How to stand out in math graduate school admissions 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?
Consider the viewpoint of the admissions committee. How would they know who to pick? One way is to trust others opinions -- this is why they seek recommendation letters from those who know your abilities. 

If you still want to "stand out", and aren't satisfied with your performance in the standard ways (grades, exams, REU programs), consider what makes you different. You should seek advice from those who know you better than us for ideas here.

In addition, if you're afraid of not getting in anywhere -- apply to a spectrum of schools, and simply pick the top one(s) you are accepted to. Having backups is a good way to be safe.