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?
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?
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.
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.
