List of Topics

Level 1

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
  • Linear and Quadratic Equations
  • Systems of Equations

Geometry

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

Combinatorics

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

Level 2

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
  • Linear and Quadratic Inequalities
  • 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
  • Special Quadrilaterals
  • Inscribed and Central Angles
  • Angle Between Tangent and Chord
  • Cyclic Quadrilaterals
  • 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

Level 3

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

  • Cyclic Quadrilaterals
  • 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