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Sixational

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.

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Telescoping Functions

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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Sums of Squares and Sums of Cubes

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

Old Nuts

Age 16 to 18 Challenge Level:

This can be solved by just working systematically with the information, thinking about which numbers are divisible by 4 and 3, and testing the possibilities.

 

Modulus arithmetic also provides an efficient way to solve the problem.