Indian Regional Mathematical Olympiad Problem 2

Algebra Level 5

Let f(x)=x3+ax2+bx+cf(x) = x^3 + ax^2 + bx + c and g(x)=x3+bx2+cx+ag(x) = x^3 + bx^2 + cx + a, where a,b,ca, b, c are integers with c0c\neq 0. Suppose that the following conditions hold: (a) f(1)=0f(1) = 0; (b) the roots of g(x)=0g(x) = 0 are the squares of the roots off(x)=0 f(x) = 0. Find the value of a2013+b2013+c2013a^{2013}+b^{2013}+c^{2013}.

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