Explanation:
There are 50 pairs of positive integers that total 101:
One integer can be selected from each pair to create a set of 50 integers in the range [1, 100] such that no two will have a sum of 101. Adding any other integer from the range [1, 100], for a total of 51 integers from that range, will complete one of the sums whose total is 101.
Thus, at most 50 integers can be selected from [1, 100] such that no two will have a sum of 101, but any set of 51 integers from that range must have at least one pair that totals 101.
