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-c01ee4dac3cca71b114555ec3c74f04e6a5003148eaf7d43950f9962331eaddc | When Carlos divides his marbles into groups of six, seven, or eight, he has one marble left over. Assuming Carlos has more than one marble, what is the smallest possible number of marbles in Carlos's collection? | [
"169"
] | [
"When Carlos divides his marbles into groups of six, seven, or eight, he has one marble left over. Assuming Carlos has more than one marble, what is the smallest possible number of marbles in Carlos's collection?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5680d3df225771c355c7927383dce2aff9b2efe20194140aef60be6f99f71e1 | When Charles traveled from Hawaii to Chicago, he moved his watch 5 hours backwards instead of 5 hours forwards. He plans to wake up at 7:00 the next morning (Chicago time). When he wakes up during the night and sees that his watch says 6:00, how many more hours should he sleep? (He has a 12-hour watch, not a 24-hour wa... | [
"3"
] | [
"When Charles traveled from Hawaii to Chicago, he moved his watch 5 hours backwards instead of 5 hours forwards. He plans to wake up at 7:00 the next morning (Chicago time). When he wakes up during the night and sees that his watch says 6:00, how many more hours should he sleep? (He has a 12-hour watch, not a 24-ho... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c79db1a13b13b44fe3365ca4fe49b293f6d1e6736083225acda8f5f0e45f3cea | When Claire divides her cupcakes into groups of 5, she has 3 remaining, and when she divides her cupcakes into groups of 7, she has 4 remaining. If Claire has fewer than 60 cupcakes, what is the sum of all possible quantities of cupcakes that she could have? | [
"71"
] | [
"When Claire divides her cupcakes into groups of 5, she has 3 remaining, and when she divides her cupcakes into groups of 7, she has 4 remaining. If Claire has fewer than 60 cupcakes, what is the sum of all possible quantities of cupcakes that she could have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b277624a886ad50bc65417227176705a89ab7b3dacba39e078a69b9d5c1c67d8 | When Darren and Eric play chess, Eric is twice as likely to win as Darren is, and the probability that they draw (the game ends in a tie) is a tenth of the probability that Eric wins. Find the probability that a game between Darren and Eric ends with Darren winning. | [
"\\dfrac{5}{16}"
] | [
"When Darren and Eric play chess, Eric is twice as likely to win as Darren is, and the probability that they draw (the game ends in a tie) is a tenth of the probability that Eric wins. Find the probability that a game between Darren and Eric ends with Darren winning."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89c69c5823343c63c467e3b819e26a60d970e167a97971bd218e65900d56bbb6 | When Dylan is one mile from his house, a robber steals his wallet and starts to ride his motorcycle in the direction opposite from Dylan’s house at $40$ miles per hour. Dylan dashes home at $10$ miles per hour and, upon reaching his house, begins driving his car at $60$ miles per hour in the direction of the robber’s m... | [
"21"
] | [
"When Dylan is one mile from his house, a robber steals his wallet and starts to ride his motorcycle in the direction opposite from Dylan’s house at $40$ miles per hour. Dylan dashes home at $10$ miles per hour and, upon reaching his house, begins driving his car at $60$ miles per hour in the direction of the robbe... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-69fc0c9ca6772ed5c52c2dc11ea0ee0704972c685b1a5cdba97e4e560bbe6d6d | When Ed sits on the left side and George, who weighs 100 pounds, sits on the right side, they are perfectly balanced. Similarly, if Fred, who weighs 400 pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in pounds? | [
"200"
] | [
"When Ed sits on the left side and George, who weighs 100 pounds, sits on the right side, they are perfectly balanced. Similarly, if Fred, who weighs 400 pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in poun... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d42e7aa05b7ba465fc83ab7049e9e0dece841dbb2f75d4af4f3648820d67b344 | When Eric gets on the bus home, he notices his 12-hour watch reads 03:30, but it isn’t working as expected. The second hand makes a full rotation in 4 seconds, then makes another in 8 seconds, then another in 12 seconds, and so on until it makes a full rotation in 60 seconds. Then it repeats this process, and again mak... | [
"14"
] | [
"When Eric gets on the bus home, he notices his 12-hour watch reads 03:30, but it isn’t working as expected. The second hand makes a full rotation in 4 seconds, then makes another in 8 seconds, then another in 12 seconds, and so on until it makes a full rotation in 60 seconds. Then it repeats this process, and agai... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f1e3024c3728f94680fa018648a377ebe8b03586d551c3d5323a5d5cce8b876 | When Erica was two years old, Leroy was four times as old as Miriam. When Miriam was twice as old as Erica, Leroy was three times as old as Miriam. How old was Erica when Leroy was twice as old as Miriam? | [
"18"
] | [
"When Erica was two years old, Leroy was four times as old as Miriam. When Miriam was twice as old as Erica, Leroy was three times as old as Miriam. How old was Erica when Leroy was twice as old as Miriam?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-87e44c357d97b4a4f85417946cc19dca9d871b59ba0481884600d63f211e960b | When Frederick was born, his grandparents gave him a gift of $\$2000$, which was invested at an interest rate of $5\%$ per year, compounded yearly. How much money will Frederick have when he collects the money at the age of $18$? Give your answer to the nearest hundredth of a dollar. | [
"\\$4813.24"
] | [
"When Frederick was born, his grandparents gave him a gift of $\\$2000$, which was invested at an interest rate of $5\\%$ per year, compounded yearly. How much money will Frederick have when he collects the money at the age of $18$? Give your answer to the nearest hundredth of a dollar."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf610341abce4996ee50c72abc50be0f9ac4555b74d507edc11d2a47b3f4162f | When Jane is one year younger than Betty will be when Jane is half as old as Betty will be when Jane is twice as old as Betty is now, Betty will be three times as old as Jane was when Betty was as old as Jane is now. One is in her teens and ages are in completed years. How old are they? | [
"Jane is 19 years old, and Betty is 25 years old.",
"Jane is 19, Betty is 25."
