Exercise 2. An East-West maritime line sees 10 ships depart each morning at distinct times, 5 ships depart from the West side and 5 from the East side. We assume that they all sail at the same speed and that as soon as two ships meet, they turn around and head back in the direction they came from, always at the same speed. What is the possible number of encounters between ships?
Solution to Exercise 2: The number of crossings does not change if we say that the boats continue straight instead of turning around when they meet. Each boat that departs from one side crosses all the other boats that depart from the other side. Therefore, there are $5 \times 5=25$ encounters.

Comment from the graders: The poorly worded statement of this exercise clearly confused the students, perhaps because it was not precise enough. Indeed, it was considered that all the boats were already in the water and had departed before the first encounter. It was then necessary to determine how many encounters there could be between the boats (not just the maximum number, but all possible numbers of encounters).

While many students gave the correct number (there are always 25), very few seemed to understand (or at least explain) that it did not actually depend on the gaps between the boats. Some students recognized that the changes in direction of the boats followed certain "phases" (the first two boats meet, head off in opposite directions, "transmit" this to the outermost boats, which head off in the opposite direction and "retransmit" the correct direction to the central boats, one by one), and tried to use this to arrive at 25. However, none justified the existence of these phases, nor why they could be treated separately, even though they do not really occur successively! To convince oneself of this, one can consider the case where the three boats coming from the west and the five boats coming from the east are very close to each other, but the last two boats coming from the west are much farther away. Other approaches were attempted, with generally insufficient justifications. It is important to remember that assertions, in an olympic exercise, are insufficient: they must be demonstrated. If one cannot demonstrate them, it is an even stronger sign that a proof is necessary. A better approach (which a few rare solutions found in various forms) was to note, as in the solution, that from the point of view of the number of encounters, the statement produces the same results as if the two boats continued their journey after a crossing. In this case, a crossing must involve two boats heading in opposite directions, and each boat coming from the west must cross each boat coming from the east, hence $5 \cdot 5=25$ crossings.