in our experience, the Mighty Eagle only appears if a level has not been solved yet. Since all levels will be unlocked, that is, have been solved already, the Mighty Eagle should not appear during the competition.
I also solved the 21 first levels but the Eagle still appears sometimes. I think it happends randomly when the game thinks you have problems solving a level. For example if you fail/retry the level at least once.
Maybe other participants should also comment their experience.
I had a discussion about this with Gary a while ago and we concluded that once levels are solved and unlocked the Mighty Eagle shouldn't appear anymore for these levels.
We have been running our setup for days non-stop with multiple agents playing at the same time and I don't think the Mighty Eagle is ever coming up. Gary can you confirm this? is it because we are logged in or run the offline version?