Question 174
For real numbers $a_{1}, a_{2}, \ldots, a_{n}$, all distinct, we compute the $\frac{n(n-1)}{2}$ sums $a_{i}+a_{j}$ with $1 \leq i\lt j \leq n$ and then sort these from small to large. Determine all integers $n \geq 3$ for which there exist $a_{1}, a_{2}, \ldots, a_{n}$ such that this sequence of $\frac{n(n-1)}{2}$ sums forms an arithmetic progression (i.e. the difference between two consecutive terms is always the same).