Question 59
Let $A = (a_{jk})$ be a $10 \times 10$ array of positive real numbers such that the sum of numbers in each row as well as in each column is $1$. Show that there exist $j \lt k$ and $l \lt m$ such that
$$
a_{jl}a_{km} + a_{jm}a_{kl} \geq \frac{1}{50}
$$
$$
a_{jl}a_{km} + a_{jm}a_{kl} \geq \frac{1}{50}
$$