1984 IMO 第 4 题

(MON 1) ${ }^{\mathrm{IMO} 5}$ Let $d$ be the sum of the lengths of all diagonals of a convex polygon of $n(n>3)$ vertices and let $p$ be its perimeter. Prove that $$\frac{n-3}{2}<\frac{d}{p}<\frac{1}{2}\left(\left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]-2\right) .$$

(MON 1) ${ }^{\mathrm{IMO} 5}$ 令 $d$ 为 $n(n>3)$ 个顶点的凸多边形的所有对角线长度之和，并令 $p$ 为其周长。证明 $$\frac{n-3}{2}<\frac{d}{p}<\frac{1}{2}\left(\left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]-2\right) 。$$