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-a9db7ba88cfbbaec4a903a61844c1ef7e5ab80bff537d808a24882443759611b | A number is called $m$-powered if it can be represented as $m^{m^{\dots^{m}}}$ (at least two $m$'s). We call $m$ and $n$ similar if there exists a $m$-powered number equal to a $n$-powered number. Find all pairs $(m,n)$ such that $m$ is similar to $n$ and $m+1$ is similar to $n+1$. | [
"\\(m = n\\)"
] | [
"A number is called $m$-powered if it can be represented as $m^{m^{\\dots^{m}}}$ (at least two $m$'s). We call $m$ and $n$ similar if there exists a $m$-powered number equal to a $n$-powered number. Find all pairs $(m,n)$ such that $m$ is similar to $n$ and $m+1$ is similar to $n+1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-c99e3dea4de1f134307521fe175f50a22d5cff0cc091e42818336ec29b2dda9d | A number is called 'Noone' if it does not have $1$ in its units digit. What is the position of $2019$ as a 'Noone' number? (Note: The position of $2$ is $1$, the position of $3$ is $2$, etc.) | [
"1817"
] | [
"A number is called 'Noone' if it does not have $1$ in its units digit. What is the position of $2019$ as a 'Noone' number? (Note: The position of $2$ is $1$, the position of $3$ is $2$, etc.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8deb00348763c3ee77641b86398d10c3cbd496cd23f3bae4a0ac516ea669b4cb | A number is called *mixed* if it is nonnegative and equals the sum of its digits plus the product of its digits. Find the remainder when the sum of all *mixed* numbers is divided by 10. | [
"1"
] | [
"A number is called *mixed* if it is nonnegative and equals the sum of its digits plus the product of its digits. Find the remainder when the sum of all *mixed* numbers is divided by 10."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67b89723707de5fd35cff5d788ec272688105605e3dc7bfa3e03197d67b6a3af | A number is called 6-composite if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is composite if it has a factor not equal to 1 or itself. In particular, 1 is not composite.) | [
"441"
] | [
"A number is called 6-composite if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is composite if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-be19c93d4b67c00f201c3d10a5ec921aaf2806cd67bcd4ed8705242d7253c722 | A number is called Good if, when prime factorized as \( p_1^{a_1} \cdot p_2^{a_2} \cdot \ldots \cdot p_k^{a_k} \), then \( a_i \leq 2 \) for all \( 1 \leq i \leq k \). Find the sum of all Good numbers from 1 to 123456789123456789 modulo 1000000007. | [
"52516352"
] | [
"A number is called Good if, when prime factorized as \\( p_1^{a_1} \\cdot p_2^{a_2} \\cdot \\ldots \\cdot p_k^{a_k} \\), then \\( a_i \\leq 2 \\) for all \\( 1 \\leq i \\leq k \\). Find the sum of all Good numbers from 1 to 123456789123456789 modulo 1000000007."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-286d4720396fb7ab88606ca3d0c20c0bf487ce2e2905bf88de2f000acbd77abe | A number is called Norwegian if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. | [
"1344"
] | [
"A number is called Norwegian if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-d0eff1349e452f005265236d901579fd89de98ead1c39d41b2d8a7f5444cdbd0 | A number is called WOW if the sum of its digits is divisible by 3. What is the sum of the largest WOW number and the smallest WOW number? | [
"1110"
] | [
"A number is called WOW if the sum of its digits is divisible by 3. What is the sum of the largest WOW number and the smallest WOW number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fee72a91cac79313a53e8aac16b72e414f15909b75b121988310d7ba5e241ad4 | A number is called [i]fun[/i] if it is composed of two digits followed by the square of those two digits, and the entire number is divisible by one less than the first two digits. For example, 12144 is [i]fun[/i] since $12^2 = 144$ and $12144$ is divisible by $11$. How many 5-digit [i]fun[/i] numbers are there? | [
"2"
] | [
"A number is called [i]fun[/i] if it is composed of two digits followed by the square of those two digits, and the entire number is divisible by one less than the first two digits. For example, 12144 is [i]fun[/i] since $12^2 = 144$ and $12144$ is divisible by $11$. How many 5-digit [i]fun[/i] numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7152a8ad5fd193be733d51762845c653a7b1524918e3e8482c389765f2ca80ba | A number is called a "red" number if it has exactly eight positive divisors and the fifth divisor in increasing order is equal to three times the third divisor minus 4. How many "red" numbers smaller than 2014 exist? | [
"7"
] | [
"A number is called a \"red\" number if it has exactly eight positive divisors and the fifth divisor in increasing order is equal to three times the third divisor minus 4. How many \"red\" numbers smaller than 2014 exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61cde46a4365b954f1c38906b3568fb6e39a9336554c913a75d6228f563a9c5d | A number is called a perfect product if it can be written as n more than the product of three consecutive integers, where n is the middle of the three integers. No perfect product can be rewritten as the sum of k consecutive positive integers. Find the smallest possible value of k. | [
"4"
] | [
"A number is called a perfect product if it can be written as n more than the product of three consecutive integers, where n is the middle of the three integers. No perfect product can be rewritten as the sum of k consecutive positive integers. Find the smallest possible value of k."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38f0d17d9a64a803f7716efc5c269cdd8ff1b0a2ae4984594053064de736129c | A number is called a trident if it is the sum of three powers of two different ones. List the tridents in ascending order. Which number occupies the 100th position in this list? | [
"577"
] | [
"A number is called a trident if it is the sum of three powers of two different ones. List the tridents in ascending order. Which number occupies the 100th position in this list?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 22
} | 23 | 0 | 0 |
omr-ws-de11ee8f493b24b5271a253094ff8abd5b8481b7766dd8b41b83a9615d4d9f50 | A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many visible factor numbers are there from 100 through 150, inclusive? | [
"19"
] | [
"A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many visible factor numbers are there from 100 through 150, inclusive?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dbc3f4c630285a0eb8bea0098649f73edb75151f81f8b0095058d82c462860e8 | A number is called capicua if it reads the same forwards and backwards (e.g., $8, 23432, 6446$). Let $x_1 < x_2 < \cdots < x_i < x_{i+1} < \cdots$ be the sequence of all capicua numbers. For each $i$, define $y_i = x_{i+1} - x_i$. How many distinct primes are in the set $\{y_1, y_2, \ldots\}$? | [
"2"
] | [
"A number is called capicua if it reads the same forwards and backwards (e.g., $8, 23432, 6446$). Let $x_1 < x_2 < \\cdots < x_i < x_{i+1} < \\cdots$ be the sequence of all capicua numbers. For each $i$, define $y_i = x_{i+1} - x_i$. How many distinct primes are in the set $\\{y_1, y_2, \\ldots\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-538e425b01994f5b3569a35d9fc870c5ccc85abc86f00ff40cb8063e11dbacd7 | A number is called complemented if it is the product of the companion to $10$ of its digits. Find all complemented numbers that do not end with zeros. | [
"\\(5\\)"
] | [
"A number is called complemented if it is the product of the companion to $10$ of its digits. Find all complemented numbers that do not end with zeros."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-835031da98092562de4cb8bb8fbc7ac52b491dad0c1c0716dd9110a8fcc348af | A number is called evil if the number of $1$s in its binary expansion is even. Find the sum of the first $2012$ evil numbers. | [
