Question 29
Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of real numbers such that $a_{1}+\cdots+a_{n}=0$ and define $b_{i}=a_{1}+\cdots+a_{i}$ for $1 \leq i \leq n$. Suppose that $b_{i}\left(a_{j+1}-a_{i+1}\right) \geq 0$ for all $1 \leq i \leq j \leq n-1$. Prove that
$$
\max _{1 \leq \ell \leq n}\left|a_{\ell}\right| \geq \max _{1 \leq m \leq n}\left|b_{m}\right|
$$
$$
\max _{1 \leq \ell \leq n}\left|a_{\ell}\right| \geq \max _{1 \leq m \leq n}\left|b_{m}\right|
$$