Harvard-MIT Math Tournament

Solution:
If you know formal power series, then this is not such a hard question, but since this is a calculus test... Use partial fractions to get
$$\frac{1}{1-3x+2x^{2}} = \frac{1/2}{1-2x} - \frac{1}{1-x}.$$
Now each of these can be expanded as a geometric series (or take derivatives and get the same result) to get
$$\frac{1}{2}\left(1 + 2x + 4x^{2} + 8x^{3} + \cdots\right) - \left(1 + x + x^{2} + x^{3} + \cdots\right),$$
so the coefficient of $x^{n}$ is $2^{n-1} - 1$. When $n=12$, that's $2047$.