Question 184
Let $n$, $k$, $r$ be positive integers. Suppose we have a collection of sets $S_{1}, S_{2}, \ldots, S_{r}$, where each $S_{i}$ is a subset of $\{1,2, \ldots, n\}$ consisting of one or more consecutive integers. We say that such a collection is a $k$-fold perfect cover of $\{1,2, \ldots, n\}$ if each element of $\{1,2, \ldots, n\}$ occurs in exactly $k$ of the sets $S_{i}$. (As an example, $S_{1}=\{1\}$, $S_{2}=\{1,2,3\}$, $S_{3}=\{2\}$, $S_{4}=\{3,4,5\}$, $S_{5}=\{4,5\}$ is a 2-fold perfect cover when $n=5$.)
Given a collection of sets which is a $k$-fold perfect cover of $\{1,2, \ldots, n\}$, show that we can partition the collection into $k$ subcollections, each of which is a 1-fold perfect cover of $\{1,2, \ldots, n\}$.
Given a collection of sets which is a $k$-fold perfect cover of $\{1,2, \ldots, n\}$, show that we can partition the collection into $k$ subcollections, each of which is a 1-fold perfect cover of $\{1,2, \ldots, n\}$.