Problem

Source: China TST 2004 Quiz

Tags: function, quadratics, algebra unsolved, algebra



Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) = af(x) + bx$ and $ \displaystyle f(x)f(y) = f(xy) + f \left( \frac {x}{y} \right)$.