## Task 2 - 301222

Determine all pairs $(x ; y)$ of integers $x$ and $y$ that satisfy the following system of inequalities (1) and (2):

$$
\begin{aligned}
2 x^{2}+2 y^{2}-12 x+20 y+65 & \leq 5
\end{aligned}
$$
By completing the square, one can also write (1) in the form

$$
(x-3)^{2}+(y+5)^{2}<\frac{3}{2}
$$

from which it immediately follows that

$$
(x, y) \in\{(3,-6),(2,-5),(3,-5),(4,-5),(3,-4)
$$

for the integer solutions $(x, y)$. However, of these pairs, only $(x, y)=(4,-5)$ also satisfies the second condition, which is thus the only solution here.