For each integer $n>1$, find a set of $n$ integers {$a_1, a_2, ..., a_n$} such that the set of numbers {$a_i + a_j | 1 \le i \le j \le n$} leave distinct remainders when divided by $n(n+1)/2$. If such set of integers does not exist, give a proof.

I know ideally I should show what I've attempted thus far but I'm completely lost and don't really know how to get started. I guess WLOG I can let $a_1 < a_2 <...<a_n$ and I also know that I should have from $0 \mod (n(n+1)/2)$ to $n(n+1)/2 - 1 \mod (n(n+1)/2)$ for $a_i + a_j$ but otherwise I'm not sure.

(I did ask this on mathematics stack exchange and I really appreciate the members' help there but I'd like more help!)

For each integer $n>1$, find a set of $n$ integers {$a_1, a_2, ..., a_n$} such that the set of numbers {$a_i + a_j | 1 \le i \le j \le n$} leave distinct remainders when divided by $n(n+1)/2$. If such set of integers does not exist, give a proof.

I know ideally I should show what I've attempted thus far but I'm completely lost and don't really know how to get started. I guess WLOG I can let $a_1 < a_2 <...<a_n$ and I also know that I should have from $0 \mod (n(n+1)/2)$ to $n(n+1)/2 - 1 \mod (n(n+1)/2)$ for $a_i + a_j$ but otherwise I'm not sure.

(I did ask this on mathematics stack exchange and I really appreciate the members' help there but I'd like more help!)