### International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

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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.
Hint: For every $n$th root of unity $\eta$, consider $$\sum_{z^n=1} z^2f(z)f(\eta z).$$