Question 3
Let $n$ be a positive integer and $a_1,a_2,\ldots,a_n$ positive real numbers. Prove that
$$\sum_{i=1}^n\frac1{2^i}\left(\frac2{1+a_i}\right)^{2^i}\ge\frac2{1+a_1a_2\cdots a_n}-\frac1{2^n}.$$
$$\sum_{i=1}^n\frac1{2^i}\left(\frac2{1+a_i}\right)^{2^i}\ge\frac2{1+a_1a_2\cdots a_n}-\frac1{2^n}.$$