SILK-ROAD 2009 Problem 3

A tourist going to visit the Complant, found that: a) in this country $1024$ cities, numbered by integers from $0$ to $1023$ , b) two cities with numbers $m$ and $n$ are connected by a straight line if and only if the binary entries of numbers $m$ and $n$ they differ exactly in one digit, c) during the stay of a tourist in that country $8$ roads will be closed for scheduled repairs. Prove that a tourist can make a closed route along the existing roads of Complant, passing through each of its cities exactly once.