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