Real numbers $x,y,z$ are chosen such  that $$\frac{1}{|x^2+2yz|} ,\frac{1}{|y^2+2zx|} ,\frac{1}{|x^2+2xy|} $$ 
are lengths of a non-degenerate triangle .
Find all possible values of $xy+yz+zx$ .

[i]Proposed by Michael Rolínek[/i]
To solve the problem, we need to find all possible values of \( xy + yz + zx \) such that the given expressions form the sides of a non-degenerate triangle. The expressions are:

\[
\frac{1}{|x^2 + 2yz|}, \quad \frac{1}{|y^2 + 2zx|}, \quad \frac{1}{|z^2 + 2xy|}
\]

For these to be the sides of a non-degenerate triangle, they must satisfy the triangle inequality:

\[
\frac{1}{|x^2 + 2yz|} + \frac{1}{|y^2 + 2zx|} > \frac{1}{|z^2 + 2xy|}
\]
\[
\frac{1}{|y^2 + 2zx|} + \frac{1}{|z^2 + 2xy|} > \frac{1}{|x^2 + 2yz|}
\]
\[
\frac{1}{|z^2 + 2xy|} + \frac{1}{|x^2 + 2yz|} > \frac{1}{|y^2 + 2zx|}
\]

We need to analyze these inequalities to find the conditions on \( xy + yz + zx \).

1. **Assume \( xy + yz + zx = 0 \)**:
   - If \( xy + yz + zx = 0 \), then we can rewrite the expressions as:
     \[
     x^2 + 2yz = x^2 - 2xy - 2zx = x^2 - 2x(y + z)
     \]
     \[
     y^2 + 2zx = y^2 - 2yz - 2xy = y^2 - 2y(z + x)
     \]
     \[
     z^2 + 2xy = z^2 - 2zx - 2yz = z^2 - 2z(x + y)
     \]

   - If two of \( x, y, z \) are equal, say \( x = y \), then:
     \[
     |x^2 + 2zx| = |x^2 + 2x^2| = |3x^2| = 3x^2
     \]
     This would imply that one of the denominators is zero, which is not possible.

2. **Assume \( x, y, z \) are distinct and \( x > y > z \)**:
   - Without loss of generality, assume \( x, y \) are positive and \( z \) is negative. Then:
     \[
     x^2 + 2yz > 0, \quad y^2 + 2zx > 0, \quad z^2 + 2xy > 0
     \]
   - The sum of the three fractions without absolute values would be:
     \[
     \sum_{\text{cyc}} \frac{1}{x^2 + 2yz} = \sum_{\text{cyc}} \frac{1}{(x-y)(x-z)} = 0
     \]
     This is a contradiction because the sum of two of the fractions with positive denominators cannot equal the other fraction, as they form a non-degenerate triangle.

Therefore, the only possible value for \( xy + yz + zx \) that satisfies the conditions is:

\[
\boxed{0}
\]