MEMO 2017 I2

Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$ -gon by $2n + 1$ 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 $n$ -gon has a label that is the arithmetic mean of the labels of all the vertices. Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$ -gon consisting of $2n + 1$ consecutive integers.