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Algebra Level 4

Let \(f(x) = x^3+ax^3+bx+c,g(x)=x^3+bx^2+cx+a \) for integers \(a,b,c\) with \(c\ne 0\). Suppose the following conditions hold:

  1. \(f(1)=0 \),

  2. The roots of \(g(x)=0\) are square of the roots of \(f(x) = 0\).

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

Source: RMO 2013.
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