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2018 BAMO E/3 2000 numbers, 1 ot -1 around a circle

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August 26, 2019
combinatorial geometrycombinatorics

Problem Statement

Suppose that 20022002 numbers, each equal to 11 or −1-1, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all 20022002 such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to 10001000. (The absolute value of xx is usually denoted by ∣x∣|x|. It is equal to xx if x≥0x \ge 0, and to −x-x if x<0x < 0. For example, ∣6∣=6,∣0∣=0|6| = 6, |0| = 0, and ∣−7∣=7|-7| = 7.)
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