TSTST 2015 Problem 1

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1 \leq k \leq n$ is good if there exists some $\ell$ with $1 \leq \ell \leq m$ such that $$a_{k}+a_{k+1}+\cdots+a_{k+\ell-1} \geq 0$$ where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $$\sum_{k \in T} a_{k} \geq 0$$