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Question Number 217690 by hardmath last updated on 18/Mar/25

Commented by hardmath last updated on 18/Mar/25

$$★\mathrm{Hard}★$$

Commented by mr W last updated on 19/Mar/25

$$★\mathrm{Hard}math★$$

Commented by hardmath last updated on 19/Mar/25

$$\mathrm{y}\mathrm{es}\mathrm{dear}\mathrm{professor}))$$

Answered by mr W last updated on 19/Mar/25

$$sayAI=x,BI=y,CI=z$$$$xyz=4{Rr}^2$$$$R⩾2r$$$$x+y+z⩾3\sqrt[3]{xyz}=3\sqrt[3]{4{Rr}^2}⩾3\sqrt[3]{8r^3}=6r$$$$...$$

Commented by hardmath last updated on 19/Mar/25

$$\mathrm{dear}\mathrm{professor},{\mathrm{that}}^{\prime }\mathrm{s}\mathrm{it}?)$$

Commented by mr W last updated on 19/Mar/25

$${what}^{\prime }sitwhatyoumean?$$

Commented by hardmath last updated on 19/Mar/25

$$$$That is, the solution (proof) is this value

Commented by mr W last updated on 19/Mar/25

$$asyoucansee,itisonlytheproof$$$$forthepart6r⩽AI+BI+CI$$

Commented by mr W last updated on 19/Mar/25

Commented by hardmath last updated on 21/Mar/25

$$\mathrm{thankyou}\mathrm{dearprofessor}$$

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