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-e91798c8462cfc08aff51a869d77111cebcbd50de5488be97edf8de79af35888 | A positive integer $N$ has a base-eleven representation $\underline{a}\,\underline{b}\,\underline{c}$ and a base-eight representation $\underline{1}\,\underline{b}\,\underline{c}\,\underline{a}$, where $a$, $b$, and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten. | [
"621"
] | [
"A positive integer $N$ has a base-eleven representation $\\underline{a}\\,\\underline{b}\\,\\underline{c}$ and a base-eight representation $\\underline{1}\\,\\underline{b}\\,\\underline{c}\\,\\underline{a}$, where $a$, $b$, and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in b... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-3bcffbebc176fc3fd593040d892e4648e8c5ff62a173d8a17f0a547bb1a5c0cc | A positive integer $N$ has base-eleven representation $\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}$ and base-eight representation $\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Find the least such... | [
"621"
] | [
"A positive integer $N$ has base-eleven representation $\\underline{a}\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}$ and base-eight representation $\\underline1\\kern 0.1em\\underline{b}\\kern 0.1em\\underline{c}\\kern 0.1em\\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Fi... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-198193e71df479608a26fb099d84d63e5eb747bb6e8d864eafd4bf04b689cba2 | A positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$. The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$. | [
"20"
] | [
"A positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \\ldots$. The smallest part is equal to $\\frac{1}{400} N$. Find the value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0d4599fa50cf50bbffbdb7b9d25a3005d91a6b30e04a32e0bc95b9037569b565 | A positive integer $N$ is given. A tree is built on $N$ vertices, and a real number is written on each vertex such that $1$ plus the number on each vertex is greater than or equal to the average of the numbers on its neighboring vertices. Let the maximum number written be $M$ and the minimum number written be $m$. What... | [
"\\((N-1)^2\\)"
] | [
"A positive integer $N$ is given. A tree is built on $N$ vertices, and a real number is written on each vertex such that $1$ plus the number on each vertex is greater than or equal to the average of the numbers on its neighboring vertices. Let the maximum number written be $M$ and the minimum number written be $m$.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 38,
"genselect": 8
} | 46 | 0 | 0 |
omr-ws-0712e5e4d610a9c85ca134244ca94eb4f53811873ede3f9a991ede5df9836091 | A positive integer $N$ is said to be nearly-square if $\sqrt{N}$ can be expressed in simplest radical form as $a\sqrt{b}$, where $a$ and $b$ are positive integers greater than $1$ with $a>b$. How many of the first $1000$ positive integers are nearly-square? | [
"54"
] | [
"A positive integer $N$ is said to be nearly-square if $\\sqrt{N}$ can be expressed in simplest radical form as $a\\sqrt{b}$, where $a$ and $b$ are positive integers greater than $1$ with $a>b$. How many of the first $1000$ positive integers are nearly-square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-cb35193586cb26f1912e84984caff215596e467e443c08e62b5726bc8bce73bc | A positive integer $N$, expressed in base $9$, has three digits. In base $6$, it has the same three digits, but in reverse order. What is the middle of those three digits? | [
"5"
] | [
"A positive integer $N$, expressed in base $9$, has three digits. In base $6$, it has the same three digits, but in reverse order. What is the middle of those three digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-00619bd46727cc22ea2430f76895df0bdbee12f57e49bd6b6603ae70fdac2ac0 | A positive integer $N=\overline{d_0 d_1 d_2 d_3 d_4 d_5 d_6 d_7 d_8 d_9}$ has exactly $d_0$ digits equal to $0$, $d_1$ digits equal to $1$, $d_2$ digits equal to $2$, ..., $d_9$ digits equal to $9$. Find the remainder when $N$ is divided by $36$. | [
"28"
] | [
"A positive integer $N=\\overline{d_0 d_1 d_2 d_3 d_4 d_5 d_6 d_7 d_8 d_9}$ has exactly $d_0$ digits equal to $0$, $d_1$ digits equal to $1$, $d_2$ digits equal to $2$, ..., $d_9$ digits equal to $9$. Find the remainder when $N$ is divided by $36$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a43f6ea235a79ef73b7b13fa3d47125e83d106aa21033a44e3a708e7e432e036 | A positive integer $\overline{a_na_{n-1}a_{n-2}...a_1a_0}$ is called steep if $a_n=2a_{n-1}$ and $a_0<a_1<a_2<...<a_{n-1}$. Find the number of four-digit steep numbers. | [
"10"
] | [
"A positive integer $\\overline{a_na_{n-1}a_{n-2}...a_1a_0}$ is called steep if $a_n=2a_{n-1}$ and $a_0<a_1<a_2<...<a_{n-1}$. Find the number of four-digit steep numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b641f575465c718c50055529d3a08be5247d97528fe000a488066ceb259acb86 | A positive integer $a$ is randomly selected from $\{1, 2, 3, 4, ..., 24\}$. What is the probability that $a^3 - a$ is a multiple of $24$? | [
"\\dfrac{5}{8}"
] | [
"A positive integer $a$ is randomly selected from $\\{1, 2, 3, 4, ..., 24\\}$. What is the probability that $a^3 - a$ is a multiple of $24$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dcb2084f326b88ffcf428b450ef3c6a44588e127d8b86777109612629b1c7d50 | A positive integer $a$ is selected, and the integers $b_1, b_2, \ldots, b_k$ are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn, he must replace some even integer $n$ on the board with $n/2$. Alice goes first and t... | [
"no_valid_answer_found"
] | [
"A positive integer $a$ is selected, and the integers $b_1, b_2, \\ldots, b_k$ are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn, he must replace some even integer $n$ on the board with $n/2$. Alice goes first... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e79eb4f30fff382b89379143e93d0a26b72e12a069e586dfe7d8ce2a5560f9a5 | A positive integer $k$ is defined as fair if the number of $2021$-digit palindromes that are multiples of $k$ is the same as the number of $2022$-digit palindromes that are multiples of $k$. Determine whether the set $M = \{1, 2, \dots, 35\}$ contains more fair numbers or non-fair numbers. | [
"more fair numbers"
] | [
"A positive integer $k$ is defined as fair if the number of $2021$-digit palindromes that are multiples of $k$ is the same as the number of $2022$-digit palindromes that are multiples of $k$. Determine whether the set $M = \\{1, 2, \\dots, 35\\}$ contains more fair numbers or non-fair numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 7
} | 16 | 0 | 0 |
omr-ws-83818e9ea23c2f6b2e6599fa010f03488ff295d42e70472150f57290fdd5ffbc | A positive integer $k$ is given. Initially, $N$ cells are marked on an infinite checkered plane. The cross of a cell $A$ is defined as the set of all cells lying in the same row or in the same column as $A$. In one turn, an unmarked cell $A$ can be marked if its cross contains at least $k$ marked cells. It is known tha... | [
"\\( k + (k-2) + (k-4) + \\cdots \\)"
] | [
