IMO Shortlist 1998 problem A3
Dodao/la:
arhiva2. travnja 2012. Let
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
and
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
be positive real numbers such that
![xyz=1](/media/m/f/c/4/fc4d25ab80408fd281a61bf02f1c976d.png)
. Prove that
%V0
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that
$$\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1998