Question

A subset B of the set of first 100 positive integers has the property that no two elements of B sum to 125 what is the maximum possible number of elements in B?

Answer 1

The universal set is A containing the first 100 positive integers.

Set B is a subset of A. Set B has the property that if m and n are elements of B then the sum of m and n is not 125.

One possibility is that B contains the first 62 positive integers. Then the greatest number that can be formed by the sum of any two elements of B is 123.

The number of elements of set B is 62.

We can also selectively interchange one or more elements of B and and add that many selected elements of the complement of B.

e.g. We can remove 62 from B and add 63

or remove 60 and 61 from B and add 64 and 65

or remove 30, 35 and 40 from B and add 85, 90 and 95, and so on.

In all these cases the number of elements in set B would be 62.