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PrintIranian Mathematical Olympiad
Iran geometry
Problem
Let , be two points on a plane and be the midpoint of . We firstly choose a point on the segment , other than , , . At step we choose a red point then choose one of and , call it , and reflect with respect to to get , then color the midpoint of red. Is it possible that after a few steps we color the by red?
Solution
We can assume that is the real line, , . At each step we choose a red point and we color one of the or . Consider the converse: we choose a red point and color the or red.
If we can color red at some point, it would be possible to start from and use inverse steps to reach , so the problem is equivalent to this: is it possible to start at and do the converse steps to reach a point .
From now on we prove this formulation and call converse steps simply steps.
If we start at after one step we are either at or . We claim that if you start at some point you will remain in this set. To see this note that if , , , and if , , . So we can not reach from to some .
If we can color red at some point, it would be possible to start from and use inverse steps to reach , so the problem is equivalent to this: is it possible to start at and do the converse steps to reach a point .
From now on we prove this formulation and call converse steps simply steps.
If we start at after one step we are either at or . We claim that if you start at some point you will remain in this set. To see this note that if , , , and if , , . So we can not reach from to some .
Final answer
No
Techniques
Cartesian coordinatesHomothetyInvariants / monovariants