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Congratulations Tom Davie, Mike Gray and Ella Ryan of Madras College for your excellent team work on this problem. This is their solution; Tom wrote up the first part, Mike solved the equation and showed that there are two possible circles, and Ella described the construction of the smallest circle. Work like this is a real pleasure to read.
A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
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.
Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?