Finding positive real numbers $x$,$y$ and $z$ IMO Shortlist 1995 A4

Question

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)

Thanks.

Answer 1

http://ohkawa.cc.it-hiroshima.ac.jp/AoPS.pdf/problem%20from%20the%20book.pdf

See page 30. There is a solution.