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