DE-MO 1999 Problem 5

Consider the following inequality for real numbers $x,y,z$ : $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ . (a) Prove that the inequality is valid for $a = 2\sqrt2$ (b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$ .