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anemone
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Show that $e^\dfrac{1}{e}_{\phantom{i}}+e^{\dfrac{1}{\pi}}_{\phantom{i}} \ge2e^{\dfrac{1}{3}}_{\phantom{i}}$.
[sp]AM-GM: $e^{1/e}+e^{1/\pi} \geqslant 2\sqrt{\mathstrut e^{1/e}e^{1/\pi}} = 2\exp\Bigl(\frac12\bigl(\frac1e + \frac1\pi\bigr)\Bigr)$. By AM-GM again, $\frac12\bigl(\frac1e + \frac1\pi\bigr) \geqslant \sqrt{\frac1{e\pi}}.$ but $e<11/4$ and $\pi < 16/5$, so $e\pi <176/20 <9.$ Thus $\sqrt{e\pi} <3$, and $\sqrt{\frac1{e\pi}} > \frac13$. Therefore $2\exp\Bigl(\frac12\bigl(\frac1e + \frac1\pi\bigr)\Bigr) > 2e^{1/3}.$[/sp]anemone said:Show that $e^{1/e}+e^{1/\pi} \geqslant 2e^{1/3}$.
"Inequality Challenge III" is a scientific research project that aims to study and address the issue of social and economic inequality in our society. It involves conducting experiments, analyzing data, and proposing solutions to reduce inequality.
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