Problems from Ukrainian Authors

Let's assume that $S=0$. Let's choose an arbitrary division of the string into two parts. It is clear that the sum of the numbers in the two parts is $0$, so one of them is not less than $0$, and the other is not greater than $0$. If they are not $0$, we get a contradiction, so the sum of the numbers of any smaller string is $0$, so all the numbers in the string are $0$, which is what we need to prove.

Let $S\ne 0$, without restriction of generality let $S>0$. Let $a$ be any number in the string. Consider an arbitrary division of the string into two parts: either both sums are not less than $0$, or not greater than $0$. It is clear that the second option is impossible, because the sum of two nonnegative integers cannot be equal to $S>0$. Then the sum of the numbers of any lesser row is not less than $0$.

Let $x$ and $y$ denote the sum of the numbers to the left and right of $a$ respectively (if there are no such numbers, then we assume the corresponding variable is $0$). Then, from the above, $x,y\ge 0$ and $x+a,y+a\ge 0$, therefore $x+y\ge 0$ and $x+y+2a\ge 0$. By definition, $S=x+y+a$, so $S-a\ge 0$ and $S+a\ge 0$, i.e. $S\le a\le S$, therefore we get $|a|\le S$.