## PROBLEM 1

Determine the integer values of $x$ for which the value of the expression $E(x)=\frac{\sqrt{(1-\sqrt{3})^{2}}+\sqrt{(\sqrt{3}-\sqrt{5})^{2}}+\sqrt{(1-\sqrt{2})^{2}}+\sqrt{14-6 \sqrt{5}}+\sqrt{6-4 \sqrt{2}}}{x-7}$ is an integer.
## Solution:

$$
\begin{aligned}
& E(x)=\frac{|1-\sqrt{3}|+|\sqrt{3}-\sqrt{5}|+|1-\sqrt{2}|+|3-\sqrt{5}|+|2-\sqrt{2}|}{x-7}=\ldots \ldots \ldots \ldots \ldots . . . . .2 \text { points } \\
& =\frac{\sqrt{3}-1+\sqrt{5}-\sqrt{3}+\sqrt{2}-1+3-\sqrt{5}+2-\sqrt{2}}{x-7} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\
& E(x)=\frac{3}{x-7} \in Z \text { if }(x-7) \mid 3 \text {, hence } x-7 \in\{-3 ;-1 ; 1 ; 3\} \ldots \ldots . \ldots . \ldots . \ldots . . . . . . . . . . . . . . . .2 \text { points } \\
& \text { it follows that } x \in\{4 ; 6 ; 8 ; 10\}
\end{aligned}
$$