"A positive integer $k$ is given. Initially, $N$ cells are marked on an infinite checkered plane. The cross of a cell $A$ is defined as the set of all cells lying in the same row or in the same column as $A$. In one turn, an unmarked cell $A$ can be marked if its cross contains at least $k$ marked cells. It is know... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-a2ef67e0bb100918d2427c8b634a01a6318b88b807861f9790f78f184b1ad102 | A positive integer $k$ is said to be good if there exists a partition of $\{1, 2, 3, \ldots, 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many good numbers are there? | [
"6"
] | [
"A positive integer $k$ is said to be good if there exists a partition of $\\{1, 2, 3, \\ldots, 20\\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many good numbers are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 8,
"tir": 27
} | 65 | 0 | 0 |
omr-ws-41c5612aa23523c67d1802eaddbdca7f44f578b8ea10aeb1d8b1422f61559f83 | A positive integer $k$ is said to be visionary if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020$. How many visionary integers are there? | [
"12"
] | [
"A positive integer $k$ is said to be visionary if there are integers $a > 0$ and $b \\geq 0$ such that $a \\cdot k + b \\cdot (k + 1) = 2020$. How many visionary integers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b83b4550c0117a9cd81887b55444737fc88a2e3b11ab02eca474fcb9c1e621de | A positive integer $k$ is said to be visionary if there exist integers $a > 0$ and $b \geq 0$ such that $ak + b(k+1) = 2020$. How many visionary integers are there? | [
"94"
] | [
"A positive integer $k$ is said to be visionary if there exist integers $a > 0$ and $b \\geq 0$ such that $ak + b(k+1) = 2020$. How many visionary integers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-cb427fa1daaa90c06a196cdaaff85738217a6bde92977daddc195e8d53ab375b | A positive integer $k$ is taken at random with $k \le 2018$. Find the probability that $k^{1009}$ has a remainder of $2$ when divided by $2018$. | [
"\\dfrac{1}{2018}"
] | [
"A positive integer $k$ is taken at random with $k \\le 2018$. Find the probability that $k^{1009}$ has a remainder of $2$ when divided by $2018$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9509f444842e51feb1559057cbdf877c6b736eaddbbffdee47a43562694d351e | A positive integer $m$ has the property that $m^2$ can be expressed in the form $4n^2 - 5n + 16$, where $n$ is an integer. Find the maximum value of $|m - n|$. | [
"14"
] | [
"A positive integer $m$ has the property that $m^2$ can be expressed in the form $4n^2 - 5n + 16$, where $n$ is an integer. Find the maximum value of $|m - n|$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-c15bf2615ba039c3437dc70303b43e7567302d4555e1c6c4af480c90dd3946e1 | A positive integer $m$ is acceptable if the sum of the squares of its proper divisors is equal to $2m+4$. Find all acceptable numbers less than $10000$. | [
"\\( m = \\{8, 15, 35, 143, 323, 899, 1763, 3599, 5183\\} \\)"
] | [
"A positive integer $m$ is acceptable if the sum of the squares of its proper divisors is equal to $2m+4$. Find all acceptable numbers less than $10000$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b12c9c790a7b0b9b8db741b1cc950f43e609349b2e3b918e6e064b2164144123 | A positive integer $m$ is expressed as the sum of two triangular numbers. If $4m+1$ can be written as a sum of two squares, find the smallest possible value of $m$ such that $4m+1 = a^2 + b^2$, where $a$ and $b$ are positive integers and $a < b$. | [
"1"
] | [
"A positive integer $m$ is expressed as the sum of two triangular numbers. If $4m+1$ can be written as a sum of two squares, find the smallest possible value of $m$ such that $4m+1 = a^2 + b^2$, where $a$ and $b$ are positive integers and $a < b$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09e0236d6e8542b93fc1e47473e41d53101c76998e055dddeaf462fc9a122786 | A positive integer $n > 1$ is prime if it has no positive integer divisors other than $1$ and $n$. The function $f(n)$ is defined as the product of all prime numbers $i \le n$. Find all $n$ such that $f(n) \le n$. | [
"2"
] | [
"A positive integer $n > 1$ is prime if it has no positive integer divisors other than $1$ and $n$. The function $f(n)$ is defined as the product of all prime numbers $i \\le n$. Find all $n$ such that $f(n) \\le n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3fd20e3efbef9bde8f75de8513b2aaa1c0a18ddc57a049ec73509a452c42e0a3 | A positive integer $n$ can be expressed as the sum of two perfect squares. Let $S(n)$ be the sum of all distinct positive integers $m$ such that $n = a^2 + b^2 = c^2 + d^2$ for some integers $a, b, c, d$ and $m = c^2 + d^2$ where $c$ and $d$ are rational numbers. Find $S(65)$. | [
"65"
] | [
"A positive integer $n$ can be expressed as the sum of two perfect squares. Let $S(n)$ be the sum of all distinct positive integers $m$ such that $n = a^2 + b^2 = c^2 + d^2$ for some integers $a, b, c, d$ and $m = c^2 + d^2$ where $c$ and $d$ are rational numbers. Find $S(65)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-2852e1a6da8d85b27a1e6b23433294df12409ffba4212e9063b1e02fb703266c | A positive integer $n$ can be expressed as the sum of two squares. If $n$ has prime factors of the form $4k+3$, let $p$ be the largest such prime factor. Find the smallest possible value of $n$ when $p=7$ and $n$ is less than $1000$. | [
"49"
] | [
"A positive integer $n$ can be expressed as the sum of two squares. If $n$ has prime factors of the form $4k+3$, let $p$ be the largest such prime factor. Find the smallest possible value of $n$ when $p=7$ and $n$ is less than $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-a6dd4e96da16aedbe936af03639fb21684bd3fdd7d57001aa403653af3f8e91b | A positive integer $n$ can be expressed in the form $3x^2 + y^2$ if and only if it can also be expressed in the form $u^2 + uv + v^2$, where $x, y, u, v$ are all positive integers. Find the smallest positive integer $n$ that can be expressed in both forms with $x, y, u, v \leq 10$. | [
"7"
] | [
"A positive integer $n$ can be expressed in the form $3x^2 + y^2$ if and only if it can also be expressed in the form $u^2 + uv + v^2$, where $x, y, u, v$ are all positive integers. Find the smallest positive integer $n$ that can be expressed in both forms with $x, y, u, v \\leq 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f802f496686f45e0b83c81558dfd7ea5e6b0a706bf4a69c344dcbe54dad9abb1 | A positive integer $n$ exists such that $2^{n - 1}$ and $2^n$ are 99 and 100 digits long, respectively. Find the last two digits of $2^n.$ | [
"12"
] | [
"A positive integer $n$ exists such that $2^{n - 1}$ and $2^n$ are 99 and 100 digits long, respectively. Find the last two digits of $2^n.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6ba20830f3ab02b3518df60aebec92a07a26bf9d030c0171f1bcd3666ba2baf5 | A positive integer $n$ has 60 divisors, and $7n$ has 80 divisors. What is the greatest integer $k$ such that $7^k$ is a factor of $n$? | [
"2"
] | [
"A positive integer $n$ has 60 divisors, and $7n$ has 80 divisors. What is the greatest integer $k$ such that $7^k$ is a factor of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6afea5a7afd9999c01455fe6e4f961eeacb41c12ac6d7c616a1fb8aa807f40a3 | A positive integer $n$ has the property $P$ if there exist integers $0<p<q<n$ such that $n$ divides $p+(p+1)+\ldots +q$. Find all positive integers with the property $P$. | [
"All positive integers except the powers of 2."
