I was wondering if anybody has some advice on how to get into top 25 programs in pure math Phd programs. After reading prospective student's applications on here and mathematicsgre, I'm not sure what grad school are looking for. I've noticed some people with high gre scores getting rejected to most places they apply, while I just finished reading about a person who bombed the math gre (with around a 570 I believe) and got accepted into northwestern!! Do schools weigh one's research interests and letter of intent heavily? Is it a matter of being interested in the right field at the right time? Or are the minimum gre cutoff scores just extremely high ( maybe > 90%).
Maybe it be better to apply to a masters program in math at a good school and then apply to a PhD program after the masters is completed. There are several good school with a master programs: UPenn, Michigan, WashU, NYU, Cambridge (Part III) etc.... Are masters programs just as hard to get into though?