57. In Ruritania, every two of the $n$ cities are connected by a direct flight operated by one of two airlines, Alpha or Beta. The antimonopoly committee wants at least $k$ flights to be operated by Alpha. To achieve this, the committee can, every day, choose any three cities and change the ownership of the three flights connecting these cities (i.e., take each of these flights from the company that operates it and transfer it to the other). For what largest $k$ can the committee definitely achieve its goal in some time, regardless of how the flights are currently distributed?
57. Answer: when $k=\frac{n(n-1)}{2}-\left[\frac{n}{2}\right]$.

If from some city $A$ there are two flights of the Beta airline - to cities $B$ and $C$, then by applying the operation to the triplet of cities $(A, B, C)$, we will reduce the total number of flights operated by the Beta airline. Acting in this way, we can achieve that from each city no more than one Beta flight departs, which means the total number of its flights does not exceed $[n / 2]$.

Next, note that for any fixed city $A$, the parity of the number of Beta airline flights departing from city $A$ does not change. In particular, in a country where initially all cities (except, perhaps, one city $X$) are paired with flights of the Beta airline, and there are no other Beta flights, the number of Beta airline flights cannot be reduced using the specified operations, since in such a country from each city, except $X$, there is guaranteed to be at least one Beta flight.