IL-GILLIS 1997 Problem 1

Find all real solutions to the system of equations $\begin{cases} x^2 +y^2 = 6z y^2 +z^2 = 6x z^2 +x^2 = 6y \end{cases}$

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.