Letting $a_1,\dots,a_{1997}$ to be our numbers, we easily see \[ \binom{1996}{96}\sum_i a_i = \sum_{S\subset [1997],|S|=97} \sum_{i\in S}a_i >0, \]yielding the conclusion.

We rearrange all the numbers, let's say that the numbers: $(b_1 \le b_2 \le b_3\le ...\le b_{97})$ are the least, We have that the sum of these is positive $\Rightarrow$ $b_{97}$ is positive $\Rightarrow$ all other numbers (that are not $b_i$ ) are positive $\Rightarrow$ The sum of all those that are not $b_i$ is positive $\Rightarrow$ The sum of all numbers is positive