2. Let $x, y, z$ be distinct non-zero digits.

a) Determine the rational values of the number $a=\sqrt{\overline{0, x(y)}+\overline{0, y(z)}+\overline{0, z(x)}}$

b) Find the rational values of the number $a=\sqrt{0,00 \ldots . .0 x(y)}+\overline{0,00 \ldots .0 y(z)}+\overline{0,00 \ldots . .0 z(x)}$, where after the decimal point in each number there are $n$ zeros, $n \in \mathbb{N}$.

Octavia Popa, Brăila
Solution. a) $a=\sqrt{\frac{10(x+y+z)}{90}}=\sqrt{\frac{x+y+z}{9}}=\frac{\sqrt{x+y+z}}{3}$

$a \in \mathbb{Q} \Leftrightarrow x+y+z=k^{2}, k \in \mathbb{N}$

Since $1 \leq x<y<z \leq 8 \Rightarrow k^{2} \in\{9,16\}$. Therefore $a \in\left\{1, \frac{4}{3}\right\}$.

b) For $n=2 k, k \geq 1, k \in \mathbb{N}$, we have $a=\sqrt{\frac{10(x+y+z)}{9 \cdot 10^{n+1}}}=\frac{\sqrt{x+y+z}}{3 \cdot 10^{k}}$ and $a \in\left\{\frac{1}{10^{k}}, \frac{4}{3 \cdot 10^{k}}\right\}$

For $n=2 k+1, k \in \mathbb{N}$, we have $a=\frac{\sqrt{10(x+y+z)}}{3 \cdot 10^{k+1}}$ for $a \in \mathbb{Q} \Rightarrow x+y+z=10$ and $a \in\left\{\frac{1}{3 \cdot 10^{k}}\right\}$.