | A bus can carry $n$ customers. Each customer has their own ticket with a seat number. The first $n-1$ customers sit in any seat, not necessarily their correct one. The $n$th customer arrives last and wants to sit in their correct seat. If their seat is occupied, they ask the person sitting there to move to their correc... | [
"\\dfrac{n-1}{2}"
] | [
"A bus can carry $n$ customers. Each customer has their own ticket with a seat number. The first $n-1$ customers sit in any seat, not necessarily their correct one. The $n$th customer arrives last and wants to sit in their correct seat. If their seat is occupied, they ask the person sitting there to move to their c... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-93f1d0f4432ec6b0fec9879a9fe5d52ef64e7defb240e87bd68e53fc7def10a7 | A bus can hold a maximum of 38 students. What is the minimum number of buses needed to transport 411 students? | [
"11"
] | [
"A bus can hold a maximum of 38 students. What is the minimum number of buses needed to transport 411 students?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff87de386fecb9a1b428582204698103baac37ce9a20723cbefc979e0cd58571 | A bus goes from Stop 1 to Stop 14. Passengers can hop on board at any stop and leave at a later stop, with the bus being empty after Stop 14. What is the maximum number of distinct tickets bought among all the passengers? (A ticket is defined by the stop a passenger boards and the stop a passenger exits.) | [
"67"
] | [
"A bus goes from Stop 1 to Stop 14. Passengers can hop on board at any stop and leave at a later stop, with the bus being empty after Stop 14. What is the maximum number of distinct tickets bought among all the passengers? (A ticket is defined by the stop a passenger boards and the stop a passenger exits.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f58e16cb5e3bbff87cc5c9f160ab0a39aa467b08c94ebeb4712c6a96e0c294b | A bus has 13 seats for passengers. Eight passengers boarded the bus before it left the terminal. Determine the number of possible seating arrangements of the passengers who boarded the bus at the terminal. | [
"51891840"
] | [
"A bus has 13 seats for passengers. Eight passengers boarded the bus before it left the terminal. Determine the number of possible seating arrangements of the passengers who boarded the bus at the terminal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d3536b33585523a758c8878be65cd1173e7baa1b2ed326cacb22cab7d3bcd65 | A bus has a maximum of 50 seats. Students sit at tables in the following configurations: some tables have 3 girls and 1 boy, some tables have 2 boys and 1 girl, and one table has 1 girl and 1 boy. If the total number of girls is 1.5 times the total number of boys, find the maximum number of students. | [
"45"
] | [
"A bus has a maximum of 50 seats. Students sit at tables in the following configurations: some tables have 3 girls and 1 boy, some tables have 2 boys and 1 girl, and one table has 1 girl and 1 boy. If the total number of girls is 1.5 times the total number of boys, find the maximum number of students."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-eca7159bfce032229bb86403c01fa831f52eac61c44182c46f5dbe0815f6dc7c | A bus is to be charted for a trip with a basic fare of $9.50 per passenger. If more than 20 people go, the fare is reduced by $0.30 for every passenger over 20. At least how many people must go to make the fare less than $7.50 per passenger? | [
"\\( x \\geq 27 \\)"
] | [
"A bus is to be charted for a trip with a basic fare of $9.50 per passenger. If more than 20 people go, the fare is reduced by $0.30 for every passenger over 20. At least how many people must go to make the fare less than $7.50 per passenger?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-61570aad4bcb69a5230e01e52d8d8d3fbfa4d0ee014bc2e3cfa15c2b066ec170 | A bus is traveling a distance of $300 \, \text{km}$ at a constant speed $v \, \text{km/h}$. The price of fuel is $\$120$ per gallon, and the fuel consumption is $\frac{v^2}{100} - 4$ gallons per hour. The driver is paid $\$8670$ for every $4$ hours. What is the minimum speed $v$ between $30$ and $100$ to minimize the t... | [
"30"
] | [
"A bus is traveling a distance of $300 \\, \\text{km}$ at a constant speed $v \\, \\text{km/h}$. The price of fuel is $\\$120$ per gallon, and the fuel consumption is $\\frac{v^2}{100} - 4$ gallons per hour. The driver is paid $\\$8670$ for every $4$ hours. What is the minimum speed $v$ between $30$ and $100$ to mi... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91e42b76329abacd2215bcae974ba3f880d0290516c1cd2b868ce56b6c1f7cef | A bus left point X for point Y. Two hours later, a car left point X for Y and arrived at Y at the same time as the bus. If the car and the bus left simultaneously from opposite ends X and Y, they would meet in 1.33 hours after the start. How much time did it take the bus to travel from X to Y? | [
"4"
] | [
"A bus left point X for point Y. Two hours later, a car left point X for Y and arrived at Y at the same time as the bus. If the car and the bus left simultaneously from opposite ends X and Y, they would meet in 1.33 hours after the start. How much time did it take the bus to travel from X to Y?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-b15baf863c631b4f50b918e3aba6e31fbfceeb9caf74e064e4d7fd10d847f7d3 | A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road. Bus No. 1 travels along the straight road, once around the circle clockwise, and returns along the straight road to the terminus. Bus No. 2, reaching t... | [
"\\dfrac{100}{3}"
] | [
"A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road. Bus No. 1 travels along the straight road, once around the circle clockwise, and returns along the straight road to the terminus. Bus No. 2, reach... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-cf26c936049d39f0df9970b42090a2be467adefa917c5164e7917cd63409a183 | A bus route from town \( A \) to town \( D \) goes along a straight line and passes first through town \( B \) and then through town \( C \). Ann has a positioning device that can tell her the distance from her current location to any of the four towns on the route. When she is somewhere between \( A \) and \( B \), fr... | [
"2.7"
] | [
"A bus route from town \\( A \\) to town \\( D \\) goes along a straight line and passes first through town \\( B \\) and then through town \\( C \\). Ann has a positioning device that can tell her the distance from her current location to any of the four towns on the route. When she is somewhere between \\( A \\) ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-56a14489a353d4977ee266fe5c6ab75dfe790fbdced01a0bb8014f8e78c4b5c5 | A bus starts from point $A$ to point $B$. Having reached $B$, it travels in the same direction. At the moment the bus reaches point $B$ a car starts from point $A$ and runs in the same direction as the bus. It takes the car 3 h 20 min less to cover the distance from $A$ to $B$ than the bus. Find those times if their su... | [
"20/3 h and 10/3 h"
] | [
