1989 CMO 第 1 题

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 $X$ )

给定两个点集 $A$ 和 $B$，它们都由单位圆上的有限不相交弧组成。此外，$B$ 中每条弧的长度等于 $\dfrac{\pi}{m}$ ( $m \in \mathbb{N}$ )。我们用 $A^j$ 表示 $A$ 绕单位圆中心逆时针旋转 $\dfrac{j\pi}{m}$ ( $j=1,2,3,\dots$ ) 所获得的集合。证明存在一个自然数 $k$ 使得 $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$ 。(这里 $l(X)$ 表示点集 $X$ 中所有不相交弧的长度之和)