Ok, for this one you only need some very basic knowledge of group theory:

Let $\displaystyle G$ be a finitenon-abeliangroup and $\displaystyle |G|=n.$ Let $\displaystyle k$ be the number of conjugacy classes of $\displaystyle G$. Prove that $\displaystyle k \leq \frac{5n}{8}$ with equality if and only if $\displaystyle |Z(G)|=\frac{n}{4}.$