## Subject 1.

a. Find the rational numbers $x$ and $y$ knowing that

$$
x \cdot\left(\sqrt{(3 \sqrt{5}-2 \sqrt{11})^{2}}+\sqrt{(6-2 \sqrt{11})^{2}}\right)+2 \sqrt{5}+y=0
$$

b. Find the numbers $\overline{a b c d}$ knowing that $\sqrt{6 \sqrt{a b c d}} \in \mathbb{N}$
## Subject 1.

a. $\sqrt{(3 \sqrt{5}-2 \sqrt{11})^{2}}+\sqrt{(6-2 \sqrt{11})^{2}}=3 \sqrt{5}-6$

.. 1p.

The relation becomes: $(3 x+2) \sqrt{5}-6 x+y=0$

.. 1p.

$x$ and $y$ are rational $\Rightarrow 3 x+2=0$ and $-6 x+y=0$

.. 1p.

Finalization $x=-\frac{2}{3}$ and $y=-4$

.. 1p.
b. $\sqrt{6 \sqrt{a b c d}} \in \mathbb{N} \Rightarrow \sqrt{\overline{a b c d}}=6 k^{2} \Rightarrow \overline{a b c d}=36 k^{4}, k \in \mathbb{N}$

.. 2p.

Finalization $\overline{a b c d} \in\{2916,9216\}$

.. 1p.