Difference between revisions of "Math books"
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− | These '''Math books''' are recommended by [[Art of Problem Solving]] administrators and members of the [http:// | + | These '''Math books''' are recommended by [[Art of Problem Solving]] administrators and members of the [http://aops.com/community AoPS Community]. |
Levels of reading and math ability are loosely defined as follows: | Levels of reading and math ability are loosely defined as follows: | ||
* Elementary is for elementary school students up through possibly early middle school. | * Elementary is for elementary school students up through possibly early middle school. | ||
− | * Getting Started is recommended for students grades | + | * Getting Started is recommended for students grades who are participating in contests like AMC 8/10 and Mathcounts. |
− | * Intermediate is recommended for students | + | * 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. | * Olympiad is recommended for high school students who are already studying math at an undergraduate level. | ||
* Collegiate is recommended for college and university students. | * Collegiate is recommended for college and university students. | ||
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More advanced topics are often left with the above levels unassigned. | More advanced topics are often left with the above levels unassigned. | ||
− | Before adding any books to this page, please review the [[ | + | Before adding any books to this page, please review the [[AoPSWiki:Linking books]] page. |
− | == Books | + | == Books By Subject == |
=== Algebra === | === Algebra === | ||
====Getting Started==== | ====Getting Started==== | ||
− | * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/ | + | * [https://www.amazon.com/After-School-Maths-100-Challenging-Problems-ebook/dp/B07QFWSTDD/ref=sr_1_2?crid=CB0XAM4P81WI&keywords=after+school+maths+kawasaki&qid=1581288606&sprefix=after+school+maths+%2Caps%2C268&sr=8-2 100 Challenging Maths Problems] |
+ | * [[AoPS]] publishes [[Richard Rusczyk]]'s, [[David Patrick]]'s, and [[Ravi Boppana]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra Prealgebra] textbook, which is recommended for advanced elementary and middle school students. | ||
+ | * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra Introduction to Algebra] textbook, which is recommended for advanced elementary, middle, and high school students. | ||
+ | |||
==== Intermediate ==== | ==== Intermediate ==== | ||
* [http://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen. | * [http://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen. | ||
− | *101 Algebra | + | * [http://www.amazon.com/Problems-Algebra-Training-Team-Enrichment/dp/187642012X/ref=sr_1_2?ie=UTF8&s=books&qid=1204029534&sr=8-2 101 Problems in Algebra from the Training of the US IMO Team] by Titu Andreescu and Zuming Feng |
− | *Complex Numbers from A to Z by Titu Andreescu | + | * [[AoPS]] publishes [[Richard Rusczyk]]'s and [[Mathew Crawford]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=interm:algebra Intermediate Algebra] textbook, which is recommended for advanced middle and high school students. |
+ | * [http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]] | ||
=== Analysis === | === Analysis === | ||
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=== Calculus === | === Calculus === | ||
==== High School ==== | ==== High School ==== | ||
+ | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus Calculus] textbook, which is recommended for advanced middle and high school students. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0914098896/artofproblems-20 Calculus] by [[Michael Spivak]]. Top students swear by this book. | * [http://www.amazon.com/exec/obidos/ASIN/0914098896/artofproblems-20 Calculus] by [[Michael Spivak]]. Top students swear by this book. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0883858126/artofproblems-20 The Hitchhiker's Guide to Calculus] by [[Michael Spivak]]. | * [http://www.amazon.com/exec/obidos/ASIN/0883858126/artofproblems-20 The Hitchhiker's Guide to Calculus] by [[Michael Spivak]]. | ||
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=== Combinatorics === | === Combinatorics === | ||
==== Getting Started ==== | ==== Getting Started ==== | ||
− | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/ | + | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:counting Introduction to Counting & Probability] textbook, which is recommended for advanced middle and high school students. |
==== Intermediate ==== | ==== Intermediate ==== | ||
− | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/ | + | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=interm:counting Intermediate Counting & Probability] textbook, which is recommended for advanced middle and high school students. |
* [http://www.amazon.com/exec/obidos/ASIN/0883856158/artofproblems-20 Mathematics of Choice] by Ivan Niven. | * [http://www.amazon.com/exec/obidos/ASIN/0883856158/artofproblems-20 Mathematics of Choice] by Ivan Niven. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]]. | * [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]]. | ||
+ | * [http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/0817642889/ref=sr_1_2?ie=UTF8&s=books&qid=1219586040&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies] by [[Titu Andreescu]] and [[Zuming Feng]]. | ||
==== Olympiad ==== | ==== Olympiad ==== | ||
* [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]]. | * [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]]. | ||
+ | * [http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology] | ||
+ | |||
==== Collegiate ==== | ==== Collegiate ==== | ||
* [http://www.amazon.com/exec/obidos/ASIN/0521663512/artofproblems-20 Enumerative Combinatorics, Volume 1] by Richard Stanley. | * [http://www.amazon.com/exec/obidos/ASIN/0521663512/artofproblems-20 Enumerative Combinatorics, Volume 1] by Richard Stanley. | ||
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=== Geometry === | === Geometry === | ||
==== Getting Started ==== | ==== Getting Started ==== | ||
− | * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/ | + | * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry Introduction to Geometry] textbook, which is recommended for advanced middle and high school students. |
==== Intermediate ==== | ==== Intermediate ==== | ||
* [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] -- A good book for students who already have a solid handle on elementary geometry. | * [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] -- A good book for students who already have a solid handle on elementary geometry. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic. | * [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic. | ||
+ | *[http://www.amazon.com/Geometry-Problems-AwesomeMath-Summer-Program/dp/0979926947 106 Geometry Problems from the AwesomeMath Summer Program] by Titu Andreescu, Michal Rolinek, and Josef Tkadlec | ||
==== Olympiad ==== | ==== Olympiad ==== | ||
+ | * [https://www.amazon.com/gp/product/0883858398?%2AVersion%2A=1&%2Aentries%2A=0&pldnSite=1 Euclidean Geometry in Mathematical Olympiads] by Evan Chen | ||
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic. | * [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0486638308/artofproblems-20 Geometry of Complex Numbers] by Hans Schwerfdtfeger. | * [http://www.amazon.com/exec/obidos/ASIN/0486638308/artofproblems-20 Geometry of Complex Numbers] by Hans Schwerfdtfeger. | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0883855224/artofproblems-20 Non-Euclidean Geometry] by [[H.S.M. Coxeter]]. | * [http://www.amazon.com/exec/obidos/ASIN/0883855224/artofproblems-20 Non-Euclidean Geometry] by [[H.S.M. Coxeter]]. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0387406239/artofproblems-20 Projective Geometry] by [[H.S.M. Coxeter]]. | * [http://www.amazon.com/exec/obidos/ASIN/0387406239/artofproblems-20 Projective Geometry] by [[H.S.M. Coxeter]]. | ||
− | * | + | * [http://www.amazon.com/Geometric-Transformations-I-Number-8/dp/0883856085/ref=sr_1_6?ie=UTF8&s=books&qid=1199807141&sr=1-6 Geometric Transformations I], [http://www.amazon.com/Geometric-Transformations-New-Mathematical-Library/dp/0883856212/ref=sr_1_5?ie=UTF8&s=books&qid=1199807203&sr=1-5 Geometric Transformations II], and [http://www.amazon.com/Geometric-Transformations-III-Mathematical-Library/dp/0883856247/ref=sr_1_1?ie=UTF8&s=books&qid=1199807249&sr=1-1 Geometric Transformations III] by I. M. Yaglom. |
+ | *[http://www.amazon.com/Geometry-Problems-Awesomemath-Year-Round-Program/dp/0979926971/ref=sr_1_1?s=books&ie=UTF8&qid=1433093202&sr=1-1&keywords=107+geometry+problems 107 Geometry Problems from the AwesomeMath Year-Round Program] Titu Andreescu, Michal Rolinek, and Josef Tkadlec | ||
==== Collegiate ==== | ==== Collegiate ==== | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0883856042/artofproblems-20 Geometric Inequalities] | * [http://www.amazon.com/exec/obidos/ASIN/0883856042/artofproblems-20 Geometric Inequalities] | ||
− | ==== Olympiad ==== | + | ==== Olympiad ==== |
+ | * [https://www.amazon.co.uk/Advanced-Olympiad-Inequalities-Algebraic-Geometric/dp/1794193928/ref=sr_1_fkmrnull_1?crid=XVQYS8R7NOL9&keywords=advanced+olympiad+inequalities&qid=1555930111&s=gateway&sprefix=advanced+ol%2Caps%2C165&sr=8-1-fkmrnull Advanced Olympiad Inequalities] by Alijadallah Belabess. | ||
