1. A fly is sitting at the vertex $A$ of a triangular room $A B C$ $\left(\angle B=60^{\circ}, \angle C=45^{\circ}, A B=5\right.$ m). At some point, it flies out in a random straight direction, and each time it reaches a wall, it turns $60^{\circ}$ and continues flying in a straight line (see figure). Can it happen that after some time, the fly has flown more than 12 meters?

![](https://cdn.mathpix.com/cropped/2024_05_06_e66cee6e1e3e95af1258g-1.jpg?height=342&width=453&top_left_y=195&top_left_x=1451)
Solution. Construct a triangle $A X C$ with $A C$ as the base and the opposite angle $60^{\circ}$ such that the fly initially flies along the ray $A X$. It is not difficult to show that the total path traveled by the fly is equal to $A X + X C$. Then, $X$ traces a certain arc of a circle, and the path length reaches its maximum when $\angle X A C = 60^{\circ}$. In fact, it is not even necessary to prove that this is the maximum: $A C = \frac{\sqrt{6}}{2} A B = 5 \sqrt{}$ $A X + X C = 2 A C = 5 \sqrt{6} > 12$.