Mock AIME 4 2006-2007 Problems/Problem 14
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Problem
Let
be the arithmetic mean of all positive integers 
such that 
.
.
Solution
We will assume that there is at least one solution, otherwise the answer would be undefined.
Using the binomial theorem it is obvious that 
. Thus the solutions come in pairs 
, and hence their average is 
, and the answer is 
.
(In this case, there are four solutions: 
, 
, 
, and 
.)