"4047138 or 4051162"
] | [
"A number is called evil if the number of $1$s in its binary expansion is even. Find the sum of the first $2012$ evil numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 13
} | 27 | 0 | 0 |
omr-ws-60cc0af5723f25d562b53aea95affe0408f323e426abb0e9b7f7c7cd682029ac | A number is called extra-constructible if it can be obtained using a compass, straightedge, and an angle trisector. Determine the smallest positive integer $n$ such that $\sqrt[3]{2}$ can be expressed as a root of a polynomial equation with integer coefficients and degree at most $n$, assuming $\sqrt[3]{2}$ is extra-co... | [
"3"
] | [
"A number is called extra-constructible if it can be obtained using a compass, straightedge, and an angle trisector. Determine the smallest positive integer $n$ such that $\\sqrt[3]{2}$ can be expressed as a root of a polynomial equation with integer coefficients and degree at most $n$, assuming $\\sqrt[3]{2}$ is e... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d78cc8e7138c722fb47bc07e0745130102c05704ee2835567cc8120c20076f97 | A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15? | [
"4"
] | [
"A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a87521712c573fbd939f9a2aa24732c5b50f03d971eedf2866f62924fbfaf277 | A number is called polite if it can be expressed as the sum of two or more consecutive positive integers. Let $g(n)$ be the smallest polite number that is greater than $2^n$. Find $g(10)$. | [
"1025"
] | [
"A number is called polite if it can be expressed as the sum of two or more consecutive positive integers. Let $g(n)$ be the smallest polite number that is greater than $2^n$. Find $g(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-315db85bd02593d7187b01841d886430592fed0e176dbc18b13c25644efad120 | A number is called wary if all its digits are different and the sum of the digits is 36. Let $N$ be the number of wary numbers. Find $\frac{N}{7!}$. | [
"106"
] | [
"A number is called wary if all its digits are different and the sum of the digits is 36. Let $N$ be the number of wary numbers. Find $\\frac{N}{7!}$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2860da261b332a13fd2af1b6d8926274324bea65f23720d8756a58c126f01a6c | A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$? | [
"\\(\\frac{9}{200}\\)"
] | [
"A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ca96023d90e146632b8f5da969329863bcfc32fa3ba50eee92034b177d117923 | A number is chosen at random from the set of consecutive natural numbers $\{1, 2, 3, \ldots, 24\}$. What is the probability that the number chosen is a factor of $4!$? Express your answer as a common fraction. | [
"\\frac{1}{3}"
] | [
"A number is chosen at random from the set of consecutive natural numbers $\\{1, 2, 3, \\ldots, 24\\}$. What is the probability that the number chosen is a factor of $4!$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2fd83fb7a34b9345a201ec344e48c27f7968cb6fffbed415b0120f0a942b2926 | A number is chosen uniformly at random from among the positive integers less than $10^8$. Given that the sum of the digits of the number is $7$, what is the probability that the number is prime? | [
"0"
] | [
"A number is chosen uniformly at random from among the positive integers less than $10^8$. Given that the sum of the digits of the number is $7$, what is the probability that the number is prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-6f9081a9abf1f144b240b267b7022152e89b68ee88f2b862d25d565e5e144249 | A number is considered beautiful if, by placing the operations $+$, $-$, $*$, $/$ and brackets between and around its digits, you can form an expression whose value is divisible by 10. What is the minimum number of digits that must be in a number to ensure that every number is beautiful? | [
"4"
] | [
"A number is considered beautiful if, by placing the operations $+$, $-$, $*$, $/$ and brackets between and around its digits, you can form an expression whose value is divisible by 10. What is the minimum number of digits that must be in a number to ensure that every number is beautiful?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-f5e096583627cccbd072be40d59b6fa402c01db8a35d6a674d10eb86921b63b6 | A number is defined as *lucky* if the repeated sum of the squares of its digits eventually leads to the number $1$. Find infinitely many pairs of consecutive numbers, each of which is lucky. | [
"(31, 32), (301, 302), (3001, 3002), \\ldots"
] | [
"A number is defined as *lucky* if the repeated sum of the squares of its digits eventually leads to the number $1$. Find infinitely many pairs of consecutive numbers, each of which is lucky."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-686e5071bb734bdfb659581719d2c246605d2bdc70f4a2bd40cfac8f0091ef4d | A number is defined as [i]additive[/i] if it has three digits, all of which are different, and the sum of two of the digits is equal to the remaining one. Find the sum of all such additive numbers. | [
"13986"
] | [
"A number is defined as [i]additive[/i] if it has three digits, all of which are different, and the sum of two of the digits is equal to the remaining one. Find the sum of all such additive numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61e5d5438b761a19bcb46041b5fa0fbd8ab3dcff7fcf579764e28961730942d7 | A number is defined as interesting if it is a 6-digit integer that contains no zeros, its first 3 digits are strictly increasing, and its last 3 digits are non-increasing. What is the average of all interesting numbers? | [
"308253"
] | [
"A number is defined as interesting if it is a 6-digit integer that contains no zeros, its first 3 digits are strictly increasing, and its last 3 digits are non-increasing. What is the average of all interesting numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-0bb4be0aef23eacbeeb270c68c21f48c6d19210b1184957c8de83e372a5f0b4d | A number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8.$ For example, the number $47\,389\,248$ is divisible by $8$ because $248$ is divisible by $8.$ However, $47\,389\,284$ is not divisible by $8$ because $284$ is not divisible by $8.$ If $992\,466\,1A6$ is divisible by $8,$ where... | [
"10"
] | [
"A number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8.$ For example, the number $47\\,389\\,248$ is divisible by $8$ because $248$ is divisible by $8.$ However, $47\\,389\\,284$ is not divisible by $8$ because $284$ is not divisible by $8.$\n\nIf $992\\,466\\,1A6$ is divisible... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-784046abb39098ef14f18f175d10bf1d8a4fc5d49f0c35134c940f08614923a1 | A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\,836$ is divisible by $9$ but $19\,825$ is not. If $D\,767\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$ | [
"25"
] | [
"A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\\,836$ is divisible by $9$ but $19\\,825$ is not.\n\nIf $D\\,767\\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b0e57a8ff4a1f884a0b65de092f561d735edb0671646db1aea5137172a94736 | A number is drawn from the set of natural numbers less than ten thousand. What is the probability that the first and last digits of the number are the same? | [
"\\(\\frac{112}{1111}\\)"
] | [
"A number is drawn from the set of natural numbers less than ten thousand. What is the probability that the first and last digits of the number are the same?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-cb33c8f3ed6732dbcedd9032644037cc936129ee0377b21af5f58ab497620674 | A number is exactly one-third of a perfect square. After one-quarter of this number is removed, the remaining number is a perfect cube. What is the smallest possible value of this number? | [
"972"
] | [
