MathNet 2009-part-03 — 175 problems (EN–VI) Question 88

For real numbers $x, y, z \in (0, 1)$, with $xyz = (1-x)(1-y)(1-z)$, show that at least one of the numbers $(1-x)y, (1-y)z, (1-z)x$ is greater than or equal to $\frac{1}{4}$.