Suppose you have 100 integers whose sum is 1. What is the maximum number of subsets whose sum is positive? via AMS puzzle corner 14 3 Responses to Maximum possible subsets Sid Hollander had this to say about that: September 25, 2013 at 10:05 am Sum 2^n as n goes from 0 to 99. The 100th sum would be the inverse of each sum in the first. Lizard had this to say about that: September 26, 2013 at 2:35 am Regardless of the numbers chosen, exactly half of the subsets will have positive sum. Pair each subset with its complement and note that if a + b = 1 then exactly one of a and b is positive. Sid Hollander had this to say about that: September 28, 2013 at 4:44 pm If S is the set are there 98 subsets with sum =1 i.e(1+0) S = (0,-97,1,1,1,......,1) Subscribe to comments

Sum 2^n as n goes from 0 to 99. The 100th sum would be the inverse of each sum in the first.

Regardless of the numbers chosen, exactly half of the subsets will have positive sum. Pair each subset with its complement and note that if a + b = 1 then exactly one of a and b is positive.

If S is the set are there 98 subsets with sum =1 i.e(1+0)

S = (0,-97,1,1,1,......,1)