About USAMO
The invitational national olympiad for top AMC 12 and AIME scorers, used to select the US team for the International Mathematical Olympiad (IMO).
USAMO Exam Format & Focus
Six proof-based problems over two days, 3 problems and 4.5 hours each day. Full written proofs are required, and answers without proof earn no credit. Calculators are not allowed.
How Skillzy Young Scholars Prepares Students for USAMO
Skillzy Young Scholars teaches USAMO in live online classes with vetted mentors who have completed 100 hours of training. The curriculum is organised into 5 units, with worksheets, quizzes, timed practice and mock exams. Progress is tracked in the LMS and shared with parents in a monthly report.
What USAMO Covers
- Writing complete mathematical proofs with clear definitions, lemmas and justified steps
- Olympiad algebra, number theory and combinatorics
- Advanced Euclidean geometry, constructions, invariants and proof strategy
USAMO Curriculum
Classes follow these units. Every student starts with a level check, and the order and pace adapt to your child. Units that go beyond the exam's core topics are stretch material.
- Target Audience
- Top-scoring AMC 12 participants (combined with their AIME index). This is a highly exclusive, invitational-only exam for the top ~250 math students in the country.
- Total Questions
- 6 proof-based problems.
- Time Limit
- Administered over two days, with 3 questions and 4.5 hours per day (9 hours total).
- Format
- Fully written, rigorous mathematical proofs are required. Answers without proof receive zero credit.
- Skill Focus
- Rigorous mathematical proof-writing across algebra, geometry, number theory, and combinatorics. Calculators are strictly prohibited.
What the program focuses on
- Rigorous mathematical proof-writing across four core Olympiad domains
- Structuring logical, gradable proofs for partial and full credit
- Advanced projective/inversive geometry, complex algebra, and number theory
- Extremal combinatorics and advanced graph theory
- Managing extreme mental endurance across a 2-day, 9-hour exam
- Interactive, hands-on learning
- Continuous progress monitoring on proof construction
Units
1 Olympiad Algebra (Extended Depth)
Students will learn to:
- Solve extremely complex functional equations using substitution techniques, injectivity/surjectivity arguments, and Cauchy's equation.
- Write rigorous proofs involving advanced inequalities: AM-GM, Cauchy-Schwarz, Power Mean, rearrangement, and Schur's inequality.
- Master polynomial theory: proving irreducibility, analyzing symmetric functions, and utilizing roots and coefficients.
- Analyze sequences defined recursively, proving closed forms and establishing rigorous analytical bounds.
USAMO Alignment - Core Algebra. USAMO algebra requires moving from computation to abstraction, proving that complex functional and algebraic relationships hold true across infinite sets using airtight logic.
Learning activities:
- Functional equation substitution mapping and proof drafting
- Inequality bounding challenges and chain-of-logic exercises
- Polynomial irreducibility proof reviews
What this means for your child. Your child will learn the language of advanced algebra, mastering how to prove generalized statements using functional substitution and advanced inequalities.
2 Olympiad Geometry (Extended Depth)
Students will learn to:
- Master synthetic proof techniques: complex angle chasing, cyclic quadrilaterals, and similar triangles.
- Utilize advanced circle theorems: radical axis, radical center, and power of a point.
- Analyze triangle centers and their properties (incenter, circumcenter, orthocenter, and the Euler line).
- Execute projective and inversive geometry (introductory theorems specifically required for USAMO-level geometric transformations).
- Apply trigonometric identities (Law of Sines/Cosines) and Barycentric/vector methods in complex proofs.
USAMO Alignment - Core Geometry. USAMO geometry is notoriously difficult, requiring students to construct brilliant auxiliary lines or utilize advanced transformations like inversion and projective geometry to map impossible shapes into solvable ones.
Learning activities:
- Projective and inversive geometry transformation exercises
- Synthetic geometry diagram construction and angle chasing
- Barycentric coordinate derivation tasks
What this means for your child. Your child will master the most advanced geometric transformations taught at the high school level, utilizing inversive and projective geometry to solve Olympiad-level problems.
3 Olympiad Number Theory (Extended Depth)
Students will learn to:
- Construct modular arithmetic proofs utilizing orders and primitive roots.
- Solve advanced Diophantine equations using methods of infinite descent and Vieta jumping.
- Master number-theoretic functions: Euler's totient, divisor functions, and multiplicative functions.
- Apply advanced techniques such as p-adic valuations and the Lifting-the-Exponent (LTE) technique.
USAMO Alignment - Core Number Theory. Number theory on the USAMO requires students to prove that no integer solutions exist for specific equations, or to prove deep divisibility properties utilizing advanced concepts like Vieta jumping and LTE.
Learning activities:
- Infinite descent and Vieta jumping logic puzzles
- Lifting-the-Exponent (LTE) and p-adic valuation application drills
- Modular arithmetic proof drafting
What this means for your child. Your child will learn to prove deep structural properties of integers, mastering advanced theorems required for international-level number theory.
4 Olympiad Combinatorics (Extended Depth)
Students will learn to:
- Construct extremal combinatorics and highly advanced pigeonhole arguments.
- Master graph theory: coloring, paths, trees, and extremal graph problems.
- Analyze combinatorial games utilizing invariants and monovariants to prove absolute winning strategies.
- Execute advanced counting methods: establishing bijections, double counting, and utilizing generating functions.
USAMO Alignment - Core Combinatorics. Combinatorics at the USAMO level often presents as an abstract game or a finite set of objects, requiring students to prove that a certain state is always reachable (or strictly impossible) using invariants.
Learning activities:
- Invariant and monovariant game strategy mapping
- Graph theory coloring and extremal bounding proofs
- Bijections and generating function systematic exercises
What this means for your child. Your child will develop the logical creativity required to prove abstract combinatorial constraints, utilizing advanced invariants and graph theory.
5 Proof-Writing Craft & Exam Strategy
Students will learn to:
- Structure a rigorous, gradable Olympiad proof from start to finish without logical leaps.
- Understand the difference between constructive vs. existence proofs.
- Master proof by contradiction and advanced mathematical induction.
- Communicate partial-credit-worthy progress clearly under extreme time pressure.
- Manage mental endurance across a 2-day, 9-hour exam.
USAMO Alignment - Test Day Readiness. A correct idea is worth 0 points if not communicated flawlessly. Olympiad grading is exceptionally strict; students must learn to write airtight arguments to secure all 7 points per problem.
Always check the official exam organizer for the current format, dates and eligibility. We do not guarantee scores or rankings.
Sample USAMO Prep Plan
- Begin with a USAMO diagnostic and check the current official format, scoring rules and eligible level.
- Weeks 3–5: learn priority concepts and solve untimed problems with written reasoning.
- Weeks 6–8: complete mixed sets under realistic time limits and review every missed question.
- Final phase: practise pacing, revisit an error log and complete a full-length mock when appropriate.
Common USAMO Mistakes We Help Students Avoid
- Preparing for USAMO using a generic or outdated format instead of checking the current official guidance.
- Using a memorised trick before checking whether it fits the problem.
- Practising only familiar questions instead of reviewing errors and weak topics.
Every student begins with a level check so we can see where the gaps are. Private 1:1 students also get a full diagnostic assessment and a personalized roadmap. We teach the concepts behind each question type, practice under timed conditions, and review every wrong answer. We cannot guarantee specific scores, but progress is tracked in monthly reports.