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Question Number 172956 by dragan91 last updated on 03/Jul/22

Commented by Shrinava last updated on 04/Jul/22

$$\mathrm{say}:\mathrm{m}=\mathrm{a}+\mathrm{b}+\mathrm{c}=2$$$$\mathrm{ab}+\mathrm{ac}+\mathrm{bc}=\frac{4-{\mathrm{n}}^2}{3}⩾0$$$$\mathrm{k}=\mathrm{abc}$$$$...\iff \mathrm{k}⩽\frac{2(\mathrm{n}-2)(\mathrm{n}+2)}{25+{2\mathrm{n}}^2}$$$$...\iff \frac{(2-\mathrm{n})}{27(25+{2\mathrm{n}}^2)}({\mathrm{n}}^2({\mathrm{n}}^2+{(\mathrm{n}-1)}^2)+{20\mathrm{n}}^2+2\mathrm{n}+4)⩾0$$$$\mathrm{the}\mathrm{last}\mathrm{inequality}\mathrm{is}\mathrm{true}\mathrm{since}\mathrm{n}⩽2\mathrm{hence}\mathrm{proven}$$

Commented by Shrinava last updated on 04/Jul/22

$$\mathrm{a}=\mathrm{b}=\mathrm{c}=\frac{2}{3}$$

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