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 the 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. You should also try to strengthen in the areas you are not as good at. This guide is intended to help you get started.

Books

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
Art of Problem Solving Volume 2 - AMC 12, AIME, USAMO, MOP

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. Alcumus is a good resource even if you do not own any of the AoPS books.

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

Here are a few more books good for preparation for higher level contests such as AMC 12, AIME, and USAMO (though some can be found online):

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

AMC 8:

Algebra: Introduction to Algebra

Geometry: Introduction to Geometry

Combinatorics: Introduction to Counting & Probability

General: Math competitions-Middle School

-Note:(Not much Number Theory is asked on the AMC 8).

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:

Algebra: Intermediate Algebra

Geometry: Introduction to Geometry

Combinatorics: Intermediate Counting & Probability, Introduction to Counting and Probability

Number Theory: Introduction to Number Theory

General: Volume 2

Extra: Precalculus

Practice Problems

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.

Clevermath provides weekly problems. (Currently, Clevermath is not available.)

Here are some old contest archives that may be useful for practicing with:

American Mathematics Competitions:

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.

American Invitational Mathematical Examination:

AIME Problems and Solutions - AIME is a contest administered to those who qualify with a high score on the AMC 10/12.

United States of America Mathematical Olympiad:

USAMO Problems and Solutions - USAMO is a proof-based contest which must be qualified for through a combination of AMC & AIME scores.

Harvard-MIT Mathematics Tournament:

HMMT is a nice contest on a hard AIME level.

More Contests.

User-Created Contests.

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

The AIME cutoff on the AMC 12 have ranged throughout (84-96) in the recent years. The top 5% of scorers qualify. In order to get a score in the range, a simple way is to answer 13 questions right and leave the rest blank which earns a score of 96. In the past, cutoffs have been around 100.5 but it’s very rare as contests are getting slightly harder year by year. This means, since the first 10 questions should be solved by nearly anyone, they are 60 “free” points. Throughout questions 10-20, answering 3-5 shall be enough.

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 (102-120) 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 they use AMC 10 scores.

Top 10 most Difficult math Competitions(National) in The USA:

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

9. AMC 10 - Intermediate Algebra, Geometry, Advanced 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

2. IMO - Very Advanced Algebra, Very Advanced Geometry, Very Advanced Number Theory, Advanced Combinatorics, Advanced Pre-Calculus

1. PUTNAM - Advanced Algebra, Geometry, Number Theory, Advanced Combinatorics, *Extremely 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 8/10-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: $[asy] draw((0,0)--(2,0)--(0,2)--cycle); label("A",(0,0),SW); label("B",(2,0),SE); label("C",(0,2),NW); [/asy]$

Cheat Sheets

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

ExploreMath's 53-page long handout has all the formulas for the AMC 10 and AMC 12. They went through thousands of past problems to make the list, and it's a great way to review and learn new topics for the AMC 10 and AMC 12. https://docs.google.com/document/d/1aRCEadlhKestAvrWe3Dz01sZlBh8ZCiMbz87ELhBbrE/edit?usp=sharing

Sohil Rathi's [1] is a great way to learn and review the topics on the AMC 8

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

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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How should I prepare?

Contents

Introduction

Books

AMC 8:

AMC 10:

AMC 12:

Practice Problems

Forums

Cheat Sheets

Classes

Summer Camps

Past AoPS Topics

Mathcounts

AMC

AIME

USAMO

See Also