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x^2y/z+y^2z/x+z^2x/y+2(y/xz+z/xy+x/yz)>= 9 if x,y,z>0

Source: Greece JBMO TST 2016 p1

April 29, 2019
algebrainequalitiesthree variable inequality

Problem Statement

a) Prove that, for any real x>0x>0, it is true that x3−3x≥−2x^3-3x\ge -2 . b) Prove that, for any real x,y,z>0x,y,z>0, it is true that x2yz+y2zx+z2xy+2(yxz+zxy+xyz)≥9\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}+2\left(\frac{y}{xz}+\frac{z}{xy}+\frac{x}{yz} \right)\ge 9 . When we have equality ?
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