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Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

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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.

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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.

Golden Eggs

Age 16 to 18 Challenge Level:

The second part of this question introduces an expression involving an infinite string of nested square roots but, as forbidding as it may seem, a little ingenuity is all that is needed to evaluate it.