] | [
"A positive integer $n$ has the property $P$ if there exist integers $0<p<q<n$ such that $n$ divides $p+(p+1)+\\ldots +q$. Find all positive integers with the property $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 12
} | 58 | 0 | 0 |
omr-ws-0c8fb23c4534b8af3ed1cf81e3a7e41ed5527a54541e627bb163c119d44d587b | A positive integer $n$ is called "consecutive-sum friendly" if it can be expressed as the sum of at least two consecutive positive integers. Let $S$ be the set of all positive integers less than $1000$ that are not "consecutive-sum friendly". Calculate the sum of all elements in $S$. | [
"1023"
] | [
"A positive integer $n$ is called \"consecutive-sum friendly\" if it can be expressed as the sum of at least two consecutive positive integers. Let $S$ be the set of all positive integers less than $1000$ that are not \"consecutive-sum friendly\". Calculate the sum of all elements in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9c4888ce7cad92d196ba8fc04b8ec251a755c19900122d9eba04b7ad605db66 | A positive integer $n$ is called "fine" if there exists a prime number $p$ such that $p|n$ and $p^2 \nmid n$. Let $S$ be the set of all "fine" numbers between $1$ and $10^{12}$. Calculate the value of $\left\lfloor\frac{|S|}{10^{12}} \times 100\right\rfloor$. | [
"99"
] | [
"A positive integer $n$ is called \"fine\" if there exists a prime number $p$ such that $p|n$ and $p^2 \\nmid n$. Let $S$ be the set of all \"fine\" numbers between $1$ and $10^{12}$. Calculate the value of $\\left\\lfloor\\frac{|S|}{10^{12}} \\times 100\\right\\rfloor$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a87362eb570ddbe21e4e454e277dac806133a81a15dd9e55d3f6047b49f9d55f | A positive integer $n$ is called "special" if the set $\{\tau(1), \tau(2), \ldots, \tau(n)\}$ is a set of all divisors of some integer. Find all "special" numbers. | [
"\\( n = 1, 2, 3 \\)"
] | [
"A positive integer $n$ is called \"special\" if the set $\\{\\tau(1), \\tau(2), \\ldots, \\tau(n)\\}$ is a set of all divisors of some integer. Find all \"special\" numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 9
} | 10 | 0 | 0 |
omr-ws-f0db9f9d88b057f6d80d4d8ebe33d7d1fe83696d464d0ac504aa96bfd4716e06 | A positive integer $n$ is called $4$-special if it can be expressed as $n = a^2 + 4b^2$ for some positive integers $a$ and $b$. Find the sum of all positive integers $k$ less than $100$ such that $k^2$ is not $4$-special. | [
"2669"
] | [
"A positive integer $n$ is called $4$-special if it can be expressed as $n = a^2 + 4b^2$ for some positive integers $a$ and $b$. Find the sum of all positive integers $k$ less than $100$ such that $k^2$ is not $4$-special."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dbef45e10de39631a52ed82645e9557473e356a457221db39411232ef3b47c0 | A positive integer $n$ is called $\emph{cheerful}$ if it can be expressed as the sum of two or more consecutive positive integers. Find the sum of all distinct $\emph{cheerful}$ numbers less than $100$ that are multiples of $4$. | [
"1076"
] | [
"A positive integer $n$ is called $\\emph{cheerful}$ if it can be expressed as the sum of two or more consecutive positive integers. Find the sum of all distinct $\\emph{cheerful}$ numbers less than $100$ that are multiples of $4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-839c0d58277d8113c846585cff65176b46175ac3f9dc536b253652349d740f82 | A positive integer $n$ is called $\emph{cheerful}$ if there exist distinct positive integers $a$, $b$, and $c$ such that $n = a^2 + b^2 + c^2$ and $a + b + c = n + 1$. Find the sum of all positive integers less than $10$ that are not $\emph{cheerful}$. | [
"45"
] | [
"A positive integer $n$ is called $\\emph{cheerful}$ if there exist distinct positive integers $a$, $b$, and $c$ such that $n = a^2 + b^2 + c^2$ and $a + b + c = n + 1$. Find the sum of all positive integers less than $10$ that are not $\\emph{cheerful}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a272bbebc33dbd83707df45dd5e2eb351d79c7871a65000b9d77e0962e19478d | A positive integer $n$ is called $\text{pretty}$ if there exist two divisors $d_1$ and $d_2$ of $n$ such that $d_2 - d_1 = d$ for each divisor $d$ of $n$ (where $1 < d < n$). Find the smallest $\text{pretty}$ number larger than $401$ that is a multiple of $401$. | [
"160400"
] | [
"A positive integer $n$ is called $\\text{pretty}$ if there exist two divisors $d_1$ and $d_2$ of $n$ such that $d_2 - d_1 = d$ for each divisor $d$ of $n$ (where $1 < d < n$). Find the smallest $\\text{pretty}$ number larger than $401$ that is a multiple of $401$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-4a6eaad307b3f5d4f67a83074500c55b402c57916cc56fe78b977ada042d4408 | A positive integer $n$ is called $k$-discerning if for any set of $k$ positive integers, there exists a subset of these integers whose sum is divisible by $n$. Find the smallest positive integer $m$ such that $8$ is $m$-discerning. | [
"8"
] | [
"A positive integer $n$ is called $k$-discerning if for any set of $k$ positive integers, there exists a subset of these integers whose sum is divisible by $n$. Find the smallest positive integer $m$ such that $8$ is $m$-discerning."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-ba9fca8ce206cae9cad8750d3303750229680038bc12f28f4f8516ac65607dc5 | A positive integer $n$ is called $k$-discerning if there exists a set of $k$ distinct positive integers $\{a_1, a_2, \ldots, a_k\}$ such that for every pair of distinct integers $a_i, a_j$ in the set, $n$ does not divide $a_i + a_j$. Find the smallest positive integer $m$ such that $9$ is not $m$-discerning. | [
"6"
] | [
"A positive integer $n$ is called $k$-discerning if there exists a set of $k$ distinct positive integers $\\{a_1, a_2, \\ldots, a_k\\}$ such that for every pair of distinct integers $a_i, a_j$ in the set, $n$ does not divide $a_i + a_j$. Find the smallest positive integer $m$ such that $9$ is not $m$-discerning."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-672bbd42ea95ccfc7c417b4c7283c4613f2da4264d9c6259e2340573ed1d48c6 | A positive integer $n$ is called $nice$ if the sum of the squares of all its positive divisors is equal to $(n+3)^2$. Given that $n=pq$ is nice, where $p$ and $q$ are not necessarily distinct primes, find the value of $n$ for which $n+2$ and $2(n+1)$ are simultaneously perfect squares. | [
"287"
] | [
"A positive integer $n$ is called $nice$ if the sum of the squares of all its positive divisors is equal to $(n+3)^2$. Given that $n=pq$ is nice, where $p$ and $q$ are not necessarily distinct primes, find the value of $n$ for which $n+2$ and $2(n+1)$ are simultaneously perfect squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-288d64b16e9b98e3caa95a6ed33dd1a555b576ba45781e5616280c8e7178021c | A positive integer $n$ is called a good number if $n^3+7n-133=m^3$ for some positive integer $m$. What is the sum of all good numbers? | [
"25"
] | [
"A positive integer $n$ is called a good number if $n^3+7n-133=m^3$ for some positive integer $m$. What is the sum of all good numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15692a4aae278766090a5b9bda5bfe8bdac314a228bc09503810ad8a21ca6da8 | A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ regardless of how its digits are rearranged. How many good numbers are there? | [
"3"
] | [
"A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ regardless of how its digits are rearranged. How many good numbers are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-044ede3869392d113bc9b944f899530b1511a0acc5d250c0f648b07b5085e31c | A positive integer $n$ is called a good number if it can be expressed as $n = a^2 + b^2 + c^2$ for some positive integers $a$, $b$, and $c$. Let $G(N)$ be the number of good numbers less than or equal to $N$. Calculate the smallest value of $N$ for which $G(N) > 1000$. | [
