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Memorable labellings of a regular n-gon

Source: MEMO 2017 I2

August 25, 2017
combinatorics

Problem Statement

Let n≥3n \geq 3 be an integer. A labelling of the nn vertices, the nn sides and the interior of a regular nn-gon by 2n+12n + 1 distinct integers is called memorable if the following conditions hold: (a) Each side has a label that is the arithmetic mean of the labels of its endpoints. (b) The interior of the nn-gon has a label that is the arithmetic mean of the labels of all the vertices. Determine all integers n≥3n \geq 3 for which there exists a memorable labelling of a regular nn-gon consisting of 2n+12n + 1 consecutive integers.
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