SILK-ROAD 2003 Problem 1

Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number. Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive. Prove, that if exist at least one $leading$ element, then sum of all $leading$ 's elements is positive.

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