I'm trying to solve some problem in the past few days(by the way, my first question here is some sort of a direction for solution - or maybe not).
Problem: Suppose that we have a list of $(n-1)^2-1$ non negative integers, where $n$ is odd number. Then there are $n$ integers from that list that their sum is dividable by $n$.
First case: where all remainders of these numbers divided by $n$ are obtained. Then, we sum these numbers, the sum is dividable by $n$ since
Second case: where not all remainders of these numbers divided by $n$ are obtained. Suppose that only $k<n$ remainders obtained. Then I thought use Pigeonhole principle to show that there exist a remainder(as a set) which contains at lesat $n$ integers from the list given. But I can't find a way showing that.