KöMaL 2006 A394

Let $a_1$,$a_2$,...,$a_N$ be nonnegative reals, not all 0. Prove that there exists a sequence 1=$n_0$<$n_1$<...<$n_k$=$N$+1 of integers such that $n_1 a_{n0}$ +$n_2 a_{n1}$ +...+$n_k a_{nk-1}$ <3($a_1$+$a_2$+...+$a_N$).