MEMO 2021 I4

Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$ -gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$ -th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$ . Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices. (Remark. For a real number $x$ , we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$ .)