"1300"
] | [
"A positive integer $n$ is called a good number if it can be expressed as $n = a^2 + b^2 + c^2$ for some positive integers $a$, $b$, and $c$. Let $G(N)$ be the number of good numbers less than or equal to $N$. Calculate the smallest value of $N$ for which $G(N) > 1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-83b5b4bfdb2852ec765c6570cae2ee787b651eeeaa8d124f961f6c1803a0c496 | A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. Find all almost perfect numbers such that some of their powers are also almost perfect. | [
"2^k"
] | [
"A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. Find all almost perfect numbers such that some of their powers are also almost perfect."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2579eff7ed41a9f4803fbe8fcb80f941c9353193d3af887c0b9e82034d053c8 | A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all untouchable numbers between $1$ and $10$ (inclusive). | [
"33"
] | [
"A positive integer $n$ is called an untouchable number if there is no positive integer $m$ for which the sum of the factors of $m$ (including $m$ itself) is $n + m$. Find the sum of all untouchable numbers between $1$ and $10$ (inclusive)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6df3ad7e43c28b6917f0a487dda1a02d2af968301ee77b318d2f9950064f7ae9 | A positive integer $n$ is called beautiful if the sum of its digits is equal to the product of its digits. Let $S(k)$ be the sum of the first $k$ beautiful numbers. Find the remainder when $S(10)$ is divided by $1000$. | [
"67"
] | [
"A positive integer $n$ is called beautiful if the sum of its digits is equal to the product of its digits. Let $S(k)$ be the sum of the first $k$ beautiful numbers. Find the remainder when $S(10)$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7e680819c4fcd9dfd0da16652f4278b8985e5069df82d9db6af7aa3a718b21a | A positive integer $n$ is called beautiful if there exist positive integers $x$ and $y$ such that \[ \frac{\varphi(x^2)}{\varphi(y^2)} = n. \] Find the smallest beautiful number $n$ such that $n > 1000$. | [
"1012"
] | [
"A positive integer $n$ is called beautiful if there exist positive integers $x$ and $y$ such that \\[ \\frac{\\varphi(x^2)}{\\varphi(y^2)} = n. \\] Find the smallest beautiful number $n$ such that $n > 1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-10e99a2dcc88465eab0c1cfc2dc357380b53499e1c71808d4c07b6bd90660028 | A positive integer $n$ is called elegant if it has a multiple whose first four digits are 2008. Find the smallest elegant number $n$ such that $n$ is a prime number greater than 100. | [
"101"
] | [
"A positive integer $n$ is called elegant if it has a multiple whose first four digits are 2008. Find the smallest elegant number $n$ such that $n$ is a prime number greater than 100."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b701db54d435af87204116efc05e7f4c47895e9f2b11ed2b52a5d5c9224abd5f | A positive integer $n$ is called good if it can be written as $5x + 8y = n$ for positive integers $x, y$. Given that 42, 43, 44, 45, and 46 are good, what is the largest $n$ that is not good? | [
"27"
] | [
"A positive integer $n$ is called good if it can be written as $5x + 8y = n$ for positive integers $x, y$. Given that 42, 43, 44, 45, and 46 are good, what is the largest $n$ that is not good?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e005b8c6d0c31610480e637358da21e2539b7282a1a53cffe9480c00d69549f3 | A positive integer $n$ is called nice if there exists a positive integer $k$ such that $n = k^2 + k + 1$. Let $S(N)$ be the sum of all nice numbers less than $N$. Find the smallest value of $N$ such that $S(N)$ is a multiple of $2023$. | [
"4094554"
] | [
"A positive integer $n$ is called nice if there exists a positive integer $k$ such that $n = k^2 + k + 1$. Let $S(N)$ be the sum of all nice numbers less than $N$. Find the smallest value of $N$ such that $S(N)$ is a multiple of $2023$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-cb4e619bd422cead13590feb153f6ccbef447983270a47711f201a241a4a3bb7 | A positive integer $n$ is called special if $\tau(n+1) = \tau(n) + 1$, where $\tau(k)$ denotes the number of divisors of $k$. How many special numbers are there which are less than $300$? | [
"7"
] | [
"A positive integer $n$ is called special if $\\tau(n+1) = \\tau(n) + 1$, where $\\tau(k)$ denotes the number of divisors of $k$. How many special numbers are there which are less than $300$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a529b64f6d08bc6b62df14a56cefb08d4ee951e9797f57f1ba81e5b65fbc4543 | A positive integer $n$ is called special if there exists a positive integer $k$ such that $n = k^2 + k + 1$. Let $S(N)$ be the sum of all special numbers less than $N$. Find the smallest value of $N$ such that $S(N)$ is divisible by $1000$. | [
"1058"
] | [
"A positive integer $n$ is called special if there exists a positive integer $k$ such that $n = k^2 + k + 1$. Let $S(N)$ be the sum of all special numbers less than $N$. Find the smallest value of $N$ such that $S(N)$ is divisible by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-4b0984188e54a94afce53192889a4d8dd7900c39aa275b0333af1cb62e23f7f9 | A positive integer $n$ is considered $litty$ if $\left(\lceil{\sqrt{\lfloor{\sqrt{n}}\rfloor}}\,\rceil \right)^4 = \lfloor{\sqrt[4]{n}}\rfloor^4.$ Find the number of $litty$ integers between $1$ and $2017.$ | [
"188"
] | [
"A positive integer $n$ is considered $litty$ if $\\left(\\lceil{\\sqrt{\\lfloor{\\sqrt{n}}\\rfloor}}\\,\\rceil \\right)^4 = \\lfloor{\\sqrt[4]{n}}\\rfloor^4.$ Find the number of $litty$ integers between $1$ and $2017.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 16
} | 39 | 0 | 0 |
omr-ws-57621e3477da9651d7104eb4db88c9364ed38c982c1dedfabb271821d07a284d | A positive integer $n$ is defined as *funny* if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors. | [
"135"
] | [
"A positive integer $n$ is defined as *funny* if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-c2a125522b0a82344f731da1d2847eccab21cfd5614b4580d55c4e4acb6c02b9 | A positive integer $n$ is defined as [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Find the smallest positive integer $n$ such that both $n$ and $2^n + n$ are Danish. | [
"1"
] | [
"A positive integer $n$ is defined as [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Find the smallest positive integer $n$ such that both $n$ and $2^n + n$ are Danish."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-7ba9b8ace3d1b4f435d64854d0b264e74e5baf25d831d8890a4350f8a73c93b1 | A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells of a $k \times n$ table. A table is called nice if for any partition of the set of rows into two nonempty subsets $R_1$ and $R_2$, there exists a nonempty set $S$ of columns such that the intersection of any row from $R_1$ with the columns fro... | [
"\\( n \\)"
] | [
"A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells of a $k \\times n$ table. A table is called nice if for any partition of the set of rows into two nonempty subsets $R_1$ and $R_2$, there exists a nonempty set $S$ of columns such that the intersection of any row from $R_1$ with the colum... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a95a03f5ac30d07fa1578cef136d5d42eaf2e1115d1c298e290056cb036df39 | A positive integer $n$ is given. On the segment $[0,n]$ of the real line, $m$ distinct segments with integer coordinates are chosen. It is impossible to select some of these segments such that their total length is $n$ and their union is $[0,n]$. Find the maximum possible value of $m$. | [
"\\dfrac{n(n-1)}{2}"
] | [
