Établissement
Lycée Emmanuel d'Alzon (Nîmes)
Année
2024-2025
Résumé
We start with a large triangle, which is divided into several smaller triangles, so that the side of one small triangle exactly matches the side of another small triangle or is on one of the sides of the large triangle (this is called ‘’triangulating’’ the original triangle).
The vertices of the triangulation are colored as follows:
\begin{itemize}
\item Each vertex of the large triangle has its own color, let’s say red, green, and blue;
\item Each vertex on the red-green side must be colored either red or green, and similarly for the red-blue and green-blue sides;
\item Other vertices inside the large triangle can be colored as desired.
\end{itemize}
It seems that, each time these rules are followed, there is always at least one small triangle whose three vertices are of different colors.
Can you prove it?
The vertices of the triangulation are colored as follows:
\begin{itemize}
\item Each vertex of the large triangle has its own color, let’s say red, green, and blue;
\item Each vertex on the red-green side must be colored either red or green, and similarly for the red-blue and green-blue sides;
\item Other vertices inside the large triangle can be colored as desired.
\end{itemize}
It seems that, each time these rules are followed, there is always at least one small triangle whose three vertices are of different colors.
Can you prove it?