KMO 2005

For a positive integer $n$ , let $f(n)$ be the number of factors of $n^2+n+1$ . Show that there are infinitely many integers $n$ which satisfy $f(n) \geq f(n+1)$ .

A group of 6 students decided to make study groups and service activity groups according to the following principle:

Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.

Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.