Question 34
In a group of 2022 students, some are friends with each other, and friendship is always mutual. We know that every subset of these students has the following property: there always exists a student in the subset who is friends with at most 100 students of the same subset.
a) Determine the smallest positive integer $N$ that guarantees the following property: it is possible to divide the students into $N$ groups (not necessarily of the same size) so that two students in the same group are never friends with each other.
b) We number the students from 1 to 2022. Let $c_i$ be the number of friends of student $i$. Determine the maximum value that the sum
$$
c_1 + c_2 + \dots + c_{2022}
$$
can take.
a) Determine the smallest positive integer $N$ that guarantees the following property: it is possible to divide the students into $N$ groups (not necessarily of the same size) so that two students in the same group are never friends with each other.
b) We number the students from 1 to 2022. Let $c_i$ be the number of friends of student $i$. Determine the maximum value that the sum
$$
c_1 + c_2 + \dots + c_{2022}
$$
can take.