1. To climb from the valley to the mountain top, one must walk 4 hours on the road, and then -4 hours on the path. On the mountain top, two fire-breathing dragons live. The first dragon spews fire for 1 hour, then sleeps for 17 hours, then spews fire for 1 hour again, and so on. The second dragon spews fire for 1 hour, then sleeps for 9 hours, then spews fire for 1 hour again, and so on. It is dangerous to walk on both the road and the path during the first dragon's fire-spewing, while it is dangerous to walk only on the path during the second dragon's fire-spewing. Both dragons start spewing fire simultaneously at midnight. Is it possible to safely climb from the valley to the mountain top and return back? (8 points)
# Solution:

The path along the road and the trail (there and back) takes 16 hours. Therefore, if you start immediately after the first dragon's eruption, this dragon will not be dangerous. The path along the trail (there and back) takes 8 hours. Therefore, if you start moving along the trail immediately after the second dragon's eruption, this dragon will not be dangerous. For a safe ascent, it is sufficient for the first dragon to stop erupting by the start of the journey along the road, and for the second dragon to stop erupting 4 hours later, by the start of the journey along the trail. The first dragon erupts in hours numbered $18 x+1$, and the second dragon in hours numbered $10 y+1$, where $x, y$ are natural numbers. We need them to erupt with a 4-hour shift. We get the following Diophantine equation:

$$
10 y+1=(18 x+1)+4,5 y=9 x+2 .
$$

The smallest solution in natural numbers is $x=2, y=4$. Therefore, for a safe ascent, you need to start at the beginning of the 38th hour ( $18 \cdot 2+1=37)$.

Answer: You need to start at the beginning of the 38th hour.

| llы | Я№ 1 |
| :---: | :---: |
| 8 | Contains all steps of the solution.  and explanation. |
| 5 | The problem is not solved, but its solution is significantly advanced, i.e.:  - a significant part of the solution is correctly performed, possibly inaccurately (for example, a Diophantine equation is formulated, or a graph of eruptions is constructed, or the possibility of a safe path is demonstrated in principle);  - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the solution to the Diophantine equation is not found, or there are errors in the graph of eruptions, or a specific moment for the start of a safe path is not indicated, or the possibility of stopping and waiting is allowed).  The solution may be incomplete. |
| 0 | All cases of the solution that do not correspond to the above criteria for scoring 5 and 8 points. |