"A positive integer $n$ is given. On the segment $[0,n]$ of the real line, $m$ distinct segments with integer coordinates are chosen. It is impossible to select some of these segments such that their total length is $n$ and their union is $[0,n]$. Find the maximum possible value of $m$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-7372033883190ae9a83252876864940ca81e92e83fe8bda8fbce835375670bec | A positive integer $n$ is said to be a good number if $(n+1)(n+2)(n+3)(n+4)$ is divisible by 11. How many good numbers are less than 2016? | [
"732"
] | [
"A positive integer $n$ is said to be a good number if $(n+1)(n+2)(n+3)(n+4)$ is divisible by 11. How many good numbers are less than 2016?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-50560ba1354eb84770e0c053e9d6d37bca12c8853bbea20b11fbd269e88009b5 | A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base $10$ is equal to $n$. For instance, $13$ is good because $7^2=49$ and $4+9=13$. How many good numbers are among $1, 2, 3, \ldots, 2007$? | [
"892"
] | [
"A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base $10$ is equal to $n$. For instance, $13$ is good because $7^2=49$ and $4+9=13$. How many good numbers are among $1, 2, 3, \\ldots, 2007$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f08af4269277c730030d216fdf478aeaefeb0ab788449a879b183f8cf8a55d05 | A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base 10 is equal to $n$. How many good numbers are there among $1, 2, 3, \ldots, 2007$? | [
"892"
] | [
"A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base 10 is equal to $n$. How many good numbers are there among $1, 2, 3, \\ldots, 2007$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-d5127b95ac7e1b48bc4c455b0d84e7d0372822626e9a201231a6e0bbc2c2c6b6 | A positive integer $n$ is special if there are exactly $2010$ positive integers smaller than $n$ and relatively prime to $n$. Compute the sum of all special numbers. | [
"6033"
] | [
"A positive integer $n$ is special if there are exactly $2010$ positive integers smaller than $n$ and relatively prime to $n$. Compute the sum of all special numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-8c0c72b074d1042ffc0a4551f9b013ca9addc0363658d230dc8360ca91e3c664 | A positive integer $n$ is tripairable if it is impossible to partition the set $\{1, 2, \ldots, n\}$ into disjoint pairs such that the sum of the two elements in each pair is a power of $3$. How many positive integers less than or equal to $2024$ are tripairable? | [
"1349"
] | [
"A positive integer $n$ is tripairable if it is impossible to partition the set $\\{1, 2, \\ldots, n\\}$ into disjoint pairs such that the sum of the two elements in each pair is a power of $3$. How many positive integers less than or equal to $2024$ are tripairable?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-fe3361670f9a7eaa20a21a6d1960b61131765b21f6e80f6fda8129fce04b8cea | A positive integer $n$ is written on a blackboard. It is replaced by the sum of any two positive integers whose product is equal to $n$. This process is repeated a finite number of times. Determine the smallest number that can appear on the board and whether it depends on the initial value of $n$. | [
"5"
] | [
"A positive integer $n$ is written on a blackboard. It is replaced by the sum of any two positive integers whose product is equal to $n$. This process is repeated a finite number of times. Determine the smallest number that can appear on the board and whether it depends on the initial value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-0305218757d5c7679af12e4215b3feea674ff7aaeef5ec479c10d1fb22ae7dde | A positive integer $n$ is written on the blackboard. In each step, we replace the number by the sum of any two positive integers whose product is equal to the number on the board. Determine the smallest number that can be obtained after a finite number of steps in terms of the initial number $n$. | [
"\\( n \\) for \\( 1 \\leq n \\leq 4 \\) and \\( 5 \\) for \\( n \\geq 5 \\)"
] | [
"A positive integer $n$ is written on the blackboard. In each step, we replace the number by the sum of any two positive integers whose product is equal to the number on the board. Determine the smallest number that can be obtained after a finite number of steps in terms of the initial number $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-b87cc01de95f82dbbe8542e85023a2ab89f3931a1f623b301756415e1afbefbb | A positive integer \( M \) with \( 2n \) digits is called [i]hypersquared[/i] if it satisfies the following conditions: 1. \( M \) is a perfect square. 2. The number formed by the first \( n \) digits of \( M \) is a perfect square. 3. The number formed by the last \( n \) digits of \( M \) is a perfect square and has ... | [
"\\( ((5 \\cdot 10^{499} - 1) \\cdot 10^{500} + 1)^2 \\)"
] | [
"A positive integer \\( M \\) with \\( 2n \\) digits is called [i]hypersquared[/i] if it satisfies the following conditions:\n1. \\( M \\) is a perfect square.\n2. The number formed by the first \\( n \\) digits of \\( M \\) is a perfect square.\n3. The number formed by the last \\( n \\) digits of \\( M \\) is a p... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5ef7399e11518fdd514a41fea51d9e2c89ccd6b559cd8bc7916cc32c5f9468b1 | A positive integer \( N \) is called "nice" if it satisfies the following conditions: - All of its digits are either \( 1 \) or \( 2 \). - All numbers formed by \( 3 \) consecutive digits of \( N \) are distinct. What is the greatest nice number? | [
"2221211122"
] | [
"A positive integer \\( N \\) is called \"nice\" if it satisfies the following conditions:\n- All of its digits are either \\( 1 \\) or \\( 2 \\).\n- All numbers formed by \\( 3 \\) consecutive digits of \\( N \\) are distinct.\nWhat is the greatest nice number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 7
} | 30 | 0 | 0 |
omr-ws-cc32265de869785700f0ec8a73275c2bdd6171c2271556e688c164c868ff9aae | A positive integer \( N \) is called "nice" if it satisfies the following conditions: - All of its digits are either \( 1 \) or \( 2 \). - All numbers formed by \( 3 \) consecutive digits of \( N \) are distinct. What is the largest possible number of digits that a nice number can have? | [
"10"
] | [
"A positive integer \\( N \\) is called \"nice\" if it satisfies the following conditions:\n- All of its digits are either \\( 1 \\) or \\( 2 \\).\n- All numbers formed by \\( 3 \\) consecutive digits of \\( N \\) are distinct.\nWhat is the largest possible number of digits that a nice number can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d4f7491acbbb74869428781140a0f5f01613f8072eb0a569804768c3ea9115d | A positive integer \( N \) is called *rioplatense* if it satisfies the following conditions: 1. There exist 34 consecutive integers such that their product is divisible by \( N \), but none of them is divisible by \( N \). 2. There do not exist 30 consecutive integers such that their product is divisible by \( N \), bu... | [
"\\( N = 31^\\alpha \\) for any integer \\(\\alpha > 1\\)"
] | [
"A positive integer \\( N \\) is called *rioplatense* if it satisfies the following conditions:\n1. There exist 34 consecutive integers such that their product is divisible by \\( N \\), but none of them is divisible by \\( N \\).\n2. There do not exist 30 consecutive integers such that their product is divisible b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-545f0b7a72f4903bcc600973e278eac9f1a9058151f15158f2f55fe0455bcf9c | A positive integer \( N \) is called [i]northern[/i] if for each digit \( d > 0 \), there exists a divisor of \( N \) whose last digit is \( d \). How many [i]northern[/i] numbers less than \( 2016 \) are there with the fewest number of divisors as possible? | [
"5"
] | [
