Question 120
Let $a_1, a_2, \dots, a_{2003}$ be a sequence of real numbers. A term $a_k$, $1 \le k \le 2003$, is said to be a *leading term*, if at least one of the expressions $a_k, a_k+a_{k+1}, \dots, a_k+a_{k+1}+\dots+a_{2003}$ is positive. Prove that the sum of all leading terms is positive provided that the sequence has at least one leading term.