2003 IMO Shortlist S13

Let $D_1$ , $D_2$ , ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$ . Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$ .

设 $D_1$ , $D_2$ , ..., $D_n$ 为平面上的闭圆盘。 (封闭圆盘是由圆限定的区域，与该圆连在一起。)假设平面上的每个点最多包含在 $2003$ 个圆盘 $D_i$ 中。证明存在一个圆盘 $D_k$ ，它最多与 $7\cdot 2003 - 1 = 14020$ 其他圆盘 $D_i$ 相交。