] | [
"When Jane is one year younger than Betty will be when Jane is half as old as Betty will be when Jane is twice as old as Betty is now, Betty will be three times as old as Jane was when Betty was as old as Jane is now. One is in her teens and ages are in completed years. How old are they?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 22,
"tir": 58
} | 80 | 0 | 0 |
omr-ws-9bb260ff10cf6905d2ea2f386b551731bc584c666b23a7dfedd23a62fd3eb0a4 | When Joyce counts the pennies in her bank by fives, she has one left over. When she counts them by threes, there are two left over. What is the least possible number of pennies in the bank? | [
"11"
] | [
"When Joyce counts the pennies in her bank by fives, she has one left over. When she counts them by threes, there are two left over. What is the least possible number of pennies in the bank?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fb7d90f0a546e8d57f99fb271ded76c2f4f646028f89e108109adb001b1d84e | When Margaret reached her 17th birthday, Meredith was four times older than Marion. When Margaret reaches her 25th birthday, eleven times the difference of Meredith and Marion’s ages will equal five times the sum of their ages. Find the ages of Meredith and Marion on Margaret’s 17th birthday. | [
"Marion is 10, Meredith is 40"
] | [
"When Margaret reached her 17th birthday, Meredith was four times older than Marion. When Margaret reaches her 25th birthday, eleven times the difference of Meredith and Marion’s ages will equal five times the sum of their ages. Find the ages of Meredith and Marion on Margaret’s 17th birthday."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 29
} | 36 | 0 | 0 |
omr-ws-698bddfd74ecd3bbe0e1ad73055381f75fd6acca20330c7c920b02ceefa923fc | When Mr. Smith cashed a check for \( x \) dollars and \( y \) cents, he received instead \( y \) dollars and \( x \) cents and found that he had two cents more than twice the proper amount. For how much was the check written? | [
"\\$65.32"
] | [
"When Mr. Smith cashed a check for \\( x \\) dollars and \\( y \\) cents, he received instead \\( y \\) dollars and \\( x \\) cents and found that he had two cents more than twice the proper amount. For how much was the check written?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-85e3361f680307d870d71b500a535b4f2f998f8bf389baa23303c45766c6145a | When P(x) is divided by x+1 and x-2, the remainders are 2 and -3, respectively. Find the remainder when P(x) is divided by (x-1)(x-2). | [
"-\\dfrac{5}{3}x + \\dfrac{1}{3}"
] | [
"When P(x) is divided by x+1 and x-2, the remainders are 2 and -3, respectively. Find the remainder when P(x) is divided by (x-1)(x-2)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bd8a07e395d9cfcbcd63532fc2efaed31941937ec49a875b1a409ad8217a7bd5 | When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.) | [
"36"
] | [
"When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc37d32edb5e657d2372cff8bbe5823c94f9caae2d39b5c0ed0eca0a570f2a9f | When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$? | [
"17"
] | [
"When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-644ee9793aeefd22cdea27b5acdd72611bf7919073c314a36970bc10745a53ad | When \( (1 - x)(a + x)^{-2} \) is expanded, the coefficient of \( x^2 \) is \( 0 \). Find the value of \( a \). | [
"\\( -\\frac{3}{2} \\)"
] | [
"When \\( (1 - x)(a + x)^{-2} \\) is expanded, the coefficient of \\( x^2 \\) is \\( 0 \\). Find the value of \\( a \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-aed2aae2fdcf3ec1161bd6a05623782afe186e02f9d45ffe381473517aac403c | When \( (5x^6 + 10x^5 + 2x^4 + 19x^3 + 2x^2 + 1)^{10000} \) is multiplied out, what is the coefficient of \( x^4 \)? | [
"200000000"
] | [
"When \\( (5x^6 + 10x^5 + 2x^4 + 19x^3 + 2x^2 + 1)^{10000} \\) is multiplied out, what is the coefficient of \\( x^4 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 25
} | 48 | 0 | 0 |
omr-ws-8d5480ee6b0fbdebb6c20656c578da3b302f3d1e17a01511c5421e25de541a0c | When \( 1 + 7 + 7^2 + \cdots + 7^{2004} \) is divided by 1000, a remainder of \( N \) is obtained. Determine the value of \( N \). | [
"801"
] | [
"When \\( 1 + 7 + 7^2 + \\cdots + 7^{2004} \\) is divided by 1000, a remainder of \\( N \\) is obtained. Determine the value of \\( N \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e84da0e8e1d03979e64fe7d0d567d5a5a98ca239caf82afc6aeaea9a6063e30f | When \( 15! \) is converted to base 8, it is expressed as \( \overline{230167356abc00} \) for some digits \( a \), \( b \), and \( c \). Find the missing string \( \overline{abc} \). | [
"540"
] | [
"When \\( 15! \\) is converted to base 8, it is expressed as \\( \\overline{230167356abc00} \\) for some digits \\( a \\), \\( b \\), and \\( c \\). Find the missing string \\( \\overline{abc} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-e4084256e1c837c0b16e12f3cd65a70c3a07a1f30f0efefe9d7eb8ed6677e3c5 | When \( n = 6 \), what is the probability that \( X = Y + 1 \), where \( X \) and \( Y \) are the number of heads obtained by two people, A and B, each throwing \( n \) fair coins? | [
"\\(\\frac{99}{512}\\)"
] | [
"When \\( n = 6 \\), what is the probability that \\( X = Y + 1 \\), where \\( X \\) and \\( Y \\) are the number of heads obtained by two people, A and B, each throwing \\( n \\) fair coins?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-29119843a613b302ec57672d045d08f9d829d613e2a6d27c489d17df94a0d878 | When \( n = 7^{3 \times 7^7} \), compute the greatest common divisor of \( 7^n - 1 \) and \( 7^n + 4949 \). | [
"6"
] | [
"When \\( n = 7^{3 \\times 7^7} \\), compute the greatest common divisor of \\( 7^n - 1 \\) and \\( 7^n + 4949 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-7e287d5ad37d1cd00fa7bda66716e269d082e9cf200416f16d32b0a04d451bb7 | When \( n \) dice are thrown at the same time, find the probability that the sum of the numbers on the dice is \( n+3 \). | [
"\\(\\frac{ \\binom{n}{3}+2 \\binom{n}{2}+\\binom{n}{1}}{6^{n}}\\)"
] | [
"When \\( n \\) dice are thrown at the same time, find the probability that the sum of the numbers on the dice is \\( n+3 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-265f6b31e160fa28af637b5122a27d6a76be9d8883ed6c0de9399520912a2679 | When \( n \) standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of \( S \). The smallest possible value of \( S \) is? | [
"337"
] | [
"When \\( n \\) standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of \\( S \\). The smallest possible value of \\( S \\) is?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-b609e272b565de2fb20f6c0d1990a671533a1142a9a143839429fafb00cf8cba | When \( n \) standard six-sided dice are rolled, the probability of obtaining a sum of \( 1994 \) is greater than zero and is the same as the probability of obtaining a sum of \( S \). What is the smallest possible value of \( S \)? | [
