Consider two integers $n \ge m \ge 4$ and $A = \{a_1, a_2, ..., a_m\}$ a subset of the set $\{1, 2, ..., n\}$ such that: for all $a, b \in A, a \ne b$, if $a + b \le n$, then $a + b \in A$. Prove that $\frac{a_1 + a_2 + ... + a_m}{m} \ge \frac{n + 1}{2}$ .