Question 156
Let $n \geqslant 3$ 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:
1. Each side has a label that is the arithmetic mean of the labels of its endpoints.
2. The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.
Determine all integers $n \geqslant 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n+1$ consecutive integers.
1. Each side has a label that is the arithmetic mean of the labels of its endpoints.
2. The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.
Determine all integers $n \geqslant 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n+1$ consecutive integers.