Tuesday, January 10th

Problem

For each side of a polygon, divide its length by the length of the other sides. Prove that the sum of all such fractions is smaller than 2. 

Solution

With out loss of generality suppose that the sum of the sides $a_1,\dots, a_n$ of the polygon is 1. Hence it suffices to prove that

$\sum_{i=1}^n \frac{a_i}{1-a_i}\leq 2$

Since $f(x)=\frac{x}{1-x}$ is a convex function, we have that

$\sum_{i=1}^n \frac{a_i}{1-a_i}\leq \frac{n}{n-1}\leq 2$