IMC 2018 Problem 5

Let $p$ and $q$ be prime numbers with $p<q$ . Suppose that in a convex polygon $P_1,P_2,…,P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that $$P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}$$ holds for every integer $k$ with $1\leqslant k\leqslant p$ .

Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin