A number of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
rectangles are drawn in the plane. Each rectangle has parallel sides and the sides of distinct rectangles lie on distinct lines. The rectangles divide the plane into a number of regions. For each region
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
let
![v(R)](/media/m/b/4/f/b4fccc85e5b08af7a5bfba322b7c0909.png)
be the number of vertices. Take the sum
![\sum v(R)](/media/m/f/d/6/fd6bc7375f9e80d83a12717dad03debe.png)
over the regions which have one or more vertices of the rectangles in their boundary. Show that this sum is less than
![40n](/media/m/4/1/9/4191b10c19786942eb7a0818e7916c04.png)
.
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A number of $n$ rectangles are drawn in the plane. Each rectangle has parallel sides and the sides of distinct rectangles lie on distinct lines. The rectangles divide the plane into a number of regions. For each region $R$ let $v(R)$ be the number of vertices. Take the sum $\sum v(R)$ over the regions which have one or more vertices of the rectangles in their boundary. Show that this sum is less than $40n$.