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Particularly General
Age 16 to 18
Challenge Level 
Expand out the brackets in a systematic, organised manner.
You will need to use a double angle trig formula and difference of two squares. Considering $n=1$ might be useful.
There is perhaps no need to provide a proof for your identities if you are very clear in your mind how such a proof would be constructed, although you might wish to produce a proof if interested.
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Binomial
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
Powerful Factors
Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.