Let $a_i$$n\ge 3$ be an integer. I would like to know if the following property $(P_n)$ holds: for all real numbers $a_i$ such that $\sum\limits_{i=1}^na_i\geq0 $, and $\sum\limits_{1\leq i<j<k\leq n}a_ia_ja_k\geq0$ and $n\geq9$. Prove that:, we have $$n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3.$$ I have a proof that $(P_n)$ holds for $3\leq n\leq8$, but for $n\geq9$ my waymethod does not work and I did not see a counter exampleany counterexample for $n=9$$n\ge 9$.
It seems that it's wrongIs the inequality $(P_n)$ true for a big value ofall $n$. Is it possible to find a biggest? Or otherwise, what is the largest value of $n$, for which this inequality is still trueit holds?
Thank you!