Clock Design

Coordinatize! Let the little circle be $x^{2}+y^{2}=1$ and the biggest circle be $x^{2}+y^{2}=r^{2}$ . We ultimately seek $d=2r$ .

The circle at 2 has the equation $(x-(r-1)\sqrt{3}/2)^{2}+(y-(r-1)/2)^{2}=1$

The point of tangency of ellipses 1 and 3 is along the line $y=x/\sqrt{3}$

The ellipse at 3 then has equation $\left( x-\frac{r+1}{2}\right)^{2} + ay^{2} = \left(\frac{r-1}{2}\right)^{2}$ for some value of $a$

For the ellipses to be tangent, the ellipse and line above must be tangent. Substituting the line into the ellipse and simplifying gives $(1+a/3)x^{2}-(r+1)x+r=0$ and since tangency requires the discriminant be zero we have $(r+1)^{2}-4(1+a/3)r=0$ or $a=\frac{3(r-1)^{2}}{4r}$ so we can use this in the ellipse equation.

From here, to complete the analytic solution, I'd need to solve the ellipse-circle equation and find the value of r for which there is a single solution to get the final point of tangency. I had trouble doing this, so I turned to Desmos .

Plug in the circle and ellipse equations. Zoom continuously on the nearest point of contact while refining $r$ manually to eventually get

$r \approx 5.642369577$ . So the answer is $\lfloor 2000r \rfloor =11284$