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DOTS Division

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}.

Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

There's Always One Isn't There

Age 14 to 16 Challenge Level:

Systematic working and recording of results help a lot here.

Conjectures are important, and should be encouraged, but along with a challenge to really explain why any claim might be true generally.

We are so familiar with numbers and what they do, or what we believe they do, that the challenge to account rigorously for the familiar can seem pedantic. Hopefully the problem expressed in this form will give students the pleasure of discerning real structure.