Hm, yeah, that's something I would use, but I'm not so sure I'd extend it to the form "to answer the problem".
I think the root of this is that sometimes a question is also a problem and vice-versa, but sometimes it's only a question or only a problem. I feel like when you're asking for "the answer to problem X", you're implying that the problem is also a question, so something like "2+2=?". But then I consider problems like this:
ax² + bx + c = 0
Solve for x.
That's not really a question, but it's a problem. And it's not something you would answer, it's something you would solve.
But I guess this is all really minor nitpicky stuff.