(i) If
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
,
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
and
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
are three real numbers, all different from
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
, such that
![xyz = 1](/media/m/c/1/9/c195fe87b9c55e41bc39cd3a4d8d2906.png)
, then 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](/media/m/1/2/b/12b5ab7da6be3857f79cab58315d84a4.png)
.
(With the
![\sum](/media/m/3/2/0/3204f02c49ce74ed7cb55fb9e437f66b.png)
sign for cyclic summation, this inequality could be rewritten as
![\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1](/media/m/8/b/4/8b4fb6c01b067d9b1e66861a7d1657a4.png)
.)
(ii) Prove that equality is achieved for infinitely many triples of rational numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
,
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
and
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
.
Author: Walther Janous, Austria
%V0
(i) If $x$, $y$ and $z$ are three real numbers, all different from $1$, such that $xyz = 1$, then prove that
$$\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$$.
(With the $\sum$ sign for cyclic summation, this inequality could be rewritten as $\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1$.)
(ii) Prove that equality is achieved for infinitely many triples of rational numbers $x$, $y$ and $z$.
Author: Walther Janous, Austria