Question 10
Eight circles of diameter $1$ are packed in the first quadrant of the coordinate plane as shown. Let region $\mathcal{R}$ be the union of the eight circular regions. Line $\ell,$ with slope $3,$ divides $\mathcal{R}$ into two regions of equal area. Line $\ell$’s equation can be expressed in the form $ax = by + c,$ where $a,$ $b,$ and $c$ are positive integers whose greatest common divisor is $1.$ Find $a^2 + b^2 + c^2.$
