National Math Olympiad

Let us divide the segment into three parts of equal length. We can find two points of the same colour in the middle part. We may assume that these two points are red. Let us call them $A$ and $B$. Let $A^{\prime }$ be the image of the point $A$ under reflection over the point $B$ and let $B^{\prime }$ be the image of the point $B$ under reflection over the point $A$.



If one of the points $A^{\prime }$ and $B^{\prime }$ is red, then we have found the three points we were looking for. Now, let us assume that both $A^{\prime }$ and $B^{\prime }$ are blue. Let $C$ be the midpoint of the segment $AB$. Then $C$ is also the midpoint of the segment $A^{\prime }{B}^{\prime }$. If $C$ is red, then $A$, $B$ and $C$ are the three points we were searching for, if not, then $A^{\prime }$, $B^{\prime }$ and $C$ are the ones we seek.