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n lamps at n vertices of a regular n-gon

Source: 2021 3nd Final Mathematical Cup Senior Division P4 FMC

October 30, 2022
combinatoricscombinatorial geometry

Problem Statement

A number of nn lamps (n≥3n\ge 3) are put at nn vertices of a regular nn-gon. Initially, all the lamps are off. In each step. Lisa will choose three lamps that are located at three vertices of an isosceles triangle and change their states (from off to on and vice versa). Her aim is to turn on all the lamps. At least how many steps are required to do so?
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