Problem

Source: ELMO Shortlist 2012, N6

Tags: floor function, algebra, polynomial, Vieta, quadratics, inequalities, number theory proposed



Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. Calvin Deng.