1994 IMO 第 2 题

A2 (FRA) ${ }^{\mathrm{IMO} 1}$ Let $m$ and $n$ be positive integers. The set $A=\left\{a_{1}, a_{2}, \ldots\right.$, $\left.a_{m}\right\}$ is a subset of $\{1,2, \ldots, n\}$. Whenever $a_{i}+a_{j} \leq n, 1 \leq i \leq j \leq m$, $a_{i}+a_{j}$ also belongs to $A$. Prove that $$\frac{a_{1}+a_{2}+\cdots+a_{m}}{m} \geq \frac{n+1}{2} .$$

A2 (FRA) ${ }^{\mathrm{IMO} 1}$ 设 $m$ 和 $n$ 为正整数。集合 $A=\left\{a_{1}, a_{2}, \ldots\right.$, $\left.a_{m}\right\}$ 是 $\{1,2, \ldots, n\}$ 的子集。每当 $a_{i}+a_{j} \leq n, 1 \leq i \leq j \leq m$ 时，$a_{i}+a_{j}$ 也属于 $A$。证明 $$\frac{a_{1}+a_{2}+\cdots+a_{m}}{m} \geq \frac{n+1}{2} 。$$