Problem:
Suppose that is a subset of such that the sum of any two (not necessarily distinct) elements of is never an element of . What is the maximum number of elements may contain?
Answer Choices:
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Solution:
Let consist of the 13 odd elements. Then the sum of two elements of is even, and hence is not in . (The set is another subset of size 13 with the given property, because the sum of any two elements of this set is greater than or equal to 26.)
To show that no larger set has the required property, suppose that the greatest element of is odd, say . Then for , both and cannot be elements of . Thus has at most elements in this case, which suffices because . Now suppose that the greatest element of is even, say . Then , and for , at most one of and is in . Thus has at most elements. Because , the size of does not exceed in this case as well.
The problems on this page are the property of the MAA's American Mathematics Competitions