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Mean Geometrically

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?

Pythagorean Golden Means

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

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

Without Calculus

Age 16 to 18
Challenge Level

Given that $u> 0$ and $v> 0$ what is the smallest possible value of $1/u + 1/v$ given that $u + v = 5$?

Can you find this value by more than one method (not involving trial and error) without using calculus?