Question 42
$n$ and $r$ are positive integers. Find the smallest $k$ for which we can construct $r$ subsets $A_1, A_2, \ldots, A_r$ of $\{0,1,2, \ldots, n-1\}$ each with $k$ elements such that each integer $0 \leq m \lt n$ can be written as a sum of one element from each of the $r$ subsets.