NO-ABEL 2004 Problem 2

(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$ . (b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$ , prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$