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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.