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Operation on n by n grid with integers

Source: China Mathematical Olympiad 2019 Q5

November 15, 2018
combinatorics

Problem Statement

Given is an n×nn\times n board, with an integer written in each grid. For each move, I can choose any grid, and add 11 to all 2n−12n-1 numbers in its row and column. Find the largest N(n)N(n), such that for any initial choice of integers, I can make a finite number of moves so that there are at least N(n)N(n) even numbers on the board.
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