Problem

Source: Indian RMO 2013 Region 1 Problem 2

Tags: algebra, RMO



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 $c\not=0$. Suppose that the following conditions hold: $f(1)=0$, the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$. Find the value of $a^{2013}+b^{2013}+c^{2013}$.