MathDB

A5

Part of 2009 Balkan MO Shortlist

Problems(1)

An inequality on the coefficients of a monic polynomial

Source: Balkan MO ShortList 2009 A5

4/6/2020
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 NN == 2n2n and having all real positive roots xix_i, for 1≤i≤N1 \le i \le N. Prove, for any cc ∈\in [0,min⁡1≤i≤N{xi})[0, \underset{1 \le i \le N}{\min} \{x_i \} ), the following inequality \begin{align*} c + \sqrt[N]{P(c)} \le \sqrt[N]{a_0} \end{align*}