There are quite a few undergrads who have done significant research in mathematics at your level. Even if you don't end up with a published paper (you shouldn't expect this, although it probably happens more often than you might expect), you will gain significant experience into what doing original research in mathematics means. However, I think expecting to find a problem to work on yourself, rather than have a mentor suggest one (or several) to you, is absolutely unrealistic. Maybe it's realistic for other fields, but in my opinion not mathematics.
Find a mentor who is willing to suggest a problem that you can tackle at your level, and (hopefully) give you ideas during the summer if you get stuck. Finding the right problems to work on is a major component of doing mathematical research, and sometimes the hardest one. If you find it on your own (rather than a mentor suggesting it), your mentor is likely not going to have any good ideas of how to attack it, it may end up a harder problem than is realistic for you to solve, and your mentor will be less motivated to help you. So my advice is to start looking for mentors now.
