Balkan MO problem Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$-element set $A$. If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x, y$, prove that $2^k\geq n$.  
This is a question from Balkan MO 1997, and I did not quite understand the question, so I could not make any attempt. Please help.

Here's a rephrasing: let $S=\{A_1,\ldots,A_k\}$ be a collection of subsets of an $n$-element set $A.$   Let $x,$ $y$ be a pair of elements of $A.$  We will say that $S$ has the XOR property for $(x,y)$ if at least one element of $S$ contains only one of $x,$ $y.$  In other words, $S$ would fail to have the XOR property for $x,$ $y$ if every element of $S$ contained either both $x$ and $y$ or neither $x$ nor $y.$
The question asks you to prove that if $S$ has the XOR property for all pairs of elements of $A,$ then $S$ can't be too small.  In particular, $S$ must consist of at least $\log_2 n$ subsets of $A.$
