Question 155
Let $R$ and $Q$ denote the sets of real numbers and rational numbers respectively. Find all functions $f : Q \rightarrow R \setminus \{0\}$ such that for all rational numbers $x, y$,
$$
(f(x))^2 f(2y) + (f(y))^2 f(2x) = 2f(x)f(y)f(x+y).
$$
$$
(f(x))^2 f(2y) + (f(y))^2 f(2x) = 2f(x)f(y)f(x+y).
$$