"A bus starts from point $A$ to point $B$. Having reached $B$, it travels in the same direction. At the moment the bus reaches point $B$ a car starts from point $A$ and runs in the same direction as the bus. It takes the car 3 h 20 min less to cover the distance from $A$ to $B$ than the bus. Find those times if t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-547ee1c6b26a7d7a5ad5df558b4d65cd965ddac197b041ac4419b7f58ad918b3 | A bus starts off empty and makes over 13 stops. At each stop, it either picks up 4 people or picks up 2 people and lets off 3 people. What is the fewest number of stops the bus makes so that it has exactly 7 people on board at the end of its run? Do not count the beginning or end of the run as stops. | [
"18"
] | [
"A bus starts off empty and makes over 13 stops. At each stop, it either picks up 4 people or picks up 2 people and lets off 3 people. What is the fewest number of stops the bus makes so that it has exactly 7 people on board at the end of its run? Do not count the beginning or end of the run as stops."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cd2fc760bfa9a2b5ba293748a5b425a0c6635d35d88ec43c50492f144212667 | A bus takes 2 minutes to drive from one stop to the next and waits 1 minute at each stop. Zia takes 5 minutes to walk from one bus stop to the next. If Zia reaches a bus stop and the bus is at the previous stop or has already left, she will wait for the bus; otherwise, she will start walking toward the next stop. Suppo... | [
"17"
] | [
"A bus takes 2 minutes to drive from one stop to the next and waits 1 minute at each stop. Zia takes 5 minutes to walk from one bus stop to the next. If Zia reaches a bus stop and the bus is at the previous stop or has already left, she will wait for the bus; otherwise, she will start walking toward the next stop. ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 6,
"tir": 3
} | 51 | 0 | 0 |
omr-ws-b617568b64dd24acb5bf79bc3d6dc07cd27f982367512ba4a32d1b7869500087 | A bus ticket is numbered from 000000 to 999999. A ticket is called a "Happy Ticket" if the sum of its first three digits is the same as the sum of its last three digits. Determine the probability of purchasing a "Happy Ticket." | [
"\\(\\frac{1}{1000}\\)"
] | [
"A bus ticket is numbered from 000000 to 999999. A ticket is called a \"Happy Ticket\" if the sum of its first three digits is the same as the sum of its last three digits. Determine the probability of purchasing a \"Happy Ticket.\""
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e2c293f5826c6f63b7d51681ff52d13ca40156058711d2393a18a4f680f58a8 | A bus travels a 100 km route. Forty minutes after departure, the computer predicts that the remaining travel time will be 1 hour, and this prediction remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? | [
"45 kilometers"
] | [
"A bus travels a 100 km route. Forty minutes after departure, the computer predicts that the remaining travel time will be 1 hour, and this prediction remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56ba8887072dd3208c3e3df2ea6ab2aea852fc669566da412f2bafc5185ec6ac | A businessman has some money in his pocket. For the first two days, he spends \(\frac{1}{3}\) of his money plus 100 YTL. For the last two days, he spends \(\frac{1}{5}\) of his remaining money plus 100 YTL. After 4 days, he has 140 YTL left. How much money did he have at the start? | [
"1500"
] | [
"A businessman has some money in his pocket. For the first two days, he spends \\(\\frac{1}{3}\\) of his money plus 100 YTL. For the last two days, he spends \\(\\frac{1}{5}\\) of his remaining money plus 100 YTL. After 4 days, he has 140 YTL left. How much money did he have at the start?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-d0ad15d9ee389b4ba3a67c3006e96d1d5bf408bd4fc0d3a396adf6b5a55b811f | A businessman visits a farm where 25 people live and stays for exactly $N$ days. Each day, he offers a math problem to solve. At the end of the visit, he realizes that for any two questions $a_1$ and $b_1$ he asks, the set of people who answered $a_1$ correctly is different from the set of people who answered $b_1$ cor... | [
"8"
] | [
"A businessman visits a farm where 25 people live and stays for exactly $N$ days. Each day, he offers a math problem to solve. At the end of the visit, he realizes that for any two questions $a_1$ and $b_1$ he asks, the set of people who answered $a_1$ correctly is different from the set of people who answered $b_1... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-2437dd9b88bd0734b90baeb348645125e1c8ae9c9e3db486cfb11835b4dfea54 | A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data: - 85 percent of the cabs in the city are blue and 15 percent are Green. - A witness identified the cab as Green. - The court tested the reliability of the witness unde... | [
"\\dfrac{12}{29}"
] | [
"A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data:\n- 85 percent of the cabs in the city are blue and 15 percent are Green.\n- A witness identified the cab as Green.\n- The court tested the reliability of the witn... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41be2269712035c050f5069ab718bf7cb1e6d05fc47d7f25c4ebdc8276f7c334 | A cable 16 meters in length hangs between two pillars that are both 15 meters high. The ends of the cable are attached to the top of the pillars. At its lowest point, the cable hangs 7 meters above the ground. How far apart are the two pillars? | [
"\\( 10\\ln(5) \\)"
] | [
"A cable 16 meters in length hangs between two pillars that are both 15 meters high. The ends of the cable are attached to the top of the pillars. At its lowest point, the cable hangs 7 meters above the ground. How far apart are the two pillars?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb65e118972efbea5799d47f8817af9a9303ae84ea3e17fb5f970c49ccaba3a5 | A cable hangs in the shape of a catenary, \( y = a \cosh \frac{x}{a} \), where \( a \) is the height of the lowest point on the cable (at \( x = 0 \)). The cable is fastened to support heights of 16 feet and 28 feet. If the lowest point of the cable is 10 feet above the ground, what is the distance between the two supp... | [
"27.3621"
] | [
"A cable hangs in the shape of a catenary, \\( y = a \\cosh \\frac{x}{a} \\), where \\( a \\) is the height of the lowest point on the cable (at \\( x = 0 \\)). The cable is fastened to support heights of 16 feet and 28 feet. If the lowest point of the cable is 10 feet above the ground, what is the distance between... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-8ff7e017b8466290f1444d3c5cf3917a84c3d6fa3552eea09084389c8ad327a2 | A cable has a diameter of 3.5 mm. Each fiber inside the cable has a diameter of 70 micrometers. How many fibers can fit inside the cable? | [
"2267"
] | [
"A cable has a diameter of 3.5 mm. Each fiber inside the cable has a diameter of 70 micrometers. How many fibers can fit inside the cable?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-4ba9016b7fd99f80cfb7ef6b6c9e25392d8443b7b0a7f1c537f552a1608d71fe | A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Approximate the required work by a Riemann Sum with 5 subintervals and calculate the total work done in foot-pounds. | [
"650000"
] | [
"A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Approximate the required work by a Riemann Sum with 5 subintervals and calculate the total work done in foot-pounds."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-ba9f2645a243dee762701b47801f67e1e577e962c3cfe2eb2f049f1e6e2f324a | A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Express the work as an integral and evaluate it. | [
