KöMaL 2008 A469

Let 0$\le k \le n$ and $m \ge$2 be integers. Consider the $k$-element subsets of $\{1,2,\ldots,n\}$; in every such subset, compute the residue of the sum of elements, modulo $m$. Prove that if the $m$ residues are distributed uniformly - i.e. every residue occurs exactly $\frac{\binom{n}{k}}{m}$ times - then $n \ge m$.