Golden ratio

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

You add 1 to the golden ratio to get its square. How do you find higher powers?

Explore the geometry of these dart and kite shapes!

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.

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.