Question 61
Let $a, b, c$ be positive real numbers. It is known that the system of equations
$$
\begin{cases} a^2x + b^2y + c^2z = 1 \\ xy + yz + zx = 1 \end{cases} \quad \text{(where }a,b,c\text{ are parameters)}
$$
has only one solution $(x, y, z)$. Prove that $a, b, c$ are the sides of a certain triangle.
$$
\begin{cases} a^2x + b^2y + c^2z = 1 \\ xy + yz + zx = 1 \end{cases} \quad \text{(where }a,b,c\text{ are parameters)}
$$
has only one solution $(x, y, z)$. Prove that $a, b, c$ are the sides of a certain triangle.