2014 AMC 12A Problems/Problem 21
Problem
For every real number , let denote the greatest integer not exceeding , and let The set of all numbers such that and is a union of disjoint intervals. What is the sum of the lengths of those intervals?
Solution
Let for some integer . Then we can rewrite as . In order for this to be less than or equal to , we need . Combining this with the fact that gives that , and so the length of the interval is . We want the sum of all possible intervals such that the inequality holds true; since all of these intervals must be disjoint, we can sum from to to get that the desired sum is