KöMaL 2013 A601

Let $q \ge$1 be an integer. Prove that there is an integer $C_q$ such that for every finite set $A$ of integers |$A$+$q^. A$|$\ge$($q$+1)|$A$|-$C_q$ holds. ($A$+$q^. A$ is the set of those integers that can be expressed as $a$+$qa$' with some $a$,$a$'$\in A$.)