IMO 1979 - Shortlisted problems Question 18

(POL 1) Let $m$ positive integers $a_1,\ldots,a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1,\ldots,b_n$ such that all sums of distinct $b_k$'s are distinct and all $a_i$ ($i\le m$) occur among them.