The ultimate goal of the selection process is to get students that are going to succeed in the program. What this means very much depends on the program: a person who would not pass comprehensive exams in Chicago may do well in a less intense school. Also the school tries to admit students that are likely to accept; I was surprised how many admitted students in my school choose to go elsewhere. Admission is a balancing acts based on past experiences of what is a good predictor of success.

My guess is that a big state school will first look at GRE scores (all of them including math subject). Solid math courses matter, e.g good grades in undergraduate analysis and algebra I/II, as well as in graduate courses. Nontrivial research experience is a big plus. Decent Putnam scores matter (above 20). Recommendation letters are read carefully, and what matters here is specific information, not generic praise, e.g. "student was among 3 top people in 20 student algebraic topology class that covered first 3 chapters Hatcher's book". It also matters how well-prepared and thoughtful the application is; one should think carefully what to put in the essay because it is a measure of student's maturity and good judgement.