Question 108
Positive numbers $a_1, a_2, \dots, a_n$ are written on the board in a row.
For every $i = 1, 2, \dots, n$, Vasya wishes to write a number $b_i \ge a_i$ so that for every $i, j \in \{1, 2, \dots, n\}$, at least one of the ratios $b_i/b_j$ and $b_j/b_i$ is an integer. Prove that Vasya can reach this goal so that
$$
b_1 b_2 \dots b_n \le 2^{(n-1)/2} a_1 a_2 \dots a_n.
$$
For every $i = 1, 2, \dots, n$, Vasya wishes to write a number $b_i \ge a_i$ so that for every $i, j \in \{1, 2, \dots, n\}$, at least one of the ratios $b_i/b_j$ and $b_j/b_i$ is an integer. Prove that Vasya can reach this goal so that
$$
b_1 b_2 \dots b_n \le 2^{(n-1)/2} a_1 a_2 \dots a_n.
$$