In less than a year from now I'll be applying to pure math PhD programs. I'm going to have a competitive application. I'm attending a well regarded school and will no doubt get good reccomendations; I took a whole slew of math undergrad courses receiving all A's; I took (and am taking) a handful of graduate courses receiving a mix of A's and B's; TA'd some undergrad courses; I have two solid REUs; participated in Budapest semesters in mathematics, etc
But these are all things that every top (20 or so) graduate school is going to expect from all it's applicants. I don't have any research independent of my REUs, I've never done notably well on the Putnam, and so on. My main concern is that I'm not going to stand out in any way to a graduate school admissions committee.
Looking at application threads, this seems quite worrisome. There are consistently applicants in a similar position as I who get almost universal rejection from good graduate programs. I love mathematics and so that I can pursue a fruitful research career someday, I want to attend a serious graduate program.
What can I do in under a year to improve my graduate application? How can I stand out?