ISL 2022 C5

Let $m, n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_{1}, X_{2}, \ldots, X_{m}$ be pairwise distinct non-empty, not necessary disjoint subsets of $X$. A function $f: X \rightarrow$ $\{1,2, \ldots, n+1\}$ is called nice if there exists an index $k$ such that $$\sum_{x \in X_{k}} f(x)>\sum_{x \in X_{i}} f(x) \text { for all } i \neq k$$ Prove that the number of nice functions is at least $n^{n}$. (Germany)