I don't think I gave this problem before.

Let $\displaystyle \sigma : \{1,\dots,n\} \to \mathbb{N}$ be an injection. Show that

$\displaystyle \sum_{j=1}^n \frac{1}{j} \leq \sum_{j=1}^n \frac{\sigma(j)}{j^2}$

with equality only if $\displaystyle \sigma$ is the embedding of $\displaystyle \{1,\dots,n\}$ in $\displaystyle \mathbb{N}$.