Let α be the positive root of the equation x^{2} \equal{} 1991x \plus{} 1. For natural numbers m and n define
m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor.
Prove that for all natural numbers p, q, and r,
(p*q)*r \equal{} p*(q*r).
floor functionIMO Shortlist