Standard Inequality

Let $f(x)=\frac{1-x}{x}$ . Since $f(x)$ is convex, applying Jensen's Inequality, we have $f(\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}) \leq \frac{f(\frac{1}{a})+f(\frac{1}{b})+f(\frac{1}{c})}{3}$ this reduces to $(a-1)+(b-1)+(c-1) \geq 6$ . Since $a,b,c>1$ , applying A.M-G.M inequality, we have $(a-1)(b-1)(c-1) \geq \boxed{8}$