3. A fly is sitting at vertex $A$ of a triangular room $A B C$ ( $\angle B=60^{\circ}, \angle C=45^{\circ}, A C=5$ m). At some point, it flies out in a random 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 9.9 meters?

![](https://cdn.mathpix.com/cropped/2024_05_06_1eac15157fb194aff838g-1.jpg?height=345&width=459&top_left_y=2229&top_left_x=1451)
Solution. Let the fly take off at an angle of 60 degrees to the line $A C$. Consider the equilateral triangle $A K C$ with side $A C$. Note that its sides $A K$ and $K C$ can be divided into parts (into infinitely many parts) such that each part equals the next segment of the fly's trajectory. The sum of these parts is $A K + K C = 10$ m, so at some point, the fly will have flown more than 10 meters.