IMO 1989 LL IRE55

Let [x] denote the greatest integer less than or equal to x. Let lpha

imolonglistmathematicsolympiad

IMO 1989 LL IRE55

Origin: IRE

Problem

Let [x] denote the greatest integer less than or equal to x. Let \alpha be the positive root of the equation x2 −1989x −1 = 0. Prove that there exist infinitely many natural numbers n that satisfy the equation [\alphan + 1989\alpha[\alphan]] = 1989n + (19892 + 1)[\alphan].