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IMO ShortList 1998, algebra problem 3

Source: IMO ShortList 1998, algebra problem 3

October 22, 2004
inequalitiesrearrangement inequality3-variable inequalityIMO ShortlistalgebraHigh School Olympiads

Problem Statement

Let x,yx,y and zz be positive real numbers such that xyz=1xyz=1. Prove that x3(1+y)(1+z)+y3(1+z)(1+x)+z3(1+x)(1+y)≥34. \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.
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