### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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### 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 $\displaystyle \Phi_q$.