How should I prepare?
A common question that regularly comes up on the forum is "How can I prepare for MathCounts/AMC/ARML/AIME/USAMO?" This page is intended to answer these questions. THIS PAGE IS VERY MUCH A WORK IN PROGRESS.
The best way to prepare for math contests is to do lots of practice problems. There are also many books and online handouts/lectures you can use to improve your problem solving skills. Depending on your current abilities, you will want to start out with different practice problems, different books, and in different areas of the forums. This guide is intended to help you get started.
AoPS has a list of books available through the website, separated by contest level, here.
The Art of Problem Solving books are an excellent resource to help prepare for math contests. They cover a broad range of topics, from algebra to geometry to number theory to combinatorics and much much more.
- Art of Problem Solving Volume 1 - MathCounts, AMC 8, AMC 10, AMC 12
- Art of Problem Solving Volume 2 - AMC 10, AMC 12, AIME, USAMO
The AoPS textbooks break down specific areas of mathematics. These books are on 3 levels, Introductory, Intermediate, and Advanced. The Advanced series, as well as part of the Intermediate series, has not yet been published. These books are indexed here. Excerpts are provided, as well as pretests and posttests to see if the books are on the right level for you.
- Introduction to Algebra - Mathcounts, AMC 8, AMC 10, AMC 12
- Introduction to Number Theory - Mathcounts, AMC 8, AMC 10, AMC 12
- Introduction to Geometry - Mathcounts, AMC 8, AMC 10, AMC 12
- Introduction to Counting & Probability - Mathcounts, AMC 8, AMC 10, AMC 12
- Intermediate Algebra - AMC 12, AIME, USAMO
- Intermediate Counting & Probability - AMC 12, AIME, USAMO
Here are a few more books good for preparation for higher level contests such as AMC 12, AIME, and USAMO:
- Art and Craft of Problem Solving by Paul Zeitz
- Problem-Solving Strategies by Arthur Engel
- Geometry Revisited by H.S.M. Coxeter & Samuel L. Greitzer
- 102 Combinatorial Problems by Titu Andreescu & Zuming Feng
- 103 Trigonometry Problems by Titu Andreescu & Zuming Feng
- 104 Number Theory Problems by Titu Andreescu & Zuming Feng
Old practice problems (with solutions) sorted by contest and year are available on the Wiki and the Resources section.
Many practice problems are also available on the forums.
Here are some old contest archives that may be useful for practicing with:
- AMC 8 is a national contest for grades 8 and younger.
- AMC 10 is a national contest for grades 10 and younger.
- AMC 12 is a national contest for grades 12 and younger.
- AIME is a contest administered to those who qualify with a high score on the AMC 10/12.
- USAMO is a proof-based contest which must be qualified for through a combination of AMC & AIME scores.
- HMMT is a nice contest on a hard AIME level.
- USC is a contest with a lot of problems based on common concepts you will see over and over.
The forums are one of the best ways to find problems to solve, get help with problems you cannot solve, and collaborate with people of all levels and abilities. The forum is divided into many subforums for problems of different difficulties.
- The MathCounts forum is for MathCounts-level problems.
- The High School Basics forum is a good place to find AMC-level and easy AIME-level problems.
- Intermediate Topics has hard AMC-level problems, as well as all levels of AIME topics. There is not a clear line between Intermediate and High School basics, but Intermediate is generally harder.
- Pre-Olympiad is a forum for problems just below olympiad level. Here you will find a lot of easy USAMO-level proofs, and problems as tough as the last few AIME problems.
- The LaTeX forum is a place to get help with LaTeX, which is what you use to type things like on the forums.
Many great reference guides are available for free on the internet.
- Coach Monk's MathCounts Playbook is a good place to start for MathCounts-level material.
- Coach Monk's High School Playbook goes a little more in depth, and is useful for all levels of high school mathematics.
- The Mandelbrot Competition maintains a nice list of topics you need to know for high school math competitions called All of Math in 3 Pages.
If you are looking for new material to learn for harder contests, many handouts on specific topics exist. CMR has a nice list of handouts that are on all levels from easy AIME to hard USAMO. They also have a list of free texts available online.