IMO 1984 Problem 5

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 $x$ .