MathNet 2014 Question 92

Consider real numbers $a_1, a_2, \dots, a_{2n}$ whose sum is $0$. Prove that among the pairs $(a_i, a_j)$, $i \lt j$, with $i, j \in \{1, 2, \dots, 2n\}$, there exist at least $2n - 1$ pairs such that $a_i + a_j \ge 0$.