When I was in my high school I studied 14 undergraduate books with almost all their exercises. I am about to begin my undergraduate in mathematics this week. I love to do research in mathematics. I met a few professors during the last few months and asked them if they can accept me as their 'unofficial' research student but they all refused even if after I become officially an undergraduate student but without taking "research for undergraduates" course or I have be their graduate students.
Trying to do research on my own, I found a book that includes many unsolved problems. My question is that if I choose to do research like most students, that is adding knowledge to mathematics by expanding it gradually and in smaller steps, I can't because I don't have a adviser to know the frontiers and if I want to be an independent researcher I just know the problems that are famous to be impossible to solve!
How can I take a win-win with both; that is how to find 'smaller' unsolved problems like the problems students publish papers on, as an independent researcher when nobody willing to share them?
Also I found out that if learn mathematics along doing research I memorize the materials easily after analyzing them. That's a good side-effect of research compared to only studying.