What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and 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.

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.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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!

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Explain how to construct a regular pentagon accurately using a straight edge and compass.

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

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

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

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

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.

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

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

A first trail through the mysterious world of the Golden Section.

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.

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