A Complete Guide To The American Mathematics Competitions (And Additional Resources To Help You Win)

The American Mathematics Competitions (AMC), organized by the Mathematical Association of America (MAA), are prestigious contests for middle and high school students. These competitions aim to develop problem-solving skills, identify mathematical talent, and foster a love of mathematics.

These competitions are extensive and require a lot of preparation. In this blog, we give you a rundown of everything you need to know about the said competitions. We will also provide a series of resources you can dive into during your preparation.

What are the American Mathematics Competitions?

The American Mathematics Competitions (AMC) program, spearheaded by MAA, or Mathematical Association of America, is an initiative in the form of a series of math competitions aimed at enhancing the mathematical skills of the next generation of problem-solvers. The MAA AMC program assists teachers in recognizing and promoting mathematical talent by offering classroom materials and arranging amicable contests. This program promotes the critical thinking abilities needed for future professions in a forward-thinking society.

The American Mathematics Competitions consist of a series of exams and curriculum materials designed to enhance problem-solving abilities and mathematical knowledge among middle and high school students. The competitions are structured as follows:

• American Mathematics Competition 8 (AMC 8)
• American Mathematics Competition 10/12 (AMC 10/12)
• American Invitational Mathematics Examination (AIME)
• United States of America Mathematical Olympiad (USAMO)
• United States of America Junior Mathematical Olympiad (USAJMO)
• The Putnam Competition

Participating in these amazing competitions, the MAA AMC program inspires a love for math and develops essential skills in students.

How do the American Mathematics Competitions work?

The American Mathematics Competitions (AMC) offer exams and resources to improve problem-solving skills and math knowledge for middle and high school students. Over 300,000 students worldwide participate in the competitions each year.

The MAA AMC oversees 14 competitions annually, including:

• American Mathematics Competition 8 (AMC 8): A middle school level competition.
• American Mathematics Competition 10/12 (AMC 10/12): A high school level competition.
• American Invitational Mathematics Examination (AIME): An invitational competition for high-scoring AMC 10/12 participants.
• United States of America Mathematical Olympiad (USAMO) and United States of America Junior Mathematical Olympiad (USAJMO): Top-tier invitational competitions for high-scoring AIME participants.
• The Putnam Competition: An annual mathematics competition for undergraduate students in the United States and Canada.

The AMC exams focus on mathematical problem-solving across various topics such as arithmetic, algebra, counting, geometry, number theory, and probability. These tests integrate these subjects more extensively than typical classroom settings.

For instance, while classrooms might touch on divisibility rules and small aspects of number theory, the AMC exams use number theory in more profound and comprehensive ways (still elementary and without advanced analysis). The difficulty of the tests varies, but they are all designed so that students do not need any background in calculus, analysis, or higher-level mathematics to participate.

The AMC exams are the first step in selecting the U.S. team for the International Mathematical Olympiad (IMO). Students who score high on the AMC 10 or 12 are invited to take the American Invitational Mathematics Examination (AIME). Those with high scores on both the AMCs and the AIME, or a high combined score, are invited to participate in the United States of America Mathematics Olympiad (USAMO).

High scorers from the USAMO attend the Math Olympiad Summer Program (MOSP), which is divided into three groups based on scores. The top group, called “black,” includes twelve students, six of whom will be chosen to represent the United States at the IMO.

American Mathematics Competition 8 (AMC 8)

The AMC 8 is a 25-question, 40-minute multiple-choice exam for middle school students, aimed at improving problem-solving skills. It encourages students to develop positive attitudes towards analytical thinking and math, which can be helpful in future careers. The exam allows students to apply their classroom knowledge to unique problem-solving situations in a low-stress and friendly environment.

The AMC 8 covers typical middle school math topics. These include counting and probability, estimation, proportional reasoning, elementary geometry (like the Pythagorean Theorem), spatial visualization, everyday math applications, and reading and interpreting graphs and tables. Some of the later questions may also cover linear or quadratic functions and equations, coordinate geometry, and other basic algebra topics.

American Mathematics Competition 10/12 (AMC 10/12)

The AMC 10 and AMC 12 are 25-question, 75-minute multiple-choice exams designed to improve problem-solving skills in high school students.

• AMC 10: For students in 10th grade or below, covering math topics up to 10th grade. Students must be 10th grade or below and under 17.5 years old on the exam day.
• AMC 12: For students in 12th grade or below, covering the entire high school math curriculum, including trigonometry, advanced algebra, and advanced geometry, but not calculus. Students must be 12th grade or below and under 19.5 years old on the exam day.

