# A Few Signs Switched

Algebra Level 5

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

Let $$S$$ be the set with elements as the terms of positive real numbers $$x,y,z$$, as shown above.

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

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