Question 167
Suppose that $2002$ numbers, each equal to $1$ or $-1$, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all $2002$ such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to $1000$. (The absolute value of $x$ is usually denoted by $|x|$. It is equal to $x$ if $x \geq 0$, and to $-x$ if $x \lt 0$. For example, $|6|=6$, $|0|=0$, and $|-7|=7$.)