(Poland 1995)

Let $n$ be a strictly positive integer. Find the minimum value that the quantity

$$
x_{1}+\frac{x_{2}^{2}}{2}+\frac{x_{3}^{3}}{3}+\ldots+\frac{x_{n}^{n}}{n}
$$

can take, where $x_{1}, \ldots, x_{n}$ are strictly positive real numbers whose sum of reciprocals is exactly $n$.
The constraint given in the statement tells us that the harmonic mean of $x_{i}$ is equal to 1. Thus, by the HM-GM inequality,

$$
x_{1} \cdots x_{n} \geq 1
$$

Once this is obtained, it suffices to apply the weighted AM-GM inequality with weights $1, \frac{1}{2}, \ldots, \frac{1}{n}$ to the $x_{i}^{i}$ to get

![](https://cdn.mathpix.com/cropped/2024_05_10_a954a7760d797f08770eg-512.jpg?height=112&width=1073&top_left_y=1646&top_left_x=500)

$$
\begin{aligned}
& \geq 1+\frac{1}{2}+\ldots+\frac{1}{n}
\end{aligned}
$$

We then see that this minimum is achieved when all $x_{i}$ are equal to 1.