How can we find all of the positive real numbers like $x$,$y$,$z$, such that :
1.) $x + y + z = a + b + c$(here $a$,$b$ and $c>0)$ and
2.) $4xyz = a^2x + b^2y + c^2z + abc$ ?(Both the conditions are simultaneously true)
Source: International Mathematics Olympiad 1995 Shortlist.
Edit: I received this problem from someone and the way it is stated, it is not quite right.I have included the condition $a$,$b$ and $c$ are also positive.I apologize for the error.(It got corrected thanks to user Phira and Puresky)