PMO 2021 Problem 2

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