Art of Problem Solving: Recommended 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 who are participating in contests like AMC 8/10 and Mathcounts.
• Intermediate is recommended for students who can expect to pass the AMC 10/12.
• 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 AoPSWiki:Linking books page.

Contents

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• 1 Books By Subject
  • 1.1 General Introduction / Multiple Topics
    • 1.1.1 Getting Started
  • 1.2 Algebra
    • 1.2.1 Getting Started
    • 1.2.2 Intermediate
  • 1.3 Abstract Algebra
    • 1.3.1 Collegiate
  • 1.4 Calculus
    • 1.4.1 Getting Started
    • 1.4.2 Single Variable (Intermediate)
    • 1.4.3 Multivariable (Collegiate)
  • 1.5 Analysis
    • 1.5.1 Collegiate
  • 1.6 Combinatorics
    • 1.6.1 Getting Started
    • 1.6.2 Intermediate
    • 1.6.3 Olympiad
    • 1.6.4 Collegiate
  • 1.7 Geometry
    • 1.7.1 Getting Started
    • 1.7.2 Intermediate
    • 1.7.3 Olympiad
    • 1.7.4 Collegiate
  • 1.8 Topology
    • 1.8.1 Collegiate
  • 1.9 Inequalities
    • 1.9.1 Intermediate
    • 1.9.2 Olympiad
    • 1.9.3 Collegiate
  • 1.10 Number Theory
    • 1.10.1 Getting Started
    • 1.10.2 Olympiad
    • 1.10.3 Collegiate
  • 1.11 Trigonometry
    • 1.11.1 Getting Started
    • 1.11.2 Intermediate
    • 1.11.3 Olympiad
  • 1.12 Problem Solving
    • 1.12.1 Getting Started
    • 1.12.2 Intermediate
    • 1.12.3 Olympiad
• 2 General Interest
• 3 Math Contest Problem Books
  • 3.1 Elementary School
  • 3.2 Getting Started
  • 3.3 Intermediate
  • 3.4 Olympiad
  • 3.5 Collegiate
• 4 See also

Books By Subject

General Introduction / Multiple Topics

Getting Started

• Getting Started with Competition Math, a textbook meant for true beginners (on-target middle school students, or advanced elementary school students). It is written by AoPS Community Member cargeek9, currently a junior in high school. It covers the basics of algebra, geometry, combinatorics, and number theory, along with sets of accompanying practice problems at the end of every section.

Algebra

Getting Started

• 100 Challenging Maths Problems
• AoPS publishes Richard Rusczyk's, David Patrick's, and Ravi Boppana's Prealgebra textbook, which is recommended for advanced elementary and middle school students.
• AoPS publishes Richard Rusczyk's Introduction to Algebra textbook, which is recommended for advanced elementary, middle, and high school students.

Intermediate

• Algebra by I.M. Gelfand and Alexander Shen.
• 101 Problems in Algebra from the Training of the US IMO Team by Titu Andreescu and Zuming Feng
• AoPS publishes Richard Rusczyk's and Mathew Crawford's Intermediate Algebra textbook, which is recommended for advanced middle and high school students.
• Complex Numbers from A to... Z by Titu Andreescu

Abstract Algebra

Collegiate

• Abstract Algebra by David S. Dummit and Richard M. Foote. This is a famous textbook, and is usually the go-to book for students wishing to learn about groups, rings, fields and their properties.
• Undergraduate Algebra by Serge Lang. Some compare it to being similar to Dummit and Foote with regards to rigor, although this text is slightly more terse.
• Algebra: Theory and Applications by Thomas Judson. One of the easiest books to get started with in the genre, and is very comprehensive.
• Algebra by Serge Lang -- Extends undergraduate Abstract Algebra to the graduate level by studying homological algebra and more.

Calculus

Getting Started

• The Hitchhiker's Guide to Calculus by Michael Spivak.
• Calculus Made Easy by Silvanus P. Thompson.

Single Variable (Intermediate)

• AoPS publishes Dr. David Patrick's Calculus textbook, which is recommended for advanced middle and high school students.
• Calculus: Volume I by Tom M. Apostol -- Provides a good transition into linear algebra which is uncommon in single variable calculus texts.
• Single Variable Calculus by James Stewart -- Contains plenty of exercises for practice and focuses on application rather than rigor.
• Calculus by Michael Spivak. Top students swear by this book.
• Honors Calculus by Charles R. MacCluer -- Uses the topological definition of the limit rather than the traditional delta-epsilon approach.

