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# Medallions

This problem by R B J T Allenby appeared in the Sunday Times on 12.11.00.

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Age 14 to 16

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I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius $4$ cm and touches one side of the box, the middle sized one has radius $9$ cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

This problem by R B J T Allenby appeared in the Sunday Times on 12.11.00.

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?