Question 162
Let $a_{1}, a_{2}, \ldots, a_{2 n}$ be real numbers such that $a_{1}+a_{2}+\cdots+a_{2 n}=0$.
Prove that there are at least $2 n-1$ pairs $\left(a_{i}, a_{j}\right)$ with $i \lt j$ such that $a_{i}+a_{j} \geqslant 0$.
Prove that there are at least $2 n-1$ pairs $\left(a_{i}, a_{j}\right)$ with $i \lt j$ such that $a_{i}+a_{j} \geqslant 0$.