"A number is exactly one-third of a perfect square. After one-quarter of this number is removed, the remaining number is a perfect cube. What is the smallest possible value of this number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d11bb032ae7b1bc9ca5a009bedc17c993c163b1b2d3cf0f331a3311681af1f4 | A number is formed by two digits whose sum is 11. If the digits are reversed, the number is increased by 27. Find the number. | [
"47"
] | [
"A number is formed by two digits whose sum is 11. If the digits are reversed, the number is increased by 27. Find the number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2da643bd78448a9cedca389f2c7174210c6bd3a262ab8e03953d8906884f86dd | A number is hole-y if at least one of its digits has a hole in them when you write them out. By this, we mean that $0$, $4$, $6$, $8$, $9$ have holes in them, and $1$, $2$, $3$, $5$, $7$ do not. How many $4$-digit integers are hole-y? | [
"8375"
] | [
"A number is hole-y if at least one of its digits has a hole in them when you write them out. By this, we mean that $0$, $4$, $6$, $8$, $9$ have holes in them, and $1$, $2$, $3$, $5$, $7$ do not. How many $4$-digit integers are hole-y?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14d44de6f964afd4b8b28cc320016c7dba3ad8b1ed624866fe8d461f306c1217 | A number is put-together if it can be written by taking two numbers that are next to each other (differ by one) and putting them together (in increasing order). For example, 1920 and 910 are put-together, and 19020 and 1312 are not put-together. How many numbers between 10 and 1000000 are put-together? | [
"998"
] | [
"A number is put-together if it can be written by taking two numbers that are next to each other (differ by one) and putting them together (in increasing order). For example, 1920 and 910 are put-together, and 19020 and 1312 are not put-together. How many numbers between 10 and 1000000 are put-together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-2b15addbc2a7bd8e218af8323d5f864012e5bbf02f0550829b7f856109ab5305 | A number is said to be "fourfriend" if it and all the numbers obtained by rearranging its digits are multiples of 4. Determine how many 7-digit numbers are fourfriends. Note: A fourfriend number cannot have the digit 0. | [
"128"
] | [
"A number is said to be \"fourfriend\" if it and all the numbers obtained by rearranging its digits are multiples of 4. Determine how many 7-digit numbers are fourfriends. Note: A fourfriend number cannot have the digit 0."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f21098ebe3c4477d7c43936dd3ceb072e31fbf91c657fb5a5a158680b7db17f0 | A number is said to be [i][color=#F1C232]GOLDEN[/color][/i] if it has an even number of digits, all digits are distinct, and the sum of the first half of the digits equals the sum of the second half, and the product of the first half of the digits equals the product of the second half. Find the smallest [i][color=#F1C2... | [
"1523"
] | [
"A number is said to be [i][color=#F1C232]GOLDEN[/color][/i] if it has an even number of digits, all digits are distinct, and the sum of the first half of the digits equals the sum of the second half, and the product of the first half of the digits equals the product of the second half. Find the smallest [i][color=... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-416b1fffe74ae440cbc69b64a60c6dd6744d43c3174c6176adae9a6b27adf470 | A number is said to be interesting if $2018$ divides $d(n)$, where $d(n)$ is the number of positive divisors of $n$. Find all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ where all its terms are interesting. | [
"2p^{1008}"
] | [
"A number is said to be interesting if $2018$ divides $d(n)$, where $d(n)$ is the number of positive divisors of $n$. Find all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ where all its terms are interesting."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-24da69cc680d6a1fff108eb3570becc3dd28af52bdeb540907fcff5f28ba01b4 | A number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43. What is the probability that this number is divisible by 11? | [
"\\(\\frac{1}{5}\\)"
] | [
"A number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43. What is the probability that this number is divisible by 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-c9c5317a29793536159c16689425fc7576073f00d42de017326e3cb80499790e | A number is selected from 1 to 59. This number is then multiplied by 27 and reduced modulo 101. If \( X \) is the expected value of the result, what is \( X \) modulo 101? | [
"2"
] | [
"A number is selected from 1 to 59. This number is then multiplied by 27 and reduced modulo 101. If \\( X \\) is the expected value of the result, what is \\( X \\) modulo 101?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b7772b8df647b0cf1cae9775fec536b2d5324b4d5efbaea83311bf3909b8ffc | A number is the product of three positive prime factors. Find the largest possible value for that number if the sum of the squares of its three prime factors is 2331. | [
"20677"
] | [
"A number is the product of three positive prime factors. Find the largest possible value for that number if the sum of the squares of its three prime factors is 2331."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-94ae96c9646e0719ccd5b8d3f78db928161107e03deb280565f019cdf858f0be | A number is written in each corner of a cube. On each step, each number is replaced with the average of the three numbers in the three adjacent corners (all numbers are replaced simultaneously). After ten such steps, every number returns to its initial value. If the initial numbers are $a, b, c, d, e, f, g, h$, calcula... | [
"0"
] | [
"A number is written in each corner of a cube. On each step, each number is replaced with the average of the three numbers in the three adjacent corners (all numbers are replaced simultaneously). After ten such steps, every number returns to its initial value. If the initial numbers are $a, b, c, d, e, f, g, h$, ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d17c47ed843020e113a600db280a670385d8d221415a42bb39c498e2a10db153 | A number is written in each square of a chessboard, so that each number not on the border is the mean of the 4 neighboring numbers. If the largest number written on the chessboard is \( N \), and \( k \) is the number of border squares that contain the number \( N \), find the minimum possible value of \( k \). | [
"1"
] | [
"A number is written in each square of a chessboard, so that each number not on the border is the mean of the 4 neighboring numbers. If the largest number written on the chessboard is \\( N \\), and \\( k \\) is the number of border squares that contain the number \\( N \\), find the minimum possible value of \\( k... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-acfb3855f7fc65061f69fb6f8bbeed655f73b85c4de1b5b3d9f5323e0ace0d2c | A number line has its 0 fixed at (1,0) on a unit circle. What central angle, in radians, is formed by the radius to (1,0) and the radius to the point that corresponds to 5.7 on the number line when it is wrapped around the circle? | [
"\\( 2\\pi - 5.7 \\)"
] | [
"A number line has its 0 fixed at (1,0) on a unit circle. What central angle, in radians, is formed by the radius to (1,0) and the radius to the point that corresponds to 5.7 on the number line when it is wrapped around the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-9f31f85f9c2e1ed266e0489022291749bd88c98a5095c80b06697de7e0615f27 | A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. On turn $i$, starting with $i=1$, the following two things happen in order: 1. The parcel is passed $i^2$ positions clockwise; and 2. The child currently holding the parcel unwraps a layer and claims the prize inside. Fo... | [
"\\( N = 2^a 3^b \\) where \\( a, b \\) are non-negative integers."
