MN-MO 2002 Problem 1

Let $n,k$ be given natural numbers. Find the smallest possible cardinality of a set $A$ with the following property: There exist subsets $A_1,A_2,\ldots,A_n$ of $A$ such that the union of any $k$ of them is $A$ , but the union of any $k-1$ of them is never $A$ .