SAEM hosts the preliminary rounds of the annual competition and ACEP hosts the final competition. These are actually pretty fun to go to, although the preliminary rounds can be of painfully diverse quality. I can't remember exact categories or the point breakdown, but they presentors are given points (roughly) for the presentation itself, the solvability of the case, the novelty of the diagnosis, etc
I agree with roja. If you want to do the competition well, you really need to practice to the point where you're sick of it.
Examples. Lemiere's syndrome comes up at least once a year. A Highland resident presented a case of Adie's pupil recently. Three years ago, a resident from my program won the competition with something like a case of post-op anaerobic infection from a C-section. My classmate presented systemic capillary leak syndrome--one of the best presentations in her group, but obviously a nearly impossible diagnosis to make. My residency's current competitor advanced to the final round with a case of malaria--not particularly novel or difficult to solve, but her presentation and case discussion were excellent.
But, reiterating roja, ideally the diagnosis is a common presentation of an uncommon disease or an uncommon presentation of a common disease. Endocarditis as a stroke? That's a really good one.