] | [
"A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. On turn $i$, starting with $i=1$, the following two things happen in order:\n1. The parcel is passed $i^2$ positions clockwise; and\n2. The child currently holding the parcel unwraps a layer and claims the prize ins... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-e9caf2339991c45ccd97f92c10f169b48e963df8893c3eca29c2264765efe3b8 | A number of $n$ lamps ($n \ge 3$) are placed at the vertices of a regular $n$-gon. Initially, all lamps are off. In each step, Lisa chooses three lamps that form an isosceles triangle and toggles their states (from off to on and vice versa). What is the minimum number of steps required for Lisa to turn on all the lamps... | [
"n"
] | [
"A number of $n$ lamps ($n \\ge 3$) are placed at the vertices of a regular $n$-gon. Initially, all lamps are off. In each step, Lisa chooses three lamps that form an isosceles triangle and toggles their states (from off to on and vice versa). What is the minimum number of steps required for Lisa to turn on all the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-c93fb0ac549edd1318177a501469af90864452d1fa62bc8a2d56fdeabdb1e03e | A number of $n$ students sit around a round table. It is known that there are as many male students as female students. If the number of pairs of 2 people sitting next to each other is counted, it turns out that the ratio between adjacent pairs of the same sex and adjacent pairs of the opposite sex is $3:2$. Find the s... | [
"10"
] | [
"A number of $n$ students sit around a round table. It is known that there are as many male students as female students. If the number of pairs of 2 people sitting next to each other is counted, it turns out that the ratio between adjacent pairs of the same sex and adjacent pairs of the opposite sex is $3:2$. Find ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-98abff2ff660578f81fb4e33a51100b1e1115acf431f97d3507d2e1b246b1441 | A number of 1 cm cubes are glued together to construct the sides and bottom of an open box. If the outside dimensions of the box are 10 cm × 10 cm × 10 cm and the sides and bottom are 1 cm thick, then how many cubes are used? | [
"424"
] | [
"A number of 1 cm cubes are glued together to construct the sides and bottom of an open box. If the outside dimensions of the box are 10 cm × 10 cm × 10 cm and the sides and bottom are 1 cm thick, then how many cubes are used?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-518e6a5d1f3ae527ffd1eb3067bddb7d07ca7d4f7b1a58ad737e7d5dd2d4a0a5 | A number of 17 workers stand in a row. Every contiguous group of at least 2 workers is a brigade. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by 4. Calculate the remainder when the number of such ways to assign the leaders i... | [
"0"
] | [
"A number of 17 workers stand in a row. Every contiguous group of at least 2 workers is a brigade. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by 4. Calculate the remainder when the number of such ways to assign the lead... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41db1cf70e3e4b0ba67f54722efa5f73827b819f3891567e7018b5015c08de10 | A number of 5 distinct digits is written down at random using the digits 1, 2, 3, 4, 5, and 6. What is the probability that the number is divisible by 3 but not by 12? | [
"\\(\\frac{3}{4}\\)"
] | [
"A number of 5 distinct digits is written down at random using the digits 1, 2, 3, 4, 5, and 6. What is the probability that the number is divisible by 3 but not by 12?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a73952c43763a6e7c83c71ac984818d8172d30841b23b682bffb27f59cec899 | A number of \( n \) tokens are arranged on the third row of a \( 3 \times n \) table. Each token can be moved one row forward or backward, within the table, provided that no other token is located on the same row at its right (either before or after the move). Find the minimal number of moves in which one can move all ... | [
"\\( 3^n - 1 \\)"
] | [
"A number of \\( n \\) tokens are arranged on the third row of a \\( 3 \\times n \\) table. Each token can be moved one row forward or backward, within the table, provided that no other token is located on the same row at its right (either before or after the move). Find the minimal number of moves in which one can... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2595fa01ad504b4c08ad46754379278d395b438f575627814dde59e1c63b4395 | A number of \(n > m \geq 1\) soccer teams play a full tournament, each team meeting (once) each other. Points are awarded: 2 for a victory, 1 for a tie, and 0 for a loss. At the end, each team has won half of its points against the \(m\) teams placed last (including each of these teams, who won half of its points again... | [
"m = \\frac{k(k + 1)}{2}"
] | [
"A number of \\(n > m \\geq 1\\) soccer teams play a full tournament, each team meeting (once) each other. Points are awarded: 2 for a victory, 1 for a tie, and 0 for a loss. At the end, each team has won half of its points against the \\(m\\) teams placed last (including each of these teams, who won half of its po... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5737a5420b26ee1d7142742d6b20a987cd5a29e9e18f67013d379868cbc418b6 | A number of aircrafts are stationed on a small island. Each aircraft can carry enough gasoline to fly exactly half the way around the world. Gasoline can be transferred between aircrafts in flight, but no more than the donor or recipient can hold. The only source of gasoline is on the island, and all planes must return... | [
"3"
] | [
"A number of aircrafts are stationed on a small island. Each aircraft can carry enough gasoline to fly exactly half the way around the world. Gasoline can be transferred between aircrafts in flight, but no more than the donor or recipient can hold. The only source of gasoline is on the island, and all planes must r... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-057008da26032f5c97073e68dcfff98796d56961a2bb458e0144da0b1bf184b8 | A number of airplanes are positioned at \( x = 0 \) on the real number line. Each plane can travel a distance of 1 unit. Planes can refuel each other instantly and can return to the carrier at \( x = 0 \) to refuel. What is the minimum number of planes required such that at least one plane can reach the island at \( x ... | [
"3"
] | [
"A number of airplanes are positioned at \\( x = 0 \\) on the real number line. Each plane can travel a distance of 1 unit. Planes can refuel each other instantly and can return to the carrier at \\( x = 0 \\) to refuel. What is the minimum number of planes required such that at least one plane can reach the island... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 28,
"genselect": 3
} | 31 | 0 | 0 |
omr-ws-e7756a2aaa0cc5a14a4b6f3a611d9c4e820cb8a70ccf6685a0f26580a490a99e | A number of chords are drawn in a circle of radius $1$ so that each diameter crosses no more than $k$ chords. If there are a total of $n$ chords, find the maximum possible value of the sum of the lengths of all the chords in terms of $k$ and $n$. | [
"2n \\sin\\left(\\frac{\\pi}{2(k + 1)}\\right)"
] | [
"A number of chords are drawn in a circle of radius $1$ so that each diameter crosses no more than $k$ chords. If there are a total of $n$ chords, find the maximum possible value of the sum of the lengths of all the chords in terms of $k$ and $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-529924ce61efb87d5381f296b15e693f31175d38789597415894d4a1f0e2c1fc | A number of coins is first divided into 200 groups and then into 300 groups. A coin is considered "special" if, in the second grouping, it is in a group with fewer coins than the group it was in during the first grouping. Determine the minimum number of "special" coins that can exist. | [
