International Mathematical Olympiad 2008 Problem 2

(a) Prove that $$\frac{x^2}{(x-1)^2} + \frac{y^2}{(y-1)^2} + \frac{z^2}{(z-1)^2} \geq 1$$ for all real numbers $x, y, z$, each different from 1, and satisfying $xyz = 1$.

(b) Prove that equality holds above for infinitely many triples of rational numbers $x, y, z$, each different from 1, and satisfying $xyz = 1$.