Found problems: 97
1978 Romania Team Selection Test, 2
Suppose that $ k,l $ are natural numbers such that $ \gcd (11m-1,k)=\gcd (11m-1, l) , $ for any natural number $ m. $
Prove that there exists an integer $ n $ such that $ k=11^nl. $
2025 Bangladesh Mathematical Olympiad, P8
Let $a, b, m, n$ be positive integers such that $gcd(a, b) = 1$ and $a > 1$. Prove that if $$a^m+b^m \mid a^n+b^n$$then $m \mid n$.
2019 Dutch IMO TST, 3
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
2025 Alborz Mathematical Olympiad, P3
For every positive integer \( n \), do there exist pairwise distinct positive integers \( a_1, a_2, \dots, a_n \) that satisfy the following condition?
For every \( 3 \leq m \leq n \), there exists an \( i \leq m-2 \) such that:
$$
a_m = a_{\gcd(m-1, i)} + \gcd(a_{m-1}, a_i).
$$
Proposed by Alireza Jannati
2019 Saudi Arabia JBMO TST, 4
Let $14$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $91$. Prove that not all numbers on the paper are different.
2016 IFYM, Sozopol, 7
We are given a non-infinite sequence $a_1,a_2…a_n$ of natural numbers. While it is possible, on each turn are chosen two arbitrary indexes $i<j$ such that $a_i \nmid a_j$, and then $a_i$ and $a_j$ are changed with their $gcd$ and $lcm$. Prove that this process is non-infinite and the created sequence doesn’t depend on the made choices.
2023 European Mathematical Cup, 1
Suppose $a,b,c$ are positive integers such that \[\gcd(a,b)+\gcd(a,c)+\gcd(b,c)=b+c+2023\] Prove that $\gcd(b,c)=2023$.
[i]Remark.[/i] For positive integers $x$ and $y$, $\gcd(x,y)$ denotes their greatest common divisor.
[i]Ivan Novak[/i]
1998 Slovenia Team Selection Test, 4
Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$
2017 Romania EGMO TST, P2
Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$
2000 Korea Junior Math Olympiad, 1
For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.
2018 Singapore Junior Math Olympiad, 3
One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.
KoMaL A Problems 2023/2024, A. 882
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
2023 Bundeswettbewerb Mathematik, 1
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
2005 Switzerland - Final Round, 4
Determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ , $$\frac{a + b}{gcd(a, b)}$$ lies in $M$.
2021 Turkey Team Selection Test, 9
For which positive integer couples $(k,n)$, the equality
$\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$
holds?
1999 Swedish Mathematical Competition, 6
$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.
2012 Dutch IMO TST, 1
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
2010 Estonia Team Selection Test, 1
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$,
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
2019 Dutch BxMO TST, 4
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
2013 Peru MO (ONEM), 2
The positive integers $a, b, c$ are such that
$$gcd \,\,\, (a, b, c) = 1,$$
$$gcd \,\,\,(a, b + c) > 1,$$
$$gcd \,\,\,(b, c + a) > 1,$$
$$gcd \,\,\,(c, a + b) > 1.$$
Determine the smallest possible value of $a + b + c$.
Clarification: gcd stands for greatest common divisor.
2004 Estonia National Olympiad, 1
Find all triples of positive integers $(x, y, z)$ satisfying $x < y < z$, $gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8$ and $lcm(x, y,z) = 2400$.
2013 Switzerland - Final Round, 1
Find all triples $(a, b, c)$ of natural numbers such that the sets
$$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and
$$\{2, 3, 5, 30, 60\}$$
are the same.
Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.
2012 Dutch IMO TST, 1
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
2016 Latvia Baltic Way TST, 16
What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$?
[hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide]
2022 Caucasus Mathematical Olympiad, 3
Pete wrote down $21$ pairwise distinct positive integers, each not greater than $1,000,000$. For every pair $(a, b)$ of numbers written down by Pete, Nick wrote the number
$$F(a;b)=a+b -\gcd(a;b)$$
on his piece of paper. Prove that one of Nick’s numbers differs from all of Pete’s numbers.