(a) Prove the identity $(p+q)^3 - 3pq(p + q) = p^3 + q^3$.
(b) The equation $2x^2 - 5x + 1 = 0$ has two real roots, $\alpha$ and $\beta$.
Consider the equation $x^2 + mx + n = 0$, where $m, n \in \mathbb{Z}$ and which has roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$.
Without solving $2x^2 - 5x + 1 = 0$, determine the values of $m$ and $n$.