"A positive integer \\( N \\) is called [i]northern[/i] if for each digit \\( d > 0 \\), there exists a divisor of \\( N \\) whose last digit is \\( d \\). How many [i]northern[/i] numbers less than \\( 2016 \\) are there with the fewest number of divisors as possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae71ab2d5f81233e9a00c55b3088bcb794a3ec800b5987068eced41dbd4e2eb7 | A positive integer \( N \) is called \emph{piquant} if there exists a positive integer \( m \) such that if \( n_i \) denotes the number of digits in \( m^i \) (in base 10), then \( n_1 + n_2 + \cdots + n_{10} = N \). Let \( p_M \) denote the fraction of the first \( M \) positive integers that are piquant. Find \( \li... | [
"\\dfrac{46}{55}"
] | [
"A positive integer \\( N \\) is called \\emph{piquant} if there exists a positive integer \\( m \\) such that if \\( n_i \\) denotes the number of digits in \\( m^i \\) (in base 10), then \\( n_1 + n_2 + \\cdots + n_{10} = N \\). Let \\( p_M \\) denote the fraction of the first \\( M \\) positive integers that are... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-5471086be1c97911dbe8c9088418cbf546dda8d5914932e20391574959eded54 | A positive integer \( N \) is called contagious if there are 1000 consecutive non-negative integers such that the sum of all their digits is \( N \). Find all contagious positive integers. | [
"\\( N \\geq 13500 \\)"
] | [
"A positive integer \\( N \\) is called contagious if there are 1000 consecutive non-negative integers such that the sum of all their digits is \\( N \\). Find all contagious positive integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-be8a8be6eed2069d85b090816e1f3f35ad0ab3f03b8e6712b35112c1ddaeee7e | A positive integer \( N \) is defined as *interoceanic* if its prime factorization \[ N = p_1^{x_1} p_2^{x_2} \cdots p_k^{x_k} \] satisfies \[ x_1 + x_2 + \cdots + x_k = p_1 + p_2 + \cdots + p_k. \] Find all interoceanic numbers less than 2020. | [
"\\( 4, 27, 48, 72, 108, 162, 320, 800, 1792, 2000 \\)"
] | [
"A positive integer \\( N \\) is defined as *interoceanic* if its prime factorization \n\n\\[ N = p_1^{x_1} p_2^{x_2} \\cdots p_k^{x_k} \\]\n\nsatisfies \n\n\\[ x_1 + x_2 + \\cdots + x_k = p_1 + p_2 + \\cdots + p_k. \\]\n\nFind all interoceanic numbers less than 2020."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 22
} | 30 | 0 | 0 |
omr-ws-74361049b995a56a3c6a4e6463afe6059c2a48f830319abf8f9ae2712b7a6ba6 | A positive integer \( N \) is written on a board. In each step, the last digit \( c \) of the number on the board is erased, and the remaining number \( m \) is replaced with \( |m - 3c| \). This process is repeated until the number on the board has only one digit. Find all positive integers \( N \) such that after a f... | [
"All multiples of 31.",
"All multiples of \\( 31 \\)"
] | [
"A positive integer \\( N \\) is written on a board. In each step, the last digit \\( c \\) of the number on the board is erased, and the remaining number \\( m \\) is replaced with \\( |m - 3c| \\). This process is repeated until the number on the board has only one digit. Find all positive integers \\( N \\) such... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 44
} | 66 | 0 | 0 |
omr-ws-8ee39398a42ac8323ebbedca26dd5290a35d488e19b0a97938c3143414889403 | A positive integer \( a > 1 \) is given (in decimal notation). We copy it twice and obtain a number \( b = \overline{aa} \) which happened to be a multiple of \( a^2 \). Find all possible values of \( \frac{b}{a^2} \). | [
"7"
] | [
"A positive integer \\( a > 1 \\) is given (in decimal notation). We copy it twice and obtain a number \\( b = \\overline{aa} \\) which happened to be a multiple of \\( a^2 \\). Find all possible values of \\( \\frac{b}{a^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-c8e7cd704afedbc43beb09b2bafed9210fe1dce0a5ce9534172e48a908e8eafb | A positive integer \( a > 1 \) is given. We form a new number \( b = \overline{aa} \) by concatenating \( a \) with itself. If \( b \) is a multiple of \( a^2 \), find all possible values of \( \frac{b}{a^2} \). | [
"7"
] | [
"A positive integer \\( a > 1 \\) is given. We form a new number \\( b = \\overline{aa} \\) by concatenating \\( a \\) with itself. If \\( b \\) is a multiple of \\( a^2 \\), find all possible values of \\( \\frac{b}{a^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-953d75e6c016b002c279cc388edada4447c7e6faf87d40a96155a7795af0617c | A positive integer \( a \) is called "good" if there exists an integer \( b > a \) such that the remainder when 2018 is divided by \( a \) plus the remainder when 2018 is divided by \( b \) equals 2018. Find the number of good integers. | [
"4"
] | [
"A positive integer \\( a \\) is called \"good\" if there exists an integer \\( b > a \\) such that the remainder when 2018 is divided by \\( a \\) plus the remainder when 2018 is divided by \\( b \\) equals 2018. Find the number of good integers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d555cd2d505b8029af036d977234d62892b8e8dff3feda617a4283c48e75058 | A positive integer \( b \) is called good if there exist positive integers \( 1 = a_1, a_2, \ldots, a_{2023} = b \) such that \( |a_{i+1} - a_i| = 2^i \). Find the number of good integers. | [
"\\( 2^{2021} \\)"
] | [
"A positive integer \\( b \\) is called good if there exist positive integers \\( 1 = a_1, a_2, \\ldots, a_{2023} = b \\) such that \\( |a_{i+1} - a_i| = 2^i \\). Find the number of good integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-174bfe31cb316f06ad5585a043b70903b140ccd5920f9b54b6736cc46722f07e | A positive integer \( b \geq 2 \) is neat if and only if there exist positive base-\( b \) digits \( x \) and \( y \) (that is, \( x \) and \( y \) are integers, \( 0 < x < b \) and \( 0 < y < b \)) such that the number \( x + \frac{y}{b} \) is an integer multiple of \( \frac{x}{y} \). Find the number of neat integers ... | [
"39"
] | [
"A positive integer \\( b \\geq 2 \\) is neat if and only if there exist positive base-\\( b \\) digits \\( x \\) and \\( y \\) (that is, \\( x \\) and \\( y \\) are integers, \\( 0 < x < b \\) and \\( 0 < y < b \\)) such that the number \\( x + \\frac{y}{b} \\) is an integer multiple of \\( \\frac{x}{y} \\). Find ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 21
} | 40 | 0 | 0 |
omr-ws-d6122230173350f704982a0d592d4c5cd99db1c520325eec6d0931526552148f | A positive integer \( k \) is called \( n \)-good if there exists a set of \( n \) distinct positive integers such that the sum of any subset of these integers is not divisible by \( k \). Find the smallest value of \( n \) for which \( 6 \) is \( n \)-good. | [
"5"
] | [
"A positive integer \\( k \\) is called \\( n \\)-good if there exists a set of \\( n \\) distinct positive integers such that the sum of any subset of these integers is not divisible by \\( k \\). Find the smallest value of \\( n \\) for which \\( 6 \\) is \\( n \\)-good."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-414939de5fabd03049c4dec5b757d63e1e6acd8608a5fefa4ed646da0d4fe470 | A positive integer \( k \) is called captivating if it satisfies the equation \( k^2 - 3k + 2 = p \), where \( p \) is a prime number. Determine the smallest positive integer \( C \) such that for any positive integer \( n \), there is at least one captivating integer in the interval \([n, n + C]\). Calculate \( C \). | [
"3"
] | [
"A positive integer \\( k \\) is called captivating if it satisfies the equation \\( k^2 - 3k + 2 = p \\), where \\( p \\) is a prime number. Determine the smallest positive integer \\( C \\) such that for any positive integer \\( n \\), there is at least one captivating integer in the interval \\([n, n + C]\\). Ca... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3678614ca76480ee978b073318673533585f754b881ca09625c598d853144856 | A positive integer \( k \) is called good if for every integer \( n \ge 4 \) such that \( 5n + 1 \) is a perfect square, there exist positive integers \( a_1, a_2, \ldots, a_k \) such that \( n + 1 = a_1^2 + a_2^2 + \cdots + a_k^2 \). Find the two smallest good positive integers. | [