Multivariable (Collegiate)

• Multivariable Calculus by James Stewart.
• Advanced Calculus by Frederick S. Woods. Advanced Calculus an iconic textbook because of how Richard Feynman learned calculus from it. Feynman later popularized a technique taught in the book in college, which is now called the "Feynman Integration Technique."
• Calculus: Volume II by Tom M. Apostol.

Analysis

Collegiate

• Understanding Analysis by Stephen Abbott.
• Principles of Mathematical Analysis by Walter Rudin. Affectionately called "Baby Rudin" by some, Principles of Mathematical Analysis is known to be very terse for the analysis layman.
• Analysis I by Terrence Tao -- An easier first read than Rudin, and provides plenty of examples with thorough explanations.
• Analysis II by Terrence Tao -- Continues off from where Volume I ended and finishes at the Lebesgue Integral.
• Real Analysis by Rami Shakarchi and Elias M. Stein.
• Complex Analysis by Rami Shakarchi and Elias M. Stein.
• Real and Complex Analysis by Walter Rudin. Called "Papa Rudin" by some, Real and Complex Analysis is typically used at the graduate level.
• Functional Analysis by Rami Shakarchi and Elias M. Stein.

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

• AoPS publishes Dr. David Patrick's Intermediate Counting & Probability textbook, which is recommended for advanced middle and high school students.
• Mathematics of Choice by Ivan Niven.
• 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.
• A Path to Combinatorics for Undergraduates: Counting Strategies by Titu Andreescu and Zuming Feng.

Olympiad

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

Collegiate

• Enumerative Combinatorics, Volume 1 by Richard Stanley.
• Enumerative Combinatorics, Volume 2 by Richard Stanley.
• A First Course in Probability by Sheldon Ross
• Introductory Combinatorics by Kenneth P. Bogart

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.
• 106 Geometry Problems from the AwesomeMath Summer Program by Titu Andreescu, Michal Rolinek, and Josef Tkadlec

Olympiad

• Euclidean Geometry in Mathematical Olympiads by Evan Chen
• Solving Problems In Geometry: Insights And Strategies For Mathematical Olympiad And Competitions by Kim Hoo Hang and Haibin Wang
• 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.
• Geometric Transformations I, Geometric Transformations II, and Geometric Transformations III by I. M. Yaglom.
• 107 Geometry Problems from the AwesomeMath Year-Round Program Titu Andreescu, Michal Rolinek, and Josef Tkadlec

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.

Topology

Collegiate

• Topology by James Munkres. Topology is arguably the most renowned topology textbook of all time. It also contains an excellent introduction to set theory and logic.

Inequalities

Intermediate

• Introduction to Inequalities
• Geometric Inequalities

Olympiad

• Advanced Olympiad Inequalities by Alijadallah Belabess.
• 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.
• Titu Andreescu's Book on Geometric Maxima and Minima
• Topics in Inequalities by Hojoo Lee
• Olympiad Inequalities by Thomas Mildorf
• A<B (A is less than B) by Kiran S. Kedlaya
• Secrets in Inequalities vol 1 and 2 by Pham Kim Hung

Collegiate

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

Number Theory

Getting Started

• The AoPS Introduction to Number Theory by Mathew Crawford.
• Number Theory by George E. Andrews.

Olympiad

• Number Theory: A Problem-Solving Approach by Titu Andreescu and Dorin Andrica.
• 104 Number Theory Problems from the Training of the USA IMO Team by Titu Andreescu, Dorin Andrica and Zuming Feng.
• Problems in Elementary Number Theory by Hojoo Lee.
• Olympiad Number Theory through Challenging Problems by Justin Stevens.
• Elementary Number theory by David M. Burton
• Modern Olympiad Number Theory by Aditya Khurmi.

Collegiate

• An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, and Andrew Wiles (6th Edition).

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 by Simon Singh.
• Godel, Escher, Bach
• Journey Through Genius by William Dunham.
• A Mathematician's Apology by G. H. Hardy.
• The Music of the Primes by Marcus du Sautoy.
• Proofs Without Words by Roger B. Nelsen.
• What is Mathematics?by Richard Courant, Herbert Robbins and Ian Stewart.

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
• Leningrad Olympiads (Published by MathProPress.com)

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
• Mathematical Olympiad Treasures
• Collections of Olympiads (APMO, China, USSR to name the harder ones) published by MathProPress.com.

Collegiate

• Three Putnam competition books are available at AoPS.

See also

• Math textbooks
• Resources for mathematics competitions
• Olympiad Books