It can't be that simple??? $n a_n^3 \ge n^2 a_n^2 \ge (a_1+...+a_n)^2$???

I'm afraid you're right. What puzzles me is not the problem itself, but how it got into our TST.

$na_n ^3 \geq a_1 ^3 + a_2 ^3 +...+a_n ^3 \geq (a_1+a_2+...+a_n)^2$

Just use $a_i \le a_n \ge n$.