International Mathematical Olympiad 1988 Problem 2

Let $n$ be a positive integer and let $A_1$, $A_2$, \ldots, $A_{2n+1}$ be subsets of a set $B$. Suppose that

(a) Each $A_i$ has exactly $2n$ elements,

(b) Each $A_i \cap A_j$ ($1 \leq i < j \leq 2n + 1$) contains exactly one element, and

(c) Every element of $B$ belongs to at least two of the $A_i$.

For which values of $n$ can one assign to every element of $B$ one of the numbers $0$ and $1$ in such a way that $A_i$ has $0$ assigned to exactly $n$ of its elements?