Suppose that $2n$ tennis players compete in a round-robin tournament. Every player has exactly one match with every other player
during $2n-1$ consecutive days. Every match has a winner and a loser. Show that it is possible to select a winning player each day without selecting the same player twice. \\ \\ \textit{Hint: Remember Hall's Theorem}
given that Suppose that $2n$ tennis players compete in a round-robin tournament. Every player has exactly one match with every other player during $2n-1$ consecutive days.
this is going to be proved by contradiction
Let there be a winning player each day where same players wins twice, let n = 3
there are 6 tennis players and match occurs for 5days
from hall's theorem, let set n days where less than n players wining a day
let on player be loser which loses every single day in n days
so, players loose to n different players in n days
if he looses to n players then , n players are winner
but, we stated less than n players are winners in n days which is contradiction.
so,
we can choose a winning players each day without selecting the same players twice.
