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Recommendations

Curriculum Recommendations

The Art of Problem Solving mathematics curriculum is specifically designed for high-performing math students. Our texts not only introduce key results of mathematics, but also teach students how to develop new ideas themselves and how to apply these ideas to challenging problems. In so doing, we help students learn the key problem solving skills needed for success in top universities and in the most competitive careers.

If you still don't know which book to choose after reading the recommendations below, then click here to receive a personalized recommendation from AoPS staff.

Elementary School

Grades 2-6, not yet ready for Prealgebra

Beast Academy is the Art of Problem Solving's new elementary school math curriculum. Grades 3 and 4 are available now, with more books coming soon.

Beast Academy 3A
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 104 pages. Practice: 128 pages.

Beast Academy Guide 3A and its companion Practice 3A are the first part in a four-part series for 3rd grade mathematics. Level 3A includes chapters on shape classification, skip-counting, and perimeter and area.

Beast Academy 3B
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 104 pages. Practice: 128 pages.

Beast Academy Guide 3B and its companion Practice 3B are the second part in a four-part series for 3rd grade mathematics. Level 3B includes chapters on multiplication, perfect squares, and the distributive property.

Beast Academy 3C
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 112 pages. Practice: 144 pages.

Beast Academy Guide 3C and its companion Practice 3C are the third part in a four-part series for 3rd grade mathematics. Level 3C includes chapters on variables, division, and units and measure.

Beast Academy 3D
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 112 pages. Practice 160 pages.

Beast Academy Guide 3D and its companion Practice 3D are the fourth part in a four-part series for 3rd grade mathematics. Level 3D includes chapters on fractions, estimation, and area.

Beast Academy 4A
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 112 pages. Practice: 176 pages.

Beast Academy Guide 4A and its companion Practice 4A are the first part in the planned four-part series for 4th grade mathematics. Level 4A includes chapters on shapes, multiplication, and exponents.

Beast Academy 4B
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 112 pages. Practice: 152 pages.

Beast Academy Guide 4B and its companion Practice 4B are the second part in the planned four-part series for 4th grade mathematics. Level 4B includes chapters on counting, division, and logic.

Beast Academy 4C
Jason Batterson. Illustrated by Erich Owen.
Paperback
Guide: 112 pages. Practice: 168 pages.

Beast Academy Guide 4C and its companion Practice 4C are the third part in the planned four-part series for 4th grade mathematics. Level 4C includes chapters on factors, fractions, and integers.

Prealgebra and Algebra

Completed elementary school, but not introductory algebra

Students who have completed an elementary school curriculum but have not taken a Prealgebra (or equivalent) course should start with Prealgebra. Students who have worked with square roots, basic variable expressions such as 2x + 3, and basic linear equations such as the equation 2x + 7 = 23, but who have not taken an Algebra 1 course or an equivalent, may be ready to move on to Introduction to Algebra.

Prealgebra
Richard Rusczyk, David Patrick, Ravi Boppana
Paperback
Text: 608 pages. Solutions: 224 pages.
Prealgebra prepares students for the rigors of algebra, and also teaches students problem-solving techniques to prepare them for prestigious middle school math contests such as MATHCOUNTS, MOEMS, and the AMC 8. The text is written to challenge students at a much deeper level than a traditional middle school prealgebra course.
Introduction to Algebra
Richard Rusczyk
Paperback
Text: 656 pages. Solutions: 312 pages.
A thorough introduction for students in grades 6-9 to algebra topics such as linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and more!

Beyond Basic Algebra

Completed an Algebra 1 course or equivalent and ready for more.

Students who have completed a basic algebra course (such as a typical school Algebra 1 course) are ready for most of our Introduction series. Consider a break from algebra with Introduction to Counting & Probability and/or Introduction to Number Theory. Our Introduction to Algebra text goes well beyond the typical algebra curriculum, so you might consult the text's diagnostic post-test to see if your student might need to spend some time with Introduction to Algebra before continuing to Introduction to Geometry. Introduction to Geometry is the most challenging of the Introduction-level books.

Introduction to Algebra
Richard Rusczyk
Paperback
Text: 656 pages. Solutions: 312 pages.
A thorough introduction for students in grades 6-9 to algebra topics such as linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and more!
Introduction to Counting and Probability
David Patrick
Paperback
Text: 256 pages. Solutions: 120 pages.
A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more.
Introduction to Number Theory
Mathew Crawford
Paperback
Text: 336 pages. Solutions: 144 pages.
A thorough introduction for students in grades 7-10 to topics in number theory such as primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and more.
Introduction to Geometry
Richard Rusczyk
Paperback
Text: 557 pages. Solutions: 226 pages.
A full course in challenging geometry for students in grades 7-10, including topics such as similar triangles, congruent triangles, quadrilaterals, polygons, circles, funky areas, power of a point, three-dimensional geometry, transformations, introductory trigonometry, and more.

