KöMaL 2012 A569

Call a set $A\subset\mathbb{Z}$ interesting if 2$y$-$x \in A$ for all pairs $x$,$y \in A$ with $x$<$y$. Let $a_1$<$a_2$<...<$a_k$ be positive integers ($k \ge$2) whose greatest common divisor is 1. Prove that if $A$ is an interesting set and $\{0,a_1,\ldots,a_k\}\subset A$ then $a_1$+$a_k$-3$\in A$.