MathDB

C4

Part of 2010 Balkan MO Shortlist

Problems(1)

A table of 2010X2010 cells and repeatedly adding 1 makes them all equal

Source: Balkan MO 2010 ShortList C4

4/5/2020
Integers are written in the cells of a table 2010×20102010 \times 2010. Adding 11 to all the numbers in a row or in a column is called a move. We say that a table is equilibrium if one can obtain after finitely many moves a table in which all the numbers are equal.
[*]Find the largest positive integer nn, for which there exists an equilibrium table containing the numbers 20,21,…,2n2^0, 2^1, \ldots , 2^n. [*] For this nn, find the maximal number that may be contained in such a table.