How to find less-difficult-than famous unsolved problems? 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.
Maybe the first thing to keep in mind is be patient.  Learning mathematics and finding good problems take time, and you're still very young.

In math, at the undergraduate level, the most important thing to becoming a good researcher is to learn a lot of math.  At this level, really the only reason to do research is for fun.  Your goal should not be to get new results--you might or you might not--but your goal should be to understand things and enjoy the process.

Thus my recommendation for doing research on your own is: just learn a bunch of things that seem interesting, and when questions naturally come to you, explore them.  You will probably stumble on to many ideas for research this way, and it doesn't matter if they have been studied/solved before or not (my experience: usually they have, but occasionally you find something new).

If you really want to do something new, it helps immensely to have an experienced mentor guiding you.  They can tell you what is known/what isn't, suggest relevant references and teach you some basic techniques.  

It's unfortunate that none of your professors seem willing to work with you now, but there could be various reasons for that, such as being busy and not knowing your abilities/background.  I can't guarantee this, but there's a good chance that if you excel in their courses, one of them may be willing to provide you some informal guidance.

Finally, to directly answer the titular question here are some specific suggestions:


Try reading articles in undergraduate-oriented journals like The American Mathematical Monthly, College Mathematics Journal or Involve.  Many of the research articles are readable by undergraduates, and implicitly or explicitly suggest projects suitable for undergraduates.
You can use Mathematics Stack Exchange and MathOverflow to try find out what is known about a problem you're wondering about.  Also browsing some of the questions and answers will introduce you to some unsolved problems.
If you read an article you find really interesting, and you have some definite ideas about a project, you can try emailing one of the authors for their thoughts.
In the US, we have summer programs like REUs (Research Experience for Undergraduates) at various institutions.  You might look to see if there are any similar opportunities in our around your country.