I don't know if this is the right place for such a question, so please redirect me to a more appropriate place if necessary.
My main question is, if you go to a small school, are you doomed to fail right from the get-go? My secondary question is, what are my options then, if I am indeed doomed to "fail"? The following is more of a personal description, so obviously it doesn't apply to everyone, which would make the question "not general enough", but I suspect that there are many others caught in a similar situation.
I'm going to be a senior (fourth-year) in the fall (August) and am currently doing my "homework" on graduate school admissions. I go to a small liberal arts college in the Midwest that you've almost certainly never heard of (we hover around 90-110 on the usnews.com ranking, and only offer a Bachelor's in math). From what I've read and heard, I'm extremely discouraged and I'd say I'm borderline depressed about my future, but the truth is, I don't really want to do anything else other than learn more math (and get a PhD in the process if I'm capable of it).
Some background information: I'm an international student (from Asia) majoring in math and computer science, and minoring in philosophy. I currently have a 4.0/4.0 GPA in each of these concentrations, and a 3.99/4.0 overall GPA, but of course, this doesn't mean much at all to admissions committees, given the obscurity of my school.
I like to think that I've worked very hard (but everyone says the same thing). In an awkwardly backwards fashion, I took topology, complex analysis, and algebra in my first year; real analysis, statistics, graph theory, and set theory in my second; and number theory and Galois theory in my third. I've been told that none of my coursework means much to graduate schools simply because of the obscurity of my school. Furthermore, much as I can get a great recommendation letter from pretty much anyone in my department, I've heard that since none of them is "relatively well-known" in their area, and everyone who applies to graduate schools gets glowing reviews anyway, I can't compete with other applicants at all.
I've taken the Putnam twice, but have only gotten a 21 each time. I've gone through all of Herstein, a little more than half (i.e., as much as I could) of Rudin, most of Hoffman/Kunze, and the first half of Munkres. Of course, none of these texts are actually used at my school. In other words, I don't think I've given any less than I'm physically capable of in the last three years, and if that isn't enough, I really shudder to think what is.
I haven't taken the GRE yet but am currently studying for it. But it's no secret that the GRE is a relatively silly measure of grad school success. A bad score would condemn me to eternal hell, but a good score isn't a direct ticket to success.
I've heard the "anything can happen ... who's to say that you can't succeed ... people from small schools have gotten into big schools" argument, but in all realisticness, there's absolutely no reason to pick me over a Princeton graduate who has the exact same credentials, and there are tons of them. So what are my best options given this reality?
Any advice/encouragement/experiences would be very much appreciated. Thank you.