Exercise 11, § 1

Let $a_{ij}$ ($1 \leq i \leq n,\ 1 \leq j \leq n$) be numbers $\geq 0$ such that for each index $i$ one has $\sum_{j=1}^n a_{ij} = 1$ and, for each index $j$ one has $\sum_{i=1}^n a_{ij} = 1$. Let $x_i$ ($1 \leq i \leq n$) be $n$ numbers > 0; put

Show that $y_1 y_2 \ldots y_n \geq x_1 x_2 \ldots x_n$ (bound $\log y_i$ below for each index $i$).