Problem 3 from the EGMO2017

Problem

Problem 1 from the EGMO2017, problem

Solution 1

The regions that a number of straight lines form in the plane can be colored with two colors so that no two regions that share a border segment are in the same color.

Assume the configuration in the problem has been so colored. For example:

Problem 3 from the EGMO2017, pic

Choose a starting point, an initial segment to move on, and note the color of the region on your, say, right. Let it be COLOR1. When you reach the end of the segment you have to switch the "road" and turn either left or right. Note that, by the condition that no three line are concurrent, however you choose to turn, a region of the COLOR1 will be on your right (it will be the same region as on the previous step, or another one, but, regardless, the color on the right will be COLOR1.) On the next crossroads of the journey you'll have again to switch roads, and, regardless of whether you move right or left, the region on your right will again be of COLOR1. And so on.

This answers the problem because during an arbitrary journey, following the rules of the problem, all the regions you pass by on the right are of the same color, COLOR1. Thus, it is impossible to traverse the segment in two directions, because every segment of the configuration, separates regions of different colors.

Acknowledgment

This is Problem 3 from the European Girls' Mathematical Olympiad 2017. I am grateful to Mike Lawler whose tweets brought the problem to my attention and later suggested a solution.

Note that the condition in the problem that stipulates that the turtle alternates her choices is a red herring.

 

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