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Geometry

Locus

• Locus Problems – application
• Geodesics on a Sphere
• Triangles on a sphere

Inequality

• Triangular Inequality
• Construction of Geometric Mean
• Geometry of AM – GM Inequality

Area problems

• Carpet Strategy
• Maximising area
• $ \textrm{Inradius} \times \textrm{semi peri meter} = \textrm{Area} $ : application

Lines, Angles and Triangle

• Angle opposite to largest side: Application
• Vectors
• Angle Bisector Theorem – application
• Extended Pythagoras Theorem
  • Application 1
  • Application 2
• Ceva’s Theorem
• Menelaus Theorem:
  • Concept
  • Application

Geometry of Circles

• Opposite angles of a cyclic quadrilateral adds up to $180^o$: Application in RMO 2008 Problem 1
• Cyclic Quadrilaterals, Napoleon Triangles: Part 1, Part 2, Part 3
• Inradius, Exradius
  • Concept
• Incenter and con-cyclic points: Application in RMO
• Orthocenter and Circles:
  • Part 1, Part 2, Part 3, Part 4
• Power of a Point
• Sine rule and Incenter
• Nine Point Circle:
  • Part 1
  • Part 2
  • Part 3
• Euler formula: $ d^2 = R^2 – 2R\times r$
• Circles in Circle

Geometric Transformation

• Translation – Concept and Application
• Reflection – Concept and Application
• Rotation – Application in Fermat Point Problem
• Inversion – Application, Concept, Part 1, Part 2

Complex Numbers in Geometry

• Complex Numbers and Geometry
• Triangles in Complex Plane

Mass Point Geometry

• Barycentric Coordinate for Math Olympiad: Part 1, Part 2

Combinatorics

Basic Combinatorics

• Multiplication and Addition Principle

Bijection Principle

• Concept and Application
• Integer Solutions

Invariance Principle

• Coloring Proof Application

Extremal Principle

• Part 1
• Part 2
• Application: Dancing Partner Problem

Combinatorial Argument

• Triangular Numbers

Partitions

• Partition of $n$ distinct objects into $2$ classes
• Partition of $n$ distinct objects into $3$ classes
• Partition and Recurrence Relation
• Partitions and Generating Functions

Number Theory

Modular Arithmetic

• Arithmetic of Remainders – the experiment
• Arithmetic of Remainders – the proof
• Theory of congruence – introduction
• Congruence relation is an equivalence relation
• Properties of Congruence Relation
• Euclid Algorithm and Modular Inverse
• GCD and Modular Inverse
• Application Problem: Part 1, Part 2, Part 3
• Application Problem: $ 7 $ divides $ 3^{2n+1} + 2^{n+2} $

Divisibility

• Pairing of Divisors
• Solving Equations with primes
• Number Theoretic Function
• Bezout’s Theorem
• Multiplicative Function: Application

Advanced Topics

• Chinese Remainder Theorem
• Geometric Excursion in Number Theory: Part 1, Part 2, Part 3

Algebra

Inequality

• AM-GM Inequality
  • Part 1
  • Part 2
  • Part 3
• Cauchy Schwarz
  • Part 1
  • Part 2
• Quadratic Mean
• Jensen Inequality – Concept