Question 38
Let $n \in \mathbb{N}$, $n \ge 2$, and $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be real positive numbers such that
$$
\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le \dots \le \frac{a_n}{b_n}.
$$
Find the largest real $c$ so that $(a_1 - b_1c)x_1 + (a_2 - b_2c)x_2 + \dots + (a_n - b_nc)x_n \ge 0$, for every $x_1, x_2, \dots, x_n \gt 0$, with $x_1 \le x_2 \le \dots \le x_n$.
$$
\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le \dots \le \frac{a_n}{b_n}.
$$
Find the largest real $c$ so that $(a_1 - b_1c)x_1 + (a_2 - b_2c)x_2 + \dots + (a_n - b_nc)x_n \ge 0$, for every $x_1, x_2, \dots, x_n \gt 0$, with $x_1 \le x_2 \le \dots \le x_n$.