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Book 2. Olympiad Inequalities

Book 2. Olympiad Inequalities

  • 1. Basic Inequality Principles
  • 2. AM-GM
  • 3. Cauchy-Schwarz
  • 4. Rearrangement and Chebyshev
  • 5. Jensen's Inequality Intro
  • 6. UVW Method and Symmetric Inequalities
  • 7. Homogeneous Inequalities
  • 8. Substitution Methods
  • 9. Inequalities with Constraints
  • 10. Hard Inequality Problems
  • 11. Mixed Inequality Sets
  • 12. Mock Olympiads: Inequalities
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Theory Examples Practice Ladders

Chapters

Chapters

Chapter

Basic Inequality Principles

The module teaches inequalities as structure: squares, ordering, extremal elements, sign control, equality cases, and invariants.
30 Problems
Theory Examples Practice

Chapter

AM-GM

The module teaches AM-GM through term choice, normalization, weights, ordering, and strict equality-case analysis rather than mechanical application.
25 Problems
Theory Examples Practice

Chapter

Cauchy-Schwarz

The module teaches the standard form, Engel form, fractional sums, cyclic denominators, and preliminary estimates before Cauchy.
20 Problems
Theory Examples Practice

Chapter

Rearrangement and Chebyshev

The module teaches students to recognize order in inequalities: rearrangement, Chebyshev, opposite order, power sums, and zero-sum problems.
24 Problems
Theory Examples Practice

Chapter

Jensen's Inequality Intro

An introductory module on convexity and concavity: Jensen, tangent lines, fixed sums, logarithms, roots, reciprocals, and choosing the function.
24 Problems
Theory Examples Practice

Chapter

UVW Method and Symmetric Inequalities

The module introduces the parameters p, q, r, symmetric sums, Schur-type inequalities, and reduction to two equal variables.
20 Problems
Theory Examples Practice

Chapter

Homogeneous Inequalities

The module teaches degree checking, choosing a normalization, homogenizing constants, and returning to the original scale.
20 Problems
Theory Examples Practice

Chapter

Substitution Methods

The module trains substitutions for ratios, product 1, triangle sides, roots, trigonometric constraints, and equality cases.
20 Problems
Theory Examples Practice

Chapter

Inequalities with Constraints

The module teaches using fixed sum, fixed product, and pairwise sum to find maxima, minima, and equality cases.
20 Problems
Theory Examples Practice

Chapter

Hard Inequality Problems

A mixed module with AM-GM, Cauchy, UVW, Jensen, normalization, SOS ideas, and rewritten olympiad methods.
20 Problems
Theory Examples Practice

Chapter

Mixed Inequality Sets

A recognition module for choosing methods: Cauchy, AM-GM, Holder, local estimates, Schur, discriminants, and pairwise differences.
20 Problems
Theory Examples Practice

Chapter

Mock Olympiads: Inequalities

Five mock sets with four problems each: from standard estimates to strong problems using Schur, discriminants, and pairwise differences.
20 Problems
Theory Examples Practice