In a school, there are $m$ teachers and $n$ students. We assume that each teacher has exactly $k$ students, and that any two students always have exactly $\ell$ teachers in common. Determine a relation between $m, n, k, \ell$.
We count $T$ = the number of student-student-teacher triplets where the teacher is common to the two students.

Method 1: Each teacher has $\binom{k}{2}$ pairs of students in their class, and thus

$$
T=m \times \binom{k}{2}
$$

Method 2: There are $\binom{n}{2}$ possible pairs of students, and thus

$$
T=\binom{n}{2} \times \ell
$$

We then obtain the relation

$$
n(n-1) \ell=m k(k-1)
$$

## 2 Power of a Point (Benoît & Martin)

In this course, we introduce the concept of the power of a point, which can be found in the first geometry course of group $C$ in the Montpellier 2014 course booklet. We then worked on the following exercises:

## Exercises