
2. Let $f(x)=x^{3}+a x^{2}+b x+c$ and $g(x)=x^{3}+b x^{2}+c x+a$, where $a, b, c$ are integers with $c \neq 0$. Suppose that the following conditions hold:

(a) $f(1)=0$;

(b) the roots of $g(x)$ are squares of the roots of $f(x)$.

Find the value of $a^{2013}+b^{2013}+c^{2013}$.


Solution. Note that $g(1)=f(1)=0$, so 1 is a root of both $f(x)$ and $g(x)$. Let $p$ and $q$ be the other two roots of $f(x)$, so $p^{2}$ and $q^{2}$ are the other two roots of $g(x)$. We then get $p q=-c$ and $p^{2} q^{2}=-a$, so $a=-c^{2}$. Also, $(-a)^{2}=(p+q+1)^{2}=p^{2}+q^{2}+1+2(p q+p+q)=-b+2 b=b$. Therefore $b=c^{4}$. Since $f(1)=0$ we therefore get $1+c-c^{2}+c^{4}=0$. Factorising, we get $(c+1)\left(c^{3}-c^{2}+1\right)=0$. Note that $c^{3}-c^{2}+1=0$ has no integer root and hence $c=-1, b=1, a=-1$. Therefore $a^{2013}+b^{2013}+c^{2013}=-1$.
