2014 IMO Problems/Problem 1
Contents
Problem
Let be an infinite sequence of positive integers, Prove that there exists a unique integer such that
Solution
Define . (In particular, ) Notice that because , we have Thus, ; i.e., is monotonic decreasing. Therefore, because , there exists a unique such that . In other words, This rearranges to give Define . Then because , we have Therefore, is also monotonic decreasing. Note that from our inequality, and so for all . Thus, the given inequality, which requires that , cannot be satisfied for , and so is the unique solution to this inequality.
--Suli 22:38, 7 February 2015 (EST)
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
2014 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |