Problem

Source: Chinese TST 2009 3rd quiz P1

Tags: ceiling function, ratio, function, floor function, logarithms, combinatorics proposed, combinatorics



Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$