\[S = \left \{\left(\frac{x+y-z}{x+y+z}\right)^2 + \left(\frac{x-y+z}{x+y+z}\right)^2 + \left(\frac{-x+y+z}{x+y+z}\right)^2: x,y,z \in \mathbb R^+ \right \}\]
Let \(S\) be a set defined as above, where \(x, y, z\) are positive real numbers.
If \(a = \text{sup } S \) \((\)supremumof \(S)\) and \(b = \text{inf } S \) \((\)infimumof \(S),\) compute \(\frac{a}{b}\).
byWorranat Pakornrat