37th Austrian Mathematical Olympiad 2006
Source: round2, problem4 - function and arithmetic sequence
February 10, 2009
functionfloor functionarithmetic sequencealgebra unsolvedalgebra
Problem Statement
Given is the function f\equal{} \lfloor x^2 \rfloor \plus{} \{ x \} for all positive reals . ( denotes the largest integer less than or equal and \{ x \} \equal{} x \minus{} \lfloor x \rfloor.)
Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator , if they are a reduced fraction, and don’t lie in the range of the function .