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An inequality

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Riemann
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An inequality

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Post by Riemann » Thu Nov 22, 2018 9:26 pm

Let $x_1, x_2, \dots, x_n$ be $n \geq 2$ positive numbers other than $1$ such that $x_1^2+x_2^2+\cdots +x_n^2=n^3$. Prove that:

$$\frac{\log_{x_1}^4 x_2}{x_1+x_2}+ \frac{\log_{x_2}^4 x_3}{x_2+x_3}+ \cdots + \frac{\log_{x_n}^4 x_1}{x_n+x_1} \geq \frac{1}{2}$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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