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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?

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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?

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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: Challenge Level:1

Make a list of Fibonnaci numbers and mark the even ones. Now $f_0$ is even and $f_1$ is odd so the sequence starts even, odd, odd, even, ... Look for a pattern in the occurrence of even Fibonnaci numbers in the sequence, then prove that your pattern must continue indefinitely in the sequence.

Again look for a pattern in the occurrences of multiples of 3 in the Fibonnaci sequence. To prove the pattern always applies use the Fibonnaci difference relation $f_{n+2}=f_{n+1}+f_n$ repeatedly to show that if a certain term is divisible by 3 then other terms further along the sequence will also be divisible by 3.