This problem was solved by Dhara Fernando, 13, from the University High School in Victoria, Australia.

Let the radii of the two inner circles be r and R, then the radius of the surrounding circle is R+ r. You are given that the area inside the largest circle surrounding the two smaller circles inside it is equal to the area of the larger of the two internal circles, so

π(R+r)2 −πR2 −πr2 = πR2.

This gives 2πRr = πR² and so 2r = R. The diameters of the 3 circles are 2r, 4r and 6r.