Question 20
(IRE 3) Let $\alpha$ be the positive root of the equation $x^2=1991x+1$. For natural numbers $m,n$ define
$$m*n=mn+[\alpha m][\alpha n],$$
where $[x]$ is the greatest integer not exceeding $x$. Prove that for all natural numbers $p,q,r$,
$$(p*q)*r=p*(q*r).$$
$$m*n=mn+[\alpha m][\alpha n],$$
where $[x]$ is the greatest integer not exceeding $x$. Prove that for all natural numbers $p,q,r$,
$$(p*q)*r=p*(q*r).$$