2022 IMO Shortlist C5

Let $m, n \geq 2$ 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)

令 $m, n \geq 2$ 为整数，令 $X$ 为包含 $n$ 个元素的集合，并令 $X_{1}, X_{2}, \ldots, X_{m}$ 为 $X$ 的成对不同非空、不必要的不相交子集。如果存在索引 $k$ 且 $$\sum_{x \in X_{k}} f(x)>\sum_{x \in X_{i}} f(x) \text { for all } i \neq k$$ ，则函数 $f: X \rightarrow$ $\{1,2, \ldots, n+1\}$ 被称为 Nice 函数。证明 Nice 函数的数量至少为 $n^{n}$。 (德国)