"101"
] | [
"A number of coins is first divided into 200 groups and then into 300 groups. A coin is considered \"special\" if, in the second grouping, it is in a group with fewer coins than the group it was in during the first grouping. Determine the minimum number of \"special\" coins that can exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08b40e407675d170b061061ce628c84dae30a83c701a5dc8b39e36b14f84d55a | A number of different objects have been distributed into $n$ boxes $B_1, B_2, \dots, B_n$. All the objects are then redistributed into $n + 1$ new boxes $B_1^\ast, B_2^\ast, \dots, B_{n+1}^\ast$ such that no new box is empty. Let $S$ be the sum of the differences between the number of objects in each old box and the nu... | [
"1"
] | [
"A number of different objects have been distributed into $n$ boxes $B_1, B_2, \\dots, B_n$. All the objects are then redistributed into $n + 1$ new boxes $B_1^\\ast, B_2^\\ast, \\dots, B_{n+1}^\\ast$ such that no new box is empty. Let $S$ be the sum of the differences between the number of objects in each old box ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-391e805f9d1939a6c34362e3ac2587ae460625b0cad084000a26cf942b2558e5 | A number of different objects have been distributed into \( n \) boxes \( B_1, B_2, \cdots, B_n \). All the objects are then redistributed into \( n + 1 \) new boxes \( B_1^*, B_2^*, \cdots, B_{n+1}^* \) such that no new box is empty. Let \( S \) be the sum of the differences between the number of objects in each new b... | [
"2"
] | [
"A number of different objects have been distributed into \\( n \\) boxes \\( B_1, B_2, \\cdots, B_n \\). All the objects are then redistributed into \\( n + 1 \\) new boxes \\( B_1^*, B_2^*, \\cdots, B_{n+1}^* \\) such that no new box is empty. Let \\( S \\) be the sum of the differences between the number of obje... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-57184caebec8c6b0df1568d97cacdd1dd32f5180306bf4642a7aa1656473570d | A number of guards were hired for the night shift at a museum. Each guard was assigned to watch a different room. Due to boredom, each guard at some point decides to take a walk around the museum before coming back to their assigned room, having visited every room of the museum exactly once. Miraculously, no two guards... | [
"3"
] | [
"A number of guards were hired for the night shift at a museum. Each guard was assigned to watch a different room. Due to boredom, each guard at some point decides to take a walk around the museum before coming back to their assigned room, having visited every room of the museum exactly once. Miraculously, no two g... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-192b1e8105af533648e1500042fb90e4c1ec79e9b9375ad3aaaef4f7064b10dc | A number of identical buildings must be constructed to yield a total living area of 40000 m$^2$. The cost of a single building depends on the cost of the foundation, which is proportional to the square root of the living area in the building, and on the cost of the surface area of the structure, which is proportional t... | [
"$n = 8$. The cost is $2.8\\sqrt 2$ min rubles approx"
] | [
"A number of identical buildings must be constructed to yield a total living area of 40000 m$^2$. The cost of a single building depends on the cost of the foundation, which is proportional to the square root of the living area in the building, and on the cost of the surface area of the structure, which is proportio... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c738100824e128e0b4963f57ae75626142b33e485e0742432769ce99be95cb92 | A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the b... | [
"173"
] | [
"A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to ... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-47f1056b42c1f4ccfb7a8f3c19a17e68395c27d5f7a0f2ba30bdced7f5b41931 | A number of men enter a disreputable establishment and each one leaves a coat and an umbrella at the door. When they leave hurriedly, no man gets both the right coat and the umbrella. If there are 5 men, calculate the number of ways in which this can happen. | [
"11844"
] | [
"A number of men enter a disreputable establishment and each one leaves a coat and an umbrella at the door. When they leave hurriedly, no man gets both the right coat and the umbrella. If there are 5 men, calculate the number of ways in which this can happen."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-def81f57d0027e97e724cb1d5fdda802dd04eda1eea60cacb6b1daaaea8ef0fb | A number of men undertook to dig a ditch and could have finished the job in 6 hours, if they had begun working simultaneously, but they began one after another, the intervals between their starting times being equal. After the last worker had begun working a time interval of the same length elapsed and the job was fini... | [
"They worked 10 hours"
] | [
"A number of men undertook to dig a ditch and could have finished the job in 6 hours, if they had begun working simultaneously, but they began one after another, the intervals between their starting times being equal. After the last worker had begun working a time interval of the same length elapsed and the job was... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e66f9548802eb21634e7c9c34cd1a4243b7a1aff9f685ee6b77d57e207ccfb7 | A number of mirrors (closed line segments off which light reflects) are placed in the plane. Each mirror has length $1$ and both endpoints at lattice points with a distance of at most $2020.5$ from the origin. A laser is fired from $(-2021, 0)$ at some angle. If the laser never passes through another lattice point, mus... | [
"Yes"
] | [
"A number of mirrors (closed line segments off which light reflects) are placed in the plane. Each mirror has length $1$ and both endpoints at lattice points with a distance of at most $2020.5$ from the origin. A laser is fired from $(-2021, 0)$ at some angle. If the laser never passes through another lattice point... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b958e81b13bb7852096a51cfd57e5e664bcc96047fe6718a84240165aabaec9 | A number of persons are seated at a round table. It is known that there are 7 women who have a woman to their right and 12 women who have a man to their right. Additionally, 3 out of every 4 men have a woman to their right. How many people are seated at the table? | [
"35"
] | [
"A number of persons are seated at a round table. It is known that there are 7 women who have a woman to their right and 12 women who have a man to their right. Additionally, 3 out of every 4 men have a woman to their right. How many people are seated at the table?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c4f39a6c8741bbbb90a234dc2626bbf37059935a19e6cfcedb0f1ac02421d5cb | A number of persons are seated at a round table. There are 7 women who have a woman to their right and 12 women who have a man to their right. Additionally, 3 out of every 4 men have a woman to their right. How many people are seated at the table? | [
"35"
] | [
"A number of persons are seated at a round table. There are 7 women who have a woman to their right and 12 women who have a man to their right. Additionally, 3 out of every 4 men have a woman to their right. How many people are seated at the table?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-d8820035ef10205699fdd4a918abcccd0da95a979438b3f0fcc375d79498c92f | A number of persons were to pay equal amounts to a total of 72 roubles. If there were 3 persons less, then each would have to contribute 4 roubles more. How many people were there? | [
"9 persons"
] | [
