3. Find all integer solutions of the equation

$$
4 x+y+4 \sqrt{x y}-28 \sqrt{x}-14 \sqrt{y}+48=0 .
$$
3. The equation can be written in the form

$$
4 x+y+49+4 \sqrt{x y}-28 \sqrt{x}-14 \sqrt{y}=1
$$

or

$$
(2 \sqrt{x}+\sqrt{y}-7)^{2}=1
$$

From here it follows that $2 \sqrt{x}+\sqrt{y}-7= \pm 1$.

6 points

$$
\begin{aligned}
& 1^{\circ} 2 \sqrt{x}+\sqrt{y}-7=1 \quad \Rightarrow \quad \sqrt{y}=8-2 \sqrt{x} \\
& \sqrt{y} \geq 0 \quad \Rightarrow \quad 8-2 \sqrt{x} \geq 0 \quad \Rightarrow \quad \sqrt{x} \leq 4 \quad \Rightarrow \\
& x=0,1,4,9,16 \quad \text{and} \quad y=64,36,16,4,0 \\
& 2^{\circ} 2 \sqrt{x}+\sqrt{y}-7=-1 \quad \Rightarrow \quad \sqrt{y}=6-2 \sqrt{x} \\
& \sqrt{y} \geq 0 \quad \Rightarrow \quad 6-2 \sqrt{x} \geq 0 \quad \Rightarrow \quad \sqrt{x} \leq 3 \quad \Rightarrow \\
& x=0,1,4,9, \quad \text{and} \quad y=36,16,4,0
\end{aligned}
$$

7 points

7 points

Thus, the solutions are contained in the set

$$
\begin{gathered}
(x, y) \in\{(0,64),(1,36),(4,16),(9,4),(16,0) \\
(0,36),(1,16),(4,4),(9,0)\}
\end{gathered}
$$

5 points