2000 IMO Shortlist S16

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is *nice* if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$ . (There can be nonconvex regions as well as regions with more than one boundary curve.)

在平面中，我们有 $n$ 个具有平行边的矩形。不同矩形的边位于不同的线上。矩形的边界将平面切割成相连的区域。如果一个区域的边界上至少有 $n$ 个矩形的顶点，则该区域是*好的*。证明所有好的区域的顶点数之和小于 $40n$ 。 (可以存在非凸区域以及具有多条边界曲线的区域。)