Found problems: 97
2004 Switzerland - Final Round, 8
A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.
2000 Korea Junior Math Olympiad, 1
For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.
2001 Saint Petersburg Mathematical Olympiad, 9.4
Let $a,b,c\in\mathbb{Z^{+}}$ such that
$$(a^2-1, b^2-1, c^2-1)=1$$
Prove that
$$(ab+c, bc+a, ca+b)=(a,b,c)$$
(As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$)
[I]Proposed by A. Golovanov[/i]
2020 MMATHS, I5
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i]
2011 Saudi Arabia IMO TST, 2
Consider the set $S= \{(a + b)^7 - a^7 - b^7 : a,b \in Z\}$. Find the greatest common divisor of all members in $S$.
2023 Indonesia TST, N
Given an integer $a>1$. Prove that there exists a sequence of positive integers
\[ n_1, n_2, n_3, \ldots \]
Such that
\[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.
2010 China Northern MO, 7
Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$
The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.
2021 Durer Math Competition (First Round), 4
Determine all triples of positive integers $a, b, c$ that satisfy
a) $[a, b] + [a, c] + [b, c] = [a, b, c]$.
b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$.
Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.
2024 Kyiv City MO Round 2, Problem 2
You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set
$\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$?
Here $(x, y)$ denotes the greatest common divisor of $x, y$.
[i]Proposed by Anton Trygub[/i]
2021 Czech-Polish-Slovak Junior Match, 3
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.
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 $n$?
2018 Saudi Arabia GMO TST, 1
Let $n$ be an odd positive integer with $n > 1$ and let $a_1, a_2,... , a_n$ be positive integers such that gcd $(a_1, a_2,... , a_n) = 1$. Let $d$ = gcd $(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)$. Show that the possible values of $d$ are $d = 1, d = 2$
2018 Turkey Team Selection Test, 7
For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?
2012 JBMO TST - Turkey, 2
Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.
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.
2025 Kosovo National Mathematical Olympiad`, P4
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously
(i) For all $m,n \in \mathbb{N}$ we have:
$$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$
(ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.
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 $a + b + c = 5n$.
2018 Peru Cono Sur TST, 8
For each pair of positive integers $m$ and $n$, we define $f_m(n)$ as follows:
$$ f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k), $$
where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$. For example,
$f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$.
$a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$.
$b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$.
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 $a + b + c = 5n$.
2008 Indonesia TST, 3
Let $n$ be an arbitrary positive integer.
(a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$.
(b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).
2023 Regional Competition For Advanced Students, 4
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
2015 Saudi Arabia IMO TST, 1
Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$.
Trần Nam Dũng
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 $d>1.$
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
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?