x^2 - 1989x - 1 = 0
Source: IMO Longlist 1989, Problem 53
September 18, 2008
floor functionpigeonhole principlealgebra unsolvedalgebra
Problem Statement
Let be the positive root of the equation x^2 \minus{} 1989x \minus{} 1 \equal{} 0. Prove that there exist infinitely many natural numbers that satisfy the equation:
\lfloor \alpha n \plus{} 1989 \alpha \lfloor \alpha n \rfloor \rfloor \equal{} 1989n \plus{} \left( 1989^2 \plus{} 1 \right) \lfloor \alpha n \rfloor.