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Lessons

25 contest topics with four difficulty tracks each (Easy → AIME / USAJMO / AMO). Plus Phase 2 exam tracks for SAT, ACT, AP Calculus, JEE, and Gaokao. Roadmap

Stars and Bars
combinatorics
Integer solutions to linear equations.
Easy
Stars and Bars (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Stars and Bars (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Stars and Bars (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Stars and Bars (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Modular Arithmetic
number theory
Congruences and power cycles.
Easy
Modular Arithmetic (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Modular Arithmetic (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Modular Arithmetic (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Modular Arithmetic (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Vieta's Formulas
algebra
Roots and coefficients.
Easy
Vieta's Formulas (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Vieta's Formulas (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Vieta's Formulas (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Vieta's Formulas (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Power of a Point
geometry
Chord and tangent-secant.
Easy
Power of a Point (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Power of a Point (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Power of a Point (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Power of a Point (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Inclusion-Exclusion
combinatorics
Unions without double-counting.
Easy
Inclusion-Exclusion (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Inclusion-Exclusion (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Inclusion-Exclusion (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Inclusion-Exclusion (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Pigeonhole Principle
combinatorics
Forced collisions.
Easy
Pigeonhole Principle (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Pigeonhole Principle (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Pigeonhole Principle (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Pigeonhole Principle (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Similar Triangles
geometry
AA and ratios.
Easy
Similar Triangles (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Similar Triangles (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Similar Triangles (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Similar Triangles (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Coordinate Geometry
geometry
Place figures on axes.
Easy
Coordinate Geometry (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Coordinate Geometry (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Coordinate Geometry (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Coordinate Geometry (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Divisibility
number theory
Factors and gcd/lcm.
Easy
Divisibility (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Divisibility (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Divisibility (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Divisibility (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Prime Factorization
number theory
Unique factorization.
Easy
Prime Factorization (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Prime Factorization (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Prime Factorization (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Prime Factorization (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Permutations
combinatorics
Ordered arrangements.
Easy
Permutations (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Permutations (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Permutations (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Permutations (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Combinations
combinatorics
Unordered selections.
Easy
Combinations (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Combinations (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Combinations (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Combinations (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Probability
combinatorics
Counting outcomes.
Easy
Probability (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Probability (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Probability (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Probability (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Arithmetic Sequences
algebra
Common differences.
Easy
Arithmetic Sequences (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Arithmetic Sequences (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Arithmetic Sequences (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Arithmetic Sequences (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Geometric Sequences
algebra
Common ratios.
Easy
Geometric Sequences (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Geometric Sequences (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Geometric Sequences (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Geometric Sequences (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Quadratics
algebra
Factor and complete square.
Easy
Quadratics (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Quadratics (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Quadratics (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Quadratics (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Systems of Equations
algebra
Linear and nonlinear systems.
Easy
Systems of Equations (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Systems of Equations (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Systems of Equations (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Systems of Equations (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Area Techniques
geometry
Triangles and polygons.
Easy
Area Techniques (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Area Techniques (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Area Techniques (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Area Techniques (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Circle Geometry
geometry
Chords, arcs, angles.
Easy
Circle Geometry (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Circle Geometry (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Circle Geometry (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Circle Geometry (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Triangle Inequality
geometry
Existence and bounds.
Easy
Triangle Inequality (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Triangle Inequality (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Triangle Inequality (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Triangle Inequality (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Base Systems
number theory
Convert and compute in bases.
Easy
Base Systems (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Base Systems (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Base Systems (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Base Systems (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Floor and Ceiling
algebra
Integer parts.
Easy
Floor and Ceiling (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Floor and Ceiling (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Floor and Ceiling (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Floor and Ceiling (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Absolute Value
algebra
Cases and graphs.
Easy
Absolute Value (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Absolute Value (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Absolute Value (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Absolute Value (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
Rates and Work
algebra
Motion and work rates.
Easy
Rates and Work (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
Rates and Work (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
Rates and Work (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
Rates and Work (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.
AM-GM Inequality
algebra
Means inequality.
Easy
AM-GM Inequality (Easy)
AMC 10 Q1–8 depth — contest entry, not school drills.
Medium
AM-GM Inequality (Medium)
AMC 10/12 mid (≈Q9–16) — multi-step.
Hard
AM-GM Inequality (Hard)
Late AMC 10/12 through early AIME — dense computation.
AIME / USAJMO / AMO
AM-GM Inequality (AIME / USAJMO / AMO)
Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Lessons

Stars and Bars

Modular Arithmetic

Vieta's Formulas

Power of a Point

Inclusion-Exclusion

Pigeonhole Principle

Similar Triangles

Coordinate Geometry

Divisibility

Prime Factorization

Permutations

Combinations

Probability

Arithmetic Sequences

Geometric Sequences

Quadratics

Systems of Equations

Area Techniques

Circle Geometry

Triangle Inequality

Base Systems

Floor and Ceiling

Absolute Value

Rates and Work

AM-GM Inequality