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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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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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Giant Holly Leaf

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).

Two Shapes & Printer Ink

Stage: 4 Challenge Level: Challenge Level:1
Some solution we were offered cleverly reasoned that the grid, offered in the Hint as an approximation, was in fact completely accurate at the pixel level - which for printer ink is what counts. Sometimes we explore mathematics in unreal perfection for the sake of the thinking that will draw us into. Other solutions did this:

"You can split the problem up into combinations of circular shapes, each of which we can find the area of. Doing this I foundthat the star is over 5% larger than the S shape."