Olympiad problems

2020 Switzerland Team Selection Test, 4

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

2019 India National OIympiad, 3

Tags: number theory , MONT

Let $m,n$ be distinct positive integers. Prove that $$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$ Further, determine when equality holds.

2016 India Regional Mathematical Olympiad, 6

Tags: arithmetic sequence , number theory , MONT

Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.

2001 All-Russian Olympiad, 4

Tags: number theory , relatively prime , Russia , Divisors , MONT

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

2017 All-Russian Olympiad, 5

Tags: number theory , MONT

$n$ is composite. $1<a_1<a_2<...<a_k<n$ - all divisors of $n$. It is known, that $a_1+1,...,a_k+1$ are all divisors for some $m$ (except $1,m$). Find all such $n$.

2017 Iran MO (2nd Round), 1

Tags: number theory , prime numbers , MONT

a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$ b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$

2001 All-Russian Olympiad, 4

Tags: number theory , relatively prime , Russia , Divisors , MONT

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.