The first five decimal places of the square root of a natural number are 44994, and the first five decimal places of the square root of the number that is 2 greater are 49444. What are these numbers?
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)
