Sorites / Ship of Theseus? Words, particularly those born of everyday usage rather than research papers, have boundaries that are fuzzy, until enough pioneers explore them and sharpen the map. You cannot say whether the ship is new because you never had to deal with this type of repairs before - Theseus' ship lies in the fuzzy boundary of "new". This is the time for you to re-draw the boundary with a sharper pen, either ahead of or behind Theseus' ship, and so establish an improved, more powerful definition of "new". i.e. Pretty much the same way you dismissed "what is art?".

I'm assuming you wouldn't consider problems that got solved by non-(directly)-philosophical means, such as Achilles and the tortoise (otherwise you could write a book just listing all the stuff that weirded out medieval philosophers). Would you consider problems that have not been solved so much as, shall we say, transcended, e.g. the liar's paradox?

One small aside I just realised: "philosophical problems" is a much less common turn of phrase than "philosophical questions". Could that be a (small) contributor to Western philosophy's notorious inertia? Having a "problem" would suggest that you are somehow worse off as long as it remains unsolved, whereas a "question" can also be an idle curiosity that you are free to tackle or ignore.