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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Making Tracks

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A bicycle passes along a path and leaves some tracks.

Is it possible to examine the curve of the individual tracks (both tyres have identical tread) and from them say which track was made by the front wheel and which by the back wheel?

Is it possible to say in which direction along the path the bicycle was travelling ?

The image above is just a picture to suggest the context, but if you'd like some accurate curves (tracks) to play with see the Hint