Question 47
Let $\alpha = \frac{1 + \sqrt{5}}{2}$. Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that, for all $x, y, z \in \mathbb{R}$,
$$
\begin{aligned}
& f(\alpha x + y) + f(\alpha y + z) + f(\alpha z + x) \\
&= \alpha f(x + y + z) + 2f(x) + 2f(y) + 2f(z).
\end{aligned}
$$
$$
\begin{aligned}
& f(\alpha x + y) + f(\alpha y + z) + f(\alpha z + x) \\
&= \alpha f(x + y + z) + 2f(x) + 2f(y) + 2f(z).
\end{aligned}
$$