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x^2+xy+y^2 <= 3(x^2-xy+y^2), so the left hand side of the statement >= (z^3+y^3)/(x^2-xy+y^2)/3 + ... ,>= {(x+y)(y+z)(z+x)}^{1/3} (applying A-G inequality, simplify) , which is no less than 2 by A-G and xyz=1