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Inequalities

Source: European Mathematical Cup Junior 2017,P4.

December 29, 2017
inequalities

Problem Statement

The real numbers x,y,zx,y,z satisfy x2+y2+z2=3.x^2+y^2+z^2=3. Prove that the inequality x3−(y2+yz+z2)x+yz(y+z)≤33.x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}. and find all triples (x,y,z)(x,y,z) for which equality holds.
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