"337"
] | [
"When \\( n \\) standard six-sided dice are rolled, the probability of obtaining a sum of \\( 1994 \\) is greater than zero and is the same as the probability of obtaining a sum of \\( S \\). What is the smallest possible value of \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-5e6c587b3efc49d51ece5fcd5e1cfb0c6f934b63fe704f913ad63a21662e7ab5 | When \( t = 0 \), point \( P \) is at \( (0,0) \) and point \( Q \) is at \( (0,1) \). Point \( P \) moves to the right along the x-axis with velocity 1, and point \( Q \) moves towards \( P \) with velocity 1. What will the distance be between points \( P \) and \( Q \) as \( t \) goes to infinity? | [
"\\(\\frac{1}{2}\\)"
] | [
"When \\( t = 0 \\), point \\( P \\) is at \\( (0,0) \\) and point \\( Q \\) is at \\( (0,1) \\). Point \\( P \\) moves to the right along the x-axis with velocity 1, and point \\( Q \\) moves towards \\( P \\) with velocity 1. What will the distance be between points \\( P \\) and \\( Q \\) as \\( t \\) goes to in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-4a19799225bef78d46cf9ffd47d3315500220a49b85d959218d77f3074de2b66 | When \( t \) moves in the range \( 0 \leq t \leq \pi \), find the minimum value of the \( x \)-coordinate of \( H \). | [
"\\(-\\frac{2}{3\\sqrt{6}}\\)"
] | [
"When \\( t \\) moves in the range \\( 0 \\leq t \\leq \\pi \\), find the minimum value of the \\( x \\)-coordinate of \\( H \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e779030bb86e49768b81e7d56f2ac48667273efb5c496c04098b6de40f694b36 | When \( t \) moves in the range \( t > 1 \), find the value of \( t \) for which \( T_n(t) - S_n(t) \) is maximized, where \( S_n \) is the area under the curve \( y = f_n(x) \) from \( x = 1 \) to \( x = t \), and \( T_n \) is the area of the rectangle with vertices \( (1, 0) \), \( (t, 0) \), \( (t, f_n(t)) \), and \... | [
"\\( e^{1/n} \\)"
] | [
"When \\( t \\) moves in the range \\( t > 1 \\), find the value of \\( t \\) for which \\( T_n(t) - S_n(t) \\) is maximized, where \\( S_n \\) is the area under the curve \\( y = f_n(x) \\) from \\( x = 1 \\) to \\( x = t \\), and \\( T_n \\) is the area of the rectangle with vertices \\( (1, 0) \\), \\( (t, 0) \\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9b732e74e21cb0458fd0af5da0a1208d06f2c501940a08191e1e6aa3cab0606 | When \( x > 0 \), find bounds for \(\frac{D_{n}(x+0.5)}{D_{n}(x+1)}\), where \( n \ge 2 \) and \( D_{n}(x) \) is the determinant of the \( n \times n \) matrix given by \[ D_{n}(x) = \begin{vmatrix} \Gamma(x) & \Gamma(x+1) & \ldots & \Gamma(x+n-1) \\ \Gamma(x+1) & \Gamma(x+2) & \ldots & \Gamma(x+n) \\ \vdots & \vdots &... | [
"\\(\\frac{1}{\\sqrt{(x+0.5)_{n}}} \\le \\frac{D_{n}(x+0.5)}{D_{n}(x+1)} \\le \\frac{1}{\\sqrt{(x)_{n}}}\\)"
] | [
"When \\( x > 0 \\), find bounds for \\(\\frac{D_{n}(x+0.5)}{D_{n}(x+1)}\\), where \\( n \\ge 2 \\) and \\( D_{n}(x) \\) is the determinant of the \\( n \\times n \\) matrix given by\n\\[ D_{n}(x) = \\begin{vmatrix}\n\\Gamma(x) & \\Gamma(x+1) & \\ldots & \\Gamma(x+n-1) \\\\\n\\Gamma(x+1) & \\Gamma(x+2) & \\ldots & ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-d871052aede6fa426f4218cd341e343ebde8f7c6f24f7fe929d1429f99d9e32c | When \( x^9 - x \) is factored as completely as possible into polynomials and monomials with integral coefficients, how many factors are there? | [
"5"
] | [
"When \\( x^9 - x \\) is factored as completely as possible into polynomials and monomials with integral coefficients, how many factors are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-efe02c2a69a9cabb84c72f9e3caedd34e3caed1f9b59bfe0cf8bb080a7cb024b | When \( x^9 - x \) is factored as completely as possible into polynomials and monomials with real integral coefficients, how many factors are there? | [
"5"
] | [
"When \\( x^9 - x \\) is factored as completely as possible into polynomials and monomials with real integral coefficients, how many factors are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-cd0a1a1f1d29a9c432ed5549d6cd859ce504566209619dee3a4f5c4b594590a1 | When \( x^{44} + x^{33} + x^{22} + x^{11} + 1 \) is divided by \( x^4 + x^3 + x^2 + x + 1 \), it gives a remainder \( R(x) \). Find \( R(1) + 2R(2) + 3R(3) \). | [
"0"
] | [
"When \\( x^{44} + x^{33} + x^{22} + x^{11} + 1 \\) is divided by \\( x^4 + x^3 + x^2 + x + 1 \\), it gives a remainder \\( R(x) \\). Find \\( R(1) + 2R(2) + 3R(3) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-0a2b3c351ea97db802e547fb735a64963019c575823c3ebafe788ad871cefc84 | When \((1-3x)^q\) is expanded, the coefficient of \(x\) is 40. Given that \(q > 0\), find the constant \(q\). | [
"\\dfrac{40}{3}"
] | [
"When \\((1-3x)^q\\) is expanded, the coefficient of \\(x\\) is 40. Given that \\(q > 0\\), find the constant \\(q\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-db443852eef381e24fcc13c8eb8157a5aa0e5376bb1d6a5aa3122d641759a270 | When \((a+b+c+d)^{10}\) is expanded and like terms combined, how many terms are in the result? | [
"286"
] | [
"When \\((a+b+c+d)^{10}\\) is expanded and like terms combined, how many terms are in the result?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2fcaf2ebb2e86c4940dcf6a1a6d4aee49af7c7c746678fecf03ec0302c0bcacc | When \((x + y + z)^{2009}\) is expanded and like terms are grouped, how many terms have coefficients that are not multiples of 5? | [
"1350"
] | [
"When \\((x + y + z)^{2009}\\) is expanded and like terms are grouped, how many terms have coefficients that are not multiples of 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 25
} | 46 | 0 | 0 |
omr-ws-0a87d5902f66a4dbdb7bbc7f6eccf401fccab54ce10140ca30e54daa13823a90 | When \((x^2 + x + 1)^{54}\) is expanded, how many coefficients of the resulting polynomial are divisible by 3? | [
"105"
] | [
"When \\((x^2 + x + 1)^{54}\\) is expanded, how many coefficients of the resulting polynomial are divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b81d5d96353c9c286115a113982128c140e714587c99d92db99d7b2111242f79 | When \(10^{20} - 1999\) is written as a single whole number, what is the sum of its digits? | [
"153"
] | [
"When \\(10^{20} - 1999\\) is written as a single whole number, what is the sum of its digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 13
} | 35 | 0 | 0 |
omr-ws-ff6898e7170145c98991b0ca2484d117969a17748c7e44637f4789f9f1e1a54d | When \(30!\) is computed, it ends in 7 zeros. Find the digit that immediately precedes those zeros. | [
"6"
] | [
