I held one in mathematics just last year at a small canadian university as a second year student (although from the sounds of it, still larger than yours). Before worrying about whether you will make the cut, find a professor first who wants to supervise you and has something you could work on. Start by talking to everyone who has taught you a course, then move on to anyone else in the department if those options are exhausted. Once you have done that, acceptance is more or less up to your professor's proposal and your grades and ability to write a resume. There is no reason at this point to not apply, so I dont see why you are worrying.
If your school is so small that there is no one interested in you, or undergrads, or even research, then the next step would be to ask the professors you know, whether they know anyone you can contact at other schools. Contacting someone at another school will go much better if you do it through a respectable professor that is familiar with you and the person at the other university. (this advice is basic logic)
You might even have a better chance of making it to the national level competition than at a big school. Science departments can only send $x$ applications so there is a local competition first. At bigger schools you might be up against more people.
You should note that these kinds of questions aren't suited for this website and you will learn much more consulting with profs at your school who are willing to give you advice.