Recommendations and advices for self study and rigorous but in-depth high school math textbooks?

Question

I apologize in advance for the long text, but I feel that I won't get a proper response without explaining my situation and my level of knowledge. There are similar questions here but none of which answers what I have properly.

To elaborate:

I am a math student who paused academic year due to private issues. I will have year or two of free time until I continue my studies. So instead of spending time on occasional play with some contest problems to satisfy my math kink, as I did until now, I decided to revisit, le-learn properly and go through everything I missed starting from basic math up to the math I know now.

What I am looking for are rigorous textbooks (only subjects marked * in the list I will present) that cover the topics from basics up to the olympiad level. I've been exposed to both contest and serious math, so the perfect textbook would be the one that introduces the topics in a rigorous way and proceeds to derive and use all the needed properties and theorems. I'm not sure such thing exists, so I would be satisfied with a textbook that has naive, intuitive approach when introducing new topics, but is still in-depth. Meaning, it starts from basics, then covers high school knowledge and contains contest level topics of the given subject.

I will list everything I plan to study so you know which topics can be excluded in the textbooks I am looking for. For example, elementary algebra usually contains some analytic geometry, trigonometry, inequalities, but I will study those separately.

This is how I organized everything:

Logic & Set theory
Elementary algebra *
Euclidean geometry *
Coordinate geometry
Trigonometry
Complex numbers
Stereometry

Analysis
Algebra
Linear algebra

Combinatorics
Probability theory
Number theory
Inequalities
Functional equations
Graph theory

Elementary algebra should contain elementary functions (only quadratic function and polynomials, rational functions and radicals, exponential function and inverses of all functions mentioned); graphs, equations and inqualities containing such functions; all other topics that fall under algebra (for example neat factoring, tricks for proving a number is irrational, etc.)

Geometry is my weak side so the larger amount of topics it covers the better.

The thing is, I am not sure where is the line between standard high school math curriculum and topics needed for high school math contests such as IMO. I want to build strong foundational knowledge, to know how things are rigorously defined and then go into learning all the theorems related to math presented, with all the things needed for olympiad. Part of the reason why I am doing this is that I plan to base my career probably around pure math, but when I'm confronted with new types of problems, I always feel that maybe I lack the knowledge needed to solve it and not skills (even if I managed to solve it in the end).

I only need textbooks, I have problem books for each subject mentioned.

What do you think of the Art of Problem Solving textbooks on these topics? — littleO, Sep 03 '21 at 20:18
On the topics that I am asking for here, I feel like AoPS textbooks are too broad. Broad in a bad way. Material is good for someone that is introduced to topics the first time. I feel like they are a bit too "playful". I am looking for something more straightforward. Besides, there are $2$ AoPS books on elementary algebra, over $2000$ pages in total. I dont have time for that and even if I did, I think that all of that could be said with many fewer words. — , Sep 03 '21 at 20:27
You might find something of interest in some of the lists of books I've posted for various purposes: advanced old-style algebra AND advanced trigonometry AND analytic geometry (see also links in this comment) AND honors level elementary calculus *(continued)* — Dave L. Renfro, Sep 03 '21 at 20:55
AND complex numbers and geometry (also worth mentioning is this book). See also the books What is Mathematics? AND Mathematics: Its Content, Methods and Meaning AND Kvant Selecta: Algebra and Analysis I AND Kvant Selecta: Algebra and Analysis II *(continued)* — Dave L. Renfro, Sep 03 '21 at 20:56
AND Elem. Math. from an Adv. Standpoint. Arith., Alg., Analysis AND Elem. Math. from an Adv. Standpoint. Geo.. Also, the Gelfand books Alg. and Functions and Graphs and The Method of Coordinates and Trig., along with the books in the Little Math. Library series. — Dave L. Renfro, Sep 03 '21 at 20:58
I would say that it's not very productive to be very concerned about contest-type material until you've learned at least rigorous single-variable calculus, analytic geometry and algebra. By studying rigorous books up to that point, you'll naturally acquire skills that go up to a level slightly below Olympiad level. The further you go into more advanced math, the easier you'll find elementary problems. — Anonymous, Sep 04 '21 at 02:10
@DaveL.Renfro thank you! I did find some helpful resources among the ones you listed, although most of them are a bit too informal (even though they do cover the topics I am interested in). I am looking for a text that is rigorous and careful about elementary topics as, lets say, standard undergraduate text is about Real analysis. — , Sep 04 '21 at 12:41
@Anonymous I am from Serbia. Among foreign languages, English only I'm afraid. And yes, that would be the case otherwise, and I would love it that way also. But tbh, I did struggle with Analytic geometry due to unfimiliarity with some basic results about coordinate geometry. It's not a problem to learn those on-hand, but I can't be comfortable with indulging into serious math while I am lacking knowledge about some elementary topics. For example, I am really good with number theory type of problems but terrible with geometry. — , Sep 04 '21 at 12:48
@Anonymous Besides, there is no reason to go that far. To learn subjects ahead of time. I have year or 2, so when I proceed with studies, I can be fully commited with facing new math, without worrying that I am missing something trivial and elementary.
About the undergraduate subjects I listed - I did study those. Maybe not on the level I wanted but I did go through standard analysis textbook, some of the modern algebra topics, etc. But I want to see it in a new light as soon as I get rid of "this should be trivial to me but will take more time because I'm missing something basic ". — , Sep 04 '21 at 12:52
is rigorous and careful about elementary topics --- See The Non-Algebraic Elementary Functions. A Rigorous Approach (and here) by André Louis Yandl (1964) AND The Anatomy of Mathematics by Richard Brandon Kershner and Lee Roy Wilcox (1950; this book is extremely rigorous and obsessively detailed) *(continued)* — Dave L. Renfro, Sep 04 '21 at 15:31
AND Algebra. A Modern Introduction by John Leroy Kelley (1965) (see also #5 review here). Also, for an *EXTREMELY RIGOROUS* approach to elementary mathematics, see the Elements of Mathematics books -- Book 1 and Book 2 *(continued)* — Dave L. Renfro, Sep 04 '21 at 15:33
and Book 3 and Book 4 and Book 5 and Book 6 and Book 7 and Book 8 and Book 9 *(continued)* — Dave L. Renfro, Sep 04 '21 at 15:36
and Book 10 and Book 11 and Book 12. Unfortunately, I don't know how to obtain copies of these books now, but maybe this search will help. I got all of the "Book 0" pamphlets and Books 1-3 back in 1988 or 1989, when you could order them by mail. — Dave L. Renfro, Sep 04 '21 at 15:39

Recommendations and advices for self study and rigorous but in-depth high school math textbooks?

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