"When \\(30!\\) is computed, it ends in 7 zeros. Find the digit that immediately precedes those zeros."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08da95657df7cbadcb35abea944c7771e63970e0380486b28e215de128c36e05 | When \(4444^{4444}\) is written in decimal notation, let \(A\) be the sum of its digits. Let \(B\) be the sum of the digits of \(A\). Find the sum of the digits of \(B\). | [
"7"
] | [
"When \\(4444^{4444}\\) is written in decimal notation, let \\(A\\) be the sum of its digits. Let \\(B\\) be the sum of the digits of \\(A\\). Find the sum of the digits of \\(B\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 14
} | 61 | 0 | 0 |
omr-ws-aafc9aa149d9da8a6c9809156ec6415e60cb9d8dee7fce31e87bfd44f4ebf55f | When \(4444^{4444}\) is written in decimal notation, the sum of its digits is \(A\). Let \(B\) be the sum of the digits of \(A\). Find the sum of the digits of \(B\). | [
"7"
] | [
"When \\(4444^{4444}\\) is written in decimal notation, the sum of its digits is \\(A\\). Let \\(B\\) be the sum of the digits of \\(A\\). Find the sum of the digits of \\(B\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bdade9bbe126e19d940eeadabf6d27cca42b7a28f684a9dfeb9e14822580d05 | When \(\gcd(a, b) > 1\), determine the possible values of \(a\) and \(b\) such that \(\overline{a.b} = \frac{b}{a}\). | [
"(9, 9)"
] | [
"When \\(\\gcd(a, b) > 1\\), determine the possible values of \\(a\\) and \\(b\\) such that \\(\\overline{a.b} = \\frac{b}{a}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-4485c8e6181e14f078a0c9d66a947179285eb3decb71196564d66334d764ad7f | When \(a\), \(b\), and \(c\) are all distinct positive integers, what is the smallest possible value of \(a + b + c\) given that \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{13}{20}\)? | [
"22"
] | [
"When \\(a\\), \\(b\\), and \\(c\\) are all distinct positive integers, what is the smallest possible value of \\(a + b + c\\) given that \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} = \\frac{13}{20}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-e7f6f748ace1624f055df5621027058379ce0ed038c09b4d7ec387d00707b9fc | When \(n \in \mathbb{N}\) and \(n \geq 3\), find the value of: $$\prod_{i=1}^{n-1} \left(1 + 2\cos{\frac{i\pi}{n}}\right)$$ | [
"\\(0\\) if \\(3 \\mid n\\), otherwise \\(\\pm 1\\) with the sign given by \\((-1)^{n-1 - \\lfloor \\frac{2n}{3} \\rfloor}\\)."
] | [
"When \\(n \\in \\mathbb{N}\\) and \\(n \\geq 3\\), find the value of:\n$$\\prod_{i=1}^{n-1} \\left(1 + 2\\cos{\\frac{i\\pi}{n}}\\right)$$"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-06ed1e19432068a2a6cafc229cd2a869ec60368be5aa2486765d42484e5a6027 | When \(x\) is large and positive, the solution to the differential equation \(\frac{d^{2}y}{dx^{2}} + 2 \frac{dy}{dx} + 17y = 17x + 36\) approximates to a linear function. Find the value of \(a + b\) where the linear function is given by \(y = ax + b\). | [
"3"
] | [
"When \\(x\\) is large and positive, the solution to the differential equation \\(\\frac{d^{2}y}{dx^{2}} + 2 \\frac{dy}{dx} + 17y = 17x + 36\\) approximates to a linear function. Find the value of \\(a + b\\) where the linear function is given by \\(y = ax + b\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bfd940114efc321cb9335dd9b762b4e5befca6d4e43238f7852b75a144f1fd3 | When \(x^{1951} - 1\) is divided by \(x^4 + x^3 + 2x^2 + x + 1\), what is the coefficient of \(x^{14}\)? | [
"0"
] | [
"When \\(x^{1951} - 1\\) is divided by \\(x^4 + x^3 + 2x^2 + x + 1\\), what is the coefficient of \\(x^{14}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-e68b93f5d3c4f54ebd2923beca4463553c3b6ab98db232f0abb24c7b00a5cfa2 | When \[ \sum_{k = 0}^{49}(-1)^k\binom{99}{2k} \] is written in the form \(a^b\) where \(a\) and \(b\) are integers and \(b\) is as large as possible, find \(a + b\). | [
"47"
] | [
"When \\[ \\sum_{k = 0}^{49}(-1)^k\\binom{99}{2k} \\] is written in the form \\(a^b\\) where \\(a\\) and \\(b\\) are integers and \\(b\\) is as large as possible, find \\(a + b\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-1bd8dd54d67a9ab9ae05f19c1750c7ef13718c4c26f5403a9d7c51838bcd674f | When a biased coin is tossed, the probability of a head is \( p \). Two players, A and B, alternately toss a coin until one of the sequences HHH or HTH occurs. Player A wins if HHH occurs first, and player B wins if HTH occurs first. For what values of \( p \) is the game fair, i.e., such that the probability that A wi... | [
"\\( \\frac{2}{3} \\)"
] | [
"When a biased coin is tossed, the probability of a head is \\( p \\). Two players, A and B, alternately toss a coin until one of the sequences HHH or HTH occurs. Player A wins if HHH occurs first, and player B wins if HTH occurs first. For what values of \\( p \\) is the game fair, i.e., such that the probability ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26d9ab9972463699bf0c0db8250f491af724684fe8ac9af5f0f44b7e599e0e91 | When a bus arrives at a bus stop, the next bus arrives an integer amount of minutes afterwards, anywhere from 1 through 20 minutes, all with equal probability. Prune James arrives at a bus stop at some fixed time. What is the expected amount of time that Prune James needs to wait for the next bus to arrive? | [
"\\(\\frac{20}{3}\\)"
] | [
"When a bus arrives at a bus stop, the next bus arrives an integer amount of minutes afterwards, anywhere from 1 through 20 minutes, all with equal probability. Prune James arrives at a bus stop at some fixed time. What is the expected amount of time that Prune James needs to wait for the next bus to arrive?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6ea24542eb2fcbee23d77582a784186885977f22b309f67c2abf7d859e09657 | When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops? | [
"70"
] | [
"When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fdc0292e22534ea746de1675cc78988005ca2900f0096c594c3ac0a79350512d | When a certain biased coin is flipped 5 times, the probability of getting heads exactly once is the same as the probability of getting heads exactly twice. What is the probability that the coin comes up heads exactly 3 times out of 5? | [
"\\(\\frac{40}{243}\\)"
] | [
"When a certain biased coin is flipped 5 times, the probability of getting heads exactly once is the same as the probability of getting heads exactly twice. What is the probability that the coin comes up heads exactly 3 times out of 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c8929edfef69d84efe198556b890c5b6e2aedb880c0beb8f287c9a813666489b | When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$. | [
"283"
] | [
"When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3206ea9f3dbcded1d513b0c6343dc0c2ea3c3abdb090830276ec98898192147d | When a certain positive integer \( N \) is divided by a positive integer \( d \), the remainder is 7. If \( 2N + 3 \) is divided by \( d \), the remainder is 1. Find all possible values of \( d \). | [
