Similar in spirit to CF 1698C, I suppose. If there are $3$ nonnegative or $2$ nonpositive numbers we are done. So suppose there are $\leq 2$ nonnegative numbers and $\leq 1$ nonpositive numbers. Then since there are $\geq 3$ numbers all inequalities turn into equality. In this case if $a_1+a_2+a_3 > 0$ we are done; otherwise remove the largest nonnegative number (which is strictly positive, since there are two of them), and the remaining sum – negative.