Encountered this question while practicing. Sadly there was no solution given. Sounds like a PHP question but with so little information given I am wondering how it can be solved. Any hints/help would be greatly appreciated. Thanks.
21 students took an exam and their scores sum to 200. If the scores are nonnegative integers, prove that there are two students with the same score.
This can be shown by a contradiction. Assume for the contradiction that no two students received the same same score. Then, a lower bound on the total sum of scores is \begin{align} \sum_{i=1}^{21} = \frac{1}{2}21\cdot 22 = 231 > 200. \end{align}
Hence at least two students received the same score.
(Or, assuming scores include 0, the lower bound on the total score is $\frac{1}{2}20\cdot 21 = 210.) $