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Degree Ceremony
Age 16 to 18
Challenge Level 
This problem gives students the chance to explore the relationships between the sine of different angles, and the symmetries of these.
As an alternative approach, students can use the graphs of trig functions to justify $\sin(90^{\circ} -x) = \cos x$ and use this to help find the value of the sum.
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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 $S_1, S_2, ... S_{r-1}$.
OK! Now Prove It
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
Overarch 2
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?