Wow. Such summation. Much variables. Very degrees.

Algebra Level pending

If the minimum value of \(c\) that satisfy the inequality for all \(x_{1},x_{2},\dots,x_{n} \geq 0\)

\[\displaystyle \large \sum\limits_{1 \leq i < j \leq n} x_{i}x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq c \left(\sum\limits_{1 \leq i \leq n} x_{i}\right)^{4}\]

can be expressed as \(\displaystyle \frac{a}{b}\) for coprime integers \(\large a,b\). What is the value of \(a^{3}+b^{3}\)?

If \(\displaystyle c <0\), then \(\displaystyle a < 0\) and \(\displaystyle b > 0\).

  • This problem is not original.

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