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-1 on a circle, after 2^k-1 steps, of multiplications

Source: Ukraine TST 2012 p8

May 1, 2020
combinatorics

Problem Statement

Call arrangement of mm number on the circle mm-negative, if all numbers are equal to −1-1. On the first step Andrew chooses one number on circle and multiplies it by −1-1. All other steps are similar: instead of the next number(clockwise) he writes its product with the number, written on the previous step. Prove that if nn-negative arrangement in kk steps becomes nn-negative again, then (2n−1)(2^n - 1)-negative after (2k−1)(2^k - 1) steps becomes (2n−1)(2^n - 1)-negative again.
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