CMO 1989 Problem 1

We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ( $m \in \mathbb{N}$ ). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ( $j=1,2,3,\dots$ ). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$ .(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$ )