IMO 1961 - Problem 1

Problem: Solve the following system of eqautions: $x+y+z=a,x^2+y^2+z^2=b^2,xy=z^2$.
where $a$ and $b$ are given real numbers. What conditions must hold on $a$ and $b$ for the solution to be positive and distinct?
Solution: This is a problem solvable using elementary manipulations, so we shall state only the final solutions. For $a=0$ we get $(x,y,z)=(0,0,0)$. For $a\ne 0$ we get $(x,y,z)\in \left\{(t_1,t_2,z_0),(t_2,t_1,z_0)\right\}$, where
$z_0=\frac{a^2-b^2}{2a}$ and $t_{1,2}=\frac{a^2+b^2\pm \sqrt{(3a^2-b^2)(3b^2-a^2)}}{4a}$.
For the solutions to be positive and distinct the following conditions are necessary and sifficient: $3b^2>a^2>b^2$ and $a>0$.