lean_workbook_plus_55974

Let $x = \frac{1}{1+a}$, $y= \frac{1}{1+b}$, and $z = \frac{1}{1+c}$. We know $x+y+z \ge 2$, and want to show $\left(\frac{1-x}{x}\right)\left(\frac{1-y}{y}\right)\left(\frac{1-z}{z}\right) \le \frac{1}{8}$. This rearranges to $(1-x)(1-y)(1-z) \le \frac18xyz$ or $(2-2x)(2-2y)(2-2z) \le xyz$