Question 10
Let $m, n$ be positive integers, $a_{ij}$ ($1 \le i \le m, 1 \le j \le n$) be nonnegative real numbers such that for any $i, j$, the inequalities
$$
a_{i,1} \ge a_{i,2} \ge \dots \ge a_{i,n}, \quad a_{1,j} \ge a_{2,j} \ge \dots \ge a_{m,j}
$$
hold. For $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$, define
$$
X_{i,j} = a_{1,j} + \dots + a_{i-1,j} + a_{i,j} + a_{i,j-1} + \dots + a_{i,1},
$$
$$
Y_{i,j} = a_{i,j} + a_{i+1,j} + \dots + a_{m,j} + a_{i,j+1} + \dots + a_{i,n}.
$$
Prove that
$$
\prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.
$$
$$
a_{i,1} \ge a_{i,2} \ge \dots \ge a_{i,n}, \quad a_{1,j} \ge a_{2,j} \ge \dots \ge a_{m,j}
$$
hold. For $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$, define
$$
X_{i,j} = a_{1,j} + \dots + a_{i-1,j} + a_{i,j} + a_{i,j-1} + \dots + a_{i,1},
$$
$$
Y_{i,j} = a_{i,j} + a_{i+1,j} + \dots + a_{m,j} + a_{i,j+1} + \dots + a_{i,n}.
$$
Prove that
$$
\prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.
$$