KöMaL 2009 A478

If $a_1$,$a_2$,...,$a_n$ are positive numbers with $a_1$+$a_2$+...+$a_n$=1, prove $a_1\cdot a_2^{2/3}+a_2\cdot a_3^{2/3}+\dots+a_{n-1}\cdot a_n^{2/3}<\frac37$.