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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Curvy Areas

Stage: 4 Challenge Level: Challenge Level:1

Take a look at the image below:

Can you see how the image was created?
Try to recreate it using a ruler and compasses.

Here are two images created in a similar way.

Can you work out the proportion of the 3-colour, 4-colour and 5-colour circles which is shaded red?
Can you make any generalisations?
Can you prove your ideas?

What about the proportion which is shaded orange? Yellow? ... 
Can you make any generalisations?
Can you prove your ideas? 

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