IMC 1999 · Problem 5

Suppose that $2n$ points of an $n \times n$ grid are marked. Show that for some $k > 1$ one can select $2k$ distinct marked points, say $a_1, \dots, a_{2k}$, such that $a_1$ and $a_2$ are in the same row, $a_2$ and $a_3$ are in the same column, $\dots$, $a_{2k-1}$ and $a_{2k}$ are in the same row, and $a_{2k}$ and $a_1$ are in the same column.