parmenides51 wrote: For each integer $n > 1$, find a set of $n$ integers $\{a_1, a_2,..., a_n\}$ such that the set of numbers $\{a_1+a_j | 1 \le i \le j \le n\}$ leave distinct remainders when divided by $n(n + 1)/2$. If such a set of integers does not exist, give a proof. Do you mean For each integer $n > 1$, find a set of $n$ integers $\{a_1, a_2,..., a_n\}$ such that the set of numbers $\{a_i+\cdots+a_j | 1 \le i \le j \le n\}$ leave distinct remainders when divided by $n(n + 1)/2$. If such a set of integers does not exist, give a proof.

I think there will be (ai) instead of a1 in above statement. I have poof like this ; first note that no two number among n numbers will give same mod n, and let assume that these numbers are a1<a2<...... <an and we also observe that no two numbers have same difference otherwise we will get simple contradiction.so It is easy to see that the only such n integers are 2C2, 3C2 ..... n+1C2 but we know that 4C2+5C2 =6C2 +2C2 which gives contradiction thus we conclude that there is no such n integers