Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Photocopiers can reduce from A3 to A4 without distorting the image. Explore the relationships between different paper sizes that make this possible.

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

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

What have Fibonacci numbers got to do with Pythagorean triples?

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

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

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

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

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?