On the Open Problem in the 2023 Simon Marais Mathematics Competition

We prove that, for $n\in {\mathbb{Z}}^{+}$ not a perfect square, the only pairs of positive integers $\{a,b\}$ for which there exists $r\in \mathbb{R}$ such that $r^a+\sqrt{n}$ and $r^b+\sqrt{n}$ are both rational numbers are those of the form $\{a,a\}$ or $\{g,2g\}$ (with $g$ odd). This resolves an open question presented in the 2023 Simon Marais Mathematics Competition.