Choose a bound, prove it and check equality. Begin at the first unfamiliar idea; the prerequisite links help you find a shorter route. You can open any lesson.

Unsure where to start? Try the starting-point check.

  1. I01

    Inequality rules and signs

    Transform inequalities while preserving equivalent conditions.

    Inside: Order and sign changes · Interval and quadratic inequalities

  2. I02

    Weierstrass product inequalities

    State the domain and prove a product bound from a two-factor case.

    Inside: Product bounds of Weierstrass type · Inductive extension and equality

  3. I03

    Absolute-value inequalities

    Translate absolute values into distance or justified cases.

    Inside: Absolute value as distance · Triangle and reverse triangle inequalities · Case splits and equality

  4. I04

    Sum of squares

    Turn a desired bound into a sum of nonnegative terms.

    Inside: Nonnegative squares · Completing squares · Designing an SOS certificate

  5. I05

    Arithmetic, geometric and harmonic means

    Choose the terms in a mean inequality and check attainable equality.

    Inside: Two-variable AM-GM · Several terms and repeated inputs · Reciprocal bounds and harmonic mean · Constraints and equality

  6. I06

    Weighted means

    Use nonnegative weights with their sum explicitly normalised.

    Inside: Rational weights · Weighted AM-GM · Matching exponents and weights

  7. I07

    Power mean inequality

    Choose a useful pair of means with valid inputs and exponents.

    Inside: Comparing means · Parameter domains and limiting cases

  8. I08

    Rearrangement inequality

    Prove why matching orders changes a sum of products.

    Inside: The two-term swap · Ordered sequences · Maximum and minimum pairings

  9. I09

    Chebyshev’s inequality

    Recognise when ordering permits a product-of-averages comparison.

    Inside: Similarly ordered sequences · Oppositely ordered sequences

  10. I10

    Cauchy–Schwarz inequality

    Recognise useful squares or denominators and state equality correctly.

    Inside: Two-term Cauchy · General sums and equality · Engel form · Choosing the two sequences

  11. I11

    Hölder’s inequality

    Select matching factors and exponents instead of recalling a formula blindly.

    Inside: Hölder in concrete forms · Matching powers · Equality and mixed applications

  12. I12

    Inequalities in geometry

    Check both the algebraic bound and the geometric attainability.

    Inside: Triangle side constraints · Algebraic bounds in geometry · Feasible equality configurations

  13. I13

    Jensen’s inequality

    Establish convexity on the relevant interval before applying Jensen.

    Inside: Convexity through chords · Jensen and weighted Jensen · Selecting an interval and equality

These original lessons introduce the methods and give practice with solutions. A single short session is not a full assessment of Olympiad readiness. Build depth through the written challenges and your country’s official past papers.