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Question Number 163718 by HongKing last updated on 09/Jan/22

$$\mathrm{let}\mathrm{m}\in \mathbb{N}\mathrm{and}0<\mathrm{x}<\mathrm{m}$$ $$\mathrm{if}\mathbf{i}=\left[\frac{(\mathrm{m}+1)\mathrm{x}}{\mathrm{x}+1}\right]$$ $$\mathrm{prove}\mathrm{that}2\left(\begin{array}{c}\mathrm{m}\\ \mathrm{i}\end{array}\right)⩾{\mathrm{x}}^{\mathbf{m}-\mathbf{i}}$$ $$\mathrm{where}\left[\mathbf{x}\right]\mathrm{is}\mathrm{integer}\mathrm{part}\mathrm{of}\mathbf{x}\mathrm{and}\left(\begin{array}{c}\mathrm{m}\\ \mathrm{i}\end{array}\right)$$ $$\mathrm{is}\mathrm{a}\mathrm{binomial}\mathrm{coefficient}$$

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