Question 123
Suppose one is given $n$ real numbers, not all zero, but such that their sum is zero. Prove that one can label these numbers $a_{1}, a_{2}, \ldots, a_{n}$ in such a manner that
$$
a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n-1} a_{n}+a_{n} a_{1}\lt 0 .
$$
$$
a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n-1} a_{n}+a_{n} a_{1}\lt 0 .
$$