* [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele. | * [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities. | * [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities. | ||
*Titu Andreescu's Book on Geometric Maxima and Minima | *Titu Andreescu's Book on Geometric Maxima and Minima | ||
+ | * [http://ultrametric.googlepages.com/tin2007.pdf Topics in Inequalities] by Hojoo Lee | ||
+ | * [http://www.artofproblemsolving.com/Resources/Papers/MildorfInequalities.pdf Olympiad Inequalities] by Thomas Mildorf | ||
+ | * [https://artofproblemsolving.com/articles/files/KedlayaInequalities.pdf A<B (A is less than B)] by Kiran S. Kedlaya | ||
+ | * [http://can-hang2007.blogspot.com/2009/12/secrets-in-inequalities-volume-1-basic.html Secrets in Inequalities vol 1 and 2] by Pham Kim Hung | ||
==== Collegiate ==== | ==== Collegiate ==== | ||
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=== Number Theory === | === Number Theory === | ||
==== Introductory ==== | ==== Introductory ==== | ||
− | * The AoPS [http://www.artofproblemsolving.com/ | + | * The AoPS [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt Introduction to Number Theory] by [[Mathew Crawford]]. |
==== Olympiad ==== | ==== Olympiad ==== | ||
* [http://www.amazon.com/exec/obidos/ASIN/081763245X/artofproblems-20 Number Theory: A Problem-Solving Approach] by [[Titu Andreescu]] and Dorin Andrica. | * [http://www.amazon.com/exec/obidos/ASIN/081763245X/artofproblems-20 Number Theory: A Problem-Solving Approach] by [[Titu Andreescu]] and Dorin Andrica. | ||
− | *104 Number Theory Problems from the Training of the USA IMO Team | + | * [http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1199806669&sr=8-1 104 Number Theory Problems from the Training of the USA IMO Team] by [[Titu Andreescu]], Dorin Andrica and Zuming Feng. |
− | + | * [http://www.problem-solving.be/pen/published/pen-20070711.pdf Problems in Elementary Number Theory] by Hojoo Lee. | |
− | + | * [https://numbertheoryguy.com/publications/olympiad-number-theory-book/ Olympiad Number Theory through Challenging Problems] by Justin Stevens. | |
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=== Trigonometry === | === Trigonometry === | ||
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* the [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=1 Art of Problem Solving Volume 1] by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 7-9. | * the [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=1 Art of Problem Solving Volume 1] by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 7-9. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0821804308/artofproblems-20 Mathematical Circles] -- A wonderful peak into Russian math training. | * [http://www.amazon.com/exec/obidos/ASIN/0821804308/artofproblems-20 Mathematical Circles] -- A wonderful peak into Russian math training. | ||
− | * [http://www.amazon.com/exec/obidos/ASIN/ | + | * [http://www.amazon.com/exec/obidos/ASIN/0486613488/artofproblems-20 100 Great Problems of Elementary Mathematics] by Heinrich Dorrie. |
==== Intermediate ==== | ==== Intermediate ==== | ||
* the [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=2 Art of Problem Solving Volume 2] by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 9-12. | * the [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=2 Art of Problem Solving Volume 2] by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 9-12. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0471135712/artofproblems-20 The Art and Craft of Problem Solving] by [[Paul Zeitz]], former coach of the U.S. math team. | * [http://www.amazon.com/exec/obidos/ASIN/0471135712/artofproblems-20 The Art and Craft of Problem Solving] by [[Paul Zeitz]], former coach of the U.S. math team. | ||
− | * [ | + | * [https://www.amazon.com/How-Solve-Mathematical-Princeton-Science-dp-069111966X/dp/069111966X/ref=dp_ob_title_bk How to Solve It] by [[George Polya]]. |
* [http://www.amazon.com/exec/obidos/ASIN/1895997046/artofproblems-20 A Mathematical Mosaic] by [[Putnam Fellow]] [[Ravi Vakil]]. | * [http://www.amazon.com/exec/obidos/ASIN/1895997046/artofproblems-20 A Mathematical Mosaic] by [[Putnam Fellow]] [[Ravi Vakil]]. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0883857006/artofproblems-20 Proofs Without Words], [http://www.amazon.com/exec/obidos/ASIN/0883857219/artofproblems-20 Proofs Without Words II] | * [http://www.amazon.com/exec/obidos/ASIN/0883857006/artofproblems-20 Proofs Without Words], [http://www.amazon.com/exec/obidos/ASIN/0883857219/artofproblems-20 Proofs Without Words II] | ||
