1. Find all pairs of natural numbers $x, y \in [1; 8]$, satisfying the equation

$$
\sqrt{x x, x x x \ldots}=y, y y y \ldots
$$

(decimal representation of each of the numbers $x x, x x x \ldots$ and $y, y y y \ldots$ consists of an infinite number of the same digits).
# Problem 1. 

Answer: $x=1, y=3$ and $x=4, y=6$.

Solution. Since an infinite decimal of the form $0, x x x \ldots$ is equal to $\frac{x}{9}$, the equation in the condition is equivalent to the relation

$$
11 x+\frac{x}{9}=\left(y+\frac{y}{9}\right)^{2} \Longleftrightarrow 9 x=y^{2}
$$

from which the answer is obtained.