1. Determine all integer solutions of the equation

$$
\sqrt{x-\frac{1}{5}}+\sqrt{y-\frac{1}{5}}=\sqrt{5}
$$
Solution. The left side of the equation is defined for $x \geq \frac{1}{5}, y \geq \frac{1}{5}$, and since $x, y \in \mathbb{Z}$, we get $x \geq 1, y \geq 1$. Also, it must hold that $x-\frac{1}{5} \leq 5, y-\frac{1}{5} \leq 5$, i.e., $x \leq 5, y \leq 5$. Now, let's find the solutions of the equation for which $x \leq y$. We introduce the substitution $y=x+k$, so we need to find the solutions of the equation

$$
\sqrt{x+k-\frac{1}{5}}=\sqrt{5}-\sqrt{x-\frac{1}{5}}
$$

for which $x \in\{1, \ldots, 5\}, k \in\{0, \ldots, 5-x\}$. The last equation, after squaring twice, reduces to the equation

$$
x=\frac{k^{2}-10 k+29}{20}
$$

which, under the given conditions, has the unique solution $x=1, k=1$. Therefore, the only possible solutions to the original equation are $x=1, y=2$ and $x=2, y=1$. It is directly verified that these are indeed solutions to the original equation.