Question 60
For a positive integer $m$, we denote by $[m]$ the set $\{1,2, \ldots, m\}$. Let $n$ be a positive integer and let $\mathcal{S}$ be a non-empty collection of subsets of $[n]$. A function $f:[n] \rightarrow [n+1]$ is called kawaii if there exists $A \in \mathcal{S}$ such that for all $B \in \mathcal{S}$ with $A \neq B$ we have
$$
\sum_{a \in A} f(a) \gt \sum_{b \in B} f(b)
$$
Prove that there are always at least $n^{n}$ kawaii functions, irrespective of $\mathcal{S}$.
$$
\sum_{a \in A} f(a) \gt \sum_{b \in B} f(b)
$$
Prove that there are always at least $n^{n}$ kawaii functions, irrespective of $\mathcal{S}$.