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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?

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Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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Cocked Hat

Sketch the graphs for this implicitly defined family of functions.

Cut Cube

Age 16 to 18 Challenge Level:

Points P, Q and R are drawn on a cube, one third of the way along the edges AB, GH and ED respectively, as shown in the diagram. The cube is then cut along the plane PQR. Which slice, if any, has the greater volume? Describe the shape and symmetries of the pieces obtained.