Difference between revisions of "2014 IMO Problems/Problem 1"
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Let <math>a__0<a_1<a_2<\cdots \quad </math> be an infinite sequence of positive integers, Prove that there exists a unique integer <math>n\ge1</math> such that | Let <math>a__0<a_1<a_2<\cdots \quad </math> be an infinite sequence of positive integers, Prove that there exists a unique integer <math>n\ge1</math> such that | ||
<cmath>a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.</cmath> | <cmath>a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.</cmath> | ||
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| + | {{alternate solutions}} | ||
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| + | ==See Also== | ||
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| + | {{IMO box|year=2014|before=First Problem|num-a=2}} | ||
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| + | [[Category:Olympiad Algebra Problems]] | ||
==Solution== | ==Solution== | ||
Revision as of 05:16, 9 October 2014
Problem
Let $a__0<a_1<a_2<\cdots \quad$ (Error compiling LaTeX. Unknown error_msg) be an infinite sequence of positive integers, Prove that there exists a unique integer 
such that
![\[a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.\]](http://latex.artofproblemsolving.com/0/c/4/0c463e53d0ab2ad520551056d74fad908d859cb5.png)
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 | ||
