Is there an unsolvable problem?
There are problems which we solve. There are many problems that have not yet been solved. Set both of these types of problems aside.
\What problems remain?
Godel’s incompleteness theorem comes to mind. In any sufficiently complex mathematical system, there will be statements (that can be expressed in that system) which cannot be shown to be true or false. Well-formed statements, expressions that make sense but expressions which cannot be evaluated. There are many enjoyable proofs.
There are no examples of a mathematical statement which cannot be shown be to true or false. Fermat’s Last Theorem was a plausible candidate for 350 years; Wiles’ proof rules that out.
Its not a computer limitation problem; its not a Turing machine problem, its not a problem that reflects the limits of human creativity.
Similarly, there may be an unsolvable problem. We may never know.