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Problem 1078

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Prove it Berkeley Math Circle: Monthly Contest 2 · United States

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.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official 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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