I realize this question is long and quite personal but any help will be greatly appreciated.
I'm a senior high school student. I've learnt maths at precalculus level. I have had little exposure to olympiad style problems. However, I pick up concepts quickly and can tackle the hardest problems in my maths books. I am confident that with some effort my skills can improve significantly.
Since I'm gоing to a small liberal arts university in the fall, I've decided to make twice the effort and use all supplementary materials and textbooks I can find but not fall behind those in top universities. I want to work in mathematics one day, and I think I have the potential to do so. The rest is hard work. For starters, I've decided to devote the spring and most of summer to making up for the lost time. Problem is I don't know where I should begin.
Should I focus on Olympiad problems in order to learn to write proofs? Or should I start some introductory course in higher maths, such as those offered on MIT OpenCourseware?
Another issue is finding the appropriate books. Many irritate me because they seem more like a set of instructions you can load into a computer and make it solve the exercises and less like a book that challenges you and teaches you to think like a mathematician - and that is exactly the kind of book I am looking for. I need books that have little prerequisites and challenging problems (with solutions included).
Thank you in advance.