Question 72
Fix an integer $n \geq 2$ and consider $n^{2}$ strictly positive real numbers $a_{ij}$, $i, j=1, \ldots, n$, which simultaneously satisfy the following two conditions:
(1) $a_{ii}=1$, $i=1, \ldots, n$; and
(2) For each $j=2, \ldots, n$, the numbers $a_{ij}$, $i=1, \ldots, j-1$, form a permutation of the numbers $\frac{1}{a_{ji}}$, $i=1, \ldots, j-1$.
Let $s_{i}=\sum_{j=1}^{n} a_{ij}$, $i=1, \ldots, n$. Determine the maximum value of the sum $\sum_{i=1}^{n} \frac{1}{s_{i}}$.
(1) $a_{ii}=1$, $i=1, \ldots, n$; and
(2) For each $j=2, \ldots, n$, the numbers $a_{ij}$, $i=1, \ldots, j-1$, form a permutation of the numbers $\frac{1}{a_{ji}}$, $i=1, \ldots, j-1$.
Let $s_{i}=\sum_{j=1}^{n} a_{ij}$, $i=1, \ldots, n$. Determine the maximum value of the sum $\sum_{i=1}^{n} \frac{1}{s_{i}}$.