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

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 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.

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 $b_n$.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.

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NRICH is part of the family of activities in the Millennium Mathematics Project.

NRICH is part of the family of activities in the Millennium Mathematics Project.