"650000 ft-lbs"
] | [
"A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. Express the work as an integral and evaluate it."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d11de8754e5d5b727a17dd744c2768e0ec683ab783a279b9d53c964e6bf896b7 | A cake has the shape of a triangle with sides 19, 20, and 21. It is allowed to cut the cake with a line into two pieces and place them on a round plate such that the pieces do not overlap and do not stick out of the plate. What is the minimal diameter of the plate? | [
"19"
] | [
"A cake has the shape of a triangle with sides 19, 20, and 21. It is allowed to cut the cake with a line into two pieces and place them on a round plate such that the pieces do not overlap and do not stick out of the plate. What is the minimal diameter of the plate?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-6f4ffa7c5d00bf4dab788375c42fd0193e8418b39558ed1a32cacfba883911dc | A cake is cut by 4 lines that intersect at a single point inside the cake, dividing it into 8 parts such that each angle at the intersection point measures 45 degrees. Let the areas of the numbered parts 1, 2, 3, and 4 be represented by $A_1, A_2, A_3,$ and $A_4$ respectively, where each numbered part consists of two e... | [
"100"
] | [
"A cake is cut by 4 lines that intersect at a single point inside the cake, dividing it into 8 parts such that each angle at the intersection point measures 45 degrees. Let the areas of the numbered parts 1, 2, 3, and 4 be represented by $A_1, A_2, A_3,$ and $A_4$ respectively, where each numbered part consists of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-574326c633a1c9c64fb4ff93cb9a0a0b382397a435e8b0c82aac6a758b5ffc46 | A cake is sliced roughly in half, with the largest piece being chosen each time and the other pieces discarded. The random cut creates pieces with proportions: - (2/3, 1/3) with probability 3/4 - (2/5, 3/5) with probability 1/4 Starting with a whole cake, how large, to first order in the exponent, is the piece of cake ... | [
"\\( V_n \\approx V_0 \\exp(n \\mu_X) \\) where \\( \\mu_X = \\frac{3}{4} \\ln\\left(\\frac{2}{3}\\right) + \\frac{1}{4} \\ln\\left(\\frac{3}{5}\\right) \\)"
] | [
"A cake is sliced roughly in half, with the largest piece being chosen each time and the other pieces discarded. The random cut creates pieces with proportions:\n- (2/3, 1/3) with probability 3/4\n- (2/5, 3/5) with probability 1/4\n\nStarting with a whole cake, how large, to first order in the exponent, is the piec... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-5fec693424b888680460c80f564e023d528668f1af3d6cf401a08970961ea5cb | A cake weighing one kilogram is cut into two pieces, and each piece is weighed separately, with the measured weights denoted by \( x \) and \( y \). Assume that the errors in obtaining \( x \) and \( y \) are independent and normally distributed with mean zero and the same (unknown) variance. Devise a test for the hypo... | [
"t = \\frac{x - y}{\\sqrt{(x - 0.5)^2 + (y - 0.5)^2}}"
] | [
"A cake weighing one kilogram is cut into two pieces, and each piece is weighed separately, with the measured weights denoted by \\( x \\) and \\( y \\). Assume that the errors in obtaining \\( x \\) and \\( y \\) are independent and normally distributed with mean zero and the same (unknown) variance. Devise a test... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-6a171f05d8ce9120de2ec54ff39e1ee565f10d4ec3e0c736b244c914b2afcbce | A cake, when placed in one pan of a scale, appears to weigh four ounces more than nine-elevenths of its true weight. When placed in the other pan, it appears to weigh three pounds more than in the first pan. What is the true weight of the cake? Assume the error is due to unequal lengths of the arms and unequal weights ... | [
"143"
] | [
"A cake, when placed in one pan of a scale, appears to weigh four ounces more than nine-elevenths of its true weight. When placed in the other pan, it appears to weigh three pounds more than in the first pan. What is the true weight of the cake? Assume the error is due to unequal lengths of the arms and unequal wei... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-f2a41d1555156221bc279b02d36a8164a212a421dd70ba3ce1295a5487661650 | A calculator displays a nonnegative integer \( N \) and has a button that replaces \( N \) with a random integer chosen uniformly from the set \(\{0, 1, \ldots, N-1\}\), provided \( N > 0 \). Initially, \( N = 2003 \). If the button is pressed repeatedly until \( N = 0 \), what is the probability that the numbers 1, 10... | [
"\\dfrac{1}{2224222}"
] | [
"A calculator displays a nonnegative integer \\( N \\) and has a button that replaces \\( N \\) with a random integer chosen uniformly from the set \\(\\{0, 1, \\ldots, N-1\\}\\), provided \\( N > 0 \\). Initially, \\( N = 2003 \\). If the button is pressed repeatedly until \\( N = 0 \\), what is the probability th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33b90abe360831d9ac1799064a3321bc7e85d8eb8a8d589ccf6989ff0507feca | A calculator has a display showing a nonnegative integer \( N \), and a button that replaces \( N \) by a random integer chosen from the set \(\{0, 1, \ldots, N-1\}\), provided that \( N > 0 \). Initially, the display holds the number \( N = 2003 \). If the button is pressed repeatedly until \( N = 0 \), what is the pr... | [
"\\(\\frac{1}{1001}\\)"
] | [
"A calculator has a display showing a nonnegative integer \\( N \\), and a button that replaces \\( N \\) by a random integer chosen from the set \\(\\{0, 1, \\ldots, N-1\\}\\), provided that \\( N > 0 \\). Initially, the display holds the number \\( N = 2003 \\). If the button is pressed repeatedly until \\( N = 0... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-449952228d637749ee57bf04da60d46d7175059509b90d06b6c89919892044d1 | A calculator is broken so that the only keys that work are sin(x), cos(x), tan(x), arcsin(x), arccos(x), and arctan(x). The display initially shows 0. Determine the minimum number of button presses required to display the rational number \( \frac{3}{5} \). | [
"5"
] | [
"A calculator is broken so that the only keys that work are sin(x), cos(x), tan(x), arcsin(x), arccos(x), and arctan(x). The display initially shows 0. Determine the minimum number of button presses required to display the rational number \\( \\frac{3}{5} \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b5de0193b906bf356f47f84cb6f512b3e9475c356e334b1e0f08462ed0b9d131 | A calculator treats angles as radians and initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? | [
"\\( 1 - \\frac{3}{4000} \\)"
] | [
"A calculator treats angles as radians and initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-b6ead4ad2a1438fa5a9e2749d41fcf21b7372a7b324a0adb2d6ca3ec22ceb930 | A calendar is a finite rectangular grid that is valid if it satisfies the following conditions: - Each square is colored white or red, with exactly 10 red squares. - If there are $N$ columns, there do not exist $N$ consecutive numbers such that the squares they are in are all white. - If there are $M$ rows, there do no... | [
"\\(10!\\)"
] | [
