# Task 1. 15 points

The distances between the transport nodes, where cargo is received and transferred, are respectively: 2 km; $\sqrt{7}$ km, and 3 km. A warehouse, located within the transport zone (a triangle with vertices at the points of cargo reception and transfer), is connected to the transport nodes by straight roads. What is the minimum distance that a truck will travel if it needs to make three trips to each transport node in sequence, each time leaving from the warehouse and returning back.
# Solution

The solution to the problem reduces to finding the minimum distance from some point inside a triangle with sides 2 km, $\sqrt{7}$ km, and 3 km, and then multiplying this distance by 2 (the truck travels to and from each node from the warehouse without repeating nodes).

Let in triangle $\mathrm{ABC}$, the base $\mathrm{AB}=3, \mathrm{AC}=2, \mathrm{BC}=\sqrt{7}$. From the cosine theorem, we get that $\angle A=60^{\circ}$. E is a point inside the triangle such that EA+EB+EC is minimized. $\triangle$ AEC is rotated 60 degrees around A outside the original triangle so that C lands at $\mathrm{C}_{1}$ and $\mathrm{E}$ lands at $\mathrm{E}_{1}$. Note that $\mathrm{CC}_{1}$ is parallel to $\mathrm{AB}$, and $\triangle \mathrm{AEE}_{1}$ is an equilateral triangle. Clearly, $\mathrm{EA}+\mathrm{EB}+\mathrm{EC}=\mathrm{C}_{1} \mathrm{E}_{1}+\mathrm{E}_{1} \mathrm{E}+\mathrm{EB}$. The minimum value of the latter sum is the length of segment $\mathrm{C}_{1} \mathrm{~B}$. Thus, the original problem reduces to finding the length of $\mathrm{C}_{1} \mathrm{~B}$, the side of $\triangle \mathrm{ABC}_{1}$, where $\mathrm{AC}_{1}=\mathrm{AC}=2, \mathrm{AB}=3, \angle C_{1} A B=120^{\circ}$. By the cosine theorem, $C_{1} B=\sqrt{A C^{2}+A B^{2}-2 A C \cdot A B \cdot \cos 120^{\circ}}=\sqrt{2^{2}+3^{2}-2 \cdot 2 \cdot 3 \cdot(-0.5)}=\sqrt{19}$.

Since the truck travels to and from each node, the obtained value needs to be multiplied by 2.

Answer: $2 \sqrt{19}$

## Criteria:

- correct solution and correct answer - 15 points,
- correct solution but incorrect answer - 14 points,
- error in the solution but correct answer - 8 points,
- incorrect solution or no solution, but correct answer - 1 point,
- incorrect solution or no solution, incorrect answer or no answer - 0 points.