Does Quadratic Forms Help?

Algebra Level 5

\[ f(a, b, c, d) = {\frac { 1 }{ { a }^{ 2 }+1 } +\frac { 1 }{ { b }^{ 2 }+1 } +\frac { 1 }{ { c }^{ 2 }+1 } +\frac { 1 }{ { d }^{ 2 }+1 } }\]

Let \(a,b,c,d\) to be non negative real numbers satisfying \(ab+ac+ad+bc+bd+cd=6\). Let \( \min f(a, b, c, d) = M \).

How many ordered quadruples of non-negative real numbers \( (a, b, c, d) \) satisfy \( f(a,b,c,d) = M \)?

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