ÖMO 1989 Problem 5

Find all real solutions of the system:

$x^2+2yz=x,$ $y^2+2zx=y,$ $z^2+2xy=z.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.