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Circles Ad Infinitum
Age 16 to 18
Challenge Level 
Can you spot your old friend the 30-60-90 triangle? Then you are
into scaling down and geometric series.
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Generally Geometric
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
Harmonically
Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?