Question 62
Let $n$ be a positive integer. For every pair of students enrolled in a certain school having $n$ students, either the pair are mutual friends or not mutual friends. Let $N$ be the smallest possible sum $a+b$ of positive integers $a$ and $b$ satisfying the following 2 conditions concerning students in this school:
(1) It is possible to divide students into $a$ teams in such a way that any pair of students belonging to a same team are mutual friends.
(2) It is possible to divide students into $b$ teams in such a way that any pair of students belonging to a same team are not mutual friends.
Assume that every student will belong to one and only one team when the students are divided to form teams to satisfy the conditions (1) and (2) above, and a team may consist of only one student, in which case this team is assumed to satisfy both of the conditions: that any pair of students in this team are mutual friends; are not mutual friends. Determine in terms of $n$ the maximum possible value that $N$ can take.
(1) It is possible to divide students into $a$ teams in such a way that any pair of students belonging to a same team are mutual friends.
(2) It is possible to divide students into $b$ teams in such a way that any pair of students belonging to a same team are not mutual friends.
Assume that every student will belong to one and only one team when the students are divided to form teams to satisfy the conditions (1) and (2) above, and a team may consist of only one student, in which case this team is assumed to satisfy both of the conditions: that any pair of students in this team are mutual friends; are not mutual friends. Determine in terms of $n$ the maximum possible value that $N$ can take.