Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Three Four Five

Age 14 to 16 Challenge Level

As shown in the diagram, two semi-circles (each of radius $1/2$)
touch each other, and a semi-circle of radius 1 touches both of
them. Find the radius of the circle which touches all three
semi-circles, and find a 3-4-5 triangle in the diagram.