Question 174
Let $\alpha$, $\beta$, $\gamma$ be positive integers such that the number
$$
A = \frac{\alpha\sqrt{2} + \beta\sqrt{3}}{\beta\sqrt{2} + \gamma\sqrt{3}}
$$
is rational. Prove that the number
$$
B = \frac{\alpha^2 + \beta^2 + \gamma^2}{\alpha + \beta + \gamma}
$$
is an integer.
$$
A = \frac{\alpha\sqrt{2} + \beta\sqrt{3}}{\beta\sqrt{2} + \gamma\sqrt{3}}
$$
is rational. Prove that the number
$$
B = \frac{\alpha^2 + \beta^2 + \gamma^2}{\alpha + \beta + \gamma}
$$
is an integer.