IMC 2018 · Problem 5

Let $p$ and $q$ be prime numbers with $p < q$. Suppose that in a convex polygon $P_1P_2 \ldots P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that

$$P_1P_2 + P_2P_3 + \cdots + P_kP_{k+1} \ge \frac{k^3 + k}{2}$$

holds for every integer $k$ with $1 \le k \le p$.