\section*{Problem 4 - 141044}

Determine all rational numbers \(r\) that satisfy the following equation:

\[
(\sqrt{2+\sqrt{3}})^{r}+(\sqrt{2-\sqrt{3}})^{r}=4
\]
}

Due to \(\sqrt{2-\sqrt{3}}=\frac{1}{\sqrt{2+\sqrt{3}}}\), \((\sqrt{2+\sqrt{3}})^{r}\) is a solution to the equation \(x+\frac{1}{x}=4\), which means that

\[
(\sqrt{2+\sqrt{3}})^{r}=2 \pm \sqrt{3}
\]

holds. On the one hand, this equation has a uniquely determined solution \(r \in \mathbb{R}\) for each choice of sign, and on the other hand, \(r= \pm 2\) are obviously solutions, which are indeed the sought rational solutions of the problem.