* [http://www.amazon.com/exec/obidos/ASIN/0486425665/artofproblems-20 Sequences, Combinations, Limits] | * [http://www.amazon.com/exec/obidos/ASIN/0486425665/artofproblems-20 Sequences, Combinations, Limits] | ||
− | * [http://www.amazon.com/exec/obidos/ASIN/ | + | * [http://www.amazon.com/exec/obidos/ASIN/0486613488/artofproblems-20 100 Great Problems of Elementary Mathematics] by Heinrich Dorrie. |
==== Olympiad ==== | ==== Olympiad ==== | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0387961712/artofproblems-20 Problem Solving Through Problems] by Loren Larson. | * [http://www.amazon.com/exec/obidos/ASIN/0387961712/artofproblems-20 Problem Solving Through Problems] by Loren Larson. | ||
− | + | == General Interest == | |
− | |||
− | == General | ||
* [http://www.amazon.com/exec/obidos/ASIN/0385495323/artofproblems-20 The Code Book] by Simon Singh. | * [http://www.amazon.com/exec/obidos/ASIN/0385495323/artofproblems-20 The Code Book] by Simon Singh. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0618251413/artofproblems-20 Count Down] by Steve Olson. | * [http://www.amazon.com/exec/obidos/ASIN/0618251413/artofproblems-20 Count Down] by Steve Olson. | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0195105192/artofproblems-20 What is Mathematics?]by Richard Courant, Herbert Robbins and Ian Stewart. | * [http://www.amazon.com/exec/obidos/ASIN/0195105192/artofproblems-20 What is Mathematics?]by Richard Courant, Herbert Robbins and Ian Stewart. | ||
− | == Math | + | == Math Contest Problem Books == |
=== Elementary School === | === Elementary School === | ||
* [[Mathematical Olympiads for Elementary and Middle Schools]] ([[MOEMS]]) publishes [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php two excellent contest problem books]. | * [[Mathematical Olympiads for Elementary and Middle Schools]] ([[MOEMS]]) publishes [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php two excellent contest problem books]. | ||
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=== Getting Started === | === Getting Started === | ||
− | * [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php | + | * [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php MATHCOUNTS books] -- Practice problems at all levels from the [[MATHCOUNTS]] competition. |
* [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php Contest Problem Books] from the [[AMC]]. | * [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php Contest Problem Books] from the [[AMC]]. | ||
* [http://www.amazon.com/exec/obidos/ASIN/0521585686/artofproblems-20 More Mathematical Challenges] by Tony Gardiner. Over 150 problems from the [[UK Junior Mathematical Olympiad]], for students ages 11-15. | * [http://www.amazon.com/exec/obidos/ASIN/0521585686/artofproblems-20 More Mathematical Challenges] by Tony Gardiner. Over 150 problems from the [[UK Junior Mathematical Olympiad]], for students ages 11-15. | ||
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=== Collegiate === | === Collegiate === | ||
* Three [[Putnam competition]] books are [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php available at AoPS]. | * Three [[Putnam competition]] books are [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php available at AoPS]. | ||
− | |||
== See also == | == See also == | ||
* [[Math textbooks]] | * [[Math textbooks]] | ||
* [[Resources for mathematics competitions]] | * [[Resources for mathematics competitions]] | ||
+ | * [[Olympiad Books]] | ||
+ | [[Category:Books]] |
Revision as of 09:17, 23 July 2021
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
Books By Subject
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
Analysis
- Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted.
Calculus
High School
- AoPS publishes Dr. David Patrick's Calculus textbook, which is recommended for advanced middle and high school students.
- 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
- 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
Collegiate
- Enumerative Combinatorics, Volume 1 by Richard Stanley.
- Enumerative Combinatorics, Volume 2 by Richard Stanley.
- A First Course in Probability by Sheldon Ross
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
- 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.
Inequalities
Intermediate
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
Introductory
- The AoPS Introduction to Number Theory by Mathew Crawford.
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.
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 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:
- 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.