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S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2 >=ra^2

Source: 2008 Cuba 2.1

August 27, 2024
algebrainequalitiestrinomial

Problem Statement

Given a polynomial of degree 22, p(x)=ax2+bx+cp(x) = ax^2 +bx+c define the function S(p)=(a−b)2+(b−c)2+(c−a)2.S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2. Determine the real numberr rsuch that, for any polynomial p(x)p(x) of degree 22 with real roots, holds S(p)≥ra2S(p) \ge ra^2
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