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 subset of \(X\). A function \(f \colon X \to \{1,2,\ldots,n+1\}\) is called nice if there exists an index \(k\) such that

\[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\]

Prove that the number of nice functions is at least \(n^n\).