2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 14 Easy

Suppose that S is a subset of \left\{ 1,2,3\cdots ..,25\right\} such that the sum of any two (not necessarily distinct) elements of S is never an element of S. What is the maximum number of element S may contain?

  • A.

    12

  • B.

    13

  • C.

    14

  • D.

    15

  • E.

    16

Answer:B

The sum of two elements must be at least 26 to avoid overlap with the original set. By this fact, 1 can only pair with 25, 2 can pair with 24 and 25, \cdots, 13 can pair with all numbers greater than itself up to 25. Therefore, the set \{13,14,\cdots, 25\} will be the desired set with 13 elements.

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