International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria


Ivan is Watching You
Ivan's Office

Day 1
    Problem 1
    Problem 2
    Problem 3
    Problem 4
    Problem 5

Day 2
    Problem 6
    Problem 7
    Problem 8
    Problem 9
    Problem 10


    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions
    Closing Ceremony

Official IMC site

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 \(\displaystyle \Phi_q\).