CL-MO 2013 Problem 3

Given a finite sequence of real numbers $a_1,a_2,...,a_n$ such that $$a_1 + a_2 + ... + a_n > 0.$$ Prove that there is at least one index $i$ such that $$a_i > 0, a_i + a_{i+1} > 0, ..., a_i + a_{i+1} + ...+ a_n > 0.$$