Problem

Source: Iran National Olympiad - 2014 Second Round - D1P3

Tags: inequalities, quadratics, function, algebra, quadratic formula, inequalities unsolved



Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]