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Problem 916

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Prove it Berkeley Math Circle: Monthly Contest 7 · United States

Mr. Fat moves around on the lattice points according to the following rules: From point $(x, y)$ he may move to any of the points $(y, x)$ , $(3x, -2y)$ , $(-2x, 3y)$ , $(x+1, y+4)$ and $(x-1, y-4)$ . Show that if he starts at $(0,1)$ he can never get to $(0,0)$ .

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:

Observe that for each of Mr. Fat's moves, the value of $x + y \pmod{5}$ is invariant. Therefore, Mr. Fat can never reach $(0,0)$ from $(0,1)$ .

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