NMO Selection Tests for the Junior Balkan Mathematical Olympiad

Denote by $T>0$ the total sum of the numbers around the circle. Clearly $S\ge T>0$. If $s\ge 0$, we are done. If $s<0$, the sum of the numbers which are not terms of $s$ is equal to $T-s$. Since $S\ge T-s$, then $S+s\ge T>0$, as needed.