International Mathematical Olympiad 1961 Problem 1

Solve the system of equations: $$\begin{aligned} x + y + z &= a \\ x^2 + y^2 + z^2 &= b^2 \\ xy &= z^2 \end{aligned}$$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$ (the solutions of the system) are distinct positive numbers.