2019 CMO 第 6 题

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

给定一个 $n\times n$ 棋盘，每个网格中写有一个整数。对于每次移动，我可以选择任何网格，并将 $1$ 添加到其行和列中的所有 $2n-1$ 数字。找到最大的 $N(n)$ ，这样对于任何整数的初始选择，我可以进行有限次数的移动，以便棋盘上至少有 $N(n)$ 偶数。