Prove that there does not exist a universal Turing machine that takes a pair $\langle M, w\rangle$ as input, where M is a Turing machine and w is a string, and that always halts, accepts if $M$ accepts $w$, and rejects if $M$ reject w.
I think assuming the existence of such a machine H could allow one to decide the halting problem. Assume the existence of such a machine. Suppose $\langle M, w\rangle$ is given as input. Then let $M'$ be the machine resulting from swapping the accept and reject states of $M$. If $H$ accepts $\langle M,w\rangle$ and $\langle M', w\rangle$ or if $H$ rejects both inputs, then either $M$ accepts w in the first case or $M$ rejects w in the second case. However, it's possible that $H$ accepts $\langle M,w\rangle$ and rejects $\langle M', w\rangle$ or vice versa, in which case $M$ can either accept or reject w. I'm not sure how to deal with this issue.