12. $\quad$ Maximum 15 points. Let $[x]$ denote the integer part of the number $x$ (i.e., the greatest integer not exceeding $x$). Solve the system of equations:

$$
\left\{\begin{array}{c}
{[x+y-3]=2-x} \\
{[x+1]+[y-7]+x=y}
\end{array}\right.
$$
# Solution:

Since for any $x \in \mathbb{R}, n \in \mathbb{Z}$ the equality holds:

$$
[x+n]=[x]+n
$$

the system is equivalent to the system:

$$
\left\{\begin{array}{c}
{[x+y]-5=-x} \\
{[x]+[y]+x-6=y}
\end{array}\right.
$$

From the first equation, it follows that $x \in \mathbb{Z}$ (it is sufficient to note that each term in the left-hand side of the first equation is an integer). Then, from the second equation, it follows that $y \in \mathbb{Z}$. Therefore, we need to solve the system in integers:

$$
\begin{gathered}
\left\{\begin{array}{c}
x+y-5=-x \\
x+y+x-6=y
\end{array}\right. \\
\left\{\begin{array}{c}
2 x+y=5 \\
x=3
\end{array}\right. \\
\left\{\begin{array}{c}
y=-1 \\
x=3
\end{array}\right.
\end{gathered}
$$

Answer: $(3, -1)$

| Criterion | Score |
| :--- | :---: |
| The correct answer is obtained through a logically justified algorithm | 15 |
| The solution contains logical flaws / the transition to an equivalent system of equations is not indicated, but the overall solution is correct and the correct answer is obtained | 10 |
| The algorithm is correct, but due to an arithmetic error, an incorrect answer is obtained | 6 |
| The answer is correct, but there is no logically justified algorithm | 2 |
| The solution does not meet any of the presented criteria | 0 |