AMC 10A 2025 Problem 21

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 $\{1,2,3,...,20\}$?

$\textbf{(A) } 8 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$

Difficulty: 43/100 — Although the construction is easy to find, it's harder to prove it's optimal.

Core Concepts: set theory

Challenges: Creating a valid construction for a set of size $10$ and proving why sets greater do not work.

Techniques: Construct a set that you know works, and convince yourself that you cannot go higher.

Error-prone Steps: Forgetting that the condition holds when the numbers are equal.

Ideal Time:

Experienced: ≤ 4 min
Intermediate: ≤ 5-6 min
Beginner: ≤ 10 min