Indian Regional Mathematical Olympiad Problem 2

Algebra Level 5

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 \(c\neq 0\). Suppose that the following conditions hold: (a) \(f(1) = 0\); (b) 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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