Be real!

The maximum number of t_i_j which are non-positive is when all of the 100 chosen real numbers but 1 are negative. (The total sum could still be positive if the only positive number is bigger then the opposite of the sum of the negative ones):

• $r_1,r_2, ... r_{99} < 0$

• $r_{100} > -\sum\limits_{i=1}^{99} r_i$

Then, the $t_{ij}\le 0$ are the ones between negative numbers, while the ones that involve $r_{100}$ are positive.

$4950 - 99 = \boxed{4851}$