MathLabs

Problem 5

Let d be the sum of lengths of all diagonals of a convex n-gon, n>3, and p its perimeter. Prove n−3<2dp<⌊n2⌋⌊n+12⌋−2n-3<\frac{2d}{p}<\left\lfloor\frac n2\right\rfloor\left\lfloor\frac{n+1}{2}\right\rfloor-2.
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n=2m:quad2d/p<m2−2n=2m:quad 2d/p<m^2-2
Detailed analysis

For n=2mn=2m, use the rr-step chains for 2≤r≤m−12\le r\le m-1, and use that each diameter is strictly shorter than half the perimeter. This gives 2d/p<m2−22d/p<m^2-2.