4. Considering $\boldsymbol{x}$ and $\boldsymbol{y}$ as integers, solve the system of equations (11 points):

$$
\left\{\begin{array}{l}
8^{x^{2}-2 x y+1}=(z+4) 5^{|y|-1} \\
\sin \frac{3 \pi z}{2}=-1
\end{array}\right.
$$
# Solution:

From the second equation of the system, we obtain that $z=\frac{4 n+3}{3}$, where $n \in \mathbb{Z}$. The number $z$ can only have 3 as its denominator. Therefore, if the number 5 were in the numerator, it would divide the right-hand side but not the left-hand side. Therefore, $5^{1-|y|}$ is an integer, i.e., $|y|-1 \leq 0$ (otherwise, the numerator of the fraction on the right-hand side of the equation would be divisible by 5, while the left-hand side would not be). Consequently, $z+4=8^{x^{2}-2 x y+1} \cdot 5^{1-|y|}$ is an integer, and it is odd; otherwise, the second equation of the original system would not hold, as $\sin \frac{3 \pi z}{2}=0$. But then the exponent of the number 8 is zero, i.e., $x^{2}-2 x y+1=0$, otherwise $z$ would be even. From $x^{2}-2 x y+1=0$, we get $(x-y)^{2}=y^{2}-1$. Considering that $|y|-1 \leq 0$, we have $y^{2}-1=0$ and $x=y$. Therefore, the solutions to the original system are the triples of numbers (-1;-1;3) and (1;1;-3).

Answer: $(-1;-1;-3),(1;1;-3)$.

| Points | Criteria for evaluating the performance of task № 4 |
| :---: | :---: |
| 12 | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are performed correctly. The correct answer is obtained. |
| 9 | A correct sequence of all steps of the solution is provided. Gaps in the justification of the estimate for the value of $\boldsymbol{y}$ or the oddness of the value of $z$ or a computational error or typo that does not affect the further course of the solution are allowed. As a result of this error or typo, an incorrect answer may be obtained. |
| 6 | The problem is not solved, but its solution is significantly advanced, i.e.: - a significant part of the solution is performed correctly, possibly inaccurately (for example, an estimate for the value of $\boldsymbol{y}$ is obtained); - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the oddness of the value of $z$ is not proven). The solution may be incomplete. |
| 3 | The problem is not solved, but there is some progress in its solution, i.e.: - the initial part of the solution is performed correctly (for example, the trigonometric equation is solved correctly); - the main part is either not performed or performed incorrectly, possibly even with logical errors (for example, an estimate for the value of $\boldsymbol{y}$ is not obtained and the oddness of the value of $z$ is not proven). The solution may be incomplete. |
| 0 | All cases of the solution that do not correspond to the above criteria for scoring 3, 6, 9, and 12 points. |