IMO 2008 Problem 2

(a) Prove that $$\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$$ 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 $xyz=1$ .

Author: Walther Janous, Austria