Mongolian Mathematical Olympiad

Let us denote the left side of the given inequality by $S$.

If $u\le \sqrt{2123}$, then $S\le u\le \sqrt{2123}$.

If $\frac{100}{v}\le \sqrt{2123}$, then $S\le \frac{100}{v}\le \sqrt{2123}$.

If $u\ge \sqrt{2123}$ and $\frac{100}{v}\ge \sqrt{2123}$, then we have $$S\le v+\frac{2023}{u}\le \frac{100}{\sqrt{2123}}+\frac{2023}{\sqrt{2123}}=\sqrt{2123}.$$ The equality holds when $u=\sqrt{2123}$ and $v=\frac{100}{\sqrt{2123}}$.