"A number of persons were to pay equal amounts to a total of 72 roubles. If there were 3 persons less, then each would have to contribute 4 roubles more. How many people were there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8dd8e20c4dd9addc0310c7aec96f8026486ef3ba8be518c2a01ac379fb00102f | A number of players form seven teams, with each pair of teams having one common member and each player being on exactly two teams. How many players are there? | [
"21"
] | [
"A number of players form seven teams, with each pair of teams having one common member and each player being on exactly two teams. How many players are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-26f6bce26e168023dc5130d232e5482dbcb0adc8981910f6aa4062fcd0028f34 | A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished be... | [
"31"
] | [
"A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finish... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-c825ba6d94571ce869156247b195fb6b8ee1272595c42c52297e54eeb7569b87 | A number of signal lights are equally spaced along a one-way railroad track, labeled in order $1, 2, \ldots, N, N \geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of tra... | [
"d \\left( \\sum_{k=1}^{K} \\frac{1}{v_k} + (N - 2) \\max_{1 \\leq k \\leq K} \\left( \\frac{1}{v_k} \\right) \\right)"
] | [
"A number of signal lights are equally spaced along a one-way railroad track, labeled in order $1, 2, \\ldots, N, N \\geq 2.$ As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-7570d27a3d299e75fe513506855b3c1f4f02b287bc4feb2b17cfec0a4d916d93 | A number of students participate in an exam which has the following composition of problems: - The first section consists of 3 problems with 2 choices (either True or False) - The second section consists of 5 problems with 5 choices (A, B, C, D, E). The minimum number of students who need to participate in the exam so ... | [
"25001"
] | [
"A number of students participate in an exam which has the following composition of problems:\n- The first section consists of 3 problems with 2 choices (either True or False)\n- The second section consists of 5 problems with 5 choices (A, B, C, D, E).\nThe minimum number of students who need to participate in the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80c8dcfc6d9580071796c31cd21df3e9df5fd4b2964279ea7d8302ac61a4884d | A number of unit discs are placed inside a square of side length $100$ such that no two discs share a common interior point, and every segment of length $10$ lying entirely within the square intersects at least one disc. Determine the minimum number of discs required to satisfy these conditions. | [
"2601"
] | [
"A number of unit discs are placed inside a square of side length $100$ such that no two discs share a common interior point, and every segment of length $10$ lying entirely within the square intersects at least one disc. Determine the minimum number of discs required to satisfy these conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-128684f9feb39ade95082a37a5330f986fc8257bea229b194c876dc095852267 | A number of unit squares are placed in a line. Let $O$ be the bottom left corner of the first square and let $P$ and $Q$ be the top right corners of the 2012th and 2013th squares, respectively. The lines $OP$ and $OQ$ intersect the right side of the first square at $X$ and $Y$, respectively. Determine the area of trian... | [
"\\(\\frac{1}{8100312}\\)"
] | [
"A number of unit squares are placed in a line. Let $O$ be the bottom left corner of the first square and let $P$ and $Q$ be the top right corners of the 2012th and 2013th squares, respectively. The lines $OP$ and $OQ$ intersect the right side of the first square at $X$ and $Y$, respectively. Determine the area of ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-1ddfb68f6b3df0dbb4efdab92a38536f3887767661582258e3aa37d4f56256d2 | A number satisfies the congruences $x \equiv 0 \pmod{9}$ and $x \equiv 4 \pmod{5}$. Find the smallest positive integer $x$ that satisfies both congruences and express it in the form $x \equiv k \pmod{45}$. What is the value of $k$? | [
"9"
] | [
"A number satisfies the congruences $x \\equiv 0 \\pmod{9}$ and $x \\equiv 4 \\pmod{5}$. Find the smallest positive integer $x$ that satisfies both congruences and express it in the form $x \\equiv k \\pmod{45}$. What is the value of $k$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e19276d47da21dece0555180863ce3d89140e8dd9ea7c72c1cc79534c8cb74c1 | A number triangle has six consecutive integers in some order in the bottom line and a top number of 2006. Provide an example of such a number triangle with the numbers 61, 62, 63, 64, 65, and 66 in some order in the bottom line. | [
"65, 62, 61, 64, 63, 66"
] | [
"A number triangle has six consecutive integers in some order in the bottom line and a top number of 2006. Provide an example of such a number triangle with the numbers 61, 62, 63, 64, 65, and 66 in some order in the bottom line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-9010fe0cd5dbe3356389333b8a6627926431ab0b53c6aa6ff26cc5528e1a2f59 | A number undergoes a process where the digits are summed repeatedly until a single digit is obtained. For example, 496 → 216 → 12 → 2 and 999 → 243 → 24 → 8, both having a perseverance of 3. What is the smallest number that has a perseverance of 12? | [
"277777788888899"
] | [
"A number undergoes a process where the digits are summed repeatedly until a single digit is obtained. For example, 496 → 216 → 12 → 2 and 999 → 243 → 24 → 8, both having a perseverance of 3. What is the smallest number that has a perseverance of 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b43cc9b7db8258e98b42aa10be7745770ae40216aa7314be56645ad900ec4633 | A number with 8 digits is a multiple of 73 and also a multiple of 137. The second digit from the left equals 7. Determine the 6th digit from the left. | [
"7"
] | [
"A number with 8 digits is a multiple of 73 and also a multiple of 137. The second digit from the left equals 7. Determine the 6th digit from the left."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eb32ac6f6bfdfc3d62def98c3982eaaa4d0e30a5ead0175be9dea9a7bacea901 | A number written in base $10$ is a string of $3^{2013}$ digit $3$s. No other digit appears. Find the highest power of $3$ which divides this number. | [
"2014"
] | [
"A number written in base $10$ is a string of $3^{2013}$ digit $3$s. No other digit appears. Find the highest power of $3$ which divides this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-116896cce211874182eb32af769d3980ee3881ac3b0acb4cea86e16566c57778 | A number written in base 10 consists of a string of \(3^{2013}\) digit 3s. Find the highest power of 3 that divides this number. | [
"\\(3^{2013}\\)"
] | [
"A number written in base 10 consists of a string of \\(3^{2013}\\) digit 3s. Find the highest power of 3 that divides this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc977d088c8262bc7ae2779b2a2fd4f87069cbcdd02696f1de7bc3e08f16cc19 | A number written in base 10 is a string of \(3^{2013}\) digit 3s. Find the highest power of 3 which divides this number. | [
"\\(3^{2014}\\)"
] | [
"A number written in base 10 is a string of \\(3^{2013}\\) digit 3s. Find the highest power of 3 which divides this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-cd163619b5f092e8e8486f0350eb89c7a6c7a42d2118dd16cc6e6738fa317e87 | A nutrition center sells two types of health food packages to mountain climbing teams. The trailblazer mix package contains one pound of corn cereal and four pounds of wheat cereal and sells for $9.75. The frontier mix package contains two pounds of corn cereal and three pounds of wheat cereal and sells for $9.50. The ... | [
"12 trailblazers, 24 frontiers, total profit $345."
