Find the maximum $n$ such that circles of radius $1, \frac12, \frac13, ..., \frac1n$ can be held immobile by a convex frame, or show that there is no maximum.
Here is an example with $n=7$.
By "immobile", I mean no circle can move without overlapping other circles or the frame, either individually or simultaneously. The frame is rigid.
It seems to get increasingly difficult to keep adding circles (and expanding the frame), while maintaining the conditions that the circles are immobile and the frame is convex. Or maybe there is a clever way to arrange the circles so that you can include all of them.
(This question was inspired by What is the minimum area of a rectangle containing all circles of radius $1/n$?)
You can start by using the Descartes Circle Theorem. For four mutually touching circles $2 (a^2+b^2+c^2+d^2) = (a+b+c+d)^2$, where the values are 1/radius (the bend). For example, circles with radii {1/2, 1/3, 1/6, 1/23} can all touch each other, but the bends are easier to work with: {2, 3, 6, 23}. Once you have 4 circles working together, you can get infinite circles, but only a third of the possible bends. Mod 12, a set of Descartes circles will only have 4 values.
So, we just need to find three Descartes circle starters that cover the 12 possible mod 12 values. Unfortunately, that may not exist. But here are some of the starter sets you can play with: {{2,3,6,23}, {1,4,12,33},{3,6,7,34}, {4,6,12,46}, {2,8,24,66}, {8,9,17,72}, {14,21,42,161},{7,28,84,231}, {21,42,49,238}, {11,44,132,363}}. Here's a set of four starters that cover all the mod 12 values, but there will be overlap: {{2, 3, 6, 23}, {1, 4, 12, 33}, {7, 10, 27, 90}, {8, 9, 17, 72}}. In the image below, circles are colored by their modulus.
Here are some starters with negative bends to work with.
{{-1,2,3,6}, {-1,2,6,11}, {-1,3,14,26}, {-1,6,11,30}, {-2,3,6,7}, {-2,3,7,10}, {-2,3,10,15}, {-2,6,7,19}, {-3,4,12,13}, {-3,4,13,16}, {-3,4,16,21}, {-3,5,8,12}, {-3,8,21,44}, {-4,8,9,17}, {-5,6,30,31}, {-7,12,17,20}, {-7,12,17,24}, {-9,14,26,27}, {-11,16,36,37}}