3000 Follower Problem 1

$\begin{cases} F^2+A^2+N^2 =3 \\ \min \left( F+A, A+N, F+N \right) > \sqrt 2 \\ \frac{F}{\left( A+N-F \right)^2}+\frac{A}{\left( N+F-A \right)^2}+\frac{N}{\left(F+A-N \right)^2} \geq \frac{S}{3000 \left( FAN \right)^2} \end{cases}$

For what maximum value of $S$ will the above conditions be always met simultaneously for positive real numbers $F,A$ and $N$ ?

Clarification:

• $FAN= F \times A \times N$ .