Saudi Arabian Mathematical Competitions

Let these numbers be $a_1, a_2, \dots, a_n$ and $S$ is the sum of these numbers. For every $i \in \{1, 2, \dots, n\}$, we have $S - a_i \mid a_i$, which means $S - a_i \nmid S$. Suppose that $S \neq 0$, without the loss of generality, $S > 0$. We investigate two cases

* If $\exists i \in \{1, 2, \dots, n\}$ such that $a_i < 0$ then $|S - a_i| > |S|$ so $S - a_i \nmid S$.

* $a_i > 0$ for all $i \in \{1, 2, \dots, n\}$. Let $a_k$ be the smallest of $a_1, a_2, \dots, a_n$. We have $S - a_m > 2a_m - a_m = a_m$, so $S - a_m \nmid a_m$.

Hence, $S \neq 0$ is impossible, which means $S = 0$.