Exercise 4. 2024 students, all of different heights, must line up in a single file. However, each student does not want to have both a shorter student in front of them and a shorter student behind them. How many ways are there to form such a line?
## Solution to Exercise 4

Let's look at the tallest student: if they are not placed at the very beginning or the very end of the line, they are between two smaller students. Thus, the tallest student must be placed at the beginning or the end.

By this reasoning, and by looking at small cases, we can conjecture that if we need to place \( n \geqslant 1 \) students all of different heights with the constraint given in the problem, there are \( 2^{n-1} \) possibilities. Let's prove this by induction on \( n \).

Initialization: for \( n=1 \) student, there is only one possible line and \( 1=2^{n-1} \).

Hereditary: suppose the hypothesis is true at rank \( n \) for some \( n \geqslant 1 \), let's show that it is true at rank \( n+1 \). The tallest student is necessarily placed either at the very beginning or at the very end. If we remove this student from the line, the line of \( n \) people still satisfies the problem statement: there are therefore potentially \( 2^{n-1} \) possibilities for the line without the tallest person. If we add the tallest person at the beginning or the end of the line, the line still satisfies the problem statement (since the person next to the tallest will not be between two smaller people), and since there are two choices for placing the tallest person, there are \( 2^n = 2^{n+1-1} \) possibilities to form the line, which concludes the induction.

Thus, by applying the property for \( \mathfrak{n}=2024 \), there are \( 2^{2023} \) possibilities.

Graders' Comment: The exercise was very well done overall, with few errors. Two main methods of proof were used: the first is similar to what the statement does, and consists of seeing how to place the students, one by one. The second consists of understanding what the "shape" of the distribution of students will be, in a V-shape, and counting these configurations by various arguments.