Comprehensive Guide to the American
Mathematics Competitions (AMC 10) and
the USAJMO
For a tenth-grade student in the United States, the pathway to the national mathematics
olympiad involves a series of increasingly rigorous examinations hosted by the Mathematical
Association of America (MAA). The journey begins with the AMC 10, advances to the AIME, and
culminates in the USA Junior Mathematical Olympiad (USAJMO).
1. The AMC 10 (American Mathematics Competition
10)
The AMC 10 is the first tier of competition for students in grade 10 and below. It is designed to
promote the development and enhancement of problem-solving skills.
●​ Format: 25-question, 75-minute multiple-choice examination.
●​ Scoring: 6 points for a correct answer, 1.5 points for an unanswered question, and 0
points for an incorrect answer. The maximum possible score is 150.
●​ Content: Covers the high school curriculum up to 10th grade. This includes elementary
algebra, basic geometry (including the Pythagorean Theorem, area, and volume),
elementary number theory, and elementary probability. It excludes trigonometry, advanced
algebra, and calculus.
●​ Qualification: Top scoring students (historically the top 2.5% of test-takers) receive an
invitation to the AIME.
2. The AIME (American Invitational Mathematics
Examination)
The AIME is the second tier, meant to provide a further challenge to high-scoring AMC students.
●​ Format: 15-question, 3-hour examination.
●​ Answers: Each answer is an integer from 000 to 999. No calculators are allowed.
●​ Scoring: 1 point for each correct answer. No penalty for incorrect answers. Maximum
score is 15.
●​ Qualification: The USAJMO qualifiers are determined by an index score: (AMC 10
Score) + 10 * (AIME Score). Top index scorers (around 200-250 students nationwide) are
invited to the USAJMO.
3. The USAJMO (USA Junior Mathematical Olympiad)
Introduced in 2010, the USAJMO serves as the pinnacle math olympiad for students in the 10th
grade and below. It bridges the gap between the computational AIME and the highly abstract
USAMO.
●​ Format: 6-question, proof-based essay examination given over two days (4.5 hours and 3
questions per day).
●​ Scoring: Each problem is worth 7 points, graded on the rigor, completeness, and
elegance of the mathematical proof. Maximum score is 42.
●​ Recognition: Top scorers are invited to the Mathematical Olympiad Summer Program
(MOP), a training camp for future International Mathematical Olympiad (IMO) participants.
4. Core Syllabus and Topics
To succeed at the AMC 10 and USAJMO level, 10th graders must master four fundamental
areas of competition mathematics.
Category
Key Concepts Covered in Grade 10
Olympiad Track
Algebra
Polynomials, Vieta's formulas, factorization
techniques, sequence and series (arithmetic,
geometric, telescoping), inequalities
(AM-GM, Cauchy-Schwarz, rearrangement),
functional equations.
Geometry
Triangle geometry (concurrency theorems
like Ceva's and Menelaus', properties of
incenters, circumcenters, orthocenters),
cyclic quadrilaterals, Power of a Point, similar
figures, 3D geometry (spheres, pyramids),
transformations.
Number Theory
Modular arithmetic, Fermat's Little Theorem,
Euler's Totient Theorem, Diophantine
equations (linear and non-linear), prime
factorization, base number systems,
divisibility rules.
Combinatorics
Permutations and combinations, Stars and
Bars (Ball and Urn), Principle of
Inclusion-Exclusion (PIE), Pigeonhole
Principle, basic graph theory, probability and
expected value, recurrence relations.
5. Preparation Strategies and Resources
Preparation requires moving beyond standard high school curricula and focusing on creative
problem-solving and rigorous proof writing.
●​ Past Papers: The most vital resource. Students should practice completing past AMC 10
and AIME papers under strict timed conditions to build stamina and identify knowledge
gaps.
●​ Art of Problem Solving (AoPS): The definitive curriculum for US Olympiad prep. The
AoPS Volume 1: The Basics and Volume 2: and Beyond books are essential for AMC 10
and AIME. The Introduction and Intermediate series offer deep dives into specific
subjects.
●​ Proof Writing: For the USAJMO, transitioning from computational math to proof writing is
critical. Students should study past USAMO/USAJMO solutions to understand how to
rigorously structure a proof without logical leaps.
6. Conclusion
Competing in the Grade 10 American Mathematics Olympiad track is a rigorous but deeply
rewarding endeavor. It develops unparalleled analytical thinking, logic, and perseverance.
Whether a student reaches the AMC 10 or qualifies for the prestigious USAJMO, the
problem-solving skills acquired will serve as a massive advantage in future STEM pursuits.