] | [
"A nutrition center sells two types of health food packages to mountain climbing teams. The trailblazer mix package contains one pound of corn cereal and four pounds of wheat cereal and sells for $9.75. The frontier mix package contains two pounds of corn cereal and three pounds of wheat cereal and sells for $9.50.... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-9dedbe1c24ad5b1cc392402bff3c44d12b66c80099d3511ced396e72d4efb0d7 | A one-dimensional ladder of length $c$ is restricted so that one endpoint must lie on the positive $x$ axis and the other endpoint on the positive $y$ axis. Let $f$ be the curve that is tangent to each of the possible arrangements of ladders. Find an equation for $f$. | [
"x^{2/3} + y^{2/3} = c^{2/3}"
] | [
"A one-dimensional ladder of length $c$ is restricted so that one endpoint must lie on the positive $x$ axis and the other endpoint on the positive $y$ axis. Let $f$ be the curve that is tangent to each of the possible arrangements of ladders. Find an equation for $f$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13de23aa99c085f9cabf8d2b5404465071d87900203be88ee4282fcfdfd7e956 | A one-high cylinder is inscribed in a unit cube. Find the volume and surface area of the union of all possible such cylinders. | [
"\\frac{9\\pi}{2} - 12"
] | [
"A one-high cylinder is inscribed in a unit cube. Find the volume and surface area of the union of all possible such cylinders."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-381c034abfd3c5d4e98b77eb8f50dddb827c2189c9060c9ff245fe0ba3767382 | A one-inch hole is drilled through a bowling ball with a two-inch radius. How much volume is left in the bowling ball? | [
"\\( 4\\pi \\sqrt{3} \\)"
] | [
"A one-inch hole is drilled through a bowling ball with a two-inch radius. How much volume is left in the bowling ball?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-3b8a9ded1576f80b37bd11e0ba55ac207f96be3cea78fb9d5e5878752805b2f9 | A pack of $2n$ cards contains $n$ different pairs of cards. Each pair consists of two identical cards. A dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the receiver, takes the card dealt, provided he does not have already its twin. If he does already have ... | [
"\\dfrac{n}{2n - 1}"
] | [
"A pack of $2n$ cards contains $n$ different pairs of cards. Each pair consists of two identical cards. A dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the receiver, takes the card dealt, provided he does not have already its twin. If he does already ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-06a101e4139e3f5f05d22ace0149dfd98cd1c1f9b047507255eb8cd5d9016cc3 | A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Calculate the value of $\frac{1}{p_{5}} - \frac{5}{p_{4}} - \frac{12}{2p_{2}}.$ | [
"402"
] | [
"A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Calculate the value of $\\frac{1}{p_{5}} - \\frac{5}{p_{4}} - \\frac{12}{2p_{2}}.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-c058375a70fc75a529c8724e9995bcee3fabbf45b0baf53753e9571a52698e6f | A pack of 13 distinct cards is shuffled in some particular manner and then repeatedly in exactly the same manner. What is the maximum number of shuffles required for the cards to return to their original positions? | [
"60"
] | [
"A pack of 13 distinct cards is shuffled in some particular manner and then repeatedly in exactly the same manner. What is the maximum number of shuffles required for the cards to return to their original positions?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-cfd739812637e5ebbc4a1d283c4c661604f0634ab27fe5b4c6e799df4be213c1 | A pack of 2008 cards, numbered from 1 to 2008, is shuffled. Each move consists of two steps: (i) the top card is placed at the bottom, and (ii) the new top card is removed. The cards are removed in the order 1, 2, ..., 2008. Which card was at the top before the game started? | [
"1883"
] | [
"A pack of 2008 cards, numbered from 1 to 2008, is shuffled. Each move consists of two steps: (i) the top card is placed at the bottom, and (ii) the new top card is removed. The cards are removed in the order 1, 2, ..., 2008. Which card was at the top before the game started?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-03f68059ff178ac09316b08ea18c14e2427f66e32114646354419b8a4f4e48e8 | A pack of cards has only 51 cards. If the first 13 cards drawn are all black, what is the probability that the missing card is red? | [
"\\(\\frac{\\binom{26}{13}}{\\binom{26}{13} + \\binom{25}{13}}\\)"
] | [
"A pack of cards has only 51 cards. If the first 13 cards drawn are all black, what is the probability that the missing card is red?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-399b0369dba31a0065e556ccf8f54546e827da34b184dd28a4eba307ef864966 | A pack of cards is cut and the suit of the exposed card is noted; the pack is then well shuffled and the process is repeated. Find the average number of cuts required to expose cards of all four suits. | [
"\\(\\frac{25}{3}\\)"
] | [
"A pack of cards is cut and the suit of the exposed card is noted; the pack is then well shuffled and the process is repeated. Find the average number of cuts required to expose cards of all four suits."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2704c63c5916cf57578d6910d647bc813c1e37ae7c67745372ed007c4ab8ac75 | A pack of cards is cut and the suit of the exposed card noted; the pack is then well shuffled and the process repeated. Find the average number of cuts required to expose cards of all four suits. | [
"\\dfrac{25}{3}"
] | [
"A pack of cards is cut and the suit of the exposed card noted; the pack is then well shuffled and the process repeated. Find the average number of cuts required to expose cards of all four suits."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94ba84fb0bab81cd0463b5b4f63a802a0e7caff4eb78a0b3c97fb23cb76a3028 | A page is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn pages? | [
"25, 26"
] | [
"A page is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn pages?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 18
} | 23 | 0 | 0 |
omr-ws-bc26f9ef5b945ead03244fe511b2490aa9584d1fec23616f6f3d08d8f9f9119b | A page is torn from the back of a novel. The sum of the remaining page numbers is 15000. What are the page numbers on the torn page? | [
"112, 113"
] | [
"A page is torn from the back of a novel. The sum of the remaining page numbers is 15000. What are the page numbers on the torn page?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 9
} | 20 | 0 | 0 |
omr-ws-9c1aa529d9fc7c22ec8f4802e02255eab09c96c15df615a1ab10837ef5cf8d78 | A paint brush is swept along both diagonals of a square to produce a symmetric painted area, such that half the area of the square is painted. What is the ratio of the side length of the square to the brush width? | [
"\\(2 + 2\\sqrt{2}\\)"
] | [
"A paint brush is swept along both diagonals of a square to produce a symmetric painted area, such that half the area of the square is painted. What is the ratio of the side length of the square to the brush width?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 6,
"tir": 6
} | 37 | 0 | 0 |
omr-ws-6f5f63d872d0ef220ae1a852ff951edd48597436854a3bdf3565358520dfd139 | A painting 3 meters high hangs on a vertical wall, with the lower edge 1 meter above the observer's eye level. At what distance from the wall should the observer stand to see the distance from the top to the bottom of the picture under the largest angle? | [
"The answer is provided in the original post as [hide=Answer]2 m [/hide]."
