Let
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
be the sum of the lengths of all the diagonals of a plane convex polygon with
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
vertices (where
![n>3](/media/m/c/b/e/cbef1f7ac5b499f3987506d46938e285.png)
). Let
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
be its perimeter. Prove that:
where
![[x]](/media/m/6/a/4/6a47dfb91475b9d5490dbb3a666604a3.png)
denotes the greatest integer not exceeding
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
.
%V0
Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices (where $n>3$). Let $p$ be its perimeter. Prove that:
$$n-3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n+1\over 2}\Bigr]-2,$$
where $[x]$ denotes the greatest integer not exceeding $x$.