6.3. On the way from city $A$ to city $B$, there are kilometer markers every kilometer. On each marker, one side shows the distance to $A$, and the other side shows the distance to $B$. In the morning, a tourist passed by a marker where one number was twice the other. After walking another 10 km, the tourist saw a marker where the two numbers differed by a factor of three. What is the distance from $A$ to $B$? Provide all possible answers and prove that there are no others.
Solution: Let $C_{1}$ and $C_{2}$ be the poles mentioned in the problem ($C_{1}$ - the pole near which the tourist was in the morning). Without loss of generality, assume the tourist was walking from $A$ to $B$. Then there are two possible situations: 1) $C_{1} A = 2 C_{1} B$ or 2) $C_{1} B = 2 C_{1} A$. Since $C_{2} A > C_{1} A$, the situation $C_{2} B = 3 C_{2} A$ is impossible, so the situation $C_{2} A = 3 C_{2} B$ must hold.
![](https://cdn.mathpix.com/cropped/2024_05_06_83d741ed4daa4f8b244bg-2.jpg?height=498&width=808&top_left_y=1902&top_left_x=630)

To the solution of problem 6.3

1) If $C_{1} A = 2 C_{1} B$, then $C_{1} A$ constitutes $\frac{2}{3}$ of the total distance $A B$. Then the distance from the second pole to $A$ is $\frac{3}{4}$ of the distance $A B$, meaning the tourist has walked $\frac{3}{4} - \frac{2}{3} = \frac{1}{12}$ of this distance. Therefore, $A B = 120$ km.
2) If $C_{1} B = 2 C_{1} A$, then $C_{1} A = \frac{1}{3} A B$. Then the tourist has walked $\frac{3}{4} - \frac{1}{3} = \frac{5}{12}$ of the entire distance in one day. Therefore, $A B = 24$ km.

Answer: either 120 km or 24 km.

Grading criteria:

| present in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| The solution process is entirely correct, but there are computational errors, possibly leading to an incorrect answer | 6 points |
| Only one of the two cases is correctly considered; (one of the two answers is correctly obtained and justified) | 3 points |
| Both answers (120 km and 24 km) are correctly stated, but either without justification or with incorrect justification | 2 points |
| Only an example that realizes one of the two cases is provided | 1 point |
| Various reasoning and calculations that did not lead to a solution | 0 points |