ILL 1990 Problem 38

Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$ . For any $m, n \in \mathbb N$ , define the operation $m*n = mn + [\alpha m][ \alpha n]$ , where $[x]$ is the largest integer no larger than $x$ . Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$