# List of Topics

### Number Theory

• Units Digits of a Number
• Tens Digits of a Number
• Decimal Representation
• Prime Factorization
• Divisibility
• Remainders
• Parity
• Modular Arithmetic
• Greatest Common Divisor
• Least Common Multiple

### Algebra

• Regrouping the Numbers in a Sum
• Regrouping the Numbers in a Product
• Gauss’ Formula
• Telescoping Products
• Telescoping Sums
• Ratios
• Percents
• Proportions
• Averages
• Powers
• Polynomial Expressions
• Special Products
• Methods of Factorization
• Systems of Equations

### Geometry

• Angle Chase
• Area of a Triangle
• Special Triangles
• Circles
• Pythagorean Theorem
• Heron’s Formula
• Shoelace Formula
• Mass Points

### Combinatorics

• Counting Techniques
• Multiplication Principle
• Permutations
• Combinations
• Complementary Counting
• Principle of Inclusion and Exclusion
• Stars and Bars
• Probability

### Number Theory

• Sets, Predicates and Quantifiers
• Divisibility Properties
• Fundamental Theorem of Arithmetic
• Infinity of Prime Numbers
• Greatest Common Divisor and Least Common Multiple
• Modular Arithmetic
• Properties of Congruences
• Diophantine Equations
• Peano Axioms
• Mathematical Induction
• Induction in Number Theory

### Algebra

• Polynomial Expressions
• Methods of Factoring
• Special Products
• Vieta's Theorem
• Induction in Algebra
• Method of Equivalent Transformations for Inequalities
• Inequality of Arithmetic and Geometric Means

### Geometry

• Congruent Triangles
• Similar Triangles
• Median Triangle
• Circumcenter and Incenter
• Centroid
• Orthocenter
• Orthic Triangle
• Excircles and their Properties
• Angle Bisector Theorem
• Inscribed and Central Angles
• Angle Between Tangent and Chord
• Simson Line
• Miquel's Point
• Fermat Point
• Nine Point Circle
• Apollonius Circle
• Locus

### Combinatorics

• Fundamental Counting Techniques
• Dirichlet Principle
• Coloring Strategies
• Invariants
• Induction in Combinatorics

### Number Theory

• Fermat's Little Theorem
• Bezout’s Identity
• Euler's Totient Function
• Euler’s Theorem
• Estimation Techniques
• Multiplicative Inverses
• Linear Congruences
• Complete Residue System
• Reduced Residue System

### Algebra

• Inequality of Arithmetic and Geometric Means
• Bernoulli’s Inequality
• Cauchy-Schwarz Inequality
• Jensen Inequality
• Rearrangement Inequality
• Functional Equations
• Cauchy Equation

### Geometry

• Analytic Geometry
• Locus
• Ceva's Theorem
• Menelaus' Theorem
• Power of a Point
• Reflection
• Rotation
• Translation
• Homothecy

### Combinatorics

• Dirichlet's Principle
• Extremal Principle
• Coloring Strategies
• Invariants
• Discrete Continuity
• Induction in Combinatorics
• Game Theory

### Sample Problems

Level 1 Sample Problems

Level 2 Sample Problems

Level 3 Sample Problems