2023 IMO Shortlist C6

Let $N$ be a positive integer, and consider an $N \times N$ grid. A right-down path is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A right-up path is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses 5 paths. (Canada)

令 $N$ 为正整数，并考虑 $N \times N$ 网格。从右向下的路径是一系列网格单元，其中每个单元格要么是序列中前一个单元格右侧的一个单元格，要么是序列中前一个单元格下方的一个单元格。右上路径是一系列网格单元，其中每个单元是序列中前一个单元右侧的一个单元或上方的一个单元。证明 $N \times N$ 网格的单元格不能划分为小于 $N$ 的右下或右上路径。例如，以下 $5 \times 5$ 网格的分区使用 5 个路径。 (加拿大)