Let
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
,
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
,
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
be real numbers satisfying
![x^2+y^2+z^2+9=4(x+y+z)](/media/m/7/b/e/7bed1e174036ff21fde0c89c8221a1c3.png)
. Prove that
![x^4+y^4+z^4+16(x^2+y^2+z^2) \ge 8(x^3+y^3+z^3)+27](/media/m/d/b/4/db44c19c99c98a87a40d57f0b8a742e2.png)
and determine when equality holds.
%V0
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.