1. ( 4 p ) a) Calculate $\left[(2+\sqrt{3})^{2014}+\frac{1}{(2-\sqrt{3})^{2014}}\right] \cdot \frac{(4-2 \cdot \sqrt{3})^{2014}}{2^{2013}}$

( 3 p ) b) Determine the rational numbers $a$ and $b$ such that

$$
2 a \sqrt{2}-b=a+b \sqrt{2}-3
$$
1. a) $\frac{(2+\sqrt{3})^{2014} \cdot(2-\sqrt{3})^{2014}+1}{(2-\sqrt{3})^{2014}} \cdot \frac{(4-2 \sqrt{3})^{2014}}{2^{2013}}=\frac{\left(2^{2}-\sqrt{3}^{2}\right)^{2014}+1}{(2-\sqrt{3})^{2014}} \cdot \frac{2^{2014}(2-\sqrt{3})^{2014}}{2^{2013}}=$

$$
\left(1^{2014}+1\right) \cdot 2=4
$$

b) $(2 a-b) \sqrt{2}=a+b-3$

$$
\begin{aligned}
& (2 a-b) \sqrt{2} \in \mathbb{R} \backslash \mathbb{Q} \text { and } a+b-3 \in \mathbb{Q} \Rightarrow \\
& \left\{\begin{array} { c } 
{ 2 a - b = 0 } \\
{ a + b - 3 = 0 }
\end{array} \Rightarrow \left\{\begin{array}{l}
a=1 \\
b=2
\end{array}\right.\right.
\end{aligned}
$$