IMO Shortlist 1974 problem 9
Dodao/la:
arhiva2. travnja 2012. Let
![x, y, z](/media/m/e/1/6/e160f3439547ca8c1afcc35a1c26f080.png)
be real numbers each of whose absolute value is different from
![\displaystyle \frac{1}{\sqrt 3}](/media/m/3/0/8/3088795eb2f1c2a36ccb7e0fbeb12973.png)
such that
![x + y + z = xyz](/media/m/a/a/c/aaca165029be0a693b40d270a0f94158.png)
. Prove that
%V0
Let $x, y, z$ be real numbers each of whose absolute value is different from $\displaystyle \frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that
$$\frac{3x-x^{3}}{1-3x^{2}}+\frac{3y-y^{3}}{1-3y^{2}}+\frac{3z-z^{3}}{1-3z^{2}}=\frac{3x-x^{3}}{1-3x^{2}}\cdot\frac{3y-y^{3}}{1-3y^{2}}\cdot\frac{3z-z^{3}}{1-3z^{2}}$$
Izvor: Međunarodna matematička olimpijada, shortlist 1974