lean_workbook_plus_69642

Let $x, y, z$ are real numbers such that $0 <x, y, z <1$ and $xyz = \left (1-x \right) \left (1-y \right) \left (1-z \right) .$ Show that at least one of the numbers $\left (1-x \right) y, \left (1-y \right) z, \left (1-z \right) x$ is greater than or equal to $\dfrac {1} {4}.$ (JBMO 2009 Shortlist A4)