Posted by **DAVID** on Thursday, April 15, 2010 at 3:52am.

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?

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