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

$$
\left\{\begin{array}{l}
4^{x^{2}+2 x y+1}=(z+2) 7^{|y|-1} \\
\sin \frac{3 \pi z}{2}=1
\end{array}\right.
$$
# Solution:

From the second equation of the system, we obtain that $z=\frac{4 n+1}{3}$, where $n \in \mathbb{Z}$. The number $\mathbf{z}$ can have only the number 3 as its denominator. Therefore, the number 7, if it were in the numerator, would divide the right-hand side, but not the left-hand side. Therefore, $7^{1-|y|}$ is an integer, i.e., $|y|-1 \leq 0$ (otherwise, the numerator of the fraction in the right-hand side of the equation would be divisible by 7, while the left-hand side would not be divisible by 7). Consequently, $z+2=4^{x^{2}+2 x y+1} \cdot 7^{1-|y|}$ is an integer, and moreover, it is odd; otherwise, the second equation of the original system would not be satisfied, since $\sin \frac{3 \pi z}{2}=0$. But then the exponent of the number 4 is zero, i.e., $x^{2}+2 x y+1=0$, otherwise $\mathbf{z}$ 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 ( $1 ;-1 ;-1$ ) and ( $(-1 ; 1 ;-1)$ ). Answer: $(1 ;-1 ;-1),(-1 ; 1 ;-1)$.

| Points | Criteria for evaluating the performance of task № 4 |
| :---: | :--- |
| $\mathbf{11}$ | 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. |
| $\mathbf{9}$ | A correct sequence of all steps of the solution is provided. Gaps in the justification of the estimate for the quantity $\boldsymbol{y}$ or the oddness of the quantity $\mathbf{z}$ 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. |
| $\mathbf{7}$ | 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 quantity $\boldsymbol{y}$ is obtained); - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the oddness of the quantity $\mathbf{z}$ is not proven). The solution may be incomplete. |
| $\mathbf{5}$ | 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 quantity $\boldsymbol{y}$ is not obtained and the oddness of the quantity $\mathbf{z}$ is not proven). The solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not correspond to the above criteria for scoring 5, 7, 9, and 11 points. |