Question 37
Let $a_1 \ge a_2 \ge \cdots \ge a_{107} \gt 0$ satisfy $\sum_{k=1}^{107} a_k \ge M$, and let $0 \lt b_1 \le b_2 \le \cdots \le b_{107}$ satisfy $\sum_{k=1}^{107} b_k \le N$.
Prove that for every $m \in \{1, 2, \dots, 107\}$, the arithmetic mean of
$$
\frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_m}{b_m}
$$
is not less than $\frac{M}{N}$.
Prove that for every $m \in \{1, 2, \dots, 107\}$, the arithmetic mean of
$$
\frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_m}{b_m}
$$
is not less than $\frac{M}{N}$.