Task 5. (5 points) Every street in a city leads either from north to south, or from east to west and is two-way. A driver traveled through the city, making 100 left turns and returned to the place where he started. How many right turns did he make, if he did not pass by any place twice?

翻译说明：虽然最后一句“if he did not pass by any place twice”在语境中是正确的，但为了保持与原文的一致性，这里直接翻译为“if he did not pass by any place twice”。
Solution: The driver has walked along the contour of a non-self-intersecting polygon. Let's first assume that he started from a point that is not a vertex of this polygon. Then he has made a total of exactly one full turn $(360^{\circ})$ around his axis. Each turn is a quarter turn $(90^{\circ})$ to the left or to the right. The sum of the left turns differs from the sum of the right turns by exactly one full turn, i.e., by 4 turns. This means that if there are $x$ left turns in the route, the right turns are $x \pm 4$. Therefore, the driver has made 104 or 96 right turns (in the first case, the full turn is clockwise, and in the second case, it is counterclockwise). Figure 4 shows example routes.

![](https://cdn.mathpix.com/cropped/2024_06_03_5d371ccc65ea9d468529g-3.jpg?height=345&width=665&top_left_y=1592&top_left_x=730)

Figure 4:

The case where the driver started from a vertex of the polygon can be reduced to the considered case: the difference is that the turn at the last vertex is "not counted." If this turn would have been a right turn, the left turns in the route are 100, and the right turns are $104-1=103$ or $96-1=95$. When the uncounted turn is left, in addition to it, there must be 100 left turns. For 101 left turns in the route, there are $101-4=97$ or $101+4=105$ right turns (Figure 5).

![](https://cdn.mathpix.com/cropped/2024_06_03_5d371ccc65ea9d468529g-4.jpg?height=383&width=1173&top_left_y=628&top_left_x=470)

Figure 5:

In conclusion, the right turns are 95, 96, 97, 103, 104, or 105.

The problem was solved by Nikolay Belukhov and Veselina Ginova from Stara Zagora, Hristo Nikolov from Ruse, and Viktor Konstantinov from AK Sofia.