There are 33 students in a class. Every pupil writes the number of other pupils in this class with the same first name as him- or herself onto the blackboard. This procedure is repeated, now with the number of students with the same surname.
In the end, of the 66 numbers on the blackboard, each of the numbers $0,1,2\ldots,9,10$ appears at least once.
Prove that at least two students in this class bear the same fore- and surname.
Remark: Every student in this class has exactly one first name and one last name (hence no middle name).
Update: I've thought about Tony's answer a bit, especially about the part "the only way this works is if there are 1 students with name A, 2 students with name B, 3 students with name C, etc up to 11 students with name K". I noticed that $1+2+\ldots+10+11=66=2\cdot33$ which means that these 1 to 11 students have to be equally distributed among first and last names (if you know what I mean). Is this just a coincidence or is this fact in any way relevant for solving the problem?