International Mathematics Competition
for University Students
2017

IMC2017: Day 1, Problem 5

5. Let $k$ and $n$ be positive integers with $n\ge k^2-3k+4$, and let $$ f(z)=z^{n-1}+c_{n-2}z^{n-2}+\ldots+c_0 $$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\ldots=c_{n-2}c_0=0.$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.

Proposed by: Vsevolod Lev and Fedor Petrov, St. Petersburg State University

  Hint    Solution