Problem 1. a) Given the number $\overline{147 a b c}$ which is a perfect square. Calculate $\sqrt{6 \sqrt{147 a b c}}$.

b) Determine the rational numbers $\mathrm{x}$ and $\mathrm{y}$ that satisfy the relation $(4+\sqrt{50}) \mathrm{y}-(2-\sqrt{8}) \mathrm{x}=5+\sqrt{32}$.
## Solution. 1p by default

a) We have $147000<\overline{147 \mathrm{abc}}<147999$ and $\overline{147 \mathrm{abc}}=\mathrm{k}^{2}, \mathrm{k} \in \mathrm{N}$ (1p)

We obtain k=384 (1p). Calculating $\sqrt{6 \sqrt{147 a b c}}=\sqrt{6 \cdot 384}=48(\mathbf{1 p})$

b) We obtain $4 \mathbf{y}+5 \sqrt{2} y-2 \mathrm{x}+2 \sqrt{2} \mathrm{x}=5+4 \sqrt{2}$ (1p) from which $(4 \mathrm{y}-2 \mathrm{x})+(5 \mathrm{y}+2 \mathrm{x}) \sqrt{2}=5+4 \sqrt{2} . \mathbf{( 0 , 5 p})$

We get $4 y-2 x=5$ and $5 y+2 x=4(\mathbf{0 , 5 p})$ from which $x=\frac{-1}{2}$ and $y=1(\mathbf{1 p})$.