Difference between revisions of "2014 IMO Problems/Problem 1"

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==Problem==
 
==Problem==
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  
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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  
 
<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>
  

Revision as of 22:57, 7 February 2015

Problem

Let $a_0<a_1<a_2<\cdots \quad$ be an infinite sequence of positive integers, Prove that there exists a unique integer $n\ge1$ such that \[a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.\]


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

Solution

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