RU-ALLRUSSIAN 2017 Problem 113

There are $n$ positive real numbers on the board $a_1,\ldots, a_n$ . Someone wants to write $n$ real numbers $b_1,\ldots,b_n$ ,such that: $b_i\geq a_i$ If $b_i \geq b_j$ then $\frac{b_i}{b_j}$ is integer. Prove that it is possible to write such numbers with the condition $$b_1 \cdots b_n \leq 2^{\frac{n-1}{2}}a_1\cdots a_n.$$