Library / Preselection Tests for the Full-time Training /6 of 10

Preselection Tests for the Full-time Training

Geometry Difficulty 7.6 National Olympiad, round 2 Prove it Saudi Arabia

The points of the plane have been colored by $2013$ different colors. We say that a triangle $\triangle ABC$ has the color $X$ if its three vertices $A$ , $B$ , $C$ have the color $X$ . Prove that there are infinitely many triangles with the same color and the same area.

Solution

Consider $2014$ parallel lines. Each line contains infinitely many points. Since the number of the colors is finite, by the pigeonhole principle, there exist on each line infinitely many points of the same color. Choose for each line one color for which there exist infinitely many points. Since there are $2013$ colors and $2014$ lines, by the pigeonhole principle there exist at least two lines $\ell_1$ , $\ell_2$ for which the same color $\mathcal{C}$ has been chosen. Choose two points $B$ , $C$ from the first line $\ell_1$ of this color $\mathcal{C}$ . Choose infinitely many points $A_1$ , $A_2$ , $\ldots$ from the second line $\ell_2$ of this color $\mathcal{C}$ . Triangles $A_1BC$ , $A_2BC$ , $\ldots$ are all of the same color $\mathcal{C}$ and have the same area since $\ell_1$ , $\ell_2$ are parallel.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.