KURSCHAK 2005 Problem 1

Let $N>1$ and let $a_1,a_2,\dots,a_N$ be nonnegative reals with sum at most $500$ . Prove that there exist integers $k\ge 1$ and $1=n_0<n_1<\dots<n_k=N$ such that $$\sum_{i=1}^k n_ia_{n_{i-1}}<2005.$$