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.

Before adding any books to this page, please review the how to link books page.

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 Niven.
• 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.

Olympiad

• 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.

Collegiate

• Enumerative Combinatorics, Volume 1 by Richard Stanley.
• Enumerative Combinatorics, Volume 2 by Richard Stanley.

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.

Collegiate

• Inequalities by G. H. Hardy, J. E. Littlewood, and G. Polya.

=== Number Theory ===]

Introductory

• Introduction to Number Theory by Mathew Crawford.

Olympiad

• Number Theory: A Problem-Solving Approach by Titu Andreescu and Dorin Andrica.

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 books:
  • ARML-NYSML 1989-1994 (see ARML).
  • ARML 1995-2004
• 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