ISL 2008 A2

(a) Prove the inequality $$\frac{x^{2}}{(x-1)^{2}}+\frac{y^{2}}{(y-1)^{2}}+\frac{z^{2}}{(z-1)^{2}} \geq 1$$ for real numbers $x, y, z \neq 1$ satisfying the condition $x y z=1$. (b) Show that there are infinitely many triples of rational numbers $x, y, z$ for which this inequality turns into equality.