KMO 2018

Let there be a figure with $9$ disks and $11$ edges, as shown below.

We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$ , and $1$ in disk $I$ . Find the minimum sum of all real numbers written in $11$ edges.

Attachments:

Let there be positive integers $a, c$ . Positive integer $b$ is a divisor of $ac-1$ . For a positive rational number $r$ which is less than $1$ , define the set $A(r)$ as follows. $$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$ Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$ .