Question 188
We fix an integer $n \geq 2$. Determine the minimum value of the expression $$
\frac{x_{1}+x_{2}+\cdots+x_{n}}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}}
$$ when $x_{1}, x_{2}, \ldots, x_{n}$ travels the set of strictly positive real numbers, subject to the $$
\frac{1}{1+x_{1}^{2}}+\frac{1}{1+x_{2}^{2}}+\cdots+\frac{1}{1+x_{n}^{2}}=1
$$ condition, and the $x_{1}, x_{2}, \ldots, x_{n}$ this minimum.
\frac{x_{1}+x_{2}+\cdots+x_{n}}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}}
$$ when $x_{1}, x_{2}, \ldots, x_{n}$ travels the set of strictly positive real numbers, subject to the $$
\frac{1}{1+x_{1}^{2}}+\frac{1}{1+x_{2}^{2}}+\cdots+\frac{1}{1+x_{n}^{2}}=1
$$ condition, and the $x_{1}, x_{2}, \ldots, x_{n}$ this minimum.