CZ-SK-MO 2011 Problem 3

Suppose that $x$ , $y$ , $z$ are real numbers satisfying $$x+y+z=12, \text{and} x^2+y^2+z^2=54.$$ Prove that:list Each of the numbers $xy$ , $yz$ , $zx$ is at least $9$ , but at most $25$ . (b) One of the numbers $x$ , $y$ , $z$ is at most $3$ , and another one is at least $5$ .[/list]