Professor proposes a question she would be extremely happy to get solved and I solved do I have to write a paper solo or with her?
You owe her an acknowledgement in the paper, but nothing more. If she wasn't involved in the solution or otherwise in the paper, then she is really not a co-author and most would agree.
In some fields, advisors are regularly given authorship when they really shouldn't be, but that is more due to things like funding a lab and providing general support that makes research possible. But nothing you describe suggests that she is an author. Editing a bit and cleaning up the presentation isn't authorship.
But put a thank you in the paper for suggesting the problem.