MathNet 2023-part-03 — 191 problems (EN–VI) Question 134

Let $a, b, c \in \mathbb{N}^{*}$ be numbers such that $\frac{a \sqrt{2023}+b}{a+c \sqrt{2023}}$ is a rational number. Prove that $\left(a^{2}+b^{2}+c^{2}\right) \vdots(a+b+c)$, i.e. $a^{2}+b^{2}+c^{2}$ is divisible by $a+b+c$.