Reasoning, convincing and proving

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Can you find the values at the vertices when you know the values on the edges?

Can you work out the number of chairs at a cafe from the number of legs?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Can you find the value of this function involving algebraic fractions for x=2000?

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Here the diagram says it all. Can you find the diagram?

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?