Question 33
Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2k$ players, where:
(1) every two players play exactly one match against each other;
(2) exactly one match is played each day;
(3) each player checks into the hotel on the day of his first match and leaves the hotel on the day of his last match.
For every day a player stays in the hotel, the organisers must pay $1$ dollar, so the committee wants to minimise the total amount paid to the hotel. Find, with proof, the minimum possible total hotel cost.
(1) every two players play exactly one match against each other;
(2) exactly one match is played each day;
(3) each player checks into the hotel on the day of his first match and leaves the hotel on the day of his last match.
For every day a player stays in the hotel, the organisers must pay $1$ dollar, so the committee wants to minimise the total amount paid to the hotel. Find, with proof, the minimum possible total hotel cost.