Found problems: 97
2017 Costa Rica - Final Round, 2
Determine the greatest common divisor of the numbers:
$$5^5-5, 7^7-7, 9^9-9 ,..., 2017^{2017}-2017,$$
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 $i \neq j$.
1995 All-Russian Olympiad Regional Round, 10.2
Natural numbers $m$ and $n$ satisfy $$gcd(m,n)+lcm(m,n) = m+n. $$Prove that one of numbers $m,n$ divides the other.
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.
2002 Estonia National Olympiad, 1
The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.
2018 Switzerland - Final Round, 2
Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take:
$$\frac{a}{gcd\,\,(a + b, a - c)}
+
\frac{b}{gcd\,\,(b + c, b - a)}
+
\frac{c}{gcd\,\,(c + a, c - b)}.$$
.
Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.
2016 Dutch Mathematical Olympiad, 3
Find all possible triples $(a, b, c)$ of positive integers with the following properties:
• $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$,
• $a$ is a divisor of $a + b + c$,
• $b$ is a divisor of $a + b + c$,
• $c$ is a divisor of $a + b + c$.
(Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)
2014 Brazil Team Selection Test, 1
For $m$ and $n$ positive integers that are prime to each other, determine the possible values of
$$\gcd (5^m + 7^m, 5^n + 7^n)$$
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).
2015 Czech-Polish-Slovak Junior Match, 4
Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$.
2020 Macedonian Nationаl Olympiad, 1
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.
2013 Saudi Arabia BMO TST, 2
For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
2012 Bogdan Stan, 2
For any $ a\in\mathbb{Z}_{\ge 0} $ make the notation $ a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . $ Prove that the following relations are equivalent:
$ \text{(1)} a\mathbb{Z}_{\ge 0} \setminus b\mathbb{Z}_{\ge 0}\subset c\mathbb{Z}_{\ge 0} \setminus d\mathbb{Z}_{\ge 0} $
$ \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d)) $
[i]Marin Tolosi[/i] and [i]Cosmin Nitu[/i]
2001 Saint Petersburg Mathematical Olympiad, 10.6
For any positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$
As usual, [x,y] denotes the least common multiply of $x,y$
[I]Proposed by A. Golovanov[/i]
2015 Romania Team Selection Tests, 1
Let $a$ be an integer and $n$ a positive integer . Show that the sum :
$$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .
2019 Saudi Arabia Pre-TST + Training Tests, 3.3
Let $d$ be a positive divisor of a positive integer $m$ and $(a_l), (b_l)$ two arithmetic sequences of positive integers. It is given that $gcd(a_i, b_j) = 1$ and $gcd(a_k, b_n) = m$ for some positive integers $i,j,k,$ and $n$. Prove that there exist positive integers $t$ and $s$ such that $gcd(a_t, b_s) = d$.
2019 Saudi Arabia BMO TST, 1
Let $19$ 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 $180$. Prove that not all numbers on the paper are different.
2025 All-Russian Olympiad, 10.5
Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained? \\
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]
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 $f_6(n) = f_n(6)$.
2024 Czech and Slovak Olympiad III A, 1
Let $a, b, c$ be positive integers such that one of the values $$gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)$$
is equal to the product of the remaining two. Prove that one of the numbers $a, b, c$ is a multiple of another of them.
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]
2019 Centers of Excellency of Suceava, 2
Let $ \left( s_n \right)_{n\ge 1 } $ be a sequence with $ s_1 $ and defined recursively as $ s_{n+1}=s_n^2-s_n+1. $
Prove that any two terms of this sequence are coprime.
[i]Dan Nedeianu[/i]
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 $d = 1, d = 2$
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 $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.