Complete Syllabus

A comprehensive guide to mathematical olympiad topics

Syllabus

1. Fundamentals

This section covers the essential building blocks of mathematical reasoning, proof-writing, and the technical skills needed to participate in the online program.

Topics:

  • • Technical Skills: Discord, PDF scanning, Google Drive, LaTeX basics
  • • Set Theory: Sets, Cartesian products, subsets, power sets, unions, intersections
  • • Logic: Statements, logical operators, conditional statements, truth tables
  • • Proof Techniques: Direct, Contrapositive, Contradiction, Induction, Counterexample

Resources:

Book of Proof by Richard Hammack Logic PlayList by Ahmed Bachir َSets PlayList by Ahmed Bachir
01

Foundation First

02

Algebraic Power

2. Algebra

This section focuses on the manipulation of expressions, solving equations, and understanding the properties of polynomials and sequences.

Topics:

  • • Algebraic Manipulations: Simplifying expressions, factoring, working with fractions
  • • Polynomials: Understanding roots, coefficients, and Vieta's formulas
  • • Sequences and Series: Arithmetic and geometric progressions, special sequences
  • • Systems of Equations & Statistics: Linear/non-linear systems, mean, median, mode

Resources:

Mastering AMC 10/12 by OmegaLearn

3. Combinatorics

Combinatorics is the art of counting. This section covers fundamental counting principles and their applications.

Topics:

  • • Basic Counting Principles: Multiplication, Addition, and Subtraction Principles
  • • Permutations and Combinations: Counting ordered and unordered arrangements
  • • The Pigeonhole Principle: A simple but powerful reasoning tool
  • • PIE & Stars and Bars: Advanced counting techniques and basic probability

Resources:

Book of Proof (Ch.3) Mastering AMC (Ch.1-5)
03

Counting Art

04

Integer Magic

4. Number Theory

This section explores the properties of integers, a core topic in mathematics competitions.

Topics:

  • • Primes and Divisibility: Prime factorization, divisors, basic divisibility rules
  • • GCD and LCM: Euclidean algorithm and their properties
  • • Modular Arithmetic: Congruences, solving linear congruences, residues
  • • Diophantine Equations & Bases: Integer solutions, number base conversions

Resources:

Mastering AMC (Ch.21-27)

5. Geometry

This section is dedicated to Euclidean geometry, focusing on building strong visual and logical problem-solving skills.

Topics:

  • • Congruence and Similarity: Criteria for congruent and similar triangles
  • • Angles and Lines: Angle chasing with transversals, parallel lines, polygons
  • • Area: Formulas and techniques for calculating area of various plane figures
  • • Circles & Triangle Centers: Chords, tangents, secants, inscribed angles, important centers

Resources:

A Beautiful Journey Through Olympiad Geometry
05

Visual Logic

Problem Solving

Olympiad problems are designed to be solvable, and often in a beautiful fashion to boot.

1. Problems from Lecture Books

Problems mentioned in the three books of lectures:

Book of Proof Mastering AMC 10/12 Olympiad Geometry

2. Coupe Animath d'Automne (PFOM)

French mathematical olympiad problems for fall season:

3. Coupe Animath de Printemps (POFM)

French mathematical olympiad problems for spring season:

4. AMC 10 (MAA)

American Mathematics Competitions:

2024 Problems 2023 Problems 2022 Problems 2021 Problems

5. Baltic Way

International mathematical olympiad for Baltic and Nordic countries:

2023 (Flensburg, Germany)
2021 (Reykjavik, Iceland)
2020 (Virtual)