"8, 16",
"\\( d = 8 \\) or \\( d = 16 \\)",
"\\( d = 8, 16 \\)"
] | [
"When a certain positive integer \\( N \\) is divided by a positive integer \\( d \\), the remainder is 7. If \\( 2N + 3 \\) is divided by \\( d \\), the remainder is 1. Find all possible values of \\( d \\)."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-4a4a7c3b27a4879c65a1bbb2153c4ec9d4156abdb94eb1e04c9ffdd7f80472bc | When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even? | [
"\\dfrac{5}{8}"
] | [
"When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-185750d04231ce307026bf2c63719e1317595593c273c2e85f02ee03ca0066de | When a chord cuts two circles in pairs of points which are harmonic conjugates, find the locus of the feet of the perpendiculars from the centers of the circles to the chord. | [
"\\text{the radical axis of the two circles}"
] | [
"When a chord cuts two circles in pairs of points which are harmonic conjugates, find the locus of the feet of the perpendiculars from the centers of the circles to the chord."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-8356068dff1071b201419c8924c5da8014d1a5fcd2f06c85037c1c37ebbb096f | When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were three students left over. What is the fewe... | [
"17"
] | [
"When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were three students left over. What is the... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffd4890086e53b52822208a631c013e8b28d005d262402a358b504812deac522 | When a coin was tossed 10 times, five heads and five tails were attained. What is the probability that all five heads came up on the first five tosses? | [
"$\\frac{5!5!}{10!}$"
] | [
"When a coin was tossed 10 times, five heads and five tails were attained. What is the probability that all five heads came up on the first five tosses?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e08f6fbd1559315ae6e98cecb53cefde969097808c5e92a9ae0f20d30bd8b9f8 | When a complex number $z$ satisfies $|z-1|=1$, determine the locus of the points $w$ in the complex plane, where $w=\frac{z-i}{z+i}$. | [
"|w + 1 + 2i| = 2"
] | [
"When a complex number $z$ satisfies $|z-1|=1$, determine the locus of the points $w$ in the complex plane, where $w=\\frac{z-i}{z+i}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b7f44183003167de7f76e8d0cddf528599685796ddb6bfbb0919dce3bdee0c59 | When a dice is thrown four times, let \( x_1, x_2, x_3, x_4 \) be the outcomes in this order. For points \( P(x_1, x_2) \), \( O(0, 0) \), and \( Q(x_3, -x_4) \), find the probability that \( \angle POQ \) is acute. | [
"\\( \\frac{605}{1296} \\)"
] | [
"When a dice is thrown four times, let \\( x_1, x_2, x_3, x_4 \\) be the outcomes in this order. For points \\( P(x_1, x_2) \\), \\( O(0, 0) \\), and \\( Q(x_3, -x_4) \\), find the probability that \\( \\angle POQ \\) is acute."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-c119c8dcff2a0c27a35eaa9fa0741cfbb094de739a413d297fff336f3eb28879 | When a die is thrown \( n \) times successively, let \( X_k \) be the number that appears in the \( k \)-th round. Find the probability that the sum \( X_1 + X_2 + \cdots + X_n \) is divisible by 7. | [
"\\(\\frac{1 + (-1)^n \\frac{1}{6^{n-1}}}{7}\\)"
] | [
"When a die is thrown \\( n \\) times successively, let \\( X_k \\) be the number that appears in the \\( k \\)-th round. Find the probability that the sum \\( X_1 + X_2 + \\cdots + X_n \\) is divisible by 7."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-de936a6a4ac575a144ca2ca536dc767012811cb21600cc70e753822ff850b69c | When a die is thrown \( n \) times, what is the probability that the product of the outcomes has a remainder of 1 when divided by 5? | [
"\\(\\frac{5^n + 3}{4 \\cdot 6^n}\\)"
] | [
"When a die is thrown \\( n \\) times, what is the probability that the product of the outcomes has a remainder of 1 when divided by 5?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-76fe7ffbf29028cb17b0a054150919baa3e7ec486e3049f67baca9f49fa5e0e8 | When a die is thrown twice, let the first number be \(x\) and the second number be \(y\). Find the probability that \(2x + y = 5\). | [
"\\(\\frac{1}{18}\\)"
] | [
"When a die is thrown twice, let the first number be \\(x\\) and the second number be \\(y\\). Find the probability that \\(2x + y = 5\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-7db09e3e2bbfa621c6ea8858ba006ebabbcb2ca3ca0d379f6e4f3d45331a3801 | When a fair coin is tossed $n$ times (with $n \geq 2$), and the outcomes determine an array of letters where Heads results in $AA$ and Tails results in $B$, find the probability that the $(n-1)$-th letter from the left in the array is $A$ and the $n$-th letter is $B$. | [
"\\dfrac{n-2}{2^n}"
] | [
"When a fair coin is tossed $n$ times (with $n \\geq 2$), and the outcomes determine an array of letters where Heads results in $AA$ and Tails results in $B$, find the probability that the $(n-1)$-th letter from the left in the array is $A$ and the $n$-th letter is $B$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df1f13637acda5d4e9e21898ab2a52d5b9b2aba0bc83caf15aedd291f1e894a0 | When a fair coin is tossed $n$ times, and the outcomes determine an array of letters where Heads results in $AA$ and Tails results in $B$, find the probability that the $n$-th letter from the left in the array is $A$. | [
"\\dfrac{2^{n+1} + (-1)^n}{3 \\cdot 2^n}"
] | [
"When a fair coin is tossed $n$ times, and the outcomes determine an array of letters where Heads results in $AA$ and Tails results in $B$, find the probability that the $n$-th letter from the left in the array is $A$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-ac412518fcc2c1b0fcf5c1bf6dee1adfeb15dd21d517d5b5ff78ad674419be47 | When a number $k$ is expressed in the octal system, $k!$ has 20 zeros at the end. What are the values of $k$ in the decimal system? | [
"64"
] | [
"When a number $k$ is expressed in the octal system, $k!$ has 20 zeros at the end. What are the values of $k$ in the decimal system?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-99551d4026303aa4fe5ac6146eb88b0e665e4f86863254ea3fdbd7b37a1b6222 | When a number is divided by $2$, it has quotient $x$ and remainder $1$. When the same number is divided by $3$, it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$? | [
"3"
] | [
"When a number is divided by $2$, it has quotient $x$ and remainder $1$. When the same number is divided by $3$, it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-79f3ad348cb314a3d9e243166b6f9af90523f67ab3f21d4509cff031993f839a | When a number is divided by 3, the result is 50 more than if the number had been divided by 4. What is the number? | [
