$(x,y,z)=(t,t,t)$ and $(t,t,-2t)$ where $t\in \mathbb{R}^+$ gives us $xy+yz+zx$ can be all real numbers except zero. To show that $xy+yz+zx$ can't be zero, suppose to the contrary, then we get $\sum_{cyc}{\frac{1}{x^2+2yz}} =\sum_{cyc}{\frac{1}{(x-y)(x-z)}} =0$. Not hard to see that $|a|,|b|,|a+b|$ can't be side lengths of non-degenerate triangle for any $a,b\in \mathbb{R}$.

Where you proved that "$xy+yz+zx$ can be all real numbers "?

@above ThE-dArK-lOrD wrote: $(x,y,z)=(t,t,t)$ and $(t,t,-2t)$ where $t\in \mathbb{R}^+$

I do not understand the above solution...

soryn wrote: I do not understand the above solution... official solution: https://www.matematickaolympiada.cz/media/3459612/brozura_a67angl_new.pdf#page=13