1. To climb from the valley to the mountain top, one must walk 6 hours on the road, and then - 6 hours on the path. On the mountain top, two fire-breathing dragons live. The first dragon spews fire for 1 hour, then sleeps for 25 hours, then spews fire for 1 hour again, and so on. The second dragon spews fire for 1 hour, then sleeps for 13 hours, then spews fire for 1 hour again, and so on. During the first dragon's fire-spewing, it is dangerous to walk both on the road and on the path, while during the second dragon's fire-spewing, it is dangerous to walk only on the path. At midnight, both dragons start spewing fire simultaneously. Is it possible to safely climb from the valley to the mountain top and return back? (6 points)
# Solution:

The path along the road and the trail (there and back) takes 24 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 12 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 necessary that by the start of movement along the road, the first dragon has stopped erupting, and 6 hours later, by the start of movement along the trail, the second dragon has stopped erupting. The first dragon erupts in hours numbered $26 x+1$, and the second dragon in hours numbered $14 y+1$, where $x, y$ are natural numbers. We need them to erupt with a 6-hour shift. We get the following Diophantine equation:

$$
14 y+1=(26 x+1)+6, \quad 7 y=13 x+3
$$

The smallest solution in natural numbers is $x=3, y=6$. Therefore, for a safe ascent, you need to start at the beginning of the 80th hour $(26 \cdot 3+1=79)$.

Answer: You need to start at the beginning of the 80th hour.

| Points | Criteria for evaluating the performance of task No. 1 |
| :---: | :--- |
| $\mathbf{6}$ | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are performed correctly. The correct answer is obtained. |
| $\mathbf{3}$ | The problem is not solved, but its solution is significantly advanced, i.e.: - a significant part of the solution is performed correctly, possibly inaccurately (for example, a Diophantine equation is formulated); - the other part is either not performed or performed incorrectly, possibly even with logical errors (for example, the solution to the Diophantine equation is not found). The solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not meet the above criteria for scoring 3 and 6 points. |