Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Prove it Slovenia

Each point on the segment $\overline{D}$ is either red or blue. Show that there exist three different points $A$ , $B$ and $C$ of the same colour such that $|AB| = |BC|$ .

Solution

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 $A'$ be the image of the point $A$ under reflection over the point $B$ and let $B'$ be the image of the point $B$ under reflection over the point $A$ .

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

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