CAPS 2008 Problem 1

Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations $$2x^3=2y(x^2+1)-(z^2+1),$$ $$2y^4=3z(y^2+1)-2(x^2+1),$$ $$2z^5=4x(z^2+1)-3(y^2+1).$$