Reasoning, convincing and proving

problem
What's a group?

Age

16 to 18

Challenge level

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.
problem
Cyclic triangles

Age

16 to 18

Challenge level

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.
problem
Discrete trends

Age

16 to 18

Challenge level

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

Age

11 to 16

Challenge level

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

Age

11 to 14

Challenge level

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
problem
A leg to stand on

Age

11 to 14

Challenge level

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

Age

14 to 16

Challenge level

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?
problem
And so on - and on - and on

Age

16 to 18

Challenge level

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

Age

11 to 14

Challenge level

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
problem
Can it be?

Age

16 to 18

Challenge level

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