Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the circles $C_i$ and $C_j$ as $$\frac {d(o_1,o_2)}{r_1+r_2},$$ where $d(x,y)$ is the Euclidean distance. So the relative distance is $\le 1$ if the two circles are intersecting, and $\ge 1$ if they are non overlapping.

What is the smallest possible number of distinct relative distances?

For unit circles this is Erdos' distinct distances problem that was famously settled by Guth and Katz.