Art of Problem Solving books
The Art of Problem Solving books are, to my mind, amongst the best maths and problem-solving textbooks there are. They are written for students who want to be stretched properly rather than moved through a syllabus, and the exercises are hard in a useful way.
I have no connection with Art of Problem Solving and they have not asked me to say any of this. I do earn a small commission if you buy through the Amazon links below.
They are American books, which makes them awkward to get hold of in the UK. Stock comes and goes quickly and some listings are marked up a long way, so do check the price and availability before you buy. If something shows as unavailable it is often back within a few weeks. If a book is unavailable or looks silly money, it is often cheaper to buy direct from Art of Problem Solving themselves.
The two classic volumes
These are the books most people mean by “AoPS”. Volume 1 covers the ground a strong student needs for junior and intermediate competitions; Volume 2 goes well beyond it. The solutions manuals are sold separately and are worth having, since the point is to read a full solution when you are stuck rather than just check an answer.
Volume 1: The Basics
Sandor Lehoczky and Richard Rusczyk
Volume 2: And Beyond
Richard Rusczyk Sandor Lehoczky
Volume 1: text and solutions set
Richard Rusczyk Sandor Lehoczky
Volume 2: text and solutions set
Sandor Lehoczky and Richard Rusczyk
The curriculum sequence
AoPS also publish a full curriculum, each book a proper course in one topic with genuinely hard exercises. Roughly in order of difficulty.
Prealgebra (text and solutions set)
Richard Rusczyk
Introduction to Algebra
Richard Rusczyk
Introduction to Counting and Probability
David Patrick
Introduction to Number Theory
Mathew Crawford
Introduction to Geometry
Richard Rusczyk
Intermediate Algebra
Richard Rusczyk and Matthew Crawford
Intermediate Counting and Probability
David Patrick and Naoki Sato
Precalculus
Richard Rusczyk
Calculus
David Patrick
Solutions manuals sold separately
Where the solutions are listed apart from the textbook.
Introduction to Algebra: Solutions Manual
Richard Rusczyk
This is what I could find on Amazon UK, which is not everything. The Volume 2 solutions in particular are only available bundled with the text, and several books come and go. If something here is unavailable or the price looks silly, it is worth buying straight from Art of Problem Solving themselves.
They also publish Beast Academy for primary-age children, roughly ages 6 to 13, in a comic-book style. UK availability on Amazon is patchy, so that is another one to buy direct.
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