I enjoy all the clever comments, but the problem of selecting appropriate students is real. My experience, after having taught in a post doctoral program for over 25 years, is that the challenge is not selecting candidates that are talented, brilliant, inquisitive, it is selecting candidates that are suited to the path they have chosen.. If they get far enough to be individually examined or interviewed all the above has been more then adequately proven. I have seen too many people struggle to be happy and productive in life, however, because they were not suited for they aspired. Programs shouldn’t concentrate on one more clever way of letting candidates show their brilliance, they should concentrate on getting brilliant candidates in the programs for which they are suited.
Briefly state your position, state why the problem you are working on is important, and indicate the important questions that need to be answered; this is your "Introduction." Push quickly through this draft--don't worry about spelling, don't search for exactly the right word, don't hassle yourself with grammar, don't worry overmuch about sequence--that's why this is called a "rough draft." Deal with these during your revisions. The point of a rough draft is to get your ideas on paper. Once they are there, you can deal with the superficial (though very important) problems.