Problem

Source: Mathematics Regional Olympiad of Mexico West 2019 P1

Tags: combinatorics



We say that a table with three rows or infinite columns is cool if it was filled with natural numbers, and also whenever the same number m appears in two or more different places in the table, the numbers that appear in the cells immediately below said places (when they exist) are equal. For example, the following table is cool: For each of the following two tables, decide whether it is possible to fill in the empty cells before the resulting tables are cool, explaining how to do this, or why it is not possible to do this. In both tables from the fifth column, the number in the third line is two units greater than the number in the first line.