Problem

Source: Mexico National Olympiad Mock Exam 2021 P5

Tags: combinatorics



Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that: No two numbers are equal. If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on them. What is the greatest value of $k$ for which this is possible?