MathNet 2013 Question 74

Let $a$, $b$ and $c$ be positive integers such that $\frac{a \sqrt{2013}+b}{b \sqrt{2013}+c}$ is a rational number. Show that $\frac{a^{2}+b^{2}+c^{2}}{a+b+c}$ and $\frac{a^{3}-2 b^{3}+c^{3}}{a+b+c}$ are both integers.