2014 AMC 12A Problems/Problem 21
Contents
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
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2014amc12a/380
~ dolphin7
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=MI3ax4WJBZA
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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