# 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 a set defined as above, where $x, y, z$ are positive real numbers.

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

×