2008 IMO Shortlist 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.

(a) 对于满足条件 $x y z=1$ 的实数 $x, y, z \neq 1$，证明不等式 $$\frac{x^{2}}{(x-1)^{2}}+\frac{y^{2}}{(y-1)^{2}}+\frac{z^{2}}{(z-1)^{2}} \geq 1$$。 (b) 证明存在无限多个有理数 $x, y, z$ 的三元组，使得这种不等式变为相等。