Advanced High School Math

Completed one or two years of algebra and a geometry course

Students without discrete math experience might start with our Introduction to Counting & Probability text before moving on to our Intermediate series of books. Intermediate Algebra should be completed before Precalculus, which should precede Calculus. The Intermediate Counting & Probability book is appropriate at any point in the sequence after Intermediate Algebra.

Introduction to Counting and Probability
David Patrick
Paperback
Text: 256 pages. Solutions: 120 pages.
A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more.
Intermediate Algebra
Richard Rusczyk and Mathew Crawford
Paperback
Text: 720 pages. Solutions: 336 pages.

A comprehensive textbook covering Algebra 2 and topics in Precalculus. This book is the follow-up to the acclaimed Introduction to Algebra textbook. In addition to offering standard Algebra 2 and Precalculus curriculum, the text includes advanced topics such as those problem solving strategies required for success on the AMC and AIME competitions.

Precalculus
Richard Rusczyk
Paperback
Text: 528 pages. Solutions: 272 pages.

A comprehensive textbook covering precalculus topics. Specific topics covered include trigonometry, complex numbers, vectors, and matrices. Includes many problems from the AIME and USAMO competitions.

Intermediate Counting & Probability
David Patrick
Paperback
Text: 400 pages. Solutions: 208 pages
An intermediate textbook in counting and probability for students in grades 9-12, containing topics such as inclusion-exclusion, recursion, conditional probability, generating functions, graph theory, and more.
Calculus
David Patrick
Paperback
Text: 336 pages. Solutions: 128 pages.

A comprehensive textbook covering single-variable calculus. Specific topics covered include limits, continuity, derivatives, integrals, power series, plane curves, and differential equations.

Contest Preparation Recommendations

Art of Problem Solving materials have been a key component in the success of many winners of major national and international math competitions. We integrate challenging problems from contests such as MATHCOUNTS, AMC 8/10/12, AIME and the USA(J)MO throughout our books. Moreover, we offer books compiled by the organizers of various major contests.

Elementary School Contests

(Late elementary school)

Math Olympiads in the Elementary and Middle Schools spans grades 4-8, and offers a nice supplement to any elementary school curriculum. Creative Problem Solving in School Mathematics offers lessons and problems, while the other two books listed below offer problems and solutions from past contests.

   $ 37.99
Creative Problem Solving in School Mathematics
Dr. George Lenchner
Paperback
284 pages.
A collection of lessons and challenging problems for beginning problem solvers in grades 4-8. Written by the creator of the popular Math Olympiads for Elementary and Middle Schools program.
   $ 35.95
MOEMS Contest Problems for Elementary and Middle Schools
Dr. George Lenchner
Paperback
280 pages.
A collection of sixteen years (400 problems!) of the Math Olympiads for Elementary and Middle Schools contests, with hints, complete solutions, and problem solving lessons. Ideal for beginners grades 4-8.
   $ 37.95
MOEMS Contest Problems: Volume 2
Richard Kalman, Ed.
Paperback
309 pages
A collection of 425 problems of the Math Olympiads for Elementary and Middle Schools contests, with hints, complete solutions, and problem solving lessons. Ideal for beginners grades 4-8.
   $ 39.95
MOEMS Contest Problems: Volume 3
Richard Kalman & Nicholas J. Restivo, Eds.
Paperback
294 pages
A collection of 400 problems of the Math Olympiads for Elementary and Middle Schools contests from 2005-2013, with hints, complete solutions, and problem solving lessons. Ideal for beginners grades 4-8.

Middle School Contests

MATHCOUNTS/AMC 8

The Art of Problem Solving Introduction series offers both a full math curriculum and problem solving training for middle school and beginning high school math contests. Over 1000 problems from major contests are included among the AoPS series of books. Our Art of Problem Solving Volume 1 and Competition Math for Middle School are designed specifically for contest preparation, and we also offer books published by MATHCOUNTS.

