How should I prepare?

Introduction

The best way to prepare for math contests is to do lots of practice problems and learn the material necessary to solve these problems. There are also many books and online handouts and lectures you can use to improve your problem-solving skills. Depending on your current abilities, you may want to start out with different practice problems, different books, and different areas of the forums. You should also try to strengthen the areas you are not as good at. This guide is intended to help you get started.

Overview

If you don't feel like going too deep and want a straightforward answer, here it is:

Beginner

To score well on the low-level competitions (like Mathcounts and AMC 8), read the following AoPS books and take their AOPS Academy/Online classes simultaneously in this order:

Prealgebra
Introduction to Algebra
Introduction to Geometry
Introduction to Number Theory
Introduction to Counting & Probability
Volume 1

Use AoPS's Alcumus tool constantly to practice all of these topics. When you answer problems incorrectly or take a long time to answer one, reread the part in the book that corresponds with that question. The more you do this, the better your skills will get.

Advanced

To score well on the high level competitions (like AMC 10 and AIME), read the following AOPS books and take their AOPS Academy/Online classes simultaneously in this order:

Intermediate Algebra
Intermediate Counting and Probability
Volume 2
Precalculus
Calculus

Use AoPS's Alcumus tool constantly to practice all of these topics. When you answer problems incorrectly or take a long time to answer one, reread the section in the book that corresponds with that question. The more you do this, the better your skills will get.

Books

Art of Problem Solving Books

AoPS books are excellent resources to help prepare for math contests. They cover a broad range of topics, mostly focusing on Algebra, Geometry, Number theory, and Combinatorics.

Art of Problem Solving Introduction series and Volume 1 - Mathcounts, AMC 8, AMC 10
Art of Problem Solving Intermediate series and Volume 2 - AMC 12, AIME, USAMO, MOP

AoPS books 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. A very important note is that the prealgebra series will cover topics from algebra, number theory, geometry, and counting & probability, but justs skims through the important parts. Theoretically, with extensive practice and going over the entire book multiple times, you could score well on the basic level competitions like Mathcounts or AMC 8.

Prealgebra - Mathcounts, MOEMS
Introduction to Algebra - Mathcounts, AMC 8
Introduction to Number Theory - Mathcounts, AMC 8, AMC 10, AMC 12, AIME
Introduction to Geometry - Mathcounts, AMC 8, AMC 10, AMC 12, AIME, HMMT
Introduction to Counting & Probability - Mathcounts, AMC 8, AMC 10, AMC 12, AIME
Intermediate Algebra - AMC 10, AMC 12, AIME, USAMO, HMMT
Intermediate Counting & Probability - AMC 12, AIME, HMMT, USAMO
Precalculus - AMC 12, AIME, USAMO
Calculus - HMMT, Putnam

Other Books

Here are a few more books good for contests:

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
Euclidean Geometry in Math Olympiads by Evan Chen
Ritvik Rustagi's ACE The AMC 10 and 12 book is a great resource to use for AMC 10 and AMC 12. The book has over 200 pages and contains 250+ problems with detailed solutions.
Free Mastering AMC 8 book is a good way to learn and review the topics on the AMC 8
Free Mastering AMC 10/12 book is a good way to learn and review the topics on the AMC 10/12

Other books can also be found online.

AMC 8

The following books are recommended for AMC 8:

Algebra: Introduction to Algebra
Geometry: Introduction to Geometry
Combinatorics: Introduction to Counting & Probability
General: Competition Math for Middle School
Number Theory: Introduction to Number Theory (Note: There is not much Number Theory on the AMC 8.)

AMC 10

The following books are recommended for AMC 10:

Algebra: Intermediate Algebra
Geometry: Introduction to Geometry
Combinatorics: Introduction to Counting & Probability
Number Theory: Introduction to Number Theory
General: Volume 1

AMC 12

The following books are recommended for AMC 12:

Algebra: Intermediate Algebra
Geometry: Introduction to Geometry
Combinatorics: Intermediate Counting & Probability (review Introduction to Counting and Probability if needed)
Number Theory: Introduction to Number Theory
General: Volume 2
Extra: Precalculus

Practice Problems

Old practice problems and solutions sorted by contest and year are available on the AoPS Wiki and the forums. Here are some old contest archives that may be useful for practicing with:

AMC 8 Problems and Solutions - AMC 8 is a national contest for grades 8 and younger.
AMC 10 Problems and Solutions - AMC 10 is a national contest for grades 10 and younger.
AMC 12 Problems and Solutions - AMC 12 is a national contest for grades 12 and younger.
AIME Problems and Solutions - The American Invitational Mathematics Examination is a contest administered to those who qualify with a high score on the AMC 10 or AMC 12.
USAMO Problems and Solutions - The United States of America Mathematical Olympiad is a proof-based contest which is qualified for through a combination of AMC and AIME scores.
HMMT is a nice contest on a hard AIME level run by Harvard and MIT.
List of mathematics competitions.
User-Created Contests.

Strategies to qualify for the AIME

There are certain strategies in preparing for the AMC 10/12- especially qualification for the AIME.

