### 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?

### Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

### OK! Now Prove It

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

# Fibonacci Factors

##### Stage: 5 Challenge Level:

The Fibonnaci sequence occurs so frequently because it is the solution of the simplest of all difference relations. It is instructive to view it in this way and perhaps to introduce the idea of difference equations with this familiar example.

Proving these results calls for considering whether or not other terms in the sequences, apart from those in the recognized patterns, can also be multiples of 2 or 3 respectively in the two cases. Are the conditions necessary as well as sufficient?