Exercise 1. Fifteen students participate in a mathematics workshop. Each evening, three of them go to eat ice cream. At the end of the workshop, it turns out that any two students have always gone to eat ice cream together exactly once. How many days did the workshop last? Justify your answer.
Solution to Exercise 1 If we choose a student $A$, then another student $B$, we thus form $15 \times 14$ pairs of students, but each pair is counted twice since to form the pair $\{A, B\}$ we can first choose $A$ then $B$, or first choose $B$ then $A$. Consequently, there are a total of $15 \times 14 / 2 = 105$ pairs of students.

If 3 students $A, B, C$ go to eat ice cream on the same evening, there are thus three pairs $\{A, B\}, \{B, C\}$, and $\{C, A\}$ of students who will simultaneously eat ice cream. We deduce that each evening, three pairs of students will eat ice cream, so there are a total of $105 / 3 = 35$ evenings.