IrMO 2013 Problem 10

Let $a,b,c$ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$ and $S=2x^3-9xy+9z .$ $(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t$ , respectively , for any real number $t$ . $(b)$ Prove that $(3y-x^2)^3\ge S^2 .$