How competitive are places for PhD positions in mathematics?
I don't know if my opinion is representative of a general consensus, but I would look for candidates who fit well for the program, in particular those who fit well with the area(s) I'd like to supervise and work on. Research is so broad, that a candidate can not be "best of the best" in all areas simultaneously. If you are trying to impress people and present yourself as the best of the best candidate, you end up writing a generic and vague statement, full of unjustified self-praise with a little evidence. Some of applications I deal with don't even have the University name in their statements (e.g. I apply to your University because...) - guess what happens with them?

I suggest you to really do your homework, research a few Departments you are really interested in, choose the place (or two or three) which suits you and then write your application very specifically for each place, to demonstrate how well your interests and skill set match their interests and research programs. It's all about the right match, not being a jack of all trades.