Question 43
Let $M = \{1, 2, \dots, 2013\}$ and let $\Gamma$ be a circle. For every nonempty subset $\mathcal{A}$ of the set $M$, denote by $S(\mathcal{A})$ the sum of elements of the set $\mathcal{A}$, and define $S(\emptyset) = 0$ ($\emptyset$ is the empty set). Is it possible to join every subset $\mathcal{A}$ of $M$ with some point $A$ on the circle $\Gamma$ so that following conditions are fulfilled:
1. Different subsets are joined with different points;
2. All joined points are vertices of a regular polygon;
3. If $A_1, A_2, \dots, A_k$ are some of the joined points, $k \gt 2$, such that $A_1A_2 \dots A_k$ is a regular $k$-gon, then 2014 divides $S(\mathcal{A}_1) + S(\mathcal{A}_2) + \dots + S(\mathcal{A}_k)$?
Supplementary schematic (AI-generated):
1. Different subsets are joined with different points;
2. All joined points are vertices of a regular polygon;
3. If $A_1, A_2, \dots, A_k$ are some of the joined points, $k \gt 2$, such that $A_1A_2 \dots A_k$ is a regular $k$-gon, then 2014 divides $S(\mathcal{A}_1) + S(\mathcal{A}_2) + \dots + S(\mathcal{A}_k)$?
Supplementary schematic (AI-generated):