Problem:
Alice chooses a set of positive integers. Then Bob lists all finite nonempty sets of positive integers with the property that the maximum element of belongs to . Bob's list has sets. Find the sum of the elements of .
Solution:
For each , the number of sets that Bob wrote with is . So the number of subsets in Bob's list is
There is only one way to write each positive integer as a sum of distinct powers of , namely its binary representation. Because
it follows that . The sum of the elements of is .
The problems on this page are the property of the MAA's American Mathematics Competitions