lean_workbook_plus_77624

Let $n \geq 1$ be a positive integer, and let $\mathcal{S} \subset {0, 1, 2, \ldots, n}$ such that $|\mathcal{S}| \geq \frac{n}{2}+1$. Show that some power of $2$ is either an element of $\mathcal{S}$ or the sum of two distinct elements of $\mathcal{S}$.