Arithmetic, geometric and harmonic means

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.
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.
problem

Classical Means

Age

16 to 18

Challenge level

Use the diagram to investigate the classical Pythagorean means.
problem
Favourite

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?
problem

AMGM

Age

14 to 16

Challenge level

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