These exams help students develop a positive attitude towards analytical thinking and mathematics, which can be beneficial for their future careers. The AMC 10/12 is the first step in a series of competitions leading up to the International Mathematical Olympiad.

American Invitational Mathematics Examination (AIME)

The American Invitational Mathematics Examination (AIME) is a competition for students who scored well on the AMC 10 or AMC 12. It consists of 15 questions, and participants have 3 hours to complete it. Each answer is an integer between 0 and 999. The questions on the AIME are significantly more difficult than those on the AMC 10 and AMC 12. High-scoring students on the AIME are invited to participate in the USAMO (United States of America Mathematical Olympiad) or USAJMO (United States of America Junior Mathematical Olympiad).

USA Mathematical Olympiad and Junior Mathematical Olympiad (USAMO and USAJMO)

The United States of America Mathematical Olympiad (USAMO) and the United States of America Junior Mathematical Olympiad (USAJMO) are rigorous competitions consisting of six questions over two days, totaling 9 hours of essay/proof-based examinations. The top-scoring participants from the AMC 12, based on combined scores from the AMC 12 and AIME, are invited to take the USAMO. Similarly, the top-scoring participants from the AMC 10, based on combined scores from the AMC 10 and AIME, are invited to take the USAJMO.

The MAA AMC promotes inclusivity within their program and competitions. To foster a mathematical community that empowers students of all genders, they invite the top-scoring female AMC participants to the USAMO and USAJMO each year. These students are placed on an ambitious path with the potential to qualify for the Mathematical Olympiad Program (MOP), the European Girls’ Mathematical Olympiad (EGMO), and ultimately the International Mathematical Olympiad (IMO).

For the USAMO, the AMC awards Gold prizes to at least approximately 6% of competitors, Silver to at least approximately 12%, and Bronze to at least approximately 18%. For the USAJMO, they will continue to name winners but will increase recognition to at least approximately 20% of contestants. For both the USAMO and USAJMO, each additional contestant with 14 points or more will receive an Honorable Mention. Each year, the Student Selection Committee will determine the exact percentages and number of awards.

Eligibility and Important Dates

Take note of the eligibility for each competition and the important dates.

Exams	Eligibility	Important Dates
AMC 8	grade 8 student or below should be under 14.5 years of age on the day of the competition	Early Registration Deadline: Jun 21, 2024 – Oct 21, 2024 Regular Registration Deadline: Oct 22, 2024 – Dec 23, 2024 Late Registration Deadline: Dec 24, 2024 – Jan 15, 2025 Competition Dates: Jan 22 – 28, 2025
AMC 10/12	AMC 10: Grade 10 or below but should be under 17.5 years old on the contest day AMC 12 Grade 12 or below but should be under 19.5 years old on the contest day	AMC 10/12 A Early Registration Deadline: Jun 21, 2024 – Sept 23, 2024 Regular Registration Deadline: Sept 24, 2024 – Oct 22, 2024 Late Registration Deadline: Oct 23, 2024 – Oct 30, 2024 Competition Date: November 6, 2024 AMC 10/12 B Early Registration Deadline: Jun 21, 2024 – Sept 30, 2024 Regular Registration Deadline: Oct 1, 2024 – Nov 1, 2024 Late Registration Deadline: Nov 2, 2024 – Nov 8, 2024 Competition Date: November 12, 2024
AIME	Top 2.5% of the AMC 10 takers Top 5.0% of the AMC 12 takers	AIME I (main AIME competition date): Feb 6, 2025 AIME II (alternate AIME competition date): Feb 12, 2025
USAMO and USAJMO	USAMO: AMC 12 Score plus 10 * AIME Score USAJMO: AMC 10 Score plus 10 * AIME Score	USAMO and USAJMO Day 1: March 19, 2025 USAMO and USAJMO Day 2: March 20, 2025

How do you register for the American Math Competitions?

To register for the AMC exams, follow these steps:

1. Create an Account: Visit AMC Registration and create an account if you don’t have one.
2. Log In: Use your account credentials to log in.
3. Register Your School/Organization: If not already registered, a teacher or administrator should register the school or organization.
4. Enroll Students: Once the school is registered, enroll students for the AMC 8, AMC 10, or AMC 12 exams.
5. Payment: Complete the payment process for the exams.
6. Administer the Exam: Follow the instructions provided to administer the exams, either digitally or via printed materials.

