The current policy is good and works. There are many people diligently looking at the contest problems and once a candidate is identified
it isEDIT its answers are quickly (soft-)deleted.
Let us consider the thought of extending this to deleting the post altogether (so including the question itself).This means that the post does not show up in many lists any more, and in particular is no longer indexed by search engines. That 10k users can still see the post is essential because the deletion might be mistaken (admittedly this would be an edge case, but it's SE philosophy to leave moderation to members and I don't see a strong case why this should be an exception). Therefore I consider the soft-deletion an appropriate measure against any attempted fraud.
Even at present (with deleting only answers) I think there is ample proof that as the MSE community we take contests seriously and do our best to protect their integrity, within reasonable limits. Regarding the suggestions in OP and the attack vectors they want to close, I consider the law of diminishing returns to be very applicable. I don't want to put our moderators (with single-vote delete power) under the extra stress and responsibility of the models advocated.
Besides the practical concerns, let me add that I also disagree with the assumption in the OP that one incident will "seriously damage the reputation of math.se". At the very least I would like this hypothesis to be supported by similar incidents and their aftermath.
As a conclusion, I propose to extend the policy to include deletion of the question (lock+delete, if possible, to avoid the deletion being reversed by users). This will make the questions harder to find, and therefore the probability lower that any deleted answers are seen by 10k users.
Of course this raises the issue what to do with such questions after the contest is over.
Personally I would be fine with just leaving them; undeleting after the contest is over is an extra maintenance burden, while actually the interesting questions from the contest can just as well be (re-)posted after the contest is actually over. If they are not worth this effort, we've saved ourselves a maintenance task.