## Task A-2.3.

Determine all triples $(x, y, z)$ of positive real numbers that satisfy the system of equations

\[
\begin{aligned}
3\lfloor x\rfloor - \{y\} + \{z\} & = 20.3 \\
3\lfloor y\rfloor + 5\lfloor z\rfloor - \{x\} & = 15.1 \\
\{y\} + \{z\} & = 0.9
\end{aligned}
\]

For a real number $t$, $\lfloor t\rfloor$ denotes the greatest integer less than or equal to $t$, and $\{t\}$ denotes its fractional part, $\{t\} = t - \lfloor t\rfloor$. For example, if $t = 15.1$, then $\lfloor t\rfloor = 15$ and $\{t\} = 0.1$.
## Solution.

By adding the first and third equations, we get

$$
3\lfloor x\rfloor+2\{z\}=21.2
$$

We have two possibilities: $\{z\}=0.1$ and $\{z\}=0.6$.

However, only for $\{z\}=0.1$ is $\lfloor x\rfloor$ an integer. Therefore, it must be $\{z\}=0.1$ and $\lfloor x\rfloor=7$. \quad 2 points

From the third equation, it follows that $\{y\}=0.8$.

1 point

From the second equation, we see that $\{x\}=0.9$ must hold.

We also have

$$
3\lfloor y\rfloor+5\lfloor z\rfloor=16
$$

Since $y$ and $z$ are positive, the only possibility is $\lfloor y\rfloor=2$ and $\lfloor z\rfloor=2$.

The system has only one solution: $x=7.9, y=2.8, z=2.1$.