Question 168
Positive real numbers $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ satisfy $a_1 \ge a_2 \ge \dots \ge a_n$ and $b_1b_2\dots b_k \ge a_1a_2\dots a_k$ for each $k = 1, 2, \dots, n$. Prove that
$$
b_1 + b_2 + \dots + b_n \ge a_1 + a_2 + \dots + a_n.
$$
$$
b_1 + b_2 + \dots + b_n \ge a_1 + a_2 + \dots + a_n.
$$