Philosophy is notorious for not answering the questions it tackles. Plato posed most of the central questions more than two millennia ago, and philosophers still haven't come to much consensus about them. Or at least, whenever philosophical questions begin to admit of answers, we start calling them scientific questions. (Astronomy, physics, chemistry, biology, and psychology all began as branches of philosophy.)
A common attitude on Less Wrong is "Too slow! Solve the problem and move on." The free will sequence argues that the free will problem has been solved.
I, for one, am bold enough to claim that some philosophical problems have been solved. Here they are:
- Is there a God? No.
- What's the solution to the mind-body problem? Materialism.
- Do we have free will? We don't have contra-causal free will, but of course we have the ability to deliberate on alternatives and have this deliberation effect the outcome.
- What is knowledge? (How do we overcome Gettier?) What is art? How do we demarcate science from non-science? If you're trying to find simple definitions that match our intuitions about the meaning of these terms in ever case, you're doing it wrong. These concepts were not invented by mathematicians for use in a formal system. They evolved in practical use among millions of humans over hundreds of years. Stipulate a coherent meaning and start using the term to successfully communicate with others.
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.