"A calendar is a finite rectangular grid that is valid if it satisfies the following conditions:\n- Each square is colored white or red, with exactly 10 red squares.\n- If there are $N$ columns, there do not exist $N$ consecutive numbers such that the squares they are in are all white.\n- If there are $M$ rows, the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-653dfa2c4ef184e8a3d6b2e77097c7de5e67ecbd29a059ed53d471b53e958203 | A camel has to cross a 1000-mile wide desert with 3000 bananas. The camel can carry a maximum of 1000 bananas at a time and eats 1 banana for every mile it travels. What is the maximum number of bananas that can be carried across the desert? | [
"533"
] | [
"A camel has to cross a 1000-mile wide desert with 3000 bananas. The camel can carry a maximum of 1000 bananas at a time and eats 1 banana for every mile it travels. What is the maximum number of bananas that can be carried across the desert?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-007f1e3c401dfc1e1881214fd4eac3018e0d30f26b08f49c437cfb7667e3ee0b | A camera tripod has three equally spaced legs, each of length 1.75m, around a circle of radius 52cm on horizontal ground. Find the angle that the legs make with the horizontal. | [
"\\(\\arccos \\left(\\frac{52}{175} \\right) \\approx 72.71^\\circ\\)"
] | [
"A camera tripod has three equally spaced legs, each of length 1.75m, around a circle of radius 52cm on horizontal ground. Find the angle that the legs make with the horizontal."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-87b7923243406724cfa509236ae83cfe6e5c7d6c9b403902fb765e5b75bb363c | A camper is lost in a rectangular forest that is $100$ km long and $1$ km wide. The camper has the energy to walk $2.83$ km. Determine the minimum distance $d$ that the camper must be able to walk in order to guarantee they can exit the forest from any starting point within the forest. | [
"2.83"
] | [
"A camper is lost in a rectangular forest that is $100$ km long and $1$ km wide. The camper has the energy to walk $2.83$ km. Determine the minimum distance $d$ that the camper must be able to walk in order to guarantee they can exit the forest from any starting point within the forest."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-5266bb35483baf93b8ad496f94a30afb06cd245d3d14dbb511abe7445b872f21 | A campus only gives out parking permits to people greater than 1 mile away from the campus. Oak Lane starts 2 miles west and 1 mile north of the campus, and ends 5 miles east from the campus. Given that Oak Lane is straight, for what distance are residents of Oak Lane not allowed to get parking permits? | [
"\\(\\sqrt{2}\\)"
] | [
"A campus only gives out parking permits to people greater than 1 mile away from the campus. Oak Lane starts 2 miles west and 1 mile north of the campus, and ends 5 miles east from the campus. Given that Oak Lane is straight, for what distance are residents of Oak Lane not allowed to get parking permits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-7b9572f3bb04059ba31703b2ae90cdb960f5038e945b1a010724517fab39af61 | A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed? | [
"6"
] | [
"A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-980183da652686bab9821710d416376bf5d1ca7a74af32eb56b0d077dc71f1a9 | A candidate travels in a plane, moving east on the first day, north on the second day, west on the third day, south on the fourth day, and so on, alternating directions in this pattern. On the $n^{\text{th}}$ day, the candidate travels $\frac{n^2}{2}$ miles. How far is the candidate from the starting point at the end o... | [
"580"
] | [
"A candidate travels in a plane, moving east on the first day, north on the second day, west on the third day, south on the fourth day, and so on, alternating directions in this pattern. On the $n^{\\text{th}}$ day, the candidate travels $\\frac{n^2}{2}$ miles. How far is the candidate from the starting point at th... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-f46579273c9c5ba0a724a0eec308f391c2e4cac0784b51f4f45518a745db2759 | A candy-seller sells lollipops for \$0.50, Snickers for \$3.00, and Crunch bars for \$10.00. You buy 100 pieces of candy for \$100. If you bought at least one Crunch bar, how many lollipops did you buy? | [
"94"
] | [
"A candy-seller sells lollipops for \\$0.50, Snickers for \\$3.00, and Crunch bars for \\$10.00. You buy 100 pieces of candy for \\$100. If you bought at least one Crunch bar, how many lollipops did you buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-023d36df40e9fdac0b2f2a05d91202e59a65076d209b5956aa69c1a02317c61d | A cannon is fired with an initial vertical velocity of $v_0=245$ meters per second and a horizontal velocity of $v_h=343$ meters per second from a hill that is $98$ meters above sea level. The acceleration due to gravity is $g=-9.8 \, \text{m/s}^2$. Using the equation $h(t)=\frac{1}{2}gt^2 + v_0 t + h_0$, find the hori... | [
"17286"
] | [
"A cannon is fired with an initial vertical velocity of $v_0=245$ meters per second and a horizontal velocity of $v_h=343$ meters per second from a hill that is $98$ meters above sea level. The acceleration due to gravity is $g=-9.8 \\, \\text{m/s}^2$. Using the equation $h(t)=\\frac{1}{2}gt^2 + v_0 t + h_0$, find ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-5e662631f681b31e14d7695c374ee19116634f87df7a67f672f3e577af74afcf | A cannon is fired with an initial vertical velocity of $v_0=245$ meters per second from a hill that is $98$ meters above sea level. The acceleration due to gravity is $g=-9.8 \, \text{m/s}^2$. Using the equation $h(t)=\frac{1}{2}gt^2 + v_0 t + h_0$, find the maximum height of the cannonball. | [
"3160.5"
] | [
"A cannon is fired with an initial vertical velocity of $v_0=245$ meters per second from a hill that is $98$ meters above sea level. The acceleration due to gravity is $g=-9.8 \\, \\text{m/s}^2$. Using the equation $h(t)=\\frac{1}{2}gt^2 + v_0 t + h_0$, find the maximum height of the cannonball."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eaa6f61fd5e7d376f1232c381144b25d185fe83d81cfba1f7c8459455c32d8ae | A cannon is fired with an initial vertical velocity of $v_0=245$ meters per second from a hill that is $98$ meters above sea level. The acceleration due to gravity is $g=-9.8 \, \text{m/s}^2$. Using the equation $h(t)=\frac{1}{2}gt^2 + v_0 t + h_0$, find the time when the cannonball hits the ground (when $h(t)=0$). | [
"50.4"
] | [
"A cannon is fired with an initial vertical velocity of $v_0=245$ meters per second from a hill that is $98$ meters above sea level. The acceleration due to gravity is $g=-9.8 \\, \\text{m/s}^2$. Using the equation $h(t)=\\frac{1}{2}gt^2 + v_0 t + h_0$, find the time when the cannonball hits the ground (when $h(t)=... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6043f959f648830bebc7b585cf409545ab4dc3b8854b9c50e5a58168c24e519e | A canoeist is paddling upstream in a river when she passes a log floating downstream. She continues upstream for a while, paddling at a constant rate. She then turns around and goes downstream, paddling twice as fast. She catches up to the same log two hours after she passed it. How long did she paddle upstream? | [
"\\dfrac{4}{3}"
] | [
"A canoeist is paddling upstream in a river when she passes a log floating downstream. She continues upstream for a while, paddling at a constant rate. She then turns around and goes downstream, paddling twice as fast. She catches up to the same log two hours after she passed it. How long did she paddle upstream?"