The digital competition will be accessible through the competition manager portal, where students can receive a unique link. Students will submit their answers directly on their digital devices, such as PCs, laptops, or tablets.

For the print and scan administration, the competition materials will not be mailed or emailed. Instead, the competitions and answer sheets will be available for printing on the competition manager (CM) portal. After the competition, CMs have the option to scan the answer sheets and upload them to the portal.

For detailed instructions, visit the AMC Registration page.

Frequently Asked Questions

1. Who can participate in AMC 8?

Students who love problem-solving, are in grade 8 or below, and are under 14.5 years old on the day of the competition are eligible to participate in the AMC 8.

2. Who can participate in AMC 10/12?

Students who enjoy problem-solving and are in grade 10 or below, and under 17.5 years old on the contest day, can take the AMC 10. Those in grade 12 or below and under 19.5 years old on the contest day can take the AMC 12.

3. Who can participate in the AIME?

The qualification criteria for the AIME depend on the AMC 10/12 results. For the AMC 10, the top 2.5% of scorers are invited. For the AMC 12, the top 5% are invited. The exact cutoff scores vary based on the difficulty of each competition.

4. Who can participate in USAMO and USAJMO?

Qualification for the USAMO and USAJMO depends on scores from the AMC 10/12 and AIME exams.

• USAMO: Selection is based on the USAMO index, calculated as AMC 12 Score + 10 * AIME Score.
• USAJMO: Selection is based on the USAJMO index, calculated as AMC 10 Score + 10 * AIME Score.

Each year, the committees adjust the difficulty levels and may set different cutoffs for each exam version based on score distributions.

5. What are the costs involved?

The costs for participating in the AMC exams vary. Schools or organizations need to register for the exams and pay associated fees, which may include the exam booklets, answer sheets, and other administrative costs. You may take note of the following registration costs as posted on their official website.

Competition	Registration Fees
AMC 8 Early Registration	$53.00
AMC 8 Regular Registration	$73.00
AMC 8 Late Registration	$113.00
AMC 10/12 A Early Registration	$56.00
AMC 10/12 A Regular Registration	$76.00
AMC 10/12 A Late Registration	$116.00
AMC 10/12 B Early Registration	$70.00
AMC 10/12 B Regular Registration	$88.00
AMC 10/12 B Late Registration	$128.00

6. What math subjects should I focus on if I want to excel in these competitions?

To excel in the AMC competitions, focus on the following math subjects:

AMC 8

• Arithmetic: Basic operations (addition, subtraction, multiplication, division), fractions, decimals, percentages, ratios, and averages.
• Basic Algebra: Simple equations, inequalities, and understanding variables.
• Counting and Probability: Basic counting principles, permutations, combinations, and simple probability.
• Elementary Geometry: Fundamental properties of shapes, area, perimeter, volume, and the Pythagorean Theorem.
• Number Theory: Divisibility rules, prime numbers, and simple modular arithmetic.
• Estimation: Approximations and rounding.
• Proportional Reasoning: Ratios, proportions, and percent problems.
• Graphs and Tables Interpretation: Reading and interpreting data from graphs and tables.

AMC 10/12

• Intermediate Algebra: Quadratic equations, polynomial functions, rational expressions, and inequalities.
• Advanced Geometry: More complex properties of shapes, coordinate geometry, and transformations.
• Trigonometry: Basic trigonometric functions, identities, and solving triangles.
• Counting and Probability: Advanced counting principles, conditional probability, and expected value.
• Number Theory: More advanced topics like Diophantine equations, Euler’s Theorem, and the Chinese Remainder Theorem.
• Logarithms and Exponential Functions: Properties and applications of logarithms and exponential functions

AIME

• Advanced Algebra: Higher degree polynomials, sequences and series, and functional equations.
• Advanced Geometry: Conic sections, advanced transformations, and geometric inequalities.
• Complex Numbers: Operations with complex numbers, polar form, and De Moivre’s Theorem.
• Advanced Counting and Probability: Advanced combinatorics, generating functions, and more intricate probability problems.
• Advanced Number Theory: Techniques like modular arithmetic, Fermat’s Little Theorem, and number theoretic functions.

USAMO and USAJMO

• Proof-based Problem Solving: Writing formal mathematical proofs, including direct proofs, contradiction, and induction.
• Advanced Algebra: Further exploration of algebraic structures, inequalities, and advanced problem-solving strategies.
• Geometry: Advanced theorems, constructions, and Olympiad-level geometry problems.
• Combinatorics: Pigeonhole principle, graph theory, and advanced counting techniques.
• Number Theory: More profound exploration of prime numbers, congruences, and divisibility.
• Functional Equations: Solving equations where the variables are functions rather than simple numbers.

