IMO 2014 #1
by Wolstenholme, Oct 26, 2014, 7:19 PM
Let 
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that
![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](//latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proof:
Note that the inequality
is equivalent to the statement that 
. This motivates us to consider the sequence 
defined by 
.
Then it suffices to show that there exists a unique
such that 
. But since 
we have that this new sequence is strictly increasing which immediately implies the desired result.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proof:
Note that the inequality
is equivalent to the statement that 
. This motivates us to consider the sequence 
defined by 
.Then it suffices to show that there exists a unique
such that 
. But since 
we have that this new sequence is strictly increasing which immediately implies the desired result.
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