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Cut Cube

Find the shape and symmetries of the two pieces of this cut cube.

Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

Cocked Hat

Sketch the graphs for this implicitly defined family of functions.


Age 16 to 18
Challenge Level

Consider figures 1 and 2 showing a 'necklace' and a 'floret' of squares. All the squares are regularly spaced around the circles and have side length 1 unit. Calculate, in each case, the inner and outer radii of the rings (or annuli, the singular is annulus) that surround the squares. If you 'merged' the two figures would one annulus lie completely inside the other?