"600"
] | [
"When a number is divided by 3, the result is 50 more than if the number had been divided by 4. What is the number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c182f90ada5fd685aa691127663f18fbc07a120e071105b0373112a5d4835c90 | When a number is divided by 7, the remainder is 2. What is the remainder when three times the number minus 7 is divided by 7? | [
"6"
] | [
"When a number is divided by 7, the remainder is 2. What is the remainder when three times the number minus 7 is divided by 7?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a9c5453ce8d06f18beac9976665b96631697d1f5cc03a6d750107e68f10f0a5 | When a number is removed from a finite collection of consecutive natural numbers, the average of the remaining numbers is 50.55. Find all possible collections of consecutive natural numbers and the number removed. | [
"(1 \\text{ to } 101, 96), (11 \\text{ to } 91, 87), (21 \\text{ to } 81, 78), (31 \\text{ to } 71, 69), (41 \\text{ to } 61, 60)",
"\\((n,n+m,k)\\in\\{(1,101,96),(11,91,87),(21,81,78),(31,71,69),(41,61,60)\\}\\)"
] | [
"When a number is removed from a finite collection of consecutive natural numbers, the average of the remaining numbers is 50.55. Find all possible collections of consecutive natural numbers and the number removed."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-5e3d97ee522efa473bf3695a8def7c60268c618593fa90bfe3fe47f0a8ee23ff | When a passenger flies with MATHS airplane, every kilo of luggage over a certain weight $P$ is taxed. Mr. and Ms. Engel's luggages weigh a total of $58$ kilos and they had to pay $11$ euros in total. Mr. and Ms. Larson's luggages also weigh a total of $58$ kilos, but they had to pay a total of $20$ euros. What is the m... | [
"23.5"
] | [
"When a passenger flies with MATHS airplane, every kilo of luggage over a certain weight $P$ is taxed. Mr. and Ms. Engel's luggages weigh a total of $58$ kilos and they had to pay $11$ euros in total. Mr. and Ms. Larson's luggages also weigh a total of $58$ kilos, but they had to pay a total of $20$ euros. What is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-119dd320d9162b09c1a982af8dbc2c5a9b711fc4b71eb4ecedc2dc636057918c | When a passenger flies with MATHS airplane, every kilo of luggage over a certain weight P is taxed. Mr. and Ms. Robinson's luggage weighs a total of 58 kilos and they paid 11 euros in total. Mr. and Ms. Larson's luggage also weighs a total of 58 kilos, but they paid 20 euros in total. What is the minimum weight P in ki... | [
"18 kilograms"
] | [
"When a passenger flies with MATHS airplane, every kilo of luggage over a certain weight P is taxed. Mr. and Ms. Robinson's luggage weighs a total of 58 kilos and they paid 11 euros in total. Mr. and Ms. Larson's luggage also weighs a total of 58 kilos, but they paid 20 euros in total. What is the minimum weight P ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-aa61812f7f66eff2227161ca9e5fc46baacb5b36223640321effea98c6e166c0 | When a person with an IQ of 104 moved from village A to village B, the average IQ of both villages increased by 1. The sum of the population of the two villages is a prime number, and the sum of the IQ of all people in both villages is 6102. Find the sum of the IQ of the people of village B including the new arrival. | [
"1886"
] | [
"When a person with an IQ of 104 moved from village A to village B, the average IQ of both villages increased by 1. The sum of the population of the two villages is a prime number, and the sum of the IQ of all people in both villages is 6102. Find the sum of the IQ of the people of village B including the new arriv... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-0a4232c5f128382f679cc3f4f20caba047bb068f2c27054b7081e80c7b0024b7 | When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 5. When $p(x)$ is divided by $x + 5,$ the remainder is $-7.$ Find the remainder when $p(x)$ is divided by $(x + 1)(x + 5).$ | [
"3x + 8"
] | [
"When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 5. When $p(x)$ is divided by $x + 5,$ the remainder is $-7.$ Find the remainder when $p(x)$ is divided by $(x + 1)(x + 5).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9e8148930a406932e2a6043aaa208beca669cf0cbe9e190d51e14b1e1c67f78 | When a polynomial \( p(x) \) is divided by: - \((x-2)(x-3)\), the remainder is \(2\) - \((x-1)(x-3)\), the remainder is \(5-x\) - \((x-1)(x-2)\), the remainder is \(6-2x\) What is the least possible degree of \( p(x) \)? | [
"2"
] | [
"When a polynomial \\( p(x) \\) is divided by:\n- \\((x-2)(x-3)\\), the remainder is \\(2\\)\n- \\((x-1)(x-3)\\), the remainder is \\(5-x\\)\n- \\((x-1)(x-2)\\), the remainder is \\(6-2x\\)\n\nWhat is the least possible degree of \\( p(x) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf5db594c19872aa165714e8c0eb1659c4aacdbb9eac0d6310cb53a9c6af357d | When a polynomial is divided by $-3x^5 + 10x - 11,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas. | [
"0,1,2,3,4"
] | [
"When a polynomial is divided by $-3x^5 + 10x - 11,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-0dd9cb1e049aab888a71ddffc62c27a6d928baba1557a1b20de084b9b4cf7ca7 | When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas. | [
"0,1"
] | [
"When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-f1a27eaf34c5f03e8c05b23ac3fbe8b1c8fe20ce0c61d92e7aeef94e2e855fcf | When a polynomial is divided by \( (x-1) \), the remainder is 1. When the quotient from this division is divided by \( (x-2) \), the remainder is 2. When the quotient from this division is divided by \( (x-3) \), the remainder is 3, and so on. Find the remainder when the given polynomial is divided by \( (x-1)(x-2)(x-3... | [
"\\(\\sum_{n=1}^{2008} n(x-1)(x-2)\\cdots(x-n+1)\\)"
] | [
"When a polynomial is divided by \\( (x-1) \\), the remainder is 1. When the quotient from this division is divided by \\( (x-2) \\), the remainder is 2. When the quotient from this division is divided by \\( (x-3) \\), the remainder is 3, and so on. Find the remainder when the given polynomial is divided by \\( (x... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-06dd3d09d1e29d73f90a2aaa4a148145299df9162455edb6af8e83156be1b982 | When a positive integer $N$ is expressed in base $9$, it has three digits. When $N$ is expressed in base $6$, it has the same digits, but in reverse order. Determine the middle digit of these three digits. | [
"5"
] | [
