$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a triangle.
This is a selection to IMO problem. I have no helpful observations so far. Perhaps someone could give me a hint?
Only two people got almost the maximum of points for this problem.