1. a) $E(\sqrt{3})=\sqrt{3}+3 ; E(-\sqrt{3})=-\sqrt{3}+3$

$m_{a}=\frac{E(\sqrt{3})+E(-\sqrt{3})}{2}$

$m_{a}=3 \in \mathrm{N}$

(1p)

b) $E(n)=n+3, E(n \sqrt{3})=n \sqrt{3}+3$

$3 \sqrt{3}-E(n)>3-E(n \sqrt{3}) \Leftrightarrow \sqrt{3}(n+3)>n+3 \Leftrightarrow(n+3)(\sqrt{3}-1)>0$.

$\sqrt{3}-1>0 \Rightarrow n+3>0 \Rightarrow n \in\{-2,-1\}$
1. Consider the expression $E(x)=x+3$, where $x$ is a real number.

(3p) a) Show that the arithmetic mean of the numbers $E(\sqrt{3})$ and $E(-\sqrt{3})$ is a natural number.

(4p) b) If $n$ is a strictly negative integer, determine the values of $n$ for which

$$
3 \sqrt{3}-E(n)>3-E(n \sqrt{3})
$$