
2. Let $n$ be a natural number. Find all real numbers $x$ satsfying the equation

$$
\sum_{k=1}^{n} \frac{k x^{k}}{1+x^{2 k}}=\frac{n(n+1)}{4}
$$


Solution: Observe that $x \neq 0$. We also have

$$
\begin{aligned}
\frac{n(n+1)}{4}=\left|\sum_{k=1}^{n} \frac{k x^{k}}{1+x^{2 k}}\right| & \leq \sum_{k=1}^{n} \frac{k|x|^{k}}{1+x^{2 k}} \\
& =\sum_{k=1}^{n} \frac{k}{\frac{1}{|x|^{k}}+|x|^{k}} \\
& \leq \sum_{k=1}^{n} \frac{k}{2}=\frac{n(n+1)}{4}
\end{aligned}
$$

Hence equality holds every where. It follows that $x=|x|$ and $|x|=1 /|x|$. We conclude that $x=1$ is the unique solution to the equation.
