MathDB

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.)

Problem Statement