Competition Pathway

AMC → AIME → USAJMO / USAMO

The competition mathematics pathway requires structural depth that goes far beyond school curriculum. This map shows the skills needed at each competition level, organized by mathematical domain.

Competition Progression

AMC 8
Grades: 6-8

25 multiple choice, 40 minutes. Foundation competition.

AMC 10
Grades: 10 and below

25 multiple choice, 75 minutes. Intermediate competition.

AMC 12
Grades: 12 and below

25 multiple choice, 75 minutes. Advanced competition.

AIME
Grades: Qualifying

15 problems, 3 hours. Integer answers 000-999.

USAJMO
Grades: AMC 10 qualifiers

6 proof problems over 2 days. Olympiad level.

USAMO
Grades: AMC 12 qualifiers

6 proof problems over 2 days. National olympiad.

Note: AMC 10 and AMC 12 both feed into AIME. Qualifying AIME scores lead to USAJMO (AMC 10 track) or USAMO (AMC 12 track). Both culminate in the Mathematical Olympiad Program.

Five Competition Domains

Click any skill to see which competition level it appears at.

Counting & Combinatorics

10 skills
Direct Counting
AMC 8-10
Case Analysis
AMC 8-10
Complementary Counting
AMC 10
Multiplication & Addition Principles
AMC 8
Permutations & Combinations
AMC 10-12
Stars and Bars
AMC 12-AIME
Inclusion-Exclusion
AMC 12-AIME
Generating Functions Intuition
AIME+
Pigeonhole Principle
AMC 12-AIME
Double Counting
AIME-USAMO

Number Theory

10 skills
Divisibility Rules & Tests
AMC 8
Prime Factorization
AMC 8-10
GCD & LCM
AMC 8-10
Modular Arithmetic
AMC 10-12
Congruences
AMC 12-AIME
Diophantine Equations
AIME
Number Bases
AMC 10
Euler's Totient Function
AIME+
Fermat's Little Theorem
AIME
Lifting the Exponent
USAMO

Competition Algebra

10 skills
Clever Substitution
AMC 10-12
Symmetric Expressions
AMC 12-AIME
Vieta's Formulas
AMC 10-12
AM-GM Inequality
AMC 12-AIME
Cauchy-Schwarz Inequality
AIME-USAMO
Functional Equations
AIME-USAMO
Telescoping
AMC 12-AIME
Sequences & Series Tricks
AMC 10-12
Floor/Ceiling Functions
AMC 12-AIME
Complex Numbers in Algebra
AIME+

Competition Geometry

10 skills
Angle Chasing
AMC 8-10
Power of a Point
AMC 12-AIME
Radical Axes
AIME-USAMO
Stewart's Theorem
AMC 12-AIME
Mass Point Geometry
AMC 10-12
Trigonometric Cevians
AIME
Area Methods
AMC 10-12
Coordinate Bashing
AMC 12-AIME
Projective Intuition
USAMO
Inversion
USAMO

Logic & Problem-Solving Strategy

10 skills
Invariants & Monovariants
AIME-USAMO
Extremal Principle
AIME-USAMO
Greedy Algorithms
AMC 10-12
Graph Theory Basics
AIME
Coloring & Parity Arguments
AMC 12-AIME
Recursion & Induction
AIME-USAMO
Proof by Contradiction
USAMO
Construction
USAMO
Working Backwards
AMC 10-12
Generalization & Specialization
AIME-USAMO

Competition Prep at LIGHT HOPE

Competition math is not just harder problems — it requires different structural thinking. Students need to see elegant solutions, not just compute. MathMap's competition track identifies exactly which structural skills are prerequisites for each competition level, so prep work is targeted rather than random.

View AMC Student: Emily Wang See the Framework