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Algebra Level 4

Let f(x)=x3+ax3+bx+c,g(x)=x3+bx2+cx+af(x) = x^3+ax^3+bx+c,g(x)=x^3+bx^2+cx+a for integers a,b,ca,b,c with c0c\ne 0. Suppose the following conditions hold:

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

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

Find the value of a2013+b2013+c2013a^{2013}+b^{2013}+c^{2013} .

Source: RMO 2013.

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