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single operation

Source: 4-th Taiwanese Mathematical Olympiad 1995

January 16, 2007
combinatorics unsolvedcombinatorics

Problem Statement

Given a sequence of eight integers x1,x2,...,x8x_{1},x_{2},...,x_{8} in a single operation one replaces these numbers with ∣x1−x2∣,∣x2−x3∣,...,∣x8−x1∣|x_{1}-x_{2}|,|x_{2}-x_{3}|,...,|x_{8}-x_{1}|. Find all the eight-term sequences of integers which reduce to a sequence with all the terms equal after finitely many single operations.
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