Apologies for the length of this post, but as one of the organizers of an external contest (the entrance quiz for the Canada/USA Mathcamp summer program) I want to add a bit of information to this discussion. Back in February, we were alerted by a math.SE user that someone had posted five of our entrance quiz questions to math.SE. The deadline was still a couple of months away (late April). Qiaochu agreed to delete those questions for us. Until a week ago I thought that was the end of the story. However...
In the past couple of weeks, another program (with a December deadline) found that its questions had been asked and answered on math.SE. In fact they were able to identify and reject an applicant whose solutions were clearly plagiarized from the math.SE answers. They notified this applicant's recommenders, one of whom is one of our instructors; indeed the applicant had attended Canada/USA Mathcamp this past summer.
Our instructor took a look at the several math.SE accounts that posted the problems from this other program, and found that back in March-April the same users also posted Mathcamp qualifying quiz questions. (The questions had been extensively re-worded, which is how we missed it at the time, but I think we've learned enough from this situation that we could hope not to miss it next time.) We've confirmed that the applicant's Mathcamp quiz solutions match the answers that were posted to math.SE. Qiaochu has confirmed for us that the user who posted the February questions matches the user who posted in March and April.
Note that since the Mathcamp deadline is long in the past, we have no objections to the problems being or remaining posted on math.SE. As I understand this thread, the request to delete the questions at this time actually came from the applicant.
First, I want to make sure it's clear to everyone that the identity of the person who posted these questions is known to us (and to the other program, but not to the math.SE moderators), so there's no need for anyone who is tempted to do detective work to try to do so. In fact, because the person is a minor, I would actively discourage anyone from trying to do so.
Second, I'd like to put in people's minds one consequence of these events. Admissions to Mathcamp is very competitive. Because of this applicant's actions, there is someone else deserving who was denied the opportunity to attend our program last summer. As you all weigh the issues about access to mathematics that are raised in this debate, please keep in mind that as a consequence of these questions being posted, it actually happened that someone was undeservedly denied access to a five-week summer program. So there are issues of loss of access on both sides of the question.
Ultimately I do think it's the responsibility of the external contest organizers to take care of the integrity of their own contests, but at the same time this obviously requires the involvement of math.SE moderators. I'd like to outline what from my point of view would be an optimal way of handling the posting of external contest questions on math.SE.
The executive summary: I would want the questions to be deleted while the contest is live, and then reinstated after the deadline.
The longer version: posts with active contest questions could be noticed by the contest organizers themselves, or by math.SE users. It's definitely not part of the moderator's job to keep an eye out for such things, but of course it's possible that they could happen to notice. My hope would be that a math.SE user or moderator who catches such a post would first notify the contest organizer. (That's what happened in February, for instance.) It would then be the responsibility of the contest organizer to ask a moderator to delete the posts, if that's what they want. I think it would be fair for the moderator, when agreeing to delete the posts, to do so only on condition that the contest organizers send a reminder about re-instating the posts once the deadline has passed.
Deleting contest questions by request of anyone except a contest organizer seems like a bad idea to me, on the other hand. Two programs have now caught this particular cheater, but it's quite plausible that they participated in other contests using answers obtained from math.SE. Among other things, deleting the rest of this person's questions denies the organizers of other programs the chance to catch possible further cheating.