<s>[INST] A few inequality problems I can't seem to get; Cauchy and the Mean Inequality Chain 
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*Prove that $ \sqrt{\frac{2x^2-2x+1}{2}}\geq\frac{1}{x+\frac{1}{x}} $ for $ 0 < x < 1. $


This one seems reminiscent of the quadratic mean on the left, maybe $\sqrt{\frac{(x-1)^2+x^2}{2}}$, but I can't find a way to compare it to the LHS, which loos a bit like the harmonic mean.


*Prove $ \frac{x+y}{x^2+y^2}+\frac{y+z}{y^2+z^2}+\frac{z+x}{z^2+x^2}\leq\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ for positive reals $x,y,z$


This one seems a LOT like Cauchy-Schwarz, but I can't quite figure out which lists to use.
I need to solve these problems, but I am pretty stuck.  Hints or even solutions would be greatly appreciated.  Thanks for the help in advance!
 [/INST] For the first question, you can try as follows.
Since 
\begin{align*}
\sqrt{\frac{2x^2-2x+1}{2}}-\frac{x}{x^2+1}=\frac{1}{2}\cdot \frac{\sqrt{4x^2-4x+2}\cdot (x^2+1)-2x}{x^2+1}=:f(x),
\end{align*}
if we set 
\begin{gather*}
a(x):=\sqrt{4x^2-4x+2}=\sqrt{4\left(x-\frac{1}{2}\right)^2+1},
\end{gather*}
the numerator of $f(x)$ is
\begin{align*}
a(x) \cdot x^2-2x+a(x)&=a(x)\cdot \left(x-\frac{1}{a(x)}\right)^2+a(x)-\frac{1}{a(x)}\\
&=a(x)\cdot \left(x-\frac{1}{a(x)}\right)^2+\frac{4\left(x-\frac{1}{2}\right)^2}{a(x)}\\
&> 0,
\end{align*}
from which we can conclude that $f(x)\geq 0, $ that is,
\begin{gather*}\tag{$\star$}
\sqrt{\frac{2x^2-2x+1}{2}}> \frac{x}{x^2+1}, \qquad \forall x\in\mathbb{R}.
\end{gather*}
From $(\star)$ we have
\begin{gather*}
\sqrt{\frac{2x^2-2x+1}{2}}>\frac{1}{x+\frac{1}{x}}, \qquad \forall x\in\mathbb{R}\backslash\{0\}.
\end{gather*}
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