Why not AM-GM? It's obvious!

Algebra Level 4

Let \(a,b,c,d\) be positive real numbers. Find the minimum value of

\[\small \frac { a+b+c }{ d } +\frac { b+c+d }{ a } +\frac { c+d+a }{ b } +\frac { d+a+b }{ c } +\frac { a }{ b+c+d } +\frac { b }{ c+d+a } +\frac { c }{ d+a+b } +\frac { d }{ a+b+c }\]

Write your answer to 3 decimal places.

Hint: When does equality hold?

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