I asked a question yesterday on MO after trying to prove a statement (that appears in a book without proof, so I assumed that the great minds on MO could answer quickly) for a day. I finally proved it myself, after a substantial amount of effort (the proof is completely inelegant, but I can't think of any other way to do it!), and I posted the answer that I figured out.

So someone just downvoted the question and answer. No comment, no nothing. I'm really annoyed.

If the person was concerned that I answered my own question, then he or she should have left a comment to the effect of "hey, you shouldn't answer your own questions", but of course, no such thing happened, because some (by no means all or even most) people on MO have no class and couldn't tell a good answer from a bad one if their lives depended on it.

Should we penalize people for answering their own questions?

I think that we should encourage such behavior, provided that it's not abused by people asking questions to which they already know the answer.