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One hundred and twenty students take an exam which is marked out of $100$ (with no fractional marks). No three students are awarded the same mark.

What is the smallest possible number of pairs of students who are awarded the same mark?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.