Middle European Mathematical Olympiad 2009 Problem T-1

Let $x, y, z$ be real numbers satisfying $x^2 + y^2 + z^2 + 9 = 4(x + y + z)$. Prove that $$x^4 + y^4 + z^4 + 16(x^2 + y^2 + z^2) \geqslant 8(x^3 + y^3 + z^3) + 27$$ and determine when equality holds.