An inequality on the coefficients of a monic polynomial
Source: Balkan MO ShortList 2009 A5
April 6, 2020
Problem Statement
Given the monic polynomial
\begin{align*} P(x) = x^N +a_{N-1}x^{N-1} + \ldots + a_1 x + a_0 \in \mathbb{R}[x] \end{align*}
of even degree and having all real positive roots , for . Prove, for any , the following inequality
\begin{align*} c + \sqrt[N]{P(c)} \le \sqrt[N]{a_0} \end{align*}