DE-MO 2000 Problem 6

A sequence ( $a_n$ ) satisfies the following conditions: (i) For each $m \in N$ it holds that $a_{2^m} = 1/m$ . (ii) For each natural $n \ge 2$ it holds that $a_{2n-1}a_{2n} = a_n$ . (iii) For all integers $m,n$ with $2m > n \ge 1$ it holds that $a_{2n}a_{2n+1} = a_{2^m+n}$ . Determine $a_{2000}$ . You may assume that such a sequence exists.