Question 146
(1) Prove that
$$
\frac{x^2}{(x-1)^2} + \frac{y^2}{(y-1)^2} + \frac{z^2}{(z-1)^2} \ge 1,
$$
for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz = 1$.
(2) Prove that the equality holds for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz = 1$.
$$
\frac{x^2}{(x-1)^2} + \frac{y^2}{(y-1)^2} + \frac{z^2}{(z-1)^2} \ge 1,
$$
for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz = 1$.
(2) Prove that the equality holds for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz = 1$.