4. Determine all pairs of real numbers $x, y$ that satisfy the system of equations

$$
\begin{aligned}
& \lfloor x+y\rfloor=2010 \\
& \lfloor x\rfloor-y=p
\end{aligned}
$$

if a) $p=2$, b) $p=3$.

The symbol $\lfloor x\rfloor$ denotes the greatest integer that is not greater than the given real number $x$ (the so-called floor function of the real number $x$).

The regional round of category C takes place

on Tuesday, March 30, 2010

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 graphical display. These details will be communicated to the students before the start of the competition.
4. Since the number $p$ is an integer, $y=\lfloor x\rfloor-p$ is also an integer and $\lfloor x+y\rfloor=\lfloor x\rfloor+y$. The original system of equations is thus equivalent to the system

$$
\begin{aligned}
& \lfloor x\rfloor+y=2010, \\
& \lfloor x\rfloor-y=p,
\end{aligned}
$$

which we can easily solve, for example, by the addition method. We obtain $\lfloor x\rfloor=\frac{1}{2}(2010+p)$ (which is valid only for even $p$) and $y=\lfloor x\rfloor-p$.

a) For $p=2$, the solution to the system is any $x \in [1006,1007)$ and $y=1004$.

b) For $p=3$, the system has no solution.

Another solution. Let $\lfloor x\rfloor=a$, then $x=a+t$, where $t \in [0,1)$.

a) For $p=2$, we rewrite the system as $y=a-2$ and $\lfloor 2a-2+t\rfloor=2010$. From the last equation, we get $2a-2=2010$, hence $a=1006$. Since $t \in [0,1)$, any $x \in [1006,1007)$ satisfies the original system, with $y=1004$.

b) For $p=3$, we get $y=a-3$ and $\lfloor 2a-3+t\rfloor=2010$. The last equation is equivalent to $2a-3=2010$, which has no integer solution for $a$. For $p=3$, the given system of equations has no solution.

For a complete solution, award 6 points, with 4 points for solving part a) and 2 points for part b).