One. (40 points) Let $n$ distinct positive integers $a_{1}$, $a_{2}, \cdots, a_{n}$ have a sum of 2000. Denote
$$
A=\max \left\{a_{1}, a_{2}, \cdots, a_{n}\right\} \text {. }
$$

Find the minimum value of $A+n$.
$$
\begin{array}{l}
\text { Let's assume } a_{1}=109. \\
\Rightarrow A+n=a_{n}+n \geqslant 110 .
\end{array}
$$

Furthermore,
$$
\begin{array}{r}
37+38+\cdots+73=\frac{(37+73) \times 37}{2}=2035, \\
\text { we can let } a_{1}=37-35, a_{i}=36+i(i=2,3, \cdots, 37) .
\end{array}
$$

Then $a_{1}+a_{2}+\cdots+a_{37}=2000$.
At this point, $A+n=73+37=110$.
Therefore, the minimum value of $A+n$ is 110.