Prealgebra
Richard Rusczyk, David Patrick, Ravi Boppana
Paperback
Text: 608 pages. Solutions: 224 pages.
Prealgebra prepares students for the rigors of algebra, and also teaches students problem-solving techniques to prepare them for prestigious middle school math contests such as MATHCOUNTS, MOEMS, and the AMC 8. The text is written to challenge students at a much deeper level than a traditional middle school prealgebra course.
Introduction to Algebra
Richard Rusczyk
Paperback
Text: 656 pages. Solutions: 312 pages.
A thorough introduction for students in grades 6-9 to algebra topics such as linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and more!
Introduction to Counting and Probability
David Patrick
Paperback
Text: 256 pages. Solutions: 120 pages.
A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more.
Introduction to Number Theory
Mathew Crawford
Paperback
Text: 336 pages. Solutions: 144 pages.
A thorough introduction for students in grades 7-10 to topics in number theory such as primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and more.
Introduction to Geometry
Richard Rusczyk
Paperback
Text: 557 pages. Solutions: 226 pages.
A full course in challenging geometry for students in grades 7-10, including topics such as similar triangles, congruent triangles, quadrilaterals, polygons, circles, funky areas, power of a point, three-dimensional geometry, transformations, introductory trigonometry, and more.
Text and Solutions   $ 42.00
Volume 1
Sandor Lehoczky and Richard Rusczyk
Paperback
Text: 288 pages. Solutions: 144 pages.
A classic problem solving textbook for students in grades 7-10 who are preparing for contests like MATHCOUNTS and the AMC 8/10/12.
Competition Math for Middle School
Jason Batterson
Paperback
390 pages.
Contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks.
   $ 35.00
Practice Competitions for MATHCOUNTS
Josh Frost
Paperback
128 pages.
A collection of 4 complete practice competitions (including Sprint, Target, and Team rounds) for MATHCOUNTS. Includes full solutions to every problem.
   $ 35.00
Practice Competitions for MATHCOUNTS Volume 2
Josh Frost
Paperback
144 pages.
A collection of 4 complete practice competitions (including Sprint, Target, and Team rounds) for MATHCOUNTS. Includes full solutions to every problem.

High School Contests

AMC 10/12, AIME, ARML, Harvard-MIT, Mandelbrot

In addition to offering a full curriculum for middle school and early high school students, the Art of Problem Solving Introduction series provides training for early high school contests such as the AMC 10. Meanwhile, our Intermediate series delivers a full curriculum for high school students alongside training for more advanced contests, such as the AIME and the Harvard-MIT Tournament.

Our classic Art of Problem Solving Volume 1 and Volume 2 have helped prepare students for major high school contests for over 20 years. We also offer books published by the Mathematical Association of America, which administers the AMC, and the Mandelbrot Competition.

Introduction to Algebra
Richard Rusczyk
Paperback
Text: 656 pages. Solutions: 312 pages.
A thorough introduction for students in grades 6-9 to algebra topics such as linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and more!
Introduction to Counting and Probability
David Patrick
Paperback
Text: 256 pages. Solutions: 120 pages.
A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more.
Introduction to Number Theory
Mathew Crawford
Paperback
Text: 336 pages. Solutions: 144 pages.
A thorough introduction for students in grades 7-10 to topics in number theory such as primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and more.
Intermediate Algebra
Richard Rusczyk and Mathew Crawford
Paperback
Text: 720 pages. Solutions: 336 pages.

A comprehensive textbook covering Algebra 2 and topics in Precalculus. This book is the follow-up to the acclaimed Introduction to Algebra textbook. In addition to offering standard Algebra 2 and Precalculus curriculum, the text includes advanced topics such as those problem solving strategies required for success on the AMC and AIME competitions.

Intermediate Counting & Probability
David Patrick
Paperback
Text: 400 pages. Solutions: 208 pages
An intermediate textbook in counting and probability for students in grades 9-12, containing topics such as inclusion-exclusion, recursion, conditional probability, generating functions, graph theory, and more.
Precalculus
Richard Rusczyk
Paperback
Text: 528 pages. Solutions: 272 pages.

A comprehensive textbook covering precalculus topics. Specific topics covered include trigonometry, complex numbers, vectors, and matrices. Includes many problems from the AIME and USAMO competitions.

