I have the following elementary problem/question that I do not know how to tackle. It comes with a "math-olympiad-flavor" but I suspect it may be much more difficult than an high-school olympiads problem.
Let $\mathscr{P}$ be a simple polygon (not necessarily convex) in the plane, and such that the number of sides of $\mathscr{P}$ is an odd number $2k+1$.
For each vertex $v$ of $P$ there is an intuitive notion of opposite side (because the number of sides is odd), just label the sides counterclockwise at $v$ from $1$ to $2k+1$, and it would be the one with the label $k+1$.
Now suppose that for each vertex and its corresponding opposite side I consider the triangle the determine in the plane. Let us call $\Delta_v$ this triangle and call it a central triangle.
Is the following:
$$ \mathscr{P} \subseteq \bigcup_{v\in \mathscr{P}} \Delta_v$$
always true? In other words, if one paints all central triangles is it true that the whole polygon is then painted?
from Hot Weekly Questions - Mathematics Stack Exchange
Luis Ferroni
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