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Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

OK! Now Prove It

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

Summats Clear

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Cyclic Triangles

Age 16 to 18
Challenge Level

Think about what stays the same and what changes if you fix $c$ and vary $a+b$.

For the first part you have a right-angled triangle so you might try to use Pythagoras' Theorem which suggests working with $(a+b)^2$. The key idea is to maximise the area of the triangle.

To generalise this go from Pythagoras' Theorem to the Cosine Rule. Again use the area of the triangle. The key idea here is that if you fix $c$ and vary $a+b$ the angle $\angle ACB$ is constant.

The last part calls for a careful argument based on four applications of the second result.