] | [
"A painting 3 meters high hangs on a vertical wall, with the lower edge 1 meter above the observer's eye level. At what distance from the wall should the observer stand to see the distance from the top to the bottom of the picture under the largest angle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f1523087c5de81f250d7e4e914267e42a70c24df31c16ee4cf643e4370838a47 | A painting job can be completed by Team A alone in $2.5$ hours or by Team B alone in $75$ minutes. On one occasion, after Team A had completed a fraction $\frac{m}{n}$ of the job, Team B took over immediately, and the whole painting job was completed in $1.5$ hours. If $m$ and $n$ are positive integers with no common f... | [
"6"
] | [
"A painting job can be completed by Team A alone in $2.5$ hours or by Team B alone in $75$ minutes. On one occasion, after Team A had completed a fraction $\\frac{m}{n}$ of the job, Team B took over immediately, and the whole painting job was completed in $1.5$ hours. If $m$ and $n$ are positive integers with no co... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-792c8c4badaf96f5ce3313ed1cec998e02126e19a2a26563fbf08efdf39692dd | A pair $(A, B)$ of positive integers is considered fancy if there exist two rectangles with integer side lengths, one with area $A$ and perimeter $B$ and the other with area $B$ and perimeter $A$. Over all fancy pairs of integers $(A, B)$, find the maximum possible value of $A + B$. | [
"164"
] | [
"A pair $(A, B)$ of positive integers is considered fancy if there exist two rectangles with integer side lengths, one with area $A$ and perimeter $B$ and the other with area $B$ and perimeter $A$. Over all fancy pairs of integers $(A, B)$, find the maximum possible value of $A + B$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 13
} | 25 | 0 | 0 |
omr-ws-b3509c314026520b556baea795191c6f7e6fce2575c6b229426a3e841006d7c1 | A pair \((a, b)\) of positive integers with \(a < 391\) is called "pupusa" if \[ \mathrm{lcm}(a, b) > \mathrm{lcm}(a, 391). \] Determine the smallest \(b\) among all possible pupusa pairs \((a, b)\). Note: \(\mathrm{lcm}(a, b)\) denotes the least common multiple of \(a\) and \(b\). | [
"18"
] | [
"A pair \\((a, b)\\) of positive integers with \\(a < 391\\) is called \"pupusa\" if\n\\[\n\\mathrm{lcm}(a, b) > \\mathrm{lcm}(a, 391).\n\\]\nDetermine the smallest \\(b\\) among all possible pupusa pairs \\((a, b)\\).\n\nNote: \\(\\mathrm{lcm}(a, b)\\) denotes the least common multiple of \\(a\\) and \\(b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-6e7643aa0f861ee07e969df4619f78840c5f583e931ae60913d9708889e7f90b | A pair \((a, b)\) of positive integers, where \(a < 391\), is called "pupusa" if \[ \text{lcm}(a, b) > \text{lcm}(a, 391) \] Find the minimum value of \(b\) across all "pupusa" pairs. | [
"18"
] | [
"A pair \\((a, b)\\) of positive integers, where \\(a < 391\\), is called \"pupusa\" if \n\n\\[\n\\text{lcm}(a, b) > \\text{lcm}(a, 391)\n\\]\n\nFind the minimum value of \\(b\\) across all \"pupusa\" pairs."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-f8ac619b89549c4052d1bb89e2e1c627033e1aaf3864cc1686144f858db857c4 | A pair of 100-gons \( M_1 \), \( M_2 \) is taken on the plane (by a polygon we mean its interior with boundary), whose vertex set is a set of 200 points, no three of which lie on one straight line, and such that the intersection \( M_1 \cap M_2 \) is the union of \( k \) pairwise disjoint polygons. Find the largest pos... | [
"9801"
] | [
"A pair of 100-gons \\( M_1 \\), \\( M_2 \\) is taken on the plane (by a polygon we mean its interior with boundary), whose vertex set is a set of 200 points, no three of which lie on one straight line, and such that the intersection \\( M_1 \\cap M_2 \\) is the union of \\( k \\) pairwise disjoint polygons. Find t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-648e806bd556f4eedcbd1d3ddac1e2c9634ba2cca3265925689c9a3e6a2e73ac | A pair of dice are rolled until the sum of either $5$ or $7$ appears. Find the probability that $5$ occurs first. | [
"$\\frac{2}{5}$"
] | [
"A pair of dice are rolled until the sum of either $5$ or $7$ appears. Find the probability that $5$ occurs first."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-d278ae293920de6d2cb90e1ed58ba188d1f353304f6b83cf3a0401e191e756f0 | A pair of dice is rolled until either the two numbers on the dice agree or the difference of the two numbers on the dice is 1. Find the probability that the two numbers agree before the difference of the two numbers is 1. | [
"\\(\\frac{3}{8}\\)"
] | [
"A pair of dice is rolled until either the two numbers on the dice agree or the difference of the two numbers on the dice is 1. Find the probability that the two numbers agree before the difference of the two numbers is 1."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
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