PROBLEM 2. If $x, y>0$ are two real numbers that satisfy the relation

$$
2 \sqrt{x}+\sqrt{y}=\sqrt{(x+1)(y+4)}
$$

calculate the geometric mean of the numbers $x$ and $y$.
PROBLEM 2. If $x, y>0$ are two real numbers that satisfy the relation

$$
2 \sqrt{x}+\sqrt{y}=\sqrt{(x+1)(y+4)}
$$

calculate the geometric mean of the numbers $x$ and $y$.

Grading criteria. We have:

(2p) $2 \sqrt{x}+\sqrt{y}=\sqrt{(x+1)(y+4)} \Leftrightarrow 4 x+y+4 \sqrt{x y}=(x+1)(y+4)$

$(2 p) \Leftrightarrow x y+4-4 \sqrt{x y}=0$

$(2 p) \Leftrightarrow(\sqrt{x y}-2)^{2}=0$

$(1 p) \Leftrightarrow \sqrt{x y}=2$.