## PROBLEM 2

Determine the rational number $a$, knowing that $\left(\frac{1}{\sqrt{9+4 \sqrt{5}}}+\frac{1}{\sqrt{9-4 \sqrt{5}}}\right) \cdot a+\sqrt{5}$ is a rational number.
## Solution:

$$
\begin{aligned}
& \left(\frac{1}{\sqrt{9+4 \sqrt{5}}}+\frac{1}{\sqrt{9-4 \sqrt{5}}}\right) \cdot a+\sqrt{5}=\frac{a}{\sqrt{(\sqrt{5}+2)^{2}}}+\frac{a}{\sqrt{(\sqrt{5}-2)^{2}}}+\sqrt{5}=\ldots \ldots \ldots \ldots \ldots . .2 \text { points } \\
& =\frac{a}{|\sqrt{5}+2|}+\frac{a}{|\sqrt{5}-2|}+\sqrt{5}=\frac{a}{\sqrt{5}+2}+\frac{a}{\sqrt{5}-2}+\sqrt{5}=\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots
\end{aligned}
$$

then \( \mathrm{a}=-\frac{1}{2} \) is a rational number.