Školjka

%V0 Let $\alpha$ be the positive root of the equation $x^{2} = 1991x + 1$. For natural numbers $m$ and $n$ define $$m*n = mn + \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor.$$ Prove that for all natural numbers $p$, $q$, and $r$, $$(p*q)*r = p*(q*r).$$