2023 IMO Shortlist A3

Let $x_{1}, x_{2}, \ldots, x_{2023}$ be distinct real positive numbers such that $$a_{n}=\sqrt{\left(x_{1}+x_{2}+\cdots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\right)}$$ is an integer for every $n=1,2, \ldots, 2023$. Prove that $a_{2023} \geq 3034$. (Netherlands)

令 $x_{1}, x_{2}, \ldots, x_{2023}$ 为不同的实正数，使得 $$a_{n}=\sqrt{\left(x_{1}+x_{2}+\cdots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\right)}$$ 是每个 $n=1,2, \ldots, 2023$ 的整数。证明 $a_{2023} \geq 3034$。 (荷兰)