IMO Shortlist 2008 problem A2


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2. travnja 2012.
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(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
Izvor: Međunarodna matematička olimpijada, shortlist 2008