Question 3
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 $a_{2023} \geqslant 3034$.
$$
a_{n}=\sqrt{\left(x_{1}+x_{2}+\cdots+x_{n}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\right)}
$$
is an integer for every $n=1,2, \ldots, 2023$. Prove that $a_{2023} \geqslant 3034$.