CGMO 2005 Problem 4

Determine all positive real numbers $a$ such that there exists a positive integer $n$ and sets $A_1, A_2, \ldots, A_n$ satisfying the following conditions:

(1) every set $A_i$ has infinitely many elements; (2) every pair of distinct sets $A_i$ and $A_j$ do not share any common element (3) the union of sets $A_1, A_2, \ldots, A_n$ is the set of all integers; (4) for every set $A_i,$ the positive difference of any pair of elements in $A_i$ is at least $a^i.$