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Question Number 155621 by mathdanisur last updated on 02/Oct/21

Answered by ghimisi last updated on 03/Oct/21

$$\frac{a+b+c}{6}=1\Rightarrow$$$${\left(\frac{a+d}{a}\right)}^{\frac{a}{6}}{\left(\frac{b+e}{b}\right)}^{\frac{b}{6}}{\left(\frac{c+f}{c}\right)}^{\frac{c}{6}}⩽\frac{a}{6}\cdot \frac{a+d}{a}+\frac{b}{6}\cdot \frac{b+e}{b}+\frac{c}{6}\cdot \frac{c+f}{c}=2\Rightarrow$$$${\left(\frac{a+d}{a}\right)}^a{\left(\frac{b+c}{b}\right)}^b{\left(\frac{c+f}{c}\right)}^c⩽64(\bullet )$$$$\frac{d+e+f}{6}=1\Rightarrow$$$${\left(\frac{a+d}{d}\right)}^{\frac{d}{6}}{\left(\frac{b+e}{e}\right)}^{\frac{e}{6}}{\left(\frac{c+f}{f}\right)}^{\frac{f}{6}}⩽\frac{d}{6}\cdot \frac{a+d}{d}+\frac{e}{6}\cdot \frac{b+e}{e}+\frac{f}{6}\cdot \frac{c+f}{f}=2$$$${\left(\frac{a+d}{d}\right)}^d{\left(\frac{b+c}{e}\right)}^e{\left(\frac{c+f}{f}\right)}^f⩽64(\bullet \bullet )$$$$(\bullet )+(\bullet \bullet )\Rightarrow \frac{{(a+d)}^{a+d}{(b+e)}^{b+e}{(c+f)}^{c+f}}{a^a{b}^b{c}^c{d}^d{e}^e{f}^f}⩽4096$$$$$$$$$$$$$$$$$$

Commented by mathdanisur last updated on 03/Oct/21

$$\mathrm{Perfect}\mathrm{dear}\mathbf{S}\mathrm{er},\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

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