An algebra problem by Mehul Chaturvedi

Let the roots of $f(x)$ be $\alpha$ , $\beta$ and $\gamma$ , then those of $g(x)$ are $\alpha^2$ , $\beta^2$ and $\gamma^2$ .

Since $\quad \alpha \beta \gamma = -c \quad$ and $\quad \alpha^2 \beta^2 \gamma^2 = -a \quad \Rightarrow a = -c^2$ .

Also $\quad \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2( \alpha \beta + \beta \gamma + \gamma \alpha)$

$\Rightarrow -b = a^2 - 2b\quad \Rightarrow b = a^2 = c^4$

Now $\quad f(1) = 1 + a + b + c = 0\quad \Rightarrow c^4 - c^2 + c +1 = 0$

$\Rightarrow c = -1$ , $a = -1$ and $b = 1$

$\Rightarrow a^{2013} + b^{2013} + c^{2013} = -1+1-1=\boxed {-1}$ .

As far as I remember, that's an R.M.O. question