2008 IMO Shortlist N2

Let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct positive integers, $n \geq 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_{i}+a_{j}$ does not divide any of the numbers $3 a_{1}, 3 a_{2}, \ldots, 3 a_{n}$.

令 $a_{1}, a_{2}, \ldots, a_{n}$ 为不同的正整数，$n \geq 3$。证明存在不同的索引 $i$ 和 $j$，使得 $a_{i}+a_{j}$ 不能整除任何数字 $3 a_{1}, 3 a_{2}, \ldots, 3 a_{n}$。