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Are there any resources ( mainly books, but other things as well :)) which involve a lot of exercises/problems.

These problems should be challenging but still solveable for a High School Student. The main idea here is are there any resources e.g. online, books, programms hwhich could be seen as 'Exercises' to improve general problem solving ability and prepare one for more rigorous mathematics. I find myself understanding concepts quite well but then failing to apply these in maths problems, also some of my peers have an amazing ability to immediately recognise what the problem is aksing and to 'fetch' a fitting formula, idea from their 'Maths Toolbox.'

First of all how does one increase their mathematical repertoire (e.g formulas, definitions) and keep this up to date ( remember them..) Secondly how do I practice challenging (but not too hard ) problems (Are there any resources for that ) and how to I improve my mathematical problem solving and prepare it for the 'next stage' ( proofs, theorems etc.)

xAly
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Charles W. Trigg — Mathematical Quickies: $270$ Stimulating Problems with Solutions.

Lucian
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At the moment I read: Daniel J. Velleman, How to Prove It: A Structured Approach.

The first 2 chapters contain propositonal logic, quantfiers and basic set operations. From then on it's all about proofs: In chapter 3 the author explains various proof techniques step by step and in chapters 4,5,6,7 on relations, functions, induction and infinite sets you get to use these. At the end of each subchapter (I'm currently at 3.7) there are 10-30 exercises in varying difficulty - till now most are of exactly the right difficulty.

I'm using it to prepare myself for university and until now I'm very happy with it, although I don't know how much the difficulty increases in the later chapters.

If you want a solid introduction to rigorous, proof based math it seems to be not too bad of a choice, your problem solving skills will definitely increase (especially if you work through the hundreds of exercises).

user2103480
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Consider "The Top-Down Approach to Problem Solving", by me, ISBN 979-8464073296. It's on amazon as paperback and ebook.

It seem you just described this book in your question. I hope it helps.

Eduardo Corpeño
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