A point charge Q is located a distance r>R from the center of a nongrounded conducting spherical shell with radius R and net charge that is also Q. If the point charge is very far from the shell, then the shell looks essentially like a point charge Q, so the force between the two objects is repulsive. But if the point charge is very close to the shell, the the excess negative charge on the near side of the shell dominates, so the force is attractive.
Show that the value of r where the force makes the transition from repulsive to attractive is r= <[R(1+5?)/2?(1.618)R]]> . The factor here is the golden ratio. (In your solution, don't panic if you end up with a quintic equation. Just show that it has a factor of the form x^2-x-1.)