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Power Quady

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

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Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)


Age 16 to 18 Challenge Level:
Note that you don't need to work out the numbers exactly to decide which is bigger!

Some possible methods:

1) Can you split up the powers into smaller pieces, or consider another quantity which your calculator can handle from which you can deduce the answer?

or 2) Can you use inequalities and logic such as 'if X > Y and Y > Z then we know that X > Z'

or 3) What happens when you raise $100^{1/100}$ to the power $9900$?