Question 6
Let $S$ be a square with sides of length 100 and let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_0A_1,A_1A_2,\ldots,A_{n-1}A_n$ with $A_0\ne A_n$. 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 $\frac12$. 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 the part of $L$ that lies between $X$ and $Y$ is not smaller than 198.