2016/093) Suppose $a_1,a_2,...,a_n$ are poistive real numbers satisfying \(a_1\cdot a_2\cdots a_n=1\).

Show that $(a_1+1)(a_2+1)\cdots(a_n+1)>=2^n$

Solution 
We have using AM GM $\frac{a_k+1}{2} >= \sqrt{a_k}$ or $a_k + 1>=2\sqrt{a_k}$
multiplying over k from 1 to n we have
$\prod_{k=1}^n(a_k+1) >= 2^n \sqrt{\prod_{k=1}^na_k} = 2^n$
or $\prod_{k=1}^n(a_k+1) >= 2^n$