Question 24
Let $a_{1}, a_{2}, \ldots, a_{n}$ be strictly positive integers. For every $k=1,2, \ldots, n$, let
$$
m_{k}=\max _{1 \leq \ell \leq k} \frac{a_{k-\ell+1}+a_{k-\ell+2}+\cdots+a_{k}}{\ell} .
$$
Show that for every $\alpha\gt 0$, the number of integers $k$ such that $m_{k}\gt \alpha$ is strictly less than $\frac{a_{1}+a_{2}+\cdots+a_{n}}{\alpha}$.
$$
m_{k}=\max _{1 \leq \ell \leq k} \frac{a_{k-\ell+1}+a_{k-\ell+2}+\cdots+a_{k}}{\ell} .
$$
Show that for every $\alpha\gt 0$, the number of integers $k$ such that $m_{k}\gt \alpha$ is strictly less than $\frac{a_{1}+a_{2}+\cdots+a_{n}}{\alpha}$.