The AIME cutoff has ranged 84-95 on AMC 12.since 2020 when the qualification was loosened to "5% of scorers". In order to get a score in the range, a simple way is to answer 13 questions right (check your work carefully!) and leave the rest blank, which earns a score of 96. In the past, since the 2020 cutoffs have been getting slightly harder each year, and new generations of competitors don't always match the new level. This means, since the first 10 questions are solvable in half the test time by most people who prepare, they are 60 easy points. Throughout questions #10-#20, answering 3-5 shall be enough to qualify.

Beware, though, that the AIME question #1 is harder than AMC question #10, so this strategy presumes that you *could* solve more than 15 AMC problems, but you are choosing to reduce your time/difficulty pressure and increase your confidence, to guarantee a qualifying score but not get your highest possible score.

Qualification for the USAMO, however, is much harder. Only 260-270 people qualify every year. USAMO qualifiers need a good combination of AMC & AIME scores. The average score on the AMC 12 for a USAMO qualifier is around 114-132. There are simple ways to do this but it takes a lot of work. Answering the first 15 right, and then getting 5 out of the 10 left would usually qualify.

The AIME cutoff on the AMC 10 have ranged throughout (94.5-104) in recent years. The top 2.5% of scorers qualify. The AMC 10 does test less topics than the AMC 12 but many questions go into much more depth. Cutoffs on the AMC 10 are higher since the testing only tests topics up to Geometry. AIME ranges from Algebra to precalculus, which means only very elite scorers make it. Though the qualifying scores are high, there is indeed a good strategy. Since you get 1.5 points for each question blank, it’s good just to do what you know. Answering 15 questions right and leaving the rest blank would earn a score of 105 while answering 20 right and leaving the rest blank would earn a score of 127.5. Since contests are getting harder as said earlier, 15-18 right should be enough.

Qualifications for the USAJMO is similar to that for the USAMO except it uses AMC 10 scores.

Top 10 most Difficult National Math Competitions in The USA

10. MATHCOUNTS - Pre-Algebra, Geometry, Number Theory, Combinatorics, Logic

9. AMC 10 - Intermediate Algebra, Geometry, Number Theory, Combinatorics

8. AMC 12 - Intermediate Algebra, Geometry, Number Theory, Combinatorics, Pre-Calculus

7. ARML - Advanced Algebra, Geometry, Number Theory, Combinatorics

6. AIME - Advanced Algebra, Advanced Geometry, Number Theory, Combinatorics, Pre-Calculus

5. USAMTS - Advanced Algebra, Advanced Geometry, Number Theory, Combinatorics, Pre-Calculus

4. USAJMO - Advanced Algebra, Advanced Geometry, Advanced Number Theory, Combinatorics

3. USAMO - Advanced Algebra, Very Advanced Geometry, Advanced Number Theory, Combinatorics, Advanced Pre-Calculus

1 (tie). IMO - Very Advanced Algebra, Very Advanced Geometry, Very Advanced Number Theory, Advanced Combinatorics, Advanced Pre-Calculus

1 (tie). PUTNAM - Advanced Algebra, Geometry, Number Theory, Advanced Combinatorics, Advanced Calculus

Forums

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 Middle School forum is for MathCounts and AMC 8-level problems.
The High School forum is a good place to find AMC10/12-level and AIME-level problems.
The Olympiad forum is a forum for problems at the olympiad level.
The LaTeX and Asymptote forum is a place to get help with $\text{\LaTeX}$ , which is what you use to type things like $2^3$ on the forums. It's also for Asymptote, which is what we use to make diagrams like this one:

$[asy] draw((0,0)--(2,0)--(0,2)--cycle); label("A",(0,0),SW); label("B",(2,0),SE); label("C",(0,2),NW); [/asy]$

Cheat Sheets

Many great reference guides are available for free on the internet.

Ritvik Rustagi's 53-page long handout has all the formulas for the AMC 10 and AMC 12. The handout was made alongside a 4 hour long review seminar for AMC 10 and AMC 12. It's a great way to review and learn new topics for the AMC 10 and AMC 12.
BOGTRO's list of theory relevant to the AIME.
Sohil Rathi's The Book of Mathematical Formulas & Strategies is a great way to review all formulas on math contests like AMCs, AIME, MATHCOUNTS, etc
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.
The Noah Sheets
The Popular Elementary Math Competitions Contests to build Math Interests

Classes

Free AMC 8 Fundamentals Class: https://www.omegalearn.org/amc8-fundamentals
Free AMC 8 Advanced/Mathcounts Class: https://www.omegalearn.org/amc8-fundamentals
Free AMC 10/12 Class: https://www.omegalearn.org/amc10-12
Free AMC 8/10 Class: https://www.youtube.com/channel/UC-Nt9Uo03VSo2QTNIzsE_cA (Some special seminars occasionally with Olympiad Winners)
If you are serious about improving your problem-solving skills, AoPS offers several online classes, available here.
WOOT is an online class offered by AoPS for olympiad training. It has one of the best peer groups in the country, and is a great way to prepare for the USAMO.

Summer Camps

Summer programs are also a great way to improve problem-solving skills. Some of these include:

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