No we should most certainly not ban a user for posting a contest question unless we have reason to believe they are deliberately and repeatedly attempting to cheat on a contest after being told not to. However, we should have a better medium in place to handle contest questions. In particular I think we need to have a special vote added to close reasons that users can cast. Now if we assume that the contest question is valid and that the user is posting in good faith then we know that the following should be done. This is from previous policy discussion if memory serves right.
The question should be closed and all answers barred from viewing aside from original authors.
Rep values should not change or be deducted as users should not be punished for being fooled by a contest question.
The question should be restored after the end of the contest.
Now considering the case of a user posting $3$ questions from the same contest, I would imagine that they know very well what they are doing. That is a special case, and really it is up to the mods and how the user reacts to decide whether or not a ban is necessary. But considering another user posted just one question from the same contest it's safe to say that my suggestion still holds water.
Basically we want a close vote that automatically hides all answers from view (except from mods and original authors) and prevents future answers until some date has passed. I don't know if the close system can be changed to accommodate such a thing but I think it is worth looking into with parent SE to see if it can be added. This would allow puzzle contests on puzzling.se to have the same protection and any other site where contest questions might come in.
Now of course the close would simply expire so then a user would be aware that it's not a punishment. It's just to prevent cheating. If a user isn't intending to cheat and is merely curious I can't see them having any issue in waiting. This is in fact why I say bans should be used leniently because can be given contest questions or obtain them outside of the contest without context and then choose to ask on here out of curiosity. The contest should ban them from the contest, but it's hardly egregious enough to warrant a stack exchange ban.
Some people mentioned as well that answering PSQ is wrong and that should factor into the problem here. However, a problem statement question can be considered good enough on its own when proper tags are given and the content itself is obviously not a homework or something with a natural application. The recreational-math tag tends to come well in showing that. After all, context can mean mathematical context (the definitions and assertions assumed when working the problem) or personal context (the actual events leading to the human being asking the question). I would argue that not all questions need the latter as Math is primarily studied without application in mind. Numerous jokes about this exist including an XKCD comic I believe.
Why do I bring this up? Because we shouldn't be scolding users for answering questions. If anything a user that is able to correctly answer a question that to others might be flawed or confusing or what-have-you shouldn't be punished. After all, while MSE has many uses at it's core level it is for a person doing some kind of Math to either get help with Math or answer a curiosity they had in Math. Punishing someone for attempting to help another user is flat out wrong and unproductive. If people wish to not be involved in the politics of the site and instead answer questions then they should not be punished for choosing to answer things that don't have a nice wrapping to them. I saw several comments here implying that users were wrong for answering them as they were PSQ's but the thing is that just because a question should be closed doesn't mean it cannot be answered or that it shouldn't be. If people just want to go around the site and provide helpful, well thought out, and carefully constructed answers then they shouldn't be punished for that. People serving the role of moderation or close voters determine what is and isn't allowed as a question, whereas people answering questions should only be required to look at what they find interesting and can answer with their abilities.
Some further points to back up my claims here:
6 Note that in this case OP seemed unaware that the question was a problem in a current journal issue. – Gerry Myerson Aug 26 '12 at 0:03
6 We've had analogous issues with problems posted from ongoing competitions (e.g. Project Euler), problems from take-home exams, etc. These too, can be passed along, with the original source being lost. Generally it is difficult if not impossible to ascertain whether the OP knows the source, and speculation about such is usually not constructive. – Bill Dubuque Aug 26 '12 at 18:28
From Should we allow ongoing journal questions?
While I have stated that the current policy is that we don't ban people for posting contest questions no different than we don't ban people for posting questions that lack context (usually assumed to be homework), I think the point that people need to understand is that stack exchange assumes the asker is asking in good faith. That is one of the things I've seen stated times. Until it is made absolutely clear that the person is violating a policy on purpose, we assume good faith. Yes, in the case of a person posting 3 contest questions it is clear they are either trying to cheat or were not aware it was an ongoing contest. However, that is a special case. Note that on the flipside we have another user who only posted one question from the same contest. We don't know if that person knew where it originated from. It could have been shared by a friend as an interesting problem. I can speak fro experience that sheets of (old) problems have been passed around in the chats before for days at a time. Now it was clear in that case because the pdf had a date on it. However, it is not hard to fake such a date for starters (just edit the pdf) and we can also assume that a person might miss the date and/or not understand that it was a contest rather than just a sheet to do for leisure.
I'm not at all saying that the user that posted three questions might be punished. However, what I am saying is that we are being very trigger-happy here and immediately jumping for ban. If someone can show there is a precedent of such bans fair enough, but as far as I am aware we do not normally ban for such things.