2020 IMO Shortlist C7

Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a saddle pair if the following two conditions are satisfied: (i) For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geq a\left(r^{\prime}, c\right)$ for all $c \in C$; (ii) For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leq a\left(r, c^{\prime}\right)$ for all $r \in R$. A saddle pair $(R, C)$ is called a minimal pair if for each saddle pair ( $R^{\prime}, C^{\prime}$ ) with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows. (Thailand)

考虑任何具有有限行和列的矩形表，行 $r$ 和列 $c$ 的单元格中有一个实数 $a(r, c)$。一对 $(R, C)$，其中 $R$ 是一组行，$C$ 是一组列，如果满足以下两个条件，则称为鞍对： (i) 对于每行 $r^{\prime}$，在 R$中存在$r \in R$，使得$a(r, c) \geq a\left(r^{\prime}, c\right)$对于所有$c \in C$； (ii) 对于每一列$c^{\prime}$，存在$c \in C$，使得$a(r, c) \leq a\left(r, c^{\prime}\right)$对于所有$r \in R$。鞍对$(R, C)$称为最小对，如果对于每个鞍对 ($R^{\prime}, C^{\prime}$) 以及$R^{\prime} \subseteq R$和$C^{\prime} \subseteq C$，我们有$R^{\prime}=R$和$C^{\prime}=C$。证明任意两个最小对包含相同的行数。 (泰国)