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All possible values

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

April 5, 2018
algebranational olympiad

Problem Statement

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