Question 34
Let the set $G = (-k, k)$, $k \gt 0$ and $x * y = \frac{k^{2}(x + y)}{k^{2} + x y}$, $x, y \in G$. Show that:
a) $(G, *)$ is an abelian group.
b) $\frac{k}{3} * \frac{k}{5} * \ldots * \frac{k}{2 n + 1} \lt \frac{k}{2} * \frac{k}{4} * \ldots * \frac{k}{2 n}$, $n \in \mathbb{N}^{*}$.
(Supplement to Gazeta Matematică no. 10/2015)
a) $(G, *)$ is an abelian group.
b) $\frac{k}{3} * \frac{k}{5} * \ldots * \frac{k}{2 n + 1} \lt \frac{k}{2} * \frac{k}{4} * \ldots * \frac{k}{2 n}$, $n \in \mathbb{N}^{*}$.
(Supplement to Gazeta Matematică no. 10/2015)