DE-BUNDESWETTBEWERB 2001 Problem 8

A square $R$ of sidelength $250$ lies inside a square $Q$ of sidelength $500$ . Prove that: One can always find two points $A$ and $B$ on the perimeter of $Q$ such that the segment $AB$ has no common point with the square $R$ , and the length of this segment $AB$ is greater than $521$ .