Course

Book 1. Foundations of Olympiad Algebra

Book 1. Foundations of Olympiad Algebra

  • 1. Algebraic Expressions and Identities
  • 2. Factorisation Methods
  • 3. Equations and Systems
  • 4. Polynomials I
  • 5. Sequences and Recurrences
  • 6. Algebraic Transformations in Problems
  • 7. Introductory Inequalities
  • 8. Introductory Functional Equations
  • 9. Algebraic Number Problems
  • 10. Mixed Algebra Problems I
  • 11. Strategy Notes
  • 12. Mock Olympiads I
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Theory Examples Practice Ladders

Chapters

Chapters

Chapter

Algebraic Expressions and Identities

The first module of the book: recognising expression structure, using difference of squares, sum and difference of cubes, grouping, symmetric substitutions, and conditions such as \(a+b+c=0\).
25 Problems
Theory Examples Practice

Chapter

Factorisation Methods

The module teaches recognition of common factors, grouping, special identities, factorisation by a root, homogeneity, and hidden factors in problems on divisibility, compositeness, and integer solutions.
28 Problems
Theory Examples Practice

Chapter

Equations and Systems

The module teaches quadratic equations, symmetric systems, systems via sum and product, equations with substitution, parameters, and first integer systems.
26 Problems
Theory Examples Practice

Chapter

Polynomials I

An introduction to roots, the factor theorem, remainders, polynomial identities, coefficient comparison, and first olympiad problems with integer coefficients.
24 Problems
Theory Examples Practice

Chapter

Sequences and Recurrences

The module introduces arithmetic and geometric progressions, recurrence sequences, telescoping sums, finite differences, invariants, and first extremal process problems.
26 Problems
Theory Examples Practice

Chapter

Algebraic Transformations in Problems

The module teaches how to choose a useful form of an expression: completing the square, making substitutions, normalizing homogeneous expressions, reducing the number of variables, and applying zero-sum identities.
25 Problems
Theory Examples Practice

Chapter

Introductory Inequalities

An introductory inequalities module: nonnegative squares, AM-GM, first applications of Cauchy-Schwarz, rearrangement intuition, and mandatory equality-case analysis.
24 Problems
Theory Examples Practice

Chapter

Introductory Functional Equations

The module introduces basic functional-equation techniques: special substitutions, finding f(0) and f(1), injectivity, surjectivity, linear functions, and the Cauchy equation on simple domains.
24 Problems
Theory Examples Practice

Chapter

Algebraic Number Problems

The module connects algebra and number theory: divisibility of expressions, integer roots, Vieta's formulas, equations in integers, and modular checks.
24 Problems
Theory Examples Practice

Chapter

Mixed Algebra Problems I

A mixed module without an announced method: factorization, systems, polynomials, sequences, inequalities, functional equations, and integer checks.
24 Problems
Theory Examples Practice

Chapter

Strategy Notes

The module teaches method recognition: when to substitute, factor, use symmetry, look at polynomial degree, and seek the equality case.
20 Problems
Theory Examples Practice

Chapter

Mock Olympiads I

Eight training variants of four problems each: equation or factorization, polynomial or sequence, inequality, functional equation or mixed algebra.
32 Problems
Theory Examples Practice