A Few Signs Switched

Algebra Level 5

S={(x+yzx+y+z)2+(xy+zx+y+z)2+(x+y+zx+y+z)2:x,y,zR+}S = \left \{\left(\frac{x+y-z}{x+y+z}\right)^2 + \left(\frac{x-y+z}{x+y+z}\right)^2 + \left(\frac{-x+y+z}{x+y+z}\right)^2: x,y,z \in \mathbb R^+ \right \}

Let SS be a set defined as above, where x,y,zx, y, z are positive real numbers.

If a=sup Sa = \text{sup } S ((supremum of S)S) and b=inf Sb = \text{inf } S ((infimum of S),S), compute ab\frac{a}{b}.

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