2017 IMO Shortlist N6

Find the smallest positive integer $n$, or show that no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers ( $a_{1}, a_{2}, \ldots, a_{n}$ ) such that both $$a_{1}+a_{2}+\cdots+a_{n} \quad \text { and } \quad \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}$$ are integers. (Singapore)

找到最小的正整数 $n$，或表明不存在这样的 $n$，具有以下属性：存在无限多个不同的正有理数 $n$ 元组 ( $a_{1}, a_{2}, \ldots, a_{n}$ )，使得 $$a_{1}+a_{2}+\cdots+a_{n} \quad \text { 和 } \quad \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}$$ 是整数。 (新加坡)