5. Islandland consists of ten islands, some of which are connected by two-way air routes. If you choose any 9 islands, you can fly around them one by one and return to the starting island at the end. Find the minimum number of air routes that can exist in this country.
Answer: 15 airlines.

Solution. Evaluation. Let islands $A$ and $B$ be connected by an airline. If we choose 9 islands for the tour, excluding $A$, then a circular path passes through $B$, meaning $B$ is connected by airlines to at least two other islands. Thus, at least three airlines depart from $B$, and a similar statement holds for each island. Therefore, there are at least $\frac{10 \cdot 3}{2}=15$ airlines.

![](https://cdn.mathpix.com/cropped/2024_05_06_420d04fb4cfc6e7f96a5g-3.jpg?height=411&width=463&top_left_y=1442&top_left_x=1456)

Example with 15 airlines - placing the islands at the vertices of a pentagonal prism. Denote the vertices of the prism as $A B C D E A_{1} B_{1} C_{1} D_{1} E_{1}$ and provide an example of a cycle that forms when one city is removed (for example, $E_{1}$, since all cities are equal): $A-A_{1}-$ $B_{1}-B-C-C_{1}-D_{1}-D-E-A$. The cycle is constructed, meaning this configuration satisfies the problem's condition.

Criteria. If the solution is incorrect - 0 points.

![](https://cdn.mathpix.com/cropped/2024_05_06_420d04fb4cfc6e7f96a5g-3.jpg?height=412&width=471&top_left_y=1867&top_left_x=1478)

Only the correct answer - 0 points.

If there is only a correct example without justification - 2 points.

If there is only a correct example with justification - 3 points.

If there is only a correct evaluation - 4 points.

If the solution is correct - 7 points.