50. In the city, 2019 metro stations have been built. Some pairs of stations are connected by tunnels, and from any station, you can reach any other station via the tunnels. The mayor ordered the organization of several metro lines: each line must include several different stations, sequentially connected by tunnels (the same tunnel can be part of several lines). Each station must lie on at least one line. To save funds, no more than $k$ lines should be created.

It turned out that the mayor's order was impossible to implement. For what largest $k$ could this have happened? (S. Berlov)
50. Answer: $k=1008$. Let's provide an example of a connected graph with 2019 vertices that cannot be covered by 1008 simple paths: one vertex is connected to 2018 vertices of degree 1. Any simple path contains no more than two pendant vertices, so at least 1009 paths are required for coverage.

We will prove that 1009 paths are always sufficient. Choose any vertex $A$ of degree at least 2 (such a vertex obviously exists), and two of its neighbors $B$ and $C$. The remaining 2016 vertices can be arbitrarily divided into 1008 pairs, and we will draw simple paths between the vertices in each pair, as well as the path $B A C$ - a total of 1009 paths.