<s>[INST] Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$ Prove that for all $x,y>0$ the following inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$ is true.
It smells like Jensen inequality, but all I can get is that $\frac{x+y}{2}ln(x) + \frac{x+y}{2} ln(y) \geq xln(\frac{x+y}{2})+yln(\frac{x+y}{2})$
 [/INST] This proof won't use Jensen. Multiply both sides by $x^y  y^x$ and regroup:
$$
\begin{alignat*}{}
&\Leftrightarrow &\ x^{x+y} y^{x+y} &\geq{}  \left(\frac{x+y}{2}\right)^{x+y} x^y  y^x\\
&\Leftrightarrow &\ \left(\frac{2xy}{x+y}\right)^{x+y} &\geq{}  x^y  y^x  \\
&\Leftrightarrow &\ \frac{2xy}{x+y} &\geq{}  x^{y/(x+y)} y^{x/(x+y)}
\end{alignat*}
$$
Now apply weighted AM-GM inequality to get:
RHS $\leq \frac{xy}{x+y} + \frac{yx}{x+y} = \frac{2xy}{x+y} =$ LHS. Done.
</s>