"4"
] | [
"A positive integer \\( k \\) is called good if for every integer \\( n \\ge 4 \\) such that \\( 5n + 1 \\) is a perfect square, there exist positive integers \\( a_1, a_2, \\ldots, a_k \\) such that \\( n + 1 = a_1^2 + a_2^2 + \\cdots + a_k^2 \\). Find the two smallest good positive integers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-f42da086d34b9bb0b56cf1cafb8a1e04b2f53e6ed7dc5b52bfe6dd1e5d3666f1 | A positive integer \( k \) is called interesting if the product of the first \( k \) primes is divisible by \( k \). For example, the product of the first two primes is \( 2 \cdot 3 = 6 \), which is divisible by 2, so 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? | [
"3"
] | [
"A positive integer \\( k \\) is called interesting if the product of the first \\( k \\) primes is divisible by \\( k \\). For example, the product of the first two primes is \\( 2 \\cdot 3 = 6 \\), which is divisible by 2, so 2 is an interesting integer. What is the maximal possible number of consecutive interest... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-3cca3af5c1be03911b076ba806593e542c77882317749e7c4a03ce2e6e50c972 | A positive integer \( k \) is said to be \((m,n)\)-tasty if there exists a set \( S \subset \mathbb{C} \) with \( |S| = k \) such that for all subsets \( A \subset S \) with \( |A| = m \), there exists a subset \( B \subset S \) with \( |B| = n \) and \(\sum_{z \in A} z = \sum_{z \in B} z\). Determine all integers \( k... | [
"k \\geq m + n"
] | [
"A positive integer \\( k \\) is said to be \\((m,n)\\)-tasty if there exists a set \\( S \\subset \\mathbb{C} \\) with \\( |S| = k \\) such that for all subsets \\( A \\subset S \\) with \\( |A| = m \\), there exists a subset \\( B \\subset S \\) with \\( |B| = n \\) and \\(\\sum_{z \\in A} z = \\sum_{z \\in B} z\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-04057b6c2e64122ac17d5c2a146743c781a1c8e767ad3a4c923af364b230fb68 | A positive integer \( k \geq 3 \) is called "fibby" if there exists a positive integer \( n \) and positive integers \( d_1 < d_2 < \ldots < d_k \) such that: - \( d_{j+2} = d_{j+1} + d_j \) for every \( j \) satisfying \( 1 \leq j \leq k-2 \), - \( d_1, d_2, \ldots, d_k \) are divisors of \( n \), - Any other divisor ... | [
"\\( k = 3 \\) and \\( k = 4 \\)"
] | [
"A positive integer \\( k \\geq 3 \\) is called \"fibby\" if there exists a positive integer \\( n \\) and positive integers \\( d_1 < d_2 < \\ldots < d_k \\) such that:\n- \\( d_{j+2} = d_{j+1} + d_j \\) for every \\( j \\) satisfying \\( 1 \\leq j \\leq k-2 \\),\n- \\( d_1, d_2, \\ldots, d_k \\) are divisors of \... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-04110721bd6f27751d459afa773cc96af620360fc359f8ae8896067fe1758453 | A positive integer \( m \) is called \( nice \) if there exist integers \( n_1, n_2, \ldots, n_m \) such that \[ m = n_1 + n_2 + \cdots + n_m = n_1 n_2 \cdots n_m. \] Find all \( nice \) numbers. | [
"\\( m \\equiv 0 \\pmod{4} \\) and \\( m > 4 \\), or \\( m \\equiv 1 \\pmod{4} \\)"
] | [
"A positive integer \\( m \\) is called \\( nice \\) if there exist integers \\( n_1, n_2, \\ldots, n_m \\) such that \n\\[ m = n_1 + n_2 + \\cdots + n_m = n_1 n_2 \\cdots n_m. \\]\nFind all \\( nice \\) numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 7,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-6f9418b8a7ac5522dd2d3579e947a369df546ed7b5415c16d408c7db92f6ce75 | A positive integer \( m \) is called self-descriptive in base \( b \) if: i) The representation of \( m \) in base \( b \) is of the form \( (a_0a_1\ldots a_{b-1})_b \) (that is \( m = a_0b^{b-1} + a_1b^{b-2} + \ldots + a_{b-2}b + a_{b-1} \), where \( 0 \le a_i \le b-1 \) are integers). ii) \( a_i \) is equal to the nu... | [
"\\( (1210)_4 \\)"
] | [
"A positive integer \\( m \\) is called self-descriptive in base \\( b \\) if:\ni) The representation of \\( m \\) in base \\( b \\) is of the form \\( (a_0a_1\\ldots a_{b-1})_b \\) (that is \\( m = a_0b^{b-1} + a_1b^{b-2} + \\ldots + a_{b-2}b + a_{b-1} \\), where \\( 0 \\le a_i \\le b-1 \\) are integers).\nii) \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-ea8865c1a579f440cd37264143d28d0cf19954065ab941a1b3b0fe2d6f947243 | A positive integer \( m \) is defined as a factorial number if there exists a positive integer \( k \) such that \( m = k(k-1)(k-2)\cdots 2 \cdot 1 \). A positive integer \( n \) is defined as a Thai number if it can be written both as the sum of two factorial numbers and as the product of two factorial numbers. What i... | [
"210"
] | [
"A positive integer \\( m \\) is defined as a factorial number if there exists a positive integer \\( k \\) such that \\( m = k(k-1)(k-2)\\cdots 2 \\cdot 1 \\). A positive integer \\( n \\) is defined as a Thai number if it can be written both as the sum of two factorial numbers and as the product of two factorial ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac992ddf3716d6afac50cd2dca51d98d9f2efcc6b0eb102278573cf30267a13d | A positive integer \( m \) is said to be a \(\emph{zero taker}\) if there exists a positive integer \( k \) such that: - \( k \) is a perfect square, - \( m \) divides \( k \), - the decimal expression of \( k \) contains at least 2021 '0' digits, but the last digit of \( k \) is not equal to 0. Find all positive integ... | [
"Positive integers not divisible by 10."
] | [
"A positive integer \\( m \\) is said to be a \\(\\emph{zero taker}\\) if there exists a positive integer \\( k \\) such that:\n- \\( k \\) is a perfect square,\n- \\( m \\) divides \\( k \\),\n- the decimal expression of \\( k \\) contains at least 2021 '0' digits, but the last digit of \\( k \\) is not equal to 0... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-fb612731162b528d0886d2cf4a0b73c45cef2d55ec3e271fc6bfe1c9258d507c | A positive integer \( n < 2017 \) is given. Exactly \( n \) vertices of a regular 2017-gon are colored red, and the remaining vertices are colored blue. Calculate the number of isosceles triangles whose vertices are all red if \( n = 1008 \). | [
"no_valid_answer_found"
] | [
"A positive integer \\( n < 2017 \\) is given. Exactly \\( n \\) vertices of a regular 2017-gon are colored red, and the remaining vertices are colored blue. Calculate the number of isosceles triangles whose vertices are all red if \\( n = 1008 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8f5dd31098563c2071001b6de384debc8055280ca02416ecac6e6dcbf0e9db9 | A positive integer \( n > 1 \) is called an "interesting" number if for every prime divisor \( p \) of \( n \), \( 2p + 1 \) is also a divisor of \( n \). Find the total number of positive divisors of the smallest "interesting" number. | [
"512"
] | [
"A positive integer \\( n > 1 \\) is called an \"interesting\" number if for every prime divisor \\( p \\) of \\( n \\), \\( 2p + 1 \\) is also a divisor of \\( n \\). Find the total number of positive divisors of the smallest \"interesting\" number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f7991cc406b7a733134bb39b6d4979db33b3eee9fbc71ed0c94902edef3c5d4 | A positive integer \( n > 1 \) is called an interesting number if for every prime divisor \( p \) of \( n \), \( 2p + 1 \) is also a divisor of \( n \). Find the total number of positive divisors of the smallest interesting number. | [
"4"
] | [
"A positive integer \\( n > 1 \\) is called an interesting number if for every prime divisor \\( p \\) of \\( n \\), \\( 2p + 1 \\) is also a divisor of \\( n \\). Find the total number of positive divisors of the smallest interesting number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2eb44b3ca9bfde45bc9a273a10b014afc3acbeca0c2ff765031dc1c2891d4c06 | A positive integer \( n > 1 \) is called beautiful if \( n \) can be written in one and only one way as \( n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k \) for some positive integers \( a_1, a_2, \ldots, a_k \), where \( k > 1 \) and \( a_1 \geq a_2 \geq \cdots \geq a_k \). Find the largest beautiful number l... | [
