Question 13
Inside a square $ABCD$ there are $2009$ points, with no three of these $2013$ points (including the four vertices) collinear. Segments joining some of the points, including square vertices, partition the square into triangles. Each segment is an edge; a walk along edges is a path.
Partition the $2013$ points into sets $X,Y$, with $A,C\in X$ and $B,D\in Y$. Prove that either there is an $A$-to-$C$ path using only vertices in $X$, or a $B$-to-$D$ path using only vertices in $Y$.
Partition the $2013$ points into sets $X,Y$, with $A,C\in X$ and $B,D\in Y$. Prove that either there is an $A$-to-$C$ path using only vertices in $X$, or a $B$-to-$D$ path using only vertices in $Y$.