A set of numbers has “the triple-sum property” (or TSP) if there exist three numbers
in the set whose sum is also in the set. [Repetitions are allowed.]
For example, the set U = {2, 3, 7} has TSP since 2 + 2 + 3 = 7, while V = {2, 3, 10} fails to have TSP.
(a) Suppose the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} is separated into two parts,
forming two subsets A and B.
Prove: Either A or B must have the triple-sum property.
[To begin the proof, suppose that statement is false and there are sets A and B as above, each without TSP.
If 1 lies in A then 3 = 1 + 1 + 1 must be in B. Complete the proof that this situation is impossible.]
(b) Is a similar result true when the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is separated into two parts?
2 answers