Competitions are fun and everything, but doing real design schemes which have gone through several iterations of refinement and presenting it in a manner that is professional, communicative of the underlying design intent and able to have a realistic shot at winning takes a MAJOR investment in time and effort/resources. That's typically why there's an "award" of some sort at the end.
It's not about ego (not for me anyway), it'd be about doing some design and getting the creative juices flowing. You'd be surprised how bloody boring the day-to-day can get in an architectural office sometimes. . . Might be nice to have the opportunity to just kick back and have some fun, although finding the time is quite difficult right now.
Sure you can do a "back of napkin" sketch in 10 minutes, but a real competition-type entry that's up to the level of expectation of most reviewers/critics? A few dozen hours - MINIMUM. More like a couple of hundred, depending on the complexity of the project. For a job like this that's so open-ended, it could actually take quite a while, since there's so much to potentially research.
Or I suppose there's VASteve's way.