What's a good book for a beginner in high school math competitions? Also, I want to make it clear: Beginner.
I'm getting really frustrated trying to study for math competitions:


*

*On the one hand, there are books teaching the high school curriculum,
but that's it. I don't need these, I know (almost) everything on your
regular high school curriculum.

*On the other hand, there are books for the IMO. But they assume you
have a really strong olympiad mathematics background, and they are
only there to prepare you for the IMO. They're basically composed of
one or two lines of theory per chapter, and then a huge list of hard
exercises and problems. I don't need these either: I'm not even near
the IMO calliber, and the solutions to the problem seems to rely on
theorems and techniques that aren't taught in high school.
What I am really looking is for a bridge between these two extrema: A book that makes a connection between high school basic math and IMO advanced math.
A book that covers this list of topics is what I am trying to find. I've never heard about any of these topics, though they seem to be essential for math competitions. Also, I've never found any book that covers them. 
I'm not sure whether a single book will contain such a broad range of topics, so I believe that I'll need a book for Algebra, a book for Number Theory, a book for Combinatorics, etc. 
But if that's the case, I also want to make it clear that I'm not looking for a book that teaches you college Algebra, college Number Theory, etc. I want something that covers topics that appear on a math competition, but doesn't assume anything other than high school maths.
Thank you in advance.
 A: It sounds like you want some of the Art of Problem Solving books. 
I personally got a lot out of Engel's Problem Solving Strategies in high school. People also seem to like Zeitz's The Art and Craft of Problem Solving, or at least they used to back in the day, but I haven't read it myself. 
A: I think that if you have time, the best you can do is find a bank of problems somewhere, try to attack them all and give yourself the solution if you feel the problem is out of your range. Each time you read a solution, try not only to understand it, but try to understand from what angle should one have attacked the problem in order to obtain the solution. That will not only show you new proof techniques, but also new ways to think about a problem whose solution you don't have. 
Unfortunately I don't know any such book as you describe it. 
That website you linked (brilliant) looks brilliant! You should try to master all the identities and techniques described there if you plan on taking on a contest, I believe they're all going to be useful. That's the best I can say.
Hope that helps,
A: Three months ago I posted a free pdf of Index to Mathematical Problems 1975-1979 edited by Stanley Rabinowitz and me.  It doesn't give solutions but has lots of challenging problems arranged by subject, with plenty of internal hyperlinks for easy navigation.
For ResearchGate and/or Academia subscribers:
https://www.researchgate.net/publication/267164498_Index_to_Mathematical_Problems_1975-1979
https://www.academia.edu/37249387/Index_to_Mathematical_Problems_1975-1979
For non-subscribers:
http://www.mathematrucker.com/problems_1975-1979.pdf
A: These pamphlets are aimed at Oxford and Cambridge entrants, so are slightly off-topic, but they are aimed at moving from high-school to proof-based, deeper maths. I use them in teaching maths clubs here in the UK. They both contain many many commented-upon problems, which is one of the best ways to approach Olympiad material, and are free!
http://www.mathshelper.co.uk/110501_Advanced_Problems_in_Mathematics.pdf
http://www.mathshelper.co.uk/112161_Advanced_Problems_in_Core_Mathematics.pdf
