Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME Prove it United States

Problem:

Consider three hockey pucks lying on a level sheet of ice; the pucks are not collinear. In a move, one may select any of the pucks and hit it so that it passes through the midpoint of the other two pucks. Determine whether it is possible, after $2019$ such moves, for all pucks to be in their original positions.

Solution

Solution:

It is not possible. Label the pucks $A$ , $B$ , $C$ . Then the triangle $ABC$ alternates between being oriented clockwise and counterclockwise between each move. Thus, the pucks cannot return to their original positions after an odd number of moves.

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