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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}.

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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.

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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?

Different by One

Age 14 to 16 Challenge Level:

In the original example, the lines were made from rods that differed in length by 2:
light green rods (3) and yellow rods (5).

You could start by creating lines made from
pink (4) and dark green (6)
yellow (5) and black (7)
dark green (6) and brown (8)
black (7) and blue (9)...

Which pairs can be used to create lines that differ by 1?