Arithmetic, geometric and harmonic means

  • Mean Geometrically
    problem
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    Mean Geometrically

    Age
    16 to 18
    Challenge level

    A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

  • Classical Means
    problem

    Classical Means

    Age
    16 to 18
    Challenge level
    Use the diagram to investigate the classical Pythagorean means.
  • AMGM
    problem

    AMGM

    Age
    14 to 16
    Challenge level

    Can you use the diagram to prove the AM-GM inequality?

  • Without Calculus
    problem

    Without Calculus

    Age
    16 to 18
    Challenge level

    Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

  • Pythagorean Golden Means
    problem

    Pythagorean Golden Means

    Age
    16 to 18
    Challenge level

    Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

  • Three Ways
    problem

    Three Ways

    Age
    16 to 18
    Challenge level

    If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

  • About Pythagorean Golden Means
    article

    About Pythagorean Golden Means

    What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?

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