MathNet 2026 — Part 09 Question 113

The set $S$ is a finite subset of $[0,1]$ with the following property: for all $x \in S$, there exist $a, b \in S \cup \{0,1\}$ with $a, b \neq x$ such that $x = (a + b)/2$. Prove that all the numbers in $S$ are rational.