An algebra problem by Mehul Chaturvedi

Algebra Level 4

Let \(f(x)=x^3+ax^2+bx+c\) and \(g(x)=x^3+bx^2+cx+a,\) where \(a,b,c\) are integers with \(c\neq0.\) Suppose that the following conditions hold:

(i) \(f(1)=0;\)

(ii) the roots of \(g(x)=0\) are the squares of the roots of \(f(x)=0.\)

Find the value of \(a^{2013}+b^{2013}+c^{2013}.\)

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