If $a_1 + a_2 + \dots + a_{100}=0$, with $a_1,\dots,a_{100}\in\mathbb{R}$, prove that $a_i+a_j \geq0, i\neq j,$ for at least $99$ $i,j$-pairs
Proving at least 99 pairwise sums of the reals are non-negative