If someone posts a task from an ongoing contest as a question, you may mention it in a comment. Or not, since the fact that the question is a contest task is a related bit of trivia but not particularly relevant in answering the question.
If the question is original from the contest, you should comment or edit to provide attribution. A question taken from somewhere else (a contest, a book, a blog…) without attribution is plagiarism. Note that contest questions are not always original — don't attribute a question to contest organizers if they took it from an earlier book!
Acknowledging the source of a question is required by academic traditions. Refusing to answer questions is very much against academic traditions.
As a (former) scientist, I am deeply appalled that a part of the Math.SE community seems to consider that it is ok to suppress a question because it is part of an ongoing contest. This amounts to giving the contest organizers a monopoly on the question.
Scientific tradition is strongly opposed to a monopoly on ideas. Even the law in most countries, which is less liberal than mathematicians tend to be, recognizes a monopoly (with limitations) on a particular way to express an idea (copyright) and on the practical use of an idea (patents). A monopoly on the ideas themselves is not acceptable to our society in general, nor the narrower community of mathematicians.
As an answerer, you are entirely free to refuse to answer a question. But you are not entitled to decide this for others. If a question is otherwise acceptable to this site (on-topic, reasonably scoped, etc.), the fact that it has also appeared elsewhere is not grounds to prevent others from answering it.
I don't see any point in flagging the question: what do you expect a Stack Exchange moderator to do with it? As a moderator on the sister site Computer Science, where a similar issue has been raised, I would have no idea what to do with such a flag, it would only be a waste of time. You may contact the contest organizers if you wish; correlating the contest participants with Stack Exchange users is their problem (and not one they can solve — even if everyone made their real identity apparent, someone could post a question through a front).
To reiterate, I find the very idea of acknowledging a monopoly on the discussion of a scientific idea unethical, and I am deeply troubled that a community of scientists even considers it.