Question

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 elements $S$ may contain?

$\ 12$

$\ 13$

$\ 14$

$\ 15$

$\ 16$

Answer: B

Concepts:subsets extremal argument pairing and grouping

Difficulty rating: 1600

Solution:

The set $S = \{13,14 \cdots ,25\}$ has $13$ elements, and every pair has sum greater than 25, so this size is attainable.

Conversely, let $m$ be the maximum element of $S.$ For every element of $S$ satisfying $i<m,$ the number $i$ and the number $m-i$ cannot both belong to $S.$

Thus, among the numbers below $m,$ at most one number can be chosen from each pair with sum $m$ ; if $m$ is even, the middle number cannot be chosen either. Hence at most $\lfloor \dfrac {m-1}2 \rfloor$ elements lie below $m,$ and including $m$ gives at most $\lfloor \dfrac{m-1}2 \rfloor +1$ elements.

The maximum value of this has $m=25,$ yielding $13.$

Thus, the answer is B .

2022 AMC 10B Problem 14

Answer: B

Difficulty rating: 1600

Solution:

Problem 14 in Other Years