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Age 16 to 18

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Dylan from Brooke Weston and Soumya from King's Maths School in the UK used diagrams and two sines to solve the problem. Here is Dylan's work:

Soumya used a slightly different diagram, but the rest of the working was the same. Here is Soumya's diagram and the first line of Soumya's working:

Joshua, Gautham and Nishad used sines and cosines with another identity. This is Nishad's work:

For the sum up to $\sin^2360^\circ$, Dylan and Joshua used graphs and symmetry. This is Joshua's work:

Dylan used a more graphical method which includes graph transformations. This is Dylan's work:

Note that Dylan's answer is slightly wrong. Can you use this diagram to see why?

Gautham and Soumya showed more terms in their working, so Dylan's lost terms are clearer. Here is Gautham's work (click on the image to see a larger version):

Soumya used sigma notation to express the same thing (click on the image to see a larger version):

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

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?