Problem

Source: IMO ShortList 1998, algebra problem 3

Tags: inequalities, rearrangement inequality, 3-variable inequality, IMO Shortlist, algebra, High School Olympiads



Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that \[ \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}. \]


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