Problem:
Let be a subset of such that the following two conditions hold:
What is the maximum possible number of elements in ?
Answer Choices:
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B.
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Solution:
If consists of the positive integers less than or equal to 2024 that are congruent to 1,4 , or 8 modulo 10 , then every pair of elements in differ by at least , and every pair of odd elements of differ by at least . This set,
satisfies the given conditions and has elements. To see that no larger set satisfies the given conditions, note that if a set satisfies the first condition and some block of 10 consecutive integers contains 4 elements of the set, then those 4 elements would need to be the th, 7 th, and 10th elements in that block, and the two odd numbers among them would be differ by 6 , in violation of the second condition. Therefore there are at most elements of among the first 2020 positive integers, and at most 2 elements of can be among .
The problems on this page are the property of the MAA's American Mathematics Competitions