The purpose of this thread is to list the mathematical competitions where math.SE based cheating is relevant. This is useful for two purposes: (1) To understand the scope of the problem and (2) To aid us users in identifying these questions when they appear. (Almost everyone in these discussions has supported at minimum identifying these questions.)
Here are how I see the appropriate guidelines for which competitions are relevant:
Competitions should take place over an extended period of time, in a non-proctored environment. While it is possible to sneak a smartphone into the ARML and post questions onto math.SE, I don't view it as a likely enough scenario to be relevant.
Competitions should have a definite ending point, so that problems are not forever locked away as competition problems. There is a complicated argument to be had about whether there are questions that should be eternally unanswered in public, but I don't want to have that argument.
Competitions should have a reasonably large scale. I'm not sure where I'd place the dividing line, but a single school's contest is too small to worry about, and a national contest is definitely ontopic.
Competitions should use original questions. Obviously, if organizers reuse classic problems, they can't suddenly expect people to stop talking about them.
Problems should be available online. We can't patrol for questions if they won't tell us what questions to patrol for.
I thought about adding a condition 6, competitions should primarily involves college age or younger competitors, because I don't think anyone takes the Monthly's problem section seriously enough to matter. But, on the other hand, if someone was posting problems from the WPC qualifying test, I would certainly be upset, and that is mostly older people (it also violates condition 5, but suppose it didn't). So I guess I'll phrase this more nebulously as "focus on contests where people would care".
My hope is that, in the future, part of organizing a major competition would be dropping by to add an answer to this question.