4. Find all triples of real numbers $x, y, z$ for which

$$
\lfloor x\rfloor-y=2 \cdot\lfloor y\rfloor-z=3 \cdot\lfloor z\rfloor-x=\frac{2004}{2005}
$$

where $\lfloor a\rfloor$ denotes the greatest integer not exceeding the number $a$.

The second round of category B takes place

on Tuesday, March 22, 2005

so that it starts in the morning and the contestants have 4 hours of pure time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphic display. This information will be communicated to the students before the start of the competition.
4. We can equivalently rewrite the system of equations in the form

$$
\begin{aligned}
& y=\lfloor x\rfloor-\alpha, \\
& z=2\lfloor y\rfloor-\alpha, \\
& x=3\lfloor z\rfloor-\alpha,
\end{aligned}
$$

where we denote by $\alpha$ the number $\frac{2004}{2005}$ from the interval $(0,1)$. From the system (1), the following equalities follow:

$$
\begin{aligned}
& \lfloor y\rfloor=\lfloor x\rfloor-1, \\
& \lfloor z\rfloor=2\lfloor y\rfloor-1=2\lfloor x\rfloor-3, \\
& \lfloor x\rfloor=3\lfloor z\rfloor-1=6\lfloor x\rfloor-10 .
\end{aligned}
$$

From the last equation, we get $\lfloor x\rfloor=2$ and from the remaining two equations, we further calculate $\lfloor y\rfloor=$ $=\lfloor z\rfloor=1$. Substituting into (1), we have $x=3-\frac{2004}{2005}=2+\frac{1}{2005}, y=2-\frac{2004}{2005}=1+\frac{1}{2005}$ and $z=2-\frac{2004}{2005}=1+\frac{1}{2005}$. The resulting non-integer numbers $x, y$ and $z$ have exactly the integer parts that we substituted into the right-hand sides of the equations (1). Thus, we have simultaneously performed the verification (which can also be done by direct substitution into the original system). The given triplet is the (unique) solution to the given problem.

Award 6 points for a complete solution, of which 4 points for deriving the relationship for one integer part. Completing the calculation is worth 2 points, and deduct 1 point for the missing mention of the verification.