12-та Европска женска математичка олимпијада (EGMO 2023)
Задача 5

 
 
Problem 5.
We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its twist $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$.
Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.