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,

High Level 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), first read the following AOPS books and take their AOPS Academy/Online classes simultaneously in this order:

 - Pre Algebra
 - Introduction to Algebra
 - Introduction to Geometry
 - Introduction to Number Theory
 - Introduction to Counting & Probability
 - Volume 1

Then head on over to AOPS's Alcumus tool and practice all of these topics constantly. When you are practicing, you will come over problems you miss. When you do, re-read 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), first 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

Then head on over to AOPS's Alcumus tool and practice all of these topics constantly. When you are practicing, you will come over problems you miss. When you do, re-read the part in the book that corresponds with that question. The more you do this, the better your skills will get.

Books

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. A very important note is that the prealgebra series will cover everything from algebra, number theory, geometry, and counting & probability, but justs skims through the important parts. Theoretically, with extensive(and we mean loads) of practice and going over the book multiple times(yes, the entire book), 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

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: (Good to review volume 1)

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 (review Introduction to Counting and Probability if needed)

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.

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 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 beenontests 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 (96-104) 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 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, 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 8/10-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 , which is what you use to type things like on the forums. It's also for Asymptote, which is what we use to make diagrams, like:

Books

• 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 page 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

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. [1]

• BOGTRO's list of theory relevant to the AIME. https://www.dropbox.com/s/icds9u5yo0xypyf/AIMElist.pdf?dl=0

• 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:

• Ross Mathematics Program
• PROMYS
• MathCamp
• MathPath
• AwesomeMath
• IdeaMath

See Also

• Resources for Math Competitions
• More Resources
• Useful Handouts
• More Useful Handouts