IMC 1999 · Problem 3

Suppose that a function $f : \mathbb{R} \to \mathbb{R}$ satisfies the inequality

$$\left| \sum_{k=1}^{n} 3^k \left( f(x + ky) - f(x - ky) \right) \right| \le 1 \tag{1}$$

for every positive integer $n$ and for all $x, y \in \mathbb{R}$. Prove that $f$ is a constant function.