Let $a_i$ be real numbers such that $\sum\limits_{i=1}^na_i\geq0 $, $\sum\limits_{1\leq i<j<k\leq n}a_ia_ja_k\geq0$ and $n\geq9$. Prove that: $$n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3.$$ I have a proof for $3\leq n\leq8$, but for $n\geq9$ my way does not work and I did not see a counter example for $n=9$.

It seems that it's wrong for big value of $n$.

Thank you!