2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 21 Easy

A set of numbers is called sum-free if whenever x and y are (not necessarily distinct) elements of the set, x+ y is not an element of the set. For example, {{1, 4,6}} and the empty set are sum-free, but {2, 4,5} is not. What is the greatest possible number of elements in a sum-free subset of {1,2,3,\cdots ,20}?

  • A.

    8

  • B.

    9

  • C.

    10

  • D.

    11

  • E.

    12

Answer:C

The set \{11, 12, \ldots, 20\} forms a sum-free subset with 10 elements.

Suppose there exists a sum-free subset \{a_1, \ldots, a_{11}\} of 11 elements, with a_{11} the largest element. Then the differences a_{11} - a_i for 1 \leq i \leq 10 are all distinct from \{a_1, \ldots, a_{11}\}.

Moreover, these differences are mutually distinct. The union of the 10 differences and the 11 original elements would form a 21-element subset of \{1, 2, \ldots, 20\}, which is a contradiction.

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