Let $\text{Max Spec}\:A$ be the subspace of maximal ideals of $\text{Spec} \:A $ for the $F$-algebra $A$ that is of finite type over a finite field $F$. Show that $\text{Max Spec}\:A$ is Hausdorff.
I don't know where to start - I'd imagine perhaps the Nullstellensatz ($F\to A/\mathfrak m$ is a finite field extension for any maximal $\mathfrak m$) could come in, but I don't know where finiteness of the field would be used other than to somehow take an intersection of finitely many opens. However, I know there are irreducible polynomials of arbitrary degree even in one variable, so there are indeed many opens.