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f40e1c5bdfd7f7cacb9fcd9a47bf2b124c2ae76af22486db756bb8f24c4309b5 | A canoeist paddled upstream for 2 hours, then downstream for 3 hours. The rate of the current was 2 mph. When she stopped, she was 20 miles downstream from her starting point. How many hours will it take her to paddle back to her starting point? | [
"\\(\\frac{5}{2}\\)"
] | [
"A canoeist paddled upstream for 2 hours, then downstream for 3 hours. The rate of the current was 2 mph. When she stopped, she was 20 miles downstream from her starting point. How many hours will it take her to paddle back to her starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-2f51e8281e392a68c83feaab130980c456ba07055a09c24c1e7765f8ef2def96 | A canoeist passes a log floating in the river's current two miles upstream from his starting point. After paddling upstream for one more hour, he turns back and reaches his starting point just as the log arrives. Find the speed of the current. | [
"1"
] | [
"A canoeist passes a log floating in the river's current two miles upstream from his starting point. After paddling upstream for one more hour, he turns back and reaches his starting point just as the log arrives. Find the speed of the current."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0bf44d7b1f6f0110e4e6a8359ffe26412a60d101dee32cfdf61f171794605ce9 | A canoeist starts from a dock and paddles 2.8 km in the direction N34°E. Then she paddles 5.2 km in the direction N65°W. What distance and direction should a second canoeist paddle to reach the same location directly, starting from the same dock? | [
"5.5\\ \\text{km}\\ \\text{N35°W}"
] | [
"A canoeist starts from a dock and paddles 2.8 km in the direction N34°E. Then she paddles 5.2 km in the direction N65°W. What distance and direction should a second canoeist paddle to reach the same location directly, starting from the same dock?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-e50358f419629d466f63c3c414bcd4b44b397fb9a0f5c177465127ac18a7e76a | A capicua number is a number that reads the same backward as forward. Let $S(n)$ represent the sum of all capicua numbers with exactly $2n$ digits that are divisible by 11. Calculate $S(3)$. | [
"495000000"
] | [
"A capicua number is a number that reads the same backward as forward. Let $S(n)$ represent the sum of all capicua numbers with exactly $2n$ digits that are divisible by 11. Calculate $S(3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-594541b7aef15cae741b425ea56fae7d9440119a4fff991f343463660f2bd854 | A captain finds his way to Treasure Island, which is circular in shape. The treasure is buried at the midpoint of the segment joining the orthocentres of triangles $ABC$ and $DEF$, where $A$, $B$, $C$, $D$, $E$, and $F$ are six palm trees on the shore of the island, not necessarily in cyclic order. What is the maximum ... | [
"20"
] | [
"A captain finds his way to Treasure Island, which is circular in shape. The treasure is buried at the midpoint of the segment joining the orthocentres of triangles $ABC$ and $DEF$, where $A$, $B$, $C$, $D$, $E$, and $F$ are six palm trees on the shore of the island, not necessarily in cyclic order. What is the max... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-401ef7ea741e67bc097e3b23a1e10f813fe123c3bcac4eb5eda9a7f4336eb841 | A car 14 ft long is passing a truck 30 ft long. The truck is traveling at 50 mi/h. How fast must the car travel to pass the truck completely in 6 seconds? | [
"55 mi/h"
] | [
"A car 14 ft long is passing a truck 30 ft long. The truck is traveling at 50 mi/h. How fast must the car travel to pass the truck completely in 6 seconds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-45f58a3fc68c2fa6ca1570ae9e33e0894ee6fca6033f9a8adc7b5f021ecd1be7 | A car and a motor-cycle left town $A$ for town $B$ at the same time, and when the motor-cycle covered a sixth of the distance, a cyclist started from $A$ in the same direction. By the moment the car arrived at town $B$, the cyclist covered a fourth of the distance. The speed of the motor-cyclist is 21 km/h lower than t... | [
"63 km/h"
] | [
"A car and a motor-cycle left town $A$ for town $B$ at the same time, and when the motor-cycle covered a sixth of the distance, a cyclist started from $A$ in the same direction. By the moment the car arrived at town $B$, the cyclist covered a fourth of the distance. The speed of the motor-cyclist is 21 km/h lower... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0044b8547476be725cea6b091cc0ef85f97f9d8d8eb78cd7f40cdf9b99c67f27 | A car and a motor-cycle start simultaneously from point $A$ a circular highway and drive in the same direction at constant speeds. The car makes two circles in the same direction without a stop and at the moment when it overtakes the motor-cycle the latter reverses its direction, increases its speed by 16 km/h and 22.5... | [
"21 km"
] | [
"A car and a motor-cycle start simultaneously from point $A$ a circular highway and drive in the same direction at constant speeds. The car makes two circles in the same direction without a stop and at the moment when it overtakes the motor-cycle the latter reverses its direction, increases its speed by 16 km/h and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-426371a48d98b3007d3ef360ef1028e8ea893324b21a6283bd104df428761bd4 | A car and a motorcyclist left point $A$ in the direction of point $B$ at 10 o'clock. Half an hour later, a cyclist left point $B$ in the direction of point $A$, and a pedestrian also left point $B$ in the direction of point $A$. The car met the cyclist at 11 o'clock and overtook the pedestrian half an hour later. The m... | [
"11:15"
] | [
"A car and a motorcyclist left point $A$ in the direction of point $B$ at 10 o'clock. Half an hour later, a cyclist left point $B$ in the direction of point $A$, and a pedestrian also left point $B$ in the direction of point $A$. The car met the cyclist at 11 o'clock and overtook the pedestrian half an hour later. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-4c400734a7dfbb6a368d2ba2dc67b365f799045f28a43f67b5913818985e9945 | A car can hold 3 people in the front seat and 4 in the back seat. In how many ways can 7 people be seated in the car if 2 particular people must sit in the back seat and 1 particular person is the driver? | [
"1152"
] | [
"A car can hold 3 people in the front seat and 4 in the back seat. In how many ways can 7 people be seated in the car if 2 particular people must sit in the back seat and 1 particular person is the driver?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f64f880211c40274b4ef7d778c2e15d539d7e7a943402ade5f508b83e962d194 | A car drives up a hill. It reaches point $A$ and then passes 30 m in the first second and 2 m less than during the preceding second for every subsequent second. Nine seconds after the car reached point $A$, a bus starts from point $B$, which is 258 m from $A$, and drives towards the car. In the first second the bus cov... | [
"20 m"
] | [
"A car drives up a hill. It reaches point $A$ and then passes 30 m in the first second and 2 m less than during the preceding second for every subsequent second. Nine seconds after the car reached point $A$, a bus starts from point $B$, which is 258 m from $A$, and drives towards the car. In the first second the b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66fe29341bbb2b4b4c85b7762322387b89717b1d29da5d90b6558e04b521456b | A car drove from one city to another. She drove the first third of the journey at a speed of $50$ km/h, the second third at $60$ km/h, and the last third at $70$ km/h. What is the average speed of the car along the entire journey? | [
"59 km/h"
] | [
"A car drove from one city to another. She drove the first third of the journey at a speed of $50$ km/h, the second third at $60$ km/h, and the last third at $70$ km/h. What is the average speed of the car along the entire journey?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-298d0f83a5b1321b705b2f0ca3148c5f8252573f44d737954c1ad6a62a6f602e | A car follows another car on a straight stretch of open highway, both cars are driving at 100 kmph and are 150 meters apart. When each car reaches a town, it slows down at the same rate to 30 kmph. Both cars travel through the town at 30 kmph. How far apart are the cars when they are both traveling at 30 kmph? | [
