Estonian Math Competitions

If the number of students is $m^2 - 1 = (m-1)(m+1)$, let us split the students to $m-1$ groups, each with $m+1$ students. Assume that each student has a unique common hobby with every other student in the same group and none of the students is practising any other hobby. Then each student has exactly $m$ hobbies but among each $m$ students at least two of them belong to the same group, i.e., have a common hobby. Hence the conditions of the problem are satisfied but there exists no hobby which is practised by more than 2 students. If there are less than $m^2 - 1$ students at the school, similar construction can be used with a suitable number of students omitted.

Let us show that if there are at least $m^2$ students at school, then there must exist a hobby that is practised by at least 3 students. Assume the contrary that every hobby is practised by 2 students at most. Let us form a group of students in which each pair of students have no hobby in common by adding in each iteration to initially empty group one student who does not have a common hobby with any of the students already assigned to the group. As every student has $m$ hobbies the most and each hobby can be common with only one other student, after $i$ iterations there are at most $i(m + 1)$ students who cannot be added to the group in the subsequent iterations. As $(m - 1)(m + 1) < m^2$, it is still possible to add another student to a group of $m - 1$ students. Therefore, there exists a group of $m$ students, in which there exists no pair who have a common hobby. That contradicts the conditions of the problem statement.