Explaining, convincing and proving

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

Can you use the diagram to prove the AM-GM inequality?

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Show that all pentagonal numbers are one third of a triangular number.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Drawing a triangle is not always as easy as you might think!

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.