For all levels, practicing past competition problems and improving problem-solving techniques is crucial.

Study Resources

One of the most effective, if not the most effective, ways to prepare is to read and watch resources available to you. This includes going through previous competition problems and solutions; reading books on different mathematical fields and more. We have listed a list of resources you can check out that will help you prepare for the said competitions.

AMC 8

Books

• The Art of Problem Solving Volume 1: The Basics by Sandor Lehoczky and Richard Rusczyk: An excellent introductory book covering a broad range of math topics, ideal for AMC 8 preparation.
• Competition Math for Middle School by Jason Batteron: A comprehensive guide focused on problem-solving strategies for middle school students.

Online Courses

• AoPS Online Courses: Offers courses specifically designed for AMC 8 preparation, focusing on problem-solving skills and techniques.

Websites

• Art of Problem Solving (AoPS): Provides resources, forums, and practice problems for AMC 8 and other math competitions.

YouTube Channels

• MathCounts: Offers a variety of math competition problems and solutions, which are beneficial for AMC 8 preparation.

AMC 10/12

Books

• The Art of Problem Solving Volume 2: And Beyond by Richard Rusczyk and Sandor Lehoczky: A follow-up to Volume 1, covering more advanced topics suitable for AMC 10/12.
• Introduction to Algebra by Richard Rusczyk: Focuses on algebraic concepts crucial for AMC 10/12 success.

Online Courses

• AoPS Online Courses: Specialized courses for AMC 10 and AMC 12, focusing on advanced problem-solving techniques.

Websites

• AoPS AMC 10/12 Resources: Includes practice problems, past exams, and forums for discussion and problem-solving.

YouTube Channels

• Richard Rusczyk’s AoPS Channel: Offers in-depth video tutorials on various math topics relevant to AMC 10/12.

AIME

Books

• AIME Problems and Solutions by Art of Problem Solving: Compilation of past AIME problems with detailed solutions.
• Intermediate Algebra by Richard Rusczyk: Covers algebra topics extensively tested on the AIME.

Online Courses

• AoPS AIME Prep Courses: Focuses on the advanced problem-solving skills needed to excel in the AIME.

Websites

• AoPS AIME Resources: Provides a collection of past AIME problems, solutions, and discussion forums.

YouTube Channels

• Evan Chen’s Math Channel: Offers problem-solving sessions and tips specifically geared towards AIME and other math competitions.

USAMO and USAJMO

Books

• Problems from the Book by Titu Andreescu: Contains advanced problems and solutions that mirror the difficulty of USAMO and USAJMO.
• Mathematical Olympiad Treasures by Titu Andreescu and Bogdan Enescu: A collection of problems and solutions from various math Olympiads, ideal for USAMO/USAJMO prep.

Online Courses

• AoPS Olympiad Courses: Advanced courses designed for students preparing for USAMO and USAJMO, focusing on proof-writing and higher-level problem-solving.

Websites

• AoPS USAMO/USAJMO Resources: Includes past problems, solutions, and forums for Olympiad-level math discussions.

YouTube Channels:

• 3Blue1Brown: While not specifically focused on Olympiads, the channel’s in-depth math explanations can be very helpful for advanced topics.
• MindYourDecisions: Offers a variety of challenging math problems and solutions that can aid in Olympiad preparation.

Should you participate in the American Mathematics Competitions? Is it worth your time and resources?

The easy answer to these questions is a resounding YES. 

Participating in the American Mathematics Competitions (AMC) brings significant prestige and recognition to the academic community. These competitions are respected nationally and internationally, showcasing your mathematical talents and problem-solving skills. High scores and participation look impressive on college applications, demonstrating your commitment and aptitude in mathematics.

Monetary awards and scholarships are another key benefit of the AMC. High-performing students may become eligible for various scholarships and prizes, both from the competition organizers and external scholarship programs. These financial incentives can provide substantial support for your higher education and ease the burden of college expenses.

Moreover, participating in the AMC promotes personal growth and self-improvement. The challenging problems encourage critical thinking, creativity, and perseverance. These skills are valuable not only in academics but also in everyday life and future careers. The experience of preparing for and competing in the AMC builds a strong foundation in mathematics, boosts confidence, and fosters a love for learning that can have lasting positive effects.