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Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$

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YCor
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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!

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 a big value of $n$. Is it possible to find a biggest value of $n$, for which this inequality is still true?

Thank you!

Let $n\ge 3$ 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 $\sum\limits_{1\leq i<j<k\leq n}a_ia_ja_k\geq0$, 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 method does not work and I did not see any counterexample for $n\ge 9$.

Is the inequality $(P_n)$ true for all $n$? Or otherwise, what is the largest value of $n$ for which it holds?

Thank you!

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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 a big value of $n$. Is it possible to find a biggest value of $n$, for which this inequality is still true?

Thank you!

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!

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 a big value of $n$. Is it possible to find a biggest value of $n$, for which this inequality is still true?

Thank you!

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