International Mathematics Competition
for University Students
2018

IMC2018: Day 1, Problem 5

Problem 5. Let \(\displaystyle p\) and \(\displaystyle q\) be prime numbers with \(\displaystyle p<q\). Suppose that in a convex polygon \(\displaystyle P_1P_2\dots P_{pq}\) all angles are equal and the side lengths are distinct positive integers. Prove that

\(\displaystyle P_1P_2+P_2P_3+\dots+P_kP_{k+1}\geq \dfrac{k^3+k}2\)

holds for every integer \(\displaystyle k\) with \(\displaystyle 1\le k\le p\).

(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin)

Hint: Use the cyclotomic polynomials \(\displaystyle \Phi_{pq}\), \(\displaystyle \Phi_p\) and \(\displaystyle \Phi_q\).

  Solution