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Golden Powers
Age 16 to 18
Challenge Level 
You will probably need to experiment a little to try to form a conjecture for the form of the coefficients. You can then use induction to try to prove your conjecture.
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Gold Again
Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
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.
Golden Triangle
Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.