Text and Solutions   $ 42.00
Volume 1
Sandor Lehoczky and Richard Rusczyk
Paperback
Text: 288 pages. Solutions: 144 pages.
A classic problem solving textbook for students in grades 7-10 who are preparing for contests like MATHCOUNTS and the AMC 8/10/12.
Text and Solutions   $ 45.00
Volume 2
Richard Rusczyk and Sandor Lehoczky
Paperback
Text: 320 pages. Solutions: 184 pages.
A classic problem solving textbook for students in grades 9-12 who are preparing for advanced high school contests like the AMC 12, AIME, and the Harvard-MIT Mathematics Tournament.
   $ 53.95
First Steps for Math Olympians
J. Douglas Faires
Hardcover
307 pages.
Lessons, examples, and exercises on topics useful for the AMC 10 and AMC 12. Ideal for self study for the AMC exams, and as a teaching resource for instructors developing lessons on various topics.
   $ 49.95
Contest Problem Book IX
David Wells and J. Douglas Faires
Paperback
230 pages.
Past problems with complete solutions from the American Mathematics Competitions 12 (AMC 12), which is one of the first tests in the series of contests that determines the United States International Math Olympiad team. This book includes all AMC 12 tests from 2001-2007.
   $ 49.95
Contest Problem Book VIII
J. Douglas Faires and David Wells
Paperback
212 pages.
Past problems with complete solutions from the American Mathematics Competitions 10 (AMC 10), which is one of the first tests in the series of contests that determines the United States International Math Olympiad team. This book includes all AMC 10 tests from 2000-2007.
   $ 25.00
Mandelbrot Morsels
Sam Vandervelde
Paperback
180 pages.
Past problems with complete solutions from The Mandelbrot Competition 2002-09. Contains problems that are excellent practice for the American Mathematics Competitions, as well as excellent problems for students first learning how to tackle challenging proof problems.
   $ 23.00
The Mandelbrot Problem Book
Sam Vandervelde
Paperback
148 pages.
Past problems with complete solutions from years 1995-2002 of The Mandelbrot Competition. Contains problems that are excellent practice for the American Mathematics Competitions, as well as excellent problems for students first learning how to tackle challenging proof problems.
   $ 20.00
The First Five Years
Sam Vandervelde
Spiralbound
112 pages.
Past problems with complete solutions from the first five years (1990-95) of The Mandelbrot Competition. Contains problems that are excellent practice for the American Mathematics Competitions, as well as excellent problems for students first learning how to tackle challenging proof problems.

High School Olympiads

USAJMO, USAMO, and other national and international olympiads

National high school olympiads target the most experienced high school students in their respective countries. For example, the USA(J)MO in the United States is an invitational exam given to roughly 500 high scorers on preliminary contests. Our Intermediate series is useful for students just getting started with olympiads, and we offer several books from Titu Andreescu and the United Kingdom Mathematics Trust that target Olympiad students.

Intermediate Algebra
Richard Rusczyk and Mathew Crawford
Paperback
Text: 720 pages. Solutions: 336 pages.

A comprehensive textbook covering Algebra 2 and topics in Precalculus. This book is the follow-up to the acclaimed Introduction to Algebra textbook. In addition to offering standard Algebra 2 and Precalculus curriculum, the text includes advanced topics such as those problem solving strategies required for success on the AMC and AIME competitions.

Intermediate Counting & Probability
David Patrick
Paperback
Text: 400 pages. Solutions: 208 pages
An intermediate textbook in counting and probability for students in grades 9-12, containing topics such as inclusion-exclusion, recursion, conditional probability, generating functions, graph theory, and more.
Precalculus
Richard Rusczyk
Paperback
Text: 528 pages. Solutions: 272 pages.

A comprehensive textbook covering precalculus topics. Specific topics covered include trigonometry, complex numbers, vectors, and matrices. Includes many problems from the AIME and USAMO competitions.

   $ 24.95
UKMT Plane Euclidean Geometry: Theory and Problems
A D Gardiner and C J Bradley
Paperback
264 pages.
This geometry text offers beginning and advanced geometric problem solving tactics, as well as numerous practice problems. The book is most appropriate for experienced geometers who are learning how to take on more challenging geometry problems, such as those offered at the high school olympiad level.
   $ 24.95
UKMT A Mathematical Olympiad Primer
Geoff Smith
Paperback
251 pages.
New second edition! A Mathematical Olympiad Primer is an excellent guide for young mathematicians preparing for competitions, such as the British Mathematical Olympiad. The book contains theory including algebra, combinatorics and geometry, and BMO1 Problems and Solutions from 1996 onwards. This is the revised and expanded 2nd Edition.
   $ 24.95
UKMT Introduction to Inequalities
C. J. Bradley
Paperback
210 pages.
This text on inequalities covers material useful for high school national and international olympiads. The text includes materials used for training the United Kingdom Olympiad team.
   $ 24.95
UKMT New Problems in Euclidean Geometry
David Monk
Paperback
128 pages.
This book should appeal to anyone who enjoys solving the kind of challenging and attractive geometry problems that have virtually vanished from the school curriculum, but which still play a central role in national and international mathematics competitions.
   $ 69.95
Problems from the Book
Titu Andreescu and Gabriel Dospinescu
Hardcover
554 pages.
A series of articles paired with collections of problems, designed for students preparing for high-level high school contests such as national olympiads.
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