Math books
These Math books are recommended by Art of Problem Solving administrators and members of the AoPS Community.
Levels of reading and math ability are loosely defined as follows:
- Elementary is for elementary school students up through possibly early middle school.
- Getting Started is recommended for students grades 6 to 9.
- Intermediate is recommended for students grades 9 to 12.
- Olympiad is recommended for high school students who are already studying math at an undergraduate level.
- Collegiate is recommended for college and university students.
More advanced topics are often left with the above levels unassigned.
Contents
Books by subject
Algebra
Intermediate
- Algebra by I.M. Gelfand and Alexander Shen.
Analysis
- Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted.
Calculus
High School
- Calculus by Michael Spivak. Top students swear by this book.
- The Hitchhiker's Guide to Calculus by Michael Spivak.
- AP Calculus Problems and Solutions Part II AB and BC -- A fantastic resource for students mastering the material required for the AP exam.
Combinatorics
Getting Started
- AoPS publishes Dr. David Patrick's Introduction to Counting & Probability textbook, which is recommended for advanced middle and high school students.
Intermediate
- Mathematics of Choice by Ivan Nevin.
- 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.
Olympiad
Geometry
Getting Started
- AoPS publishes Richard Rusczyk's Introduction to Geometry textbook, which is recommended for advanced middle and high school students.
Intermediate
- Challenging Problems in Geometry -- A good book for students who already have a solid handle on elementary geometry.
- Geometry Revisited -- A classic.
Olympiad
- Geometry Revisited -- A classic.
- Geometry of Complex Numbers by Hans Schwerfdtfeger.
- Geometry: A Comprehensive Course by Dan Pedoe.
- Non-Euclidean Geometry by H.S.M. Coxeter.
- Projective Geometry by H.S.M. Coxeter.
Collegiate
- Geometry of Complex Numbers by Hans Schwerfdtfeger.
- Geometry: A Comprehensive Course by Dan Pedoe.
- Non-Euclidean Geometry by H.S.M. Coxeter.
- Projective Geometry by H.S.M. Coxeter.
Probability
- AoPS publishes Dr. David Patrick's Introduction to Counting & Probability textbook, which is recommended for advanced middle and high school students.
Trigonometry
Getting Started
- Trigonometry by I.M. Gelfand and Mark Saul.
Intermediate
- Trigonometry by I.M. Gelfand and Mark Saul.
- 103 Trigonometry Problems by Titu Andreescu and Zuming Feng.
Olympiad
Problem Solving
Getting Started
- the Art of Problem Solving Volume 1 by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 7-9.
- Mathematical Circles -- A wonderful peak into Russian math training.
- 100 Great Problems of Elementary Mathematics by Heinrich Dorrie.
Intermediate
- the Art of Problem Solving Volume 2 by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 9-12.
- The Art and Craft of Problem Solving by Paul Zeitz, former coach of the U.S. math team.
- How to Solve It by George Polya.
- A Mathematical Mosaic by Putnam Fellow Ravi Vakil.
- Proofs Without Words, Proofs Without Words II
- Sequences, Combinations, Limits
- 100 Great Problems of Elementary Mathematics by Heinrich Dorrie.
Olympiad
- Mathematical Olympiad Challenges
- Problem Solving Strategies by Arthur Engel.
- Problem Solving Through Problems by Loren Larson.
General interest
- The Code Book by Simon Singh.
- Count Down by Steve Olson.
- Fermat's Enigma
- Godel, Escher, Bach
- Journey Through Genius by William Dunham.
- A Mathematician's Apology
- The Music of the Primes by Marcus du Sautoy.
- Proofs Without Words
- What is Mathematics?
Math contest problem books
Elementary School
- Mathematical Olympiads for Elementary and Middle Schools (MOEMS) publishes two excellent contest problem books.
Getting Started
- MathCounts books -- Practice problems at all levels from the MathCounts competition.
- Contest Problem Books from the AMC.
- More Mathematical Challenges by Tony Gardiner. Over 150 problems from the UK Junior Mathematical Olympiad, for students ages 11-15.
Intermediate
- The Mandelbrot competition has two problem books for sale at AoPS.
- ARML-NYSML 1989-1994
- Five Hundred Mathematical Challenges -- An excellent collection of problems (with solutions).
- The USSR Problem Book
Olympiad
- USAMO 1972-1986 -- Problems from the United States of America Mathematical Olympiad.
- The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004
- Mathematical Olympiad Challenges
- Problem Solving Strategies by Arthur Engel.
- Problem Solving Through Problems by Loren Larson.
- Hungarian Problem Book III
- Mathematical Miniatures
Collegiate
- Three Putnam competition books are available at AoPS.
