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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4d>2(n−3)p⟹n−3<2dp4d>2(n-3)p\Longrightarrow n-3<\frac{2d}{p}
Detailed analysis

For each fixed AA, sum over the n−3n-3 choices of XX, then sum over AA. Every diagonal is counted four times and every side 2(n−3)2(n-3) times, giving 4d>2(n−3)p4d>2(n-3)p.