Nineteen is the smallest possible number of pairs of students who
must be awarded the same mark.
This occurs when every possible mark from $0$ to $100$ is awarded
to at least one student. This accounts for $101$ students and
therefore the remaining $19$ students must all be awarded the same
mark as exactly one of their colleagues.
The $120$ students are made up of $19$ pairs of students who are
awarded the same mark and $82$ students who are all awarded a
different mark from everyone else.
This problem is taken from the UKMT Mathematical Challenges.