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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.

Golden Powers

Age 16 to 18
Challenge Level

The famous golden ratio is $g={\sqrt5 + 1 \over 2}$. Prove that $g^2=g+1$.

Let $g^n = a_ng + b_n$. Find the sequences of coefficients $a_n$ and $b_n$.