ISL 1989 Problem 53

Let $\alpha$ be the positive root of the equation $x^2 - 1989x - 1 = 0.$ Prove that there exist infinitely many natural numbers $n$ that satisfy the equation:

$$\lfloor \alpha n + 1989 \alpha \lfloor \alpha n \rfloor \rfloor = 1989n + \left( 1989^2 + 1 \right) \lfloor \alpha n \rfloor.$$