EGMO 2023 | Problem 5 - 17 April 2023

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 $s$ .