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+1:2d/p<m2+m−2n=2m+1:\quad 2d/p<m^2+m-2
Detailed analysis

For n=2m+1n=2m+1, bound each rr-step diagonal (2≤r≤m2\le r\le m) by the rr-side boundary chain. Cyclic summation gives 2d/p≤m2+m−22d/p\le m^2+m-2.