"When a positive integer $N$ is expressed in base $9$, it has three digits. When $N$ is expressed in base $6$, it has the same digits, but in reverse order. Determine the middle digit of these three digits."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0843ea8a05d5b263c4f848001f5bf6ed2dfa0aded6fd4d82e5802372211aeb2 | When a positive integer \( N \) is fed into a machine, the output is a number calculated according to the rule shown below. For example, starting with an input of \( N = 7 \), the machine will output \( 3 \cdot 7 + 1 = 22 \). Then if the output is repeatedly inserted into the machine five more times, the final output i... | [
"83"
] | [
"When a positive integer \\( N \\) is fed into a machine, the output is a number calculated according to the rule shown below. For example, starting with an input of \\( N = 7 \\), the machine will output \\( 3 \\cdot 7 + 1 = 22 \\). Then if the output is repeatedly inserted into the machine five more times, the fi... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-81ab8d1fccffc925dbbd81535ed4418a1944fbd85e051740a990ffc8eaf065b3 | When a positive integer \( N \) is fed into a machine, the output is calculated according to the following rule: - If \( N \) is even, the output is \( \frac{N}{2} \). - If \( N \) is odd, the output is \( 3N + 1 \). For example, starting with \( N = 7 \), the machine will output \( 3 \cdot 7 + 1 = 22 \). Then if the o... | [
"83"
] | [
"When a positive integer \\( N \\) is fed into a machine, the output is calculated according to the following rule:\n- If \\( N \\) is even, the output is \\( \\frac{N}{2} \\).\n- If \\( N \\) is odd, the output is \\( 3N + 1 \\).\n\nFor example, starting with \\( N = 7 \\), the machine will output \\( 3 \\cdot 7 +... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-1fb55c195bba697c9e19eebbb986f5ac46c62680b662a3fa6ebe4f3ddadd7064 | When a positive integer \( N \) is fed into a machine, the output is calculated according to the following rule: - If \( N \) is even, the output is \( \frac{N}{2} \). - If \( N \) is odd, the output is \( 3N + 1 \). For example, starting with \( N = 7 \), the sequence of outputs after 6 steps is: \[ 7 \to 22 \to 11 \t... | [
"83"
] | [
"When a positive integer \\( N \\) is fed into a machine, the output is calculated according to the following rule:\n- If \\( N \\) is even, the output is \\( \\frac{N}{2} \\).\n- If \\( N \\) is odd, the output is \\( 3N + 1 \\).\n\nFor example, starting with \\( N = 7 \\), the sequence of outputs after 6 steps is... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 16
} | 37 | 0 | 0 |
omr-ws-5dda32207729754eed0a997a9caf9ae012dfd1f5f200746b0b77a99bf44074db | When a positive integer \( N \) is written in base 9, it is a two-digit number. When \( 6N \) is written in base 7, it is a three-digit number obtained by writing the digit 4 to the right of the two-digit number. Find the decimal representations of all such numbers \( N \). | [
"\\( N = 10 \\) or \\( N = 24 \\)"
] | [
"When a positive integer \\( N \\) is written in base 9, it is a two-digit number. When \\( 6N \\) is written in base 7, it is a three-digit number obtained by writing the digit 4 to the right of the two-digit number. Find the decimal representations of all such numbers \\( N \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-0fd01fd840ada28d3a52e5acfbbe7fbe0d91d682abd19d47ac9a81d91de79a51 | When a positive integer \( N \) is written in base 9, it is a two-digit number. When \( 6N \) is written in base 7, it is a three-digit number obtained from the two-digit number by writing digit 4 to its right. Find the decimal representations of all such numbers \( N \). | [
"24"
] | [
"When a positive integer \\( N \\) is written in base 9, it is a two-digit number. When \\( 6N \\) is written in base 7, it is a three-digit number obtained from the two-digit number by writing digit 4 to its right. Find the decimal representations of all such numbers \\( N \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92ce639da090d88dc6ce8d29ed6a9d467cbd933b457ca0288725fc3922be7903 | When a positive integer \( x \) is divided by a positive integer \( y \), the quotient is \( u \) and the remainder is \( v \), where \( u \) and \( v \) are integers. What is the remainder when \( x + 2uy \) is divided by \( y \)? | [
"v"
] | [
"When a positive integer \\( x \\) is divided by a positive integer \\( y \\), the quotient is \\( u \\) and the remainder is \\( v \\), where \\( u \\) and \\( v \\) are integers. What is the remainder when \\( x + 2uy \\) is divided by \\( y \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b573571bad3fb3b2d4245a275df82fbcbd8c2155551a3dfddc722286a0373c52 | When a positive integer is expressed in base 7, it is $AB_7$, and when it is expressed in base 5, it is $BA_5$. What is the positive integer in decimal? | [
"17"
] | [
"When a positive integer is expressed in base 7, it is $AB_7$, and when it is expressed in base 5, it is $BA_5$. What is the positive integer in decimal?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1520ceee4ebfc7f818c5a8e4ad3ab68bc4acaae1ad500ff2675f5985dd05392 | When a positive three-digit dividend is divided by a positive one-digit divisor, the result is a whole number quotient with a remainder of 1. How many distinct values are possible for this three-digit dividend? | [
"694"
] | [
"When a positive three-digit dividend is divided by a positive one-digit divisor, the result is a whole number quotient with a remainder of 1. How many distinct values are possible for this three-digit dividend?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-ed6fbacdcf01d7c2afe8ea4ba7c8ea477f9e9b7341f5ea67ebc073483eca6bde | When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle? | [
"26"
] | [
"When a right triangle is rotated about one leg, the volume of the cone produced is $800 \\pi$ $\\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \\pi$ $\\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-d2e52a0236bf3fda24483c07a842866deadc5e8c8fb872bcf45f24359c6c951a | When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of the triangle? | [
"26"
] | [
"When a right triangle is rotated about one leg, the volume of the cone produced is $800\\pi \\;\\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\\pi \\;\\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?",
"When a right triang... | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-d0916ac178c74b7a5ca37c63d297abbc8eafa2f034638d6d0abe158a9f3feb4b | When a right triangle is rotated about one leg, the volume of the cone produced is \(800 \pi \text{ cm}^3\). When the triangle is rotated about the other leg, the volume of the cone produced is \(1920 \pi \text{ cm}^3\). What is the length (in cm) of the hypotenuse of the triangle? | [
"26"
] | [
