1) A holomorphic function $f$ on the disc $\{z \in \mathbb{C}: |z|< 2\}$ such that $f(1/n) = (-1)^n/n$ for every positive integer $n$.
2) A rational function $f$ having a pole at 0 such that the residue of $f$ at 0 equals 2 and the residue of the derivative $f'$ at 0 equals 1.
A group of us have plugged away at these for a while, but have no idea what we're missing.