Below is a list of books of interest to developing problem solvers. The difficulty levels listed below are somewhat fluid; some of the advanced books have intermediate concepts and so on. The books are grouped within each level into textbooks, problem books, and general interest books. The general interest books are those that, while educational, are designed for enjoyment or to inspire deeper interest rather than for rigorous study.

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Texts

• the Art of Problem Solving, Volume 1: the Basics by Lehoczky & Rusczyk. Since you're here, we're guessing you already know about this one.
• Introduction to Algebra by Richard Rusczyk. A full introduction algebra book. Ideal for homeschoolers and students preparing for programs such as MATHCOUNTS and the AMC. Covers all algebraic topics of algebra I and many of the topics of algebra II.
• Introduction to Counting & Probability by Dr. Dave Patrick. A very thorough book teaching counting techniques and probability through a problem solving approach. Can be used as a textbook, and for preparation for MATHCOUNTS and the AMC.
• Introduction to Geometry by Richard Rusczyk. A full course in introductory geometry. Ideal for homeschoolers and students preparing for programs such as MATHCOUNTS and the AMC.
• Introduction to Number Theory by Mathew Crawford. A broad introduction to the fundamentals of number theory. Can be used as a textbook, and for preparation for MATHCOUNTS and the AMC.
• Mathematical Circles by Fomin, Genkin, Itenberg. An excellent introduction to various problem solving priciples, as well as a good guide to teaching students using a series of questions of increasing difficulty. Developed through the experiences of educators in the former Soviet Union teaching gifted students 'extracurricular mathematics'.

Problems

• MATHCOUNTS. Past problems and training booklets for the premier American middle school contest.
• More Mathematical Challenges by Gardiner. Over 150 problems from the UK Junior Mathematical Olympiad, for students ages 11-15.

General Interest

• Anything by Martin Gardner. Martin Gardner wrote a column for Scientific American for many years. The puzzles and mathematical curiosities he used in his columns and elsewhere are complied into many books which have amused and intrigued readers all over the world.
• Fermat's Enigma by Simon Singh. A delightful trip through the history of number theory, including some of the classic problems and solutions in the field.

Texts

• Intermediate Counting & Probability by Dr. David Patrick. A continuation of Introduction to Counting & Probability that covers more advanced discrete math topics like those found on the AMC, AIME, and USAMO.
• the Art of Problem Solving, Volume 2: and Beyond by Lehoczky and Rusczyk.
• First Steps for Math Olympians by J. Douglas Faires. Comprehensive problem solving textbook based on AMC problems written by AMC 10 Director.
• The Art and Craft of Problem Solving by Paul Zeitz. A mix of intermediate and advanced concepts, this is the book we recommend as the follow up to the Art of Problem Solving. An excellent book for students who are making the step to Olympiad-level problems.
• Mathematics of Choice by Ivan Niven. An excellent book on beginning, intermediate, and advanced counting techniques.

Problem Books

• Past American Mathematics Competitions. The AMC has produced several books containing their past contests, which are excellent sources of problems. This link is to one of many such books.
• The Mandelbrot Competition. Past Mandelbrot Competition tests are particularly suitable for those students who are moving from basic problems to advanced applications and mathematical writing. Two compilations are available at the linked site.
• ARML-NYSML Contests 1989-1994 by Zimmerman and Kessler. Past tests with solutions from the American Regions Mathematics League. As with the Mandelbrot competition, these problems are excellent preparation for the AIME and the USAMO.
• ARML 1995-2003 by Barry and Kilkelly. ARML contests and Power Contests from 1995-2003. Email J. Bryan Sullivan at jbsully@earthlink.net.
• 500 Mathematical Challenges by Barbeau, Klamkin, and Moser. A collection of problems and solutions in a variety of subject areas ranging from intermediate difficulty to very challenging.
• Challenging Problems in Geometry by Posamentier, Salkind. A collection of geometry problems.

General Interest

• Count Down: Six Kids Vie for Glory at the World's Toughest Math Competition by Steve Olson. National Book Award Finalist Steve Olson chronicles the efforts of the members of the 2001 United States IMO team while discussing many topics relevant to avid students and educators.
• Music of the Primes (and in paperback) by Marcus du Sautoy. A history of number theory culminating in the description of one of math's greatest mysteries, the Riemann Hypothesis.
• The Code Book by Simon Singh. An entertaining history of cryptography, which has been a field of study for mathematicians since the first time a person tried to encrypt a message.
• Proofs Without Words by Nelsen. A collection of visual proofs which encourage the reader to think differently about problems in a variety of subjects such as geometry, series, number theory, calculus, inequalities, and more. (Note: the first volume, linked here, is much better than the second.)
• A Mathematical Mosaic by Ravi Vakil. A delightful collection of problems, puzzles, methods, and curiosities designed to instruct and inspire deeper study.
• Godel, Escher, Bach by Hofstadter. This Pulitzer Prize winner is the starting point for many who become entranced by the beauty and complexity of mathematics.
• How to Solve It by Polya. The classic book about how to think about problems. Countless experienced problem solvers have found this a useful description of how they approach problems. As such, it's very useful for teachers trying to dissect their own thinking so they can describe it to students.
• Journey Through Genius by William Dunham. An entertaining blend of history and mathematics.

Texts

• Geometry Revisited by Coxeter and Greitzer. For anyone serious about developing a deep understanding of geometry, this book is a must.
• Mathematical Olympiad Challenges by Andreescu and Gelca. While this book might be better described as a problem book, each section of problems is preceded by a useful brief discussion on some advanced problem solving technique. This book is for lovers of advanced problem solving, as are the several other math olympiad books authored or co-authored by Titu Andreescu, coach of the USA Olympiad team.
• Problem-Solving Strategies by Engel. Like the previous book, it might also be called a problem book. The instruction in this book is more detailed and written in a style appropriate for an experienced reader of mathematics (as opposed to books which are written at a more introductory level).
• Complex Numbers from A to ... Z by Andreescu and Andrica. Challenging compilation of complex number techniques and problems.
• Geometric Problems on Maxima and Minima by Andreescu, Mushkarov, and Stoyanov. Challenging compilation of geometric optimization techniques and problems.
• Problem Solving Through Problems by Larson. A wide survey of problem solving techniques and problems. A little more approachable for the beginning student in its text than the Engel book, but it doesn't have solutions to the problems in the text (unlike the Engel book, which has solutions for most problems).
• Advanced books from Gil Publishing. Assorted Olympiad-level text and problem books.
• Find more Olympiad level books and videos at Highperception.

Problems

• Past USAMO tests. The USA Mathematical Olympiads from the late 70s and early 80s are compiled here.
• Any of the Andreescu/Feng Olympiad books. Here are a couple: 2000, 2001.
• The William Lowell Putnam Mathematics Examination 1985-2000 by Kedlaya, Poonen, Vakil. The Putnam is the primary mathematics competition for undergraduates.
• USSR Olympiad Problem Book. A challenging collection of problems from number theory and algebra.
• 102 Combinatorial Problems by Andreescu and Feng. A collection of very challenging counting problems.
• Hungarian Problem Book III by Liu. Intermediate and challenging problems from an old Hungarian competition.

General Interest

• What is Mathematics? by Courant, Robbins, Stewart. A mathematician's coffee table book. Open to any page and start reading and see all your old friends - the math concepts that have intrigued you ever since you first learned them.
• A Mathematician's Apology by G. H. Hardy. While this book is easy to read, perhaps only an experienced problem solver can fully appreciate Hardy's lament.