"When a right triangle is rotated about one leg, the volume of the cone produced is \\(800 \\pi \\text{ cm}^3\\). When the triangle is rotated about the other leg, the volume of the cone produced is \\(1920 \\pi \\text{ cm}^3\\). What is the length (in cm) of the hypotenuse of the triangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62da6789744aba3fe4807265d870c5bd86dfe8ec49f4e27f3756fac86e356638 | When a school play charges $2 for admission, an average of 100 people attend. For each 10-cent increase in admission price, the average number of attendees decreases by 1. What admission price would maximize the revenue? Formulate the equation in the form \(a(x-h)^2 + k\). | [
"\\(-\\frac{1}{10}(x+40)^2 + 40\\)"
] | [
"When a school play charges $2 for admission, an average of 100 people attend. For each 10-cent increase in admission price, the average number of attendees decreases by 1. What admission price would maximize the revenue? Formulate the equation in the form \\(a(x-h)^2 + k\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-feadc95a86b31db07493dee0765a1b88f7663e958fb4740b858c9c0372319672 | When a student multiplied the number $66$ by the repeating decimal $$1.\overline{ab}$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.54$ less than the correct answer. What is the $2$-digit integer $ab$? | [
"81"
] | [
"When a student multiplied the number $66$ by the repeating decimal\n$$1.\\overline{ab}$$\nwhere $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.54$ less than the correct answer. What is the $2$-digit integer $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07b9223eda7e3402ba88ed8d367e47c87568dee0f33e00930a2888b2a0e06185 | When a student multiplied the number $66$ by the repeating decimal $$\underline{1}.\underline{a} \underline{b} \underline{a} \underline{b} \cdots = \underline{1}.\overline{\underline{ab}}$$ Where $a$ and $b$ are digits. He did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a}\underline... | [
"75"
] | [
"When a student multiplied the number $66$ by the repeating decimal\n$$\\underline{1}.\\underline{a} \\underline{b} \\underline{a} \\underline{b} \\cdots = \n \\underline{1}.\\overline{\\underline{ab}}$$\nWhere $a$ and $b$ are digits. He did not notice the notation and just multiplied $66$ times $\\underline{1}.... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b31049d74f395fb63a3004cbb15077dcd14db0d68ea9bac85e168a0c1dfccde9 | When a student multiplied the number $66$ by the repeating decimal $1.\overline{ab}$, later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit integer $ab$? | [
"75"
] | [
"When a student multiplied the number $66$ by the repeating decimal $1.\\overline{ab}$, later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit integer $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4c9f07f39856ebc757f5336fe5b9fdd8fb724ce7034460b935adf660235ca294 | When a student multiplied the number $66$ by the repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ by $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the two-digit integer $ab$? | [
"75"
] | [
"When a student multiplied the number $66$ by the repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ by $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the two-digit integer $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f4f6c4bd77667c8f3b57cfc6d1b90972042176ef4b03882afdb6aaa9b52ee6b9 | When a student multiplied the number $66$ by the repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit integer $ab$? | [
"75"
] | [
"When a student multiplied the number $66$ by the repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit integer $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-043651e580dc23cb717bc4106a567ce31d6055665df77c826c11bed7ffdce822 | When a student multiplied the number $66$ by the repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$? | [
"75"
] | [
"When a student multiplied the number $66$ by the repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-3fb119d4fa0a7391cb12cf83bdafdd44737fa2d1a4d82f802ec5c153d567e4f7 | When a student multiplied the number $66$ by the repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.abab$. Later he found out his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$? | [
"75"
] | [
"When a student multiplied the number $66$ by the repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.abab$. Later he found out his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 19,
"tir": 16
} | 35 | 0 | 0 |
omr-ws-2b6fec236d6f68c6190acf0547b75c7d33e6af1553dde423404854a5b8b1efb4 | When a student multiplied the number 66 by the repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied 66 times $1.ab$. Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit integer $ab$? | [
"75"
] | [
"When a student multiplied the number 66 by the repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, he did not notice the notation and just multiplied 66 times $1.ab$. Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit integer $ab$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6669f5512d4e539eb52b1f0f0b19b6f9a34ef01bf8f4d5297d13c3fbb8d94a7c | When a student multiplied the number 66 by the repeating decimal \(1.\overline{ab}\), where \(a\) and \(b\) are digits, he did not notice the notation and just multiplied 66 times \(1.ab\). Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit number \(ab\)? | [
"75"
] | [
"When a student multiplied the number 66 by the repeating decimal \\(1.\\overline{ab}\\), where \\(a\\) and \\(b\\) are digits, he did not notice the notation and just multiplied 66 times \\(1.ab\\). Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit number \\(ab\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4823aadbf23f970f2eabd29198e9b0b0dc9dd9e43151b4841c4eba925bcaecb | When a subgroup $K$ of a group $G$ acts on $G/H$ by left multiplication, what is the necessary condition for the action to have a fixed point? | [
"K \\subseteq gHg^{-1} \\text{ for some } g \\in G"
] | [
"When a subgroup $K$ of a group $G$ acts on $G/H$ by left multiplication, what is the necessary condition for the action to have a fixed point?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-75ae9f6d34964f1f58bf5d86ae9a0ddea2d7cf0406fb6227ebfd7b590e145895 | When a three-digit number in nonary notation (base-$9$) is expressed in septenary notation (base-$7$), the order of the digits is reversed. Express this number in octal notation (base-$8$). | [
"$370_8$"
] | [
"When a three-digit number in nonary notation (base-$9$) is expressed in septenary notation (base-$7$), the order of the digits is reversed. Express this number in octal notation (base-$8$)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
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