2022 AMC 10B Problems/Problem 14

Problem

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?

Solution (Pigeonhole Principle)

Denote by $M$ the largest number in $S$ . We categorize numbers $\left\{ 1, 2, \cdots , M-1 \right\}$ (except $\frac{M}{2}$ if $M$ is even) into $\lfloor \frac{M-1}{2} \rfloor$ groups, such that the $i$ th group contains two numbers $i$ and $M-i$ .

Recall that $M \in S$ and the sum of two numbers in $S$ cannot be equal to $M$ , and the sum of numbers in each group above is equal to $S$ . Thus, each of the above $\lfloor \frac{M-1}{2} \rfloor$ groups can have at most one number in $S$ . Therefore, $\begin{align*} |S| & \leq 1 + \left\lfloor \frac{M-1}{2} \right\rfloor \\ & \leq 1 + \left\lfloor \frac{25}{2} \right\rfloor \\ & = 13 . \end{align*}$

Next, we construct an instance of $S$ with $|S| = 13$ . Let $S = \left\{ 13, 14, \cdots , 25 \right\}$ . Thus, this set is feasible. Therefore, the most number of elements in $S$ is \boxed{\textbf{(B) 13}}.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution

https://youtu.be/_K29sOequlY

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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2022 AMC 10B Problems/Problem 14

Problem

Solution (Pigeonhole Principle)

Video Solution