Finding $a^{2014} + b^{2014} + c^{2014}$ given some conditions on $a,b,c$.

Question

I came across this problem:

"Let $a$, $b$, $c$ be nonzero real numbers that satisfy the conditions :

$$a + b + c = 9,\\\mathrm{and}~ab + bc + ca = 27 $$ Calculate

$$a^{2014} + b^{2014} + c^{2014}."$$

Answer 1

Hint: $$(a-b)^2+(b-c)^2+(c-a)^2= 2(a+b+c)^2-6(ab+bc+ca)=0$$