Question 144
There are $n$ islands, where $n \ge 3$. A ferry company sets up routes between some pairs of islands such that whenever we split the islands into two non-empty groups, we can always find one island from each group such that there is a route between these two islands. At the end of every year, the company chooses two islands $X$ and $Y$ connected by a route and closes the route between them. At the same time, for every other island $A$: if $A$ has a route to $X$ but not to $Y$, the company adds a route between $A$ and $Y$; conversely, if $A$ has a route to $Y$ but not to $X$, the company adds a route between $A$ and $X$.
For two groups of islands $\mathcal{A}$ and $\mathcal{B}$, if $A \in \mathcal{A}$ and $B \in \mathcal{B}$, we say that the route linking $A$ and $B$ connects $\mathcal{A}$ and $\mathcal{B}$. Suppose we know that at any moment, if we split the islands into two non-empty groups, then the company will certainly close a route connecting these two groups within some number of years. Prove that after some years there will be an island that has routes to all other islands.
For two groups of islands $\mathcal{A}$ and $\mathcal{B}$, if $A \in \mathcal{A}$ and $B \in \mathcal{B}$, we say that the route linking $A$ and $B$ connects $\mathcal{A}$ and $\mathcal{B}$. Suppose we know that at any moment, if we split the islands into two non-empty groups, then the company will certainly close a route connecting these two groups within some number of years. Prove that after some years there will be an island that has routes to all other islands.