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Inequality [Mediterranean Mathematics Olympiad 2017, P4]

Source: Mediterranean Mathematics Olympiad 2017, Problem 4

June 15, 2017
inequalities

Problem Statement

Let x,y,zx,y,z and a,b,ca,b,c be positive real numbers with a+b+c=1a+b+c=1. Prove that (x2+y2+z2)(a3x2+2y2+b3y2+2z2+c3z2+2x2)≥19.\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.
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