5. Twenty students participating in a math camp have decided to send messages to each other, with each of them sending messages to exactly ten of the remaining students. Determine the minimum possible number of mutual messages, i.e., find an example of a message-sending schedule in which the number of mutual messages is the smallest possible and prove that a smaller number of mutual messages cannot be achieved.

(A message between students $\mathrm{A}$ and $\mathrm{B}$ is considered mutual if student $\mathrm{A}$ has sent a message to student B and student B has sent a message to student A.)
5. A total of $20 \cdot 10=200$ messages were sent, and the number of pairs of students is $\frac{20 \cdot 19}{2}=190$.

Therefore, at least $200-190=10$ messages must be sent between the same pairs of students, so the minimum possible number of mutual messages is at least 10.

Now, let's construct a schedule of sent messages such that the number of mutual messages is exactly 10.

Let the students be arranged in a circle, and let each student send a message to the ten students who are located after them in the clockwise direction. Note that in this arrangement, only students who are directly opposite each other (diametrically opposite) will send mutual messages.

![](https://cdn.mathpix.com/cropped/2024_05_30_8091e4f2781742023a76g-21.jpg?height=686&width=696&top_left_y=1727&top_left_x=680)

(Each point represents one student, and the directed line segment represents a one-way message.)

This constructs the desired message schedule, where the number of mutual messages is exactly 10. From the above, we can conclude that the minimum possible number of mutual messages is 10.