"95"
] | [
"A positive integer \\( n > 1 \\) is called beautiful if \\( n \\) can be written in one and only one way as \\( n = a_1 + a_2 + \\cdots + a_k = a_1 \\cdot a_2 \\cdots a_k \\) for some positive integers \\( a_1, a_2, \\ldots, a_k \\), where \\( k > 1 \\) and \\( a_1 \\geq a_2 \\geq \\cdots \\geq a_k \\). Find the l... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-05fbf4c9015ff37fe44fd5ed7f53ae59e96842614365576f7a1092442980c854 | A positive integer \( n > 2 \) is chosen, and each of the numbers \( 1, 2, \dots, n \) is colored red or blue. Each subset of \(\{1,2,\ldots,n\}\) is then colored either red or blue such that each red number lies in more red subsets than blue subsets, and each blue number lies in more blue subsets than red subsets. Let... | [
"536"
] | [
"A positive integer \\( n > 2 \\) is chosen, and each of the numbers \\( 1, 2, \\dots, n \\) is colored red or blue. Each subset of \\(\\{1,2,\\ldots,n\\}\\) is then colored either red or blue such that each red number lies in more red subsets than blue subsets, and each blue number lies in more blue subsets than r... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-1b55cd612cb415d40a72ad545153e898112c1825f0393b8f0ef38857501a57aa | A positive integer \( n \) (\( n \geq 2 \)) is defined as "good" if for every \( (n-1) \)-tuple \((a_2, a_3, \ldots, a_n)\), there exists an integer \( x \) such that \( i \nmid (x - a_i) \) for all \( 2 \leq i \leq n \). Is there a largest good integer? If so, find it. | [
"4"
] | [
"A positive integer \\( n \\) (\\( n \\geq 2 \\)) is defined as \"good\" if for every \\( (n-1) \\)-tuple \\((a_2, a_3, \\ldots, a_n)\\), there exists an integer \\( x \\) such that \\( i \\nmid (x - a_i) \\) for all \\( 2 \\leq i \\leq n \\). Is there a largest good integer? If so, find it."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-7e1ef82fcbb4cd54b60ab5ba53af26c8dc9641b330d1f846c80599c2b9159d57 | A positive integer \( n \) between \( 1 \) and \( N = 2007^{2007} \) inclusive is selected at random. If \( a \) and \( b \) are natural numbers such that \( \frac{a}{b} \) is the probability that \( N \) and \( n^3 - 36n \) are relatively prime, find the value of \( a + b \). | [
"1109"
] | [
"A positive integer \\( n \\) between \\( 1 \\) and \\( N = 2007^{2007} \\) inclusive is selected at random. If \\( a \\) and \\( b \\) are natural numbers such that \\( \\frac{a}{b} \\) is the probability that \\( N \\) and \\( n^3 - 36n \\) are relatively prime, find the value of \\( a + b \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 3,
"tir": 19
} | 38 | 0 | 0 |
omr-ws-63ad303675650c8df60f4b0d2eccb26a0a51c2ac558ef2c93fabfdb4440d591b | A positive integer \( n \) has 60 divisors and \( 7n \) has 80 divisors. What is the greatest integer \( k \) such that \( 7^k \) divides \( n \)? | [
"2"
] | [
"A positive integer \\( n \\) has 60 divisors and \\( 7n \\) has 80 divisors. What is the greatest integer \\( k \\) such that \\( 7^k \\) divides \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-dbe99e4a47cd73ad8f705e1c5ba461fa77fa997d11139ffc4280c5308cc4d02c | A positive integer \( n \) has exactly 12 positive divisors \( 1 = d_1 < d_2 < \cdots < d_{12} = n \). Let \( m = d_4 - 1 \). We have \( d_m = (d_1 + d_2 + d_4)d_8 \). Find \( n \). | [
"1989"
] | [
"A positive integer \\( n \\) has exactly 12 positive divisors \\( 1 = d_1 < d_2 < \\cdots < d_{12} = n \\). Let \\( m = d_4 - 1 \\). We have \\( d_m = (d_1 + d_2 + d_4)d_8 \\). Find \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-ec5644a3cfad9693e3eb9c63fddad51b7eda2ff6abc38e3c2d7ede67a58683b8 | A positive integer \( n \) has exactly 12 positive divisors \( 1 = d_{1} < d_{2} < d_{3} < \ldots < d_{12} = n \). Let \( m = d_{4} - 1 \). Given that \( d_{m} = (d_{1} + d_{2} + d_{4}) d_{8} \), find \( n \). | [
"1989"
] | [
"A positive integer \\( n \\) has exactly 12 positive divisors \\( 1 = d_{1} < d_{2} < d_{3} < \\ldots < d_{12} = n \\). Let \\( m = d_{4} - 1 \\). Given that \\( d_{m} = (d_{1} + d_{2} + d_{4}) d_{8} \\), find \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-14be73d3f0d308377c84cc2717237cb0129be8452adb53d94b42ecc7760f43d8 | A positive integer \( n \) has exactly 4 positive divisors, including 1 and \( n \). Furthermore, \( n+1 \) is four times the sum of the other two divisors. Find \( n \). | [
"95"
] | [
"A positive integer \\( n \\) has exactly 4 positive divisors, including 1 and \\( n \\). Furthermore, \\( n+1 \\) is four times the sum of the other two divisors. Find \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d3563df8ff9cc7abb1c02e52626bd073bbddefb84f9c9d0a481bd6c3e25b501 | A positive integer \( n \) has the property that \( 2n \) has 28 divisors and \( 3n \) has 30 divisors. How many divisors does \( 6n \) have? | [
"35"
] | [
"A positive integer \\( n \\) has the property that \\( 2n \\) has 28 divisors and \\( 3n \\) has 30 divisors. How many divisors does \\( 6n \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-25a62da763ad2802acf5a4f918d722be907f65332b5734adf6dfe4835d01428d | A positive integer \( n \) has the property that when divided by exactly 9 different positive integers, it has a remainder of 5. What is the smallest possible value of \( n \)? | [
"131"
] | [
"A positive integer \\( n \\) has the property that when divided by exactly 9 different positive integers, it has a remainder of 5. What is the smallest possible value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-9378a9a714bd2a14dbde596b589dbcfbc11399a0b78e46726feac9d4137c273c | A positive integer \( n \) having distinct digits is called super if either it is a 1-digit number or there exists a divisor of \( n \) which can be obtained by removing a single digit from \( n \) and this divisor is also super. Find the largest super number. | [
"987654321"
] | [
"A positive integer \\( n \\) having distinct digits is called super if either it is a 1-digit number or there exists a divisor of \\( n \\) which can be obtained by removing a single digit from \\( n \\) and this divisor is also super. Find the largest super number."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cf052a36f4378939c404a2604ea544db424fc592d562bcf61a8edbe6031f0f5 | A positive integer \( n \) is "olympic" if there exist \( n \) non-negative integers \( x_1, x_2, \ldots, x_n \) such that: - At least one \( x_j \) is positive. - For any choice of \( n \) numbers \( c_1, c_2, \ldots, c_n \) from the set \(\{-1, 0, 1\}\) where not all \( c_i \) are zero, the sum \( c_1x_1 + c_2x_2 + \... | [
"9"
] | [
"A positive integer \\( n \\) is \"olympic\" if there exist \\( n \\) non-negative integers \\( x_1, x_2, \\ldots, x_n \\) such that:\n- At least one \\( x_j \\) is positive.\n- For any choice of \\( n \\) numbers \\( c_1, c_2, \\ldots, c_n \\) from the set \\(\\{-1, 0, 1\\}\\) where not all \\( c_i \\) are zero, t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f9bc7fe3c0f5186732a679ca9e733c10fb90f723d99df488a4de557766f0a336 | A positive integer \( n \) is "orz" if it has \( p \) positive divisors where \( p \) is a prime number greater than 2. How many "orz" numbers are there less than 1000? | [
"16"
] | [
"A positive integer \\( n \\) is \"orz\" if it has \\( p \\) positive divisors where \\( p \\) is a prime number greater than 2. How many \"orz\" numbers are there less than 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
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