Question 81
Let $S$ be a subset of $\{1,2,\ldots,9\}$ such that the sums of every two elements of $S$ are distinct. For example, the set $\{1,2,3,5\}$ has that property but the set $\{1,2,3,4,5\}$ does not, because $\{2,3\}$ and $\{1,4\}$ both have sum $5$. How many elements can $S$ contain at most? Explain your answer.