"45 meters"
] | [
"A car follows another car on a straight stretch of open highway, both cars are driving at 100 kmph and are 150 meters apart. When each car reaches a town, it slows down at the same rate to 30 kmph. Both cars travel through the town at 30 kmph. How far apart are the cars when they are both traveling at 30 kmph?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-8d329b807de97ee6a98db6abd70de3817662b56b3e225337e4c9127ccbb5e099 | A car has 4 identical wheels and 4 positions (2 at the front and 2 at the rear) to place the wheels. If the wheels are placed at the 2 front positions, they can last for at most 300 km. If the wheels are placed at the 2 rear positions, they can last for at most 450 km. How many kilometers can the car cover at most, giv... | [
"360"
] | [
"A car has 4 identical wheels and 4 positions (2 at the front and 2 at the rear) to place the wheels. If the wheels are placed at the 2 front positions, they can last for at most 300 km. If the wheels are placed at the 2 rear positions, they can last for at most 450 km. How many kilometers can the car cover at most... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-ddf95220cfeba30d40d950b47b567de04a98e10bcc4b1e0f10136daca80cbcaf | A car has a 4-digit integer price. When the price is turned upside down, it reads as a different 4-digit number that is 1626 less than the original price. What was the original price of the car? | [
"6815"
] | [
"A car has a 4-digit integer price. When the price is turned upside down, it reads as a different 4-digit number that is 1626 less than the original price. What was the original price of the car?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-7d5ee755d4fd714a365b5241762f6ec348f3d6adc286ec9ecf7b42206d4d1360 | A car has a defected odometer that skips the digit $4$. For example, it goes from $39$ to $50$. If the odometer reads $2005$, what is the actual distance traveled by the car? | [
"1462"
] | [
"A car has a defected odometer that skips the digit $4$. For example, it goes from $39$ to $50$. If the odometer reads $2005$, what is the actual distance traveled by the car?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a21344bb4c56e86ad05a91dd837c629967fed1fcceae6c685414b4d0fb2b6583 | A car has a defected odometer that skips the digit 4. For example, it goes from 39 to 50 after traveling one more kilometer. If the odometer reads 2005, what is the actual distance traveled by the car? | [
"1463"
] | [
"A car has a defected odometer that skips the digit 4. For example, it goes from 39 to 50 after traveling one more kilometer. If the odometer reads 2005, what is the actual distance traveled by the car?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c32ef63dd9e3a57d99dfd0652f65c62e92bb31190ed2def0e869ca211b9bb526 | A car has a faulty odometer that skips the digit $4$. For example, it proceeds from $000339$ to $000350$ and from $000353$ to $000355$. If the odometer shows $002051$, what is the actual distance travelled by the car? | [
"1495"
] | [
"A car has a faulty odometer that skips the digit $4$. For example, it proceeds from $000339$ to $000350$ and from $000353$ to $000355$. If the odometer shows $002051$, what is the actual distance travelled by the car?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 2
} | 21 | 0 | 0 |
omr-ws-741b3e77bf967431b081845105878cc2d0704c665ee8b882c81c33fe62f87831 | A car has traveled 60 km in 60 minutes, not necessarily at a constant speed. For which values of \( T \) can we be sure that there exists an interval of exactly \( T \) minutes within the 60 minutes during which the car has traveled exactly \( T \) km? | [
"[0, 60]"
] | [
"A car has traveled 60 km in 60 minutes, not necessarily at a constant speed. For which values of \\( T \\) can we be sure that there exists an interval of exactly \\( T \\) minutes within the 60 minutes during which the car has traveled exactly \\( T \\) km?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-881c37a30e350d9850585211c0dde615f4491a6cb60cbbcb80bbe6914970cbab | A car is at the intersection of \( k+1 \) roads. The driver knows that he has enough fuel to cover the distance to a city placed on one of the roads \( 2 \cdot \frac{(k+1)^{k+1}}{k^k} + 1 \) times the distance to the city, but he doesn't know on which road the city is placed or how much fuel he has in his tank. The dri... | [
"2 \\cdot \\frac{(k+1)^{k+1}}{k^k} + 1"
] | [
"A car is at the intersection of \\( k+1 \\) roads. The driver knows that he has enough fuel to cover the distance to a city placed on one of the roads \\( 2 \\cdot \\frac{(k+1)^{k+1}}{k^k} + 1 \\) times the distance to the city, but he doesn't know on which road the city is placed or how much fuel he has in his ta... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-3124c4cdbd607b4b2c1455a10f98cd649faa5f8356443897b67b8be6cb8fd925 | A car is averaging 50 miles per hour. If the car maintains this speed, how many minutes less would a 450-mile trip take than a 475-mile trip? | [
"30"
] | [
"A car is averaging 50 miles per hour. If the car maintains this speed, how many minutes less would a 450-mile trip take than a 475-mile trip?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5bfd2e21ea69fcebbb3212db134adca60952aed9714eb92e0227d479a9ddfce | A car is moving at 80 kilometers per hour to the north, and the wind is moving at 50 kilometers per hour from the west. Calculate the resultant wind speed relative to the car. Show steps. | [
"94.339"
] | [
"A car is moving at 80 kilometers per hour to the north, and the wind is moving at 50 kilometers per hour from the west. Calculate the resultant wind speed relative to the car. Show steps."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 25
} | 37 | 0 | 0 |
omr-ws-44c22a0cf7addcbdc98b8ed827334782e9fa828730b2393d9692e08062e183da | A car is traveling along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100m west and 100 m north of the origin and travels in an easterly direction. There is a statue at 100 m east and 50 m north of the origin. At what point on the highway will the car's headlights illuminate... | [
"\\((100 - 50\\sqrt{2}, 150 - 100\\sqrt{2})\\)"
] | [
"A car is traveling along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100m west and 100 m north of the origin and travels in an easterly direction. There is a statue at 100 m east and 50 m north of the origin. At what point on the highway will the car's headlights illum... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-1aeed7eae996d640d566d449e39703cd265de431fc92adee05bcb779dafd985c | A car is traveling along a parabolic curve opening northward. A deer is located 100 feet east and 50 feet north of the vertex of the parabola. The car starts 100 feet west and 100 feet north of the vertex. How far away is the car from the deer when the car's headlights illuminate the deer? | [
"50\\sqrt{14 - 8\\sqrt{2}}"
] | [
"A car is traveling along a parabolic curve opening northward. A deer is located 100 feet east and 50 feet north of the vertex of the parabola. The car starts 100 feet west and 100 feet north of the vertex. How far away is the car from the deer when the car's headlights illuminate the deer?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ac9b33ed1560ccb348777b925fb707265078b27d89a460e8efdab2850d391f9a | A car leaves Safeway for the airport, and thirty minutes later, another car leaves the airport for Safeway. Both cars arrive at their respective locations 5 hours after the first car leaves. How many hours does it take for the two cars to meet? | [
"\\(\\frac{50}{19}\\)"
] | [
