About USAJMO
The invitational national olympiad for top AMC 10 and AIME scorers. It has the same proof format as the USAMO, written for younger students.
USAJMO 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 USAJMO
Skillzy Young Scholars teaches USAJMO 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 USAJMO Covers
- Writing complete mathematical proofs with clear definitions, lemmas and justified steps
- Olympiad algebra, number theory and combinatorics
- Euclidean geometry, constructions, invariants and proof strategy
USAJMO 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 10 participants (combined with their AIME index). This is a highly exclusive, invitational-only exam.
- 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 credit
- Advanced synthetic geometry, algebra, and number theory
- Combinatorial logic and 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 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 linking roots and coefficients.
- Analyze sequences defined recursively, proving closed forms and establishing rigorous bounds.
USAJMO Alignment - Core Algebra. USAJMO algebra requires moving from finding an answer to proving that a relationship holds true for all infinite possibilities 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).
- Apply trigonometric identities in geometric proofs (e.g., using the Law of Sines/Cosines as a proof mechanism).
- Execute Barycentric and vector methods for complex geometry proofs.
USAJMO Alignment - Core Geometry. USAJMO geometry abandons the coordinate plane in favor of pure synthetic geometry, requiring students to construct auxiliary lines and prove collinearity or concyclicity.
Learning activities:
- Synthetic geometry diagram construction and angle chasing
- Triangle center and radical axis mapping
- Barycentric coordinate derivation exercises
What this means for your child. Your child will move beyond calculating areas and lengths, learning to write rigorous geometric proofs that establish absolute spatial truths.
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) lemma.
USAJMO Alignment - Core Number Theory. Number theory on the Olympiad level requires students to prove that no integer solutions exist for specific equations, or to prove divisibility properties for infinite sets.
Learning activities:
- Infinite descent and Vieta jumping logic puzzles
- Lifting-the-Exponent (LTE) 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 Olympiad-level number theory.
4 Olympiad Combinatorics (Extended Depth)
Students will learn to:
- Construct extremal combinatorics and advanced pigeonhole arguments.
- Master graph theory basics: coloring, paths, trees, and extremal graph problems.
- Analyze combinatorial games utilizing invariants and monovariants to prove winning strategies.
- Execute advanced counting methods: establishing bijections, double counting, and using generating functions.
USAJMO Alignment - Core Combinatorics. Combinatorics at this level often presents as a game or a finite set of objects, requiring students to prove that a certain state is always reachable (or impossible).
Learning activities:
- Invariant and monovariant game strategy mapping
- Graph theory coloring and path-finding proofs
- Bijections and double-counting systematic exercises
What this means for your child. Your child will develop the logical creativity required to prove abstract combinatorial constraints, utilizing 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.
USAJMO Alignment - Test Day Readiness. A correct idea is worth 0 points if not communicated properly. Olympiad grading is notoriously 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 USAJMO Prep Plan
- Begin with a USAJMO 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 USAJMO Mistakes We Help Students Avoid
- Preparing for USAJMO 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.