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Answer by eyeballfrog for An unusual inequality problem involving real...

The form $4\omega^\alpha+\omega^{n-4\alpha}$ strongly suggests it's AGM time:$$\frac{4\omega^\alpha+\omega^{n-4\alpha}}{5}\ge\sqrt[5]{\omega^{4\alpha}\omega^{n-4\alpha}} = \omega^{n/5}.$$So we...

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An unusual inequality problem involving real exponential power

Problem.Let $\alpha\in\mathbb R,\, \omega \in\mathbb R_{+}$ and $n\in\mathbb Z_{\geq 5}$, then prove that$$4\omega ^{\alpha }+\omega^{n-4\alpha}-n\omega+n\geq 5.$$The source of the problem is a student...

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