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}<0
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