International Mathematical Olympiad 1982 Problem 6

Let $S$ be a square with sides of length 100, and let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1, A_1A_2, \ldots, A_{n-1}A_n$ with $A_0 \neq A_n$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $1/2$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than 1, and the length of that part of $L$ which lies between $X$ and $Y$ is not smaller than 198.