Making up for wasted high school years - where should I begin? 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.
 A: I think What is Mathematics? by Richard Courant and Herbert Robbins is a great book for self-study. It covers very many topics (Number Theory, Algebra, Topology, Calculus, Projective Geometry) with enough to give you an idea of what they are about. This is guaranteed to keep you occupied for the most part of this summer.
A: My personal experience is that MIT OCW that gives reading and homework assignments, but the notes are mostly sketchy.
Not knowing exactly where you are, if you want to really have a go at algebra with a real math perspective, take a look at Gelfand's "Algebra." It was written by a great mathematician for students in your situation.
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
Abstract algebra at the university level is one of the great branches of math. It is a big transition from high school type math (mostly mechanical) to mathematics as it is done in higher academic contexts.
Berkeley offers podcasts of some past calculus courses. You might want to find one that is less mechanical and focuses more on concepts.
http://webcast.berkeley.edu/series.html#c,d,Mathematics
Take a look at the Math 53 offerings at the bottom and see if it's something you find beneficial. It may be a big leap for now, but it's something to have for the hopefully near future.
I just checked and the prerequisite for Math 53 - something you want to do, is Math 1 B, so you can look at 1 A & B, and get rolling.
A: I haven't found complete courses on MIT OCW, or complete enough to say they would be equivalent to the full subject that you would learn in college. The best ones (imo) were the walter lewin course but those are physics and not math. 
Your best choice is to get books. If you are going to learn on your own, you will have to go slow, don't try to jump or skip things, go slowly and understand every concept, proof, etc, even if you don't remember them (it's not about memorizing millions of proofs, but you will have to have done them once at least).
If you want some intuitive approach that you won't find in rigorous books, you may look for online courses, but don't make them central. Apparently Coursera has a couple of them in Algebra an Calculus that are really good, try them if you want.
A: First review your highschool skills using this game:
https://www.khanacademy.org/exercisedashboard
This book learned me the basics of mathematics:
http://books.google.nl/books/about/Reading_Writing_and_Proving.html?id=AhVCXPE5yukC&redir_esc=y
I think that many prefer to learn the basics rules of logic and set theory, before going into analysis, linear algebra, abstract algebra etc.
Videos are great way to learn concept. I learned a lot of the videos of the khanacademy,patrickjmt and especially this course:
http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra
This is the way I like to learn math:


*

*I start with reading a paragraph of the book.

*I try to memorize definitions and theorems.

*I watch math videos to get more intuition behind those definitions and theorems.

*I start making the exercises.

*Everything I still don't understand after reading/watching and thinking I ask at stackexchange. 


Especially step 5 is something I would absolutely recommend doing.
A: Don't waste more time on olympiad problems or following online courses. Take some real mathematics books and dig in. I would suggest you start by reading textbooks on algebra, analysis, and set theory/logic. There are many good texts, and you should look for those that do things rigorously, prove everything, and define everything explicitly and rigorously.
Here are some free online textbooks (the first two are not necessarily the best books but they are free and available to you immediately. They are good enough though):
Analysis: Zakon's book
Algebra: LADW by S. Treil
Logic and set theory: Jaap van Oosten's webpage has a link to a book "Sets, Models, and Proofs", by Moerdijk and van Oosten.
A: Take a look at William Chen's lecture notes. Without knowing what you want to study, and some hint on the syllabus of the classes you'll be taking, it is hard to give more specific guidance. Take a look at the university's webpages, check if there are lecture notes on line. If not, track down faculty/teaching assitants/students and ask them for guidance/hints.
Good luck!
A: Don't worry, you caught yourself just in time.  You have plenty of time ahead of you.
If I were you, I'd focus on getting A's in all your math classes in your freshman year.  If you find that easy in the first quarters/semester, go ahead and pick out a book appropriate to your "mathematical maturity" and read that too.  Otherwise, just stick it out.
Also, join a nice online "community" like this one, or Usenet's sci.math.  Don't be afraid to ask questions.  We're here to help.  Joining a community of mathy people will help you learn how to ask good questions.
Finally, sooner or later, you're going to get lost.  It might happen as an undergrad or graduate.  But there will come a point where you're dealing with "abstract nonsense".  Memorize it, work the problems, try to put what the symbols mean into words, and when you've worked it enough, you will have earned the intuition.  This is how we learn "mathematical maturity".
