Are there undecidable problems for which a solution has been found? I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an instance of the problem. And yet human mathematicians have come up with such a solution.

In a certain sense made rigorous elsewhere in a mathematical logic conjecture whose origin I can't remember, every mathematician's proof is expressible in terms of basic semantics that a computer can understand given basic axioms and deduction rules. If you accept that, then no, there is no computer undecidable problem that a mathematician can figure out a deterministic decision procedure for.
