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≥0, and to −x if x<0. For example, ∣6∣=6,∣0∣=0, and ∣−7∣=7.) combinatorial geometrycombinatorics