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

Answered by ghimisi last updated on 03/Oct/21

((a+b+c)/6)=1⇒  (((a+d)/a))^(a/6) (((b+e)/b))^(b/6) (((c+f)/c))^(c/6) ≤(a/6)∙((a+d)/a)+(b/6)∙((b+e)/b)+(c/6)∙((c+f)/c)=2⇒  (((a+d)/a))^a (((b+c)/b))^b (((c+f)/c))^c ≤64  (•)  ((d+e+f)/6)=1⇒  (((a+d)/d))^(d/6) (((b+e)/e))^(e/6) (((c+f)/f))^(f/6) ≤(d/6)∙((a+d)/d)+(e/6)∙((b+e)/e)+(f/6)∙((c+f)/f)=2  (((a+d)/d))^d (((b+c)/e))^e (((c+f)/f))^f ≤64  (••)  (•)+(••)⇒(((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

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

Commented by mathdanisur last updated on 03/Oct/21

Perfect dear Ser, thank you so much

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

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