Question 31
A number of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a single and that between a boy and a girl was called a mixed single. The total number of boys differed from the total number of girls by at most $1$. The total number of singles differed from the total number of mixed singles by at most $1$. At most how many schools were represented by an odd number of players?