"A car leaves Safeway for the airport, and thirty minutes later, another car leaves the airport for Safeway. Both cars arrive at their respective locations 5 hours after the first car leaves. How many hours does it take for the two cars to meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c88d2156304ee7bf8f72b24d283fb1df22fa8730cceaae6830a16bbefa8e297e | A car leaves a starting point and drives at 60 km/hour. A second car leaves the same starting point some time later following the first car at a speed of 75 km/hour. It catches the first car after traveling a distance of 30 km. How many minutes later did the second car leave? | [
"6 minutes"
] | [
"A car leaves a starting point and drives at 60 km/hour. A second car leaves the same starting point some time later following the first car at a speed of 75 km/hour. It catches the first car after traveling a distance of 30 km. How many minutes later did the second car leave?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-172771aad187d4b45d3d8403c2c19971e3a94646b7fc38999be0e886af485420 | A car leaves point $A$ for $B$. At the same time, a cyclist starts out in the opposite direction (from $B$). Three minutes after they meet, the car turns around and follows the cyclist; after catching up with the cyclist, it turns around and goes to $B$. If the car had turned around 1 minute after the meeting, and the ... | [
"$1:3$"
] | [
"A car leaves point $A$ for $B$. At the same time, a cyclist starts out in the opposite direction (from $B$). Three minutes after they meet, the car turns around and follows the cyclist; after catching up with the cyclist, it turns around and goes to $B$. If the car had turned around 1 minute after the meeting, and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c00c49f2f927c0d93ae1ce62387cf0f5f6f0c446155720d3f19643488fe7644 | A car left city A towards city B at a certain average speed. After covering a distance of 60 km, due to bad road conditions, it had to travel at 2/3 of its original speed and reached city B 40 minutes late. Had the car met bad road conditions after covering 90 km, it would have reached city B 20 minutes late. What was ... | [
"45 km/h"
] | [
"A car left city A towards city B at a certain average speed. After covering a distance of 60 km, due to bad road conditions, it had to travel at 2/3 of its original speed and reached city B 40 minutes late. Had the car met bad road conditions after covering 90 km, it would have reached city B 20 minutes late. What... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-fa9e01a529285724fc7075a2d0550687506808b75de15f28d0e88f71f6a3a3de | A car left town $A$ for town $B$ and two hours later stopped for 45 min. Then it continued traveling to town $B$, having increased the initial speed by 20 km/h, and later arrived at $B$. If the car had traveled at its initial speed without stopping, it would have spent the same time to travel from $A$ to $B$. Find the ... | [
"60 km/h"
] | [
"A car left town $A$ for town $B$ and two hours later stopped for 45 min. Then it continued traveling to town $B$, having increased the initial speed by 20 km/h, and later arrived at $B$. If the car had traveled at its initial speed without stopping, it would have spent the same time to travel from $A$ to $B$. Find... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffc2f2f28ac41b77c28ed20413920d4543553cc41b4fc8ba4b0a6112db646497 | A car rental agency has 2 locations. The probability that a car rented at location 1 will be returned there next week is 0.7, and the probability that a car rented at location 2 will be returned there is 0.5. Initially, half of the available cars are at each location. What is the probability that a randomly chosen car ... | [
"0.62"
] | [
"A car rental agency has 2 locations. The probability that a car rented at location 1 will be returned there next week is 0.7, and the probability that a car rented at location 2 will be returned there is 0.5. Initially, half of the available cars are at each location. What is the probability that a randomly chosen... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-309687f09f9f53c3891e4df4f831b8f7425423609b51fb8bf8f8eb76ff21a7de | A car salesman sells two types of cars: Alfa and Beta. An Alfa is sold at a 40% profit, whereas a Beta is sold at a 60% profit. If he sells the same number of each car, his overall profit is 48%. In reality, he sells 50% more Betas than Alfas. What is his profit in this scenario? | [
"50\\%"
] | [
"A car salesman sells two types of cars: Alfa and Beta. An Alfa is sold at a 40% profit, whereas a Beta is sold at a 60% profit. If he sells the same number of each car, his overall profit is 48%. In reality, he sells 50% more Betas than Alfas. What is his profit in this scenario?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96365887ca8337ba1e7037f62fe91aecad5e91b8ff0325b5d164452502d604b8 | A car starts from city X in country A with a full tank of gasoline and drives towards the frontier with country B. The price of gasoline increases linearly from city X to the frontier. The car's fuel consumption rate is constant. Determine the optimal strategy for the car to minimize the total cost of the trip, includi... | [
"\\text{Refuel at every } \\frac{C}{r} \\text{ km intervals, filling the tank each time.}"
] | [
"A car starts from city X in country A with a full tank of gasoline and drives towards the frontier with country B. The price of gasoline increases linearly from city X to the frontier. The car's fuel consumption rate is constant. Determine the optimal strategy for the car to minimize the total cost of the trip, in... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15fe0df4cf2b89823607c8d1c3b2b1161836a82bda46eb84bed6aa2d8258bc25 | A car starts from point A towards point B with some initial velocity. Due to an engine problem, after traveling 30 km, the car's velocity decreases to 4/5th of its initial velocity, and it reaches point B 45 minutes later than the expected time. If the engine problem occurs after traveling 45 km, the car reaches point ... | [
"Initial velocity = 25 km/hr, Distance between A and B = 105 km"
] | [
"A car starts from point A towards point B with some initial velocity. Due to an engine problem, after traveling 30 km, the car's velocity decreases to 4/5th of its initial velocity, and it reaches point B 45 minutes later than the expected time. If the engine problem occurs after traveling 45 km, the car reaches p... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-b1e83e983cba9bfb130a285c26a891327b4425fd0a3874498412671868ea5d88 | A car traveled 75% of the way from town A to town B at an average speed of 50 mph. The car travels at an average speed of \( S \) mph for the remaining part of the trip. The average speed for the entire trip was 40 mph. What is \( S \)? | [
"25"
] | [
"A car traveled 75% of the way from town A to town B at an average speed of 50 mph. The car travels at an average speed of \\( S \\) mph for the remaining part of the trip. The average speed for the entire trip was 40 mph. What is \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c03d920193911b41f99609fa9e1dcac530bc9cfd6be8d600ee1e37e4cb5024ca | A car traveling at 84 ft/sec begins to decelerate at a constant rate of 14 ft/sec^2. After how many seconds does the car come to a stop and how far will it travel before stopping? | [
"t = 6 seconds, s = 252 feet"
] | [
"A car traveling at 84 ft/sec begins to decelerate at a constant rate of 14 ft/sec^2. After how many seconds does the car come to a stop and how far will it travel before stopping?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-965aab6ec795dab24ac217122b06c60dc29dd3033f885d3601eddcc89cbc94e5 | A car travels 120 miles from A to B at 30 miles per hour but returns the same distance at 40 miles per hour. What is the average speed for the round trip? | [
"34 mph"
] | [
"A car travels 120 miles from A to B at 30 miles per hour but returns the same distance at 40 miles per hour. What is the average speed for the round trip?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-dd893595ea559f00af71f51ac23ea934043f9df9008f2308147c33aa08d7b63e | A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? | [
"256"
] | [
