Question 69
Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma:\{1,2, \ldots, n\} \rightarrow \{1,2, \ldots, n\}$ such that
$$
\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^{2}}\lt \frac{1}{2}
$$
$$
\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^{2}}\lt \frac{1}{2}
$$