Let the two integers in question be $n$ and $n+2$. The integer parts of their square roots must be the same, otherwise the difference between the two square roots would be more than 1, but

$$\sqrt{n+2}-\sqrt{n}=\frac{2}{\sqrt{n+2}+\sqrt{n}}\frac{1}{0.0446}-1>\frac{1}{0.045}-1=\frac{191}{9}>21$$

Since $A$ is an integer, only 22 can fit. Then $n$ is the integer between $22.44^2$ and $22.45^2$, that is,

$$503.5536<n<504.0025$$

so $n=504$, and indeed

$$\begin{aligned} & \sqrt{504}=22.449944 \ldots \\ & \sqrt{506}=22.494443 \ldots \end{aligned}$$

Agnes Baranyai (Budapest, Martos F. Secondary School II. grade)