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Three Balls
Age 14 to 16
Challenge Level 
Here is a hint for this tough nut. Just think about the angle between a chord and a tangent to a circle and think about cyclic quadrilaterals.
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Doodles
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Russian Cubes
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
Polycircles
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?