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I'm a high school student who just finished elementary school.Though since I was into math I started going through advanced math while I was in elementary school and I pretty much finished most of the high school material and I would like to continue with more advanced material like ones on the math olympiads,more advanced and harder problems,and more advanced theorem material.As for now I mostly did trigonometric functions,number theory,math induction,extrema(using derivatives),logarithmic equations,polynomials(in a small portion),series(convergence and recurrence),limits(properties,few theorems),diophante(well only those on math competitions).But I would like to know about things like that odd polynomials must have real roots,or Euler and Fermats theorem,properties of rings and fields and etc. I just don't know where I could find such material,that would expand my knowledge on those themes taught in high school but on a more advanced level that is still fit for a high school student.

ADDED: I did read a lot of material my self,I couldn't find a fitting book.I find most of the problems just easy to solve,I do not even have to use my pencil for doing it and feel pretty bored reading theory material that I all ready know and which is too long for such a simple thing.I would like just a simple stating of the theorem and one a bit harder example done using the theorem,I liked this page for functional equations maybe with a little bit more theory material(not necessarily),though that site is quite alright I feel lack of material,as that only good material they got is functional equations

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Related. – Git Gud Apr 17 '14 at 15:40
I looked through G.H. Hardy's A Course of Pure Mathematics towards the end of high school, and I felt it was useful. So that's a suggestion for a book to look at. – nigelvr Apr 17 '14 at 16:35

Here is some relevant advice. If you are not yet in highschool, or still have 4 years of highschool left, then you could potentially train very hard to do well on olympiads. Check out the site for such resources.

However, this means literally spending every free time you have practicing since the competition is very fierce. On the other hand, you can take classes in a local community college/university which means you could really advance your studies and get you math major (if you want) very early.

Or, you could do a combination of both which is definitely the most fun.

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I tried that site but all I could find was number theory( and inequalities which really seem kinda not usable and not written in a good way.Though yeah I all ready competed on a country competition won 3rd prize,solved few IMO problems from number theory(though a little bit older problems) – kingW3 Apr 17 '14 at 16:39

Read textbooks.

This sounds simple, but a lot of people think that they can learn advanced math without ever leaving the internet. You can certainly explore and get some ideas on what to study next, but a good textbook is irreplaceable. If there are any libraries, especially university libraries, look in their math section and find out how to borrow books from them.

Sometimes you can also buy textbooks online, but often math textbooks are prohibitively expensive, and since there are a lot of terrible math textbooks, this is to be avoided unless you've already had a chance to read some of the book yourself. Since you mentioned algebraic structures, I think A Book Of Abstract Algebra is a great introduction, and it costs about 10 bucks. You can find textbook recommendations for any subject that interests you by searching for questions on this site.

As for knowing what subjects to study, you seem to have a fairly good idea. Wikipedia will help you trace the paths from specific objects or theorems you find interesting back to their prerequisites.

A note on reading textbooks: I mentioned that most are terrible, and I meant it. Most math textbooks are apparantly written by robots. Don't give up, though, if you find the book you're reading to be impenetrable, just look for a better one.

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