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.

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.

Collegiate

• Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus by Michael Spivak.

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

• 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.

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.

Inequalities

Intermediate

• Introduction to Inequalities
• Geometric Inequalities

Olympiad

• The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
• Problem Solving Strategies by Arthur Engel contains significant material on inequalities.

Probability

Getting Started

• 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

• 103 Trigonometry Problems by Titu Andreescu and Zuming Feng.

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.

See also

• Math textbooks
• Resources for mathematics competitions