CMO 2025 Problem 5

A rectangle $R$ is divided into a set $S$ of finitely many smaller rectangles with sides parallel to the sides of $R$ such that no three rectangles in $S$ share a common corner. An ant is initially located at the bottom-left corner of $R$ . In one operation, we can choose a rectangle $r \in S$ such that the ant is currently located at one of the corners of $r$ , say $c$ , and move the ant to one of the two corners of $r$ adjacent to $c$ .

Suppose that after a finite number of operations, the ant ends up at the top-right corner of $R$ . Prove that some rectangle $r \in S$ was chosen in at least two operations.