Mock AIME 4 2006-2007 Problems/Problem 14
Problem
Let be the arithmetic mean of all positive integers
such that
.
Find the greatest integer less than or equal to .
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
.)