XLVIII OM - III - Problem 2

Find all triples of real numbers $ x $, $ y $, $ z $ satisfying the system of equations
Suppose that the numbers $ x $, $ y $, $ z $ satisfy the given system; from the second equation it follows that






Further, we have






which, by the first equation of the system, gives:






There is also the inequality






that is,






By property (1), we can multiply inequalities (3) and (4) side by side:






According to the second equation of the system, inequality (5) must be an equality. This means that either both multiplied inequalities (3) and (4) are equalities, or the expressions on both sides of (4) are equal to zero. In the first case, the equality sign in relations (2) and (3) leads to the conclusion that $ x = y = z = \pm 1/3 $. In the second case, the products $ zy $, $ yz $, $ zx $ are equal to zero, which means that two of the numbers $ z $, $ y $, $ z $ are equal to zero; the third must be equal to $ \pm 1/\sqrt{3} $, in accordance with the first equation of the system. This gives eight triples $ (x, y, z) $ that satisfy the given system of equations:






where $ \varepsilon = \pm 1 $.