MEMO 2009 Problem 5

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