Problem

Source: Czech and Slovak Olympiad 2018, National Round, Problem 2

Tags: algebra, national olympiad



Let $x,y,z$ be real numbers such that the numbers $$\frac{1}{|x^2+2yz|},\quad\frac{1}{|y^2+2zx|},\quad\frac{1}{|z^2+2xy|}$$are lengths of sides of a (non-degenerate) triangle. Determine all possible values of $xy+yz+zx$.