"A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-171fe26c72d5613a50d1a521e2c0eb8fa0585742d34030700ff4e520db574d86 | A car travels 40 kph for 20 kilometers, 50 kph for 25 kilometers, 60 kph for 45 minutes and 48 kph for 15 minutes. What is the average speed of the car, in kph? | [
"51"
] | [
"A car travels 40 kph for 20 kilometers, 50 kph for 25 kilometers, 60 kph for 45 minutes and 48 kph for 15 minutes. What is the average speed of the car, in kph?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93191ed7b6d0881ab3e4bc7c444ac7d9d8d80afbd3d516326a1562de464ff44f | A car travels along a straight road, and a vulture flies off in a direction that maximizes the distance of closest approach to the car. If the angle between the vulture's flight path and the road is $A$, and the speed of the car is $v_c$, find the speed $v_v$ of the vulture in terms of $v_c$ and $A$. Express your answe... | [
"\\cos A"
] | [
"A car travels along a straight road, and a vulture flies off in a direction that maximizes the distance of closest approach to the car. If the angle between the vulture's flight path and the road is $A$, and the speed of the car is $v_c$, find the speed $v_v$ of the vulture in terms of $v_c$ and $A$. Express your ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-198c3dd50ae8525403084cc03bba48559f48e1f3a857e60e03036820882c07fc | A car travels at 63 km/h on level ground, 56 km/h uphill, and 72 km/h downhill. A trip from A to B takes 4 hours, while the return trip from B to A takes 4 hours and 40 minutes. What is the distance between A and B? | [
"273"
] | [
"A car travels at 63 km/h on level ground, 56 km/h uphill, and 72 km/h downhill. A trip from A to B takes 4 hours, while the return trip from B to A takes 4 hours and 40 minutes. What is the distance between A and B?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6fde6f1249aac604d9da38e408f5f47b01b782e154d89a18bbe2f84efa51bae | A car travels at a speed of 60 km per hour along a straight highway. A 100-meter long fence is parallel to the highway. Every second, the passenger measures the angle of vision of the fence. Calculate the maximum possible sum of all angles measured by the passenger in degrees. | [
"1080"
] | [
"A car travels at a speed of 60 km per hour along a straight highway. A 100-meter long fence is parallel to the highway. Every second, the passenger measures the angle of vision of the fence. Calculate the maximum possible sum of all angles measured by the passenger in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4a1cdfed106537d3930d54296b0c8f730df8e5eb39372dcaed2e1b1e6c9cee4e | A car travels at speeds of 5, 10, 15, and 20 miles per hour for one hour each. Determine the total distance traveled. | [
"25"
] | [
"A car travels at speeds of 5, 10, 15, and 20 miles per hour for one hour each. Determine the total distance traveled."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d23a64195b8ba8fd2557500189a461dc0f94c0a1431e7fa726fcae0d5c9c80a | A car travels downhill at 72 mph, on the level at 63 mph, and uphill at 56 mph. The car takes 4 hours to travel from town A to town B and 4 hours 40 minutes to return. Find the distance between the two towns. | [
"273"
] | [
"A car travels downhill at 72 mph, on the level at 63 mph, and uphill at 56 mph. The car takes 4 hours to travel from town A to town B and 4 hours 40 minutes to return. Find the distance between the two towns."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-232d72a7856a86117f0863c60514feba35b9e593d6f3e96972459c6c205ad8e0 | A car travels downhill at 72 mph, on the level at 63 mph, and uphill at 56 mph. The car takes 4 hours to travel from town A to town B. The return trip takes 4 hours and 40 minutes. Find the distance (in miles) between the two towns. | [
"273"
] | [
"A car travels downhill at 72 mph, on the level at 63 mph, and uphill at 56 mph. The car takes 4 hours to travel from town A to town B. The return trip takes 4 hours and 40 minutes. Find the distance (in miles) between the two towns."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-d7deae3d810089aea113d48a7d32c5fc3b1faf3643fe0e9126d90c8a0382c3b1 | A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the st... | [
"198"
] | [
"A car travels due east at $\\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\\frac 12\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enter... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8270003cbc3b5e34bf74e19627aaadcce08f42585d15d39882efc0460bdc9e78 | A car travels due east at $\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm with a radius of 51 miles moves southeast at $\frac{1}{2} \sqrt{2}$ mile per minute. At time $t = 0$, the center of the storm is 110 miles due north of the car. At time $t = t_1$ minutes, the car enters ... | [
"198"
] | [
"A car travels due east at $\\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm with a radius of 51 miles moves southeast at $\\frac{1}{2} \\sqrt{2}$ mile per minute. At time $t = 0$, the center of the storm is 110 miles due north of the car. At time $t = t_1$ minutes, the car... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f6e294464eaa6f432a9cd6af49ccb07195cd5f0d048e3b9b58aa5be4704d8de7 | A car travels due east at $\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm with a radius of 51 miles moves southeast at $\frac{1}{2}\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the s... | [
"198"
] | [
"A car travels due east at $\\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm with a radius of 51 miles moves southeast at $\\frac{1}{2}\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car ente... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-0d1bdf580e60db285dd5a3439d674ae8cba2b594c051bc6f14f3a089d02e3275 | A car travels due east at $\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac{\sqrt{2}}{2}$ mile per minute. At time $t = 0$, the center of the storm is 110 miles due north of the car. At time $t = t_1$, the car enters the storm... | [
"198"
] | [
"A car travels due east at $\\frac{2}{3}$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\\frac{\\sqrt{2}}{2}$ mile per minute. At time $t = 0$, the center of the storm is 110 miles due north of the car. At time $t = t_1$, the car enters t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-acda15c374db2df8e04f11f238c7cfe46805fd56f7d62d5f59efc524a71df0ae | A car travels due east at \( \frac{2}{3} \) mile per minute on a long, straight road. At the same time, a storm in the shape of a circle with a radius of 51 miles moves southeast at \( \frac{\sqrt{2}}{2} \) miles per minute. At time \( t = 0 \), the center of the storm is 110 miles due north of the car. At time \( t = ... | [
"92.4"
] | [
"A car travels due east at \\( \\frac{2}{3} \\) mile per minute on a long, straight road. At the same time, a storm in the shape of a circle with a radius of 51 miles moves southeast at \\( \\frac{\\sqrt{2}}{2} \\) miles per minute. At time \\( t = 0 \\), the center of the storm is 110 miles due north of the car. A... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3e573097df99411e842bc85eefb035bd378eb9af404759df48ccc3f4c59ed00 | A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$? | [
"36"
] | [
"A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5214fd8d53929beb04b1c135d476aeefcb0c62436129dd48ce2fe66a957bf6c | A car with tires having a radius of 33 cm was driven on a trip, and the odometer indicated that the distance traveled was 640 km. Two weeks later, with snow tires installed, the odometer indicated that the distance for the return trip over the same route was 625 km. Find the radius of the snow tires. | [
"33.8"
] | [
"A car with tires having a radius of 33 cm was driven on a trip, and the odometer indicated that the distance traveled was 640 km. Two weeks later, with snow tires installed, the odometer indicated that the distance for the return trip over the same route was 625 km. Find the radius of the snow tires."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
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