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Part of 2016 Bosnia and Herzegovina Junior BMO TST

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Bosnia and Herzegovina JBMO TST 2016 Problem 4

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2016

9/16/2018
Let xx, yy and zz be positive real numbers such that xy+yz+zx=3\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3. Prove that x3+x+y3+y+z3+z≥6(x+y+z)\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}
algebrainequalities