BMO 2019 Shortlist

Let $a_0$ be an arbitrary positive integer. Let $\{a_n\}$ be an infinite sequence of positive integers such that for every positive integer $n$ the term $a_n$ is the smallest positive integer such that $a_0+a_1+\cdots +a_n$ is divisible by $n$. Prove that there is a positive integer $N$ such that $a_{n+1}=a_n$ for all $n\ge N$.

Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n{)}_{n\ge 1}$, defined inductively as follows: given $a_0,a_1,\dots ,a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+\cdots +a_n$ is divisible by $n$. Prove that there exists a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\ge M$.