Question 53
Let $n \geqslant 3$ 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 $\left\lceil\frac{n}{2}\right\rceil$ jumps Zagi has visited $\left\lceil\frac{n}{2}\right\rceil+1$ 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 $x$.)
(Remark. For a real number $x$, we denote by $\lceil x\rceil$ the smallest integer larger or equal to $x$.)