Question 133
Let $m$ and $n$ be positive integers, while $p$ is a prime number. Find the maximal $s \in \mathbb{N}$ (as a function of $m, n$, and $p$) such that from an arbitrary group of $mnp$ positive integers, one can always choose $snp$ among them, satisfying the following property: The numbers can be split into $s$ disjoint subsets of $np$ elements, such that the sum of the elements in each of the subsets gives the same residue modulo $p$.