Question 1
Let $m$ and $n$ be positive integers. The set $A=\{a_1,a_2,\ldots,a_m\}$ is a subset of $\{1,2,\ldots,n\}$. Whenever $a_i+a_j\le n$, $1\le i\le j\le m$, $a_i+a_j$ also belongs to $A$. Prove that
$$\frac{a_1+a_2+\cdots+a_m}{m}\ge\frac{n+1}{2}.$$
$$\frac{a_1+a_2+\cdots+a_m}{m}\ge\frac{n+1}{2}.$$