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Question Number 199847 by cortano12 last updated on 10/Nov/23

$$\mathrm{Find}\mathrm{all}\mathrm{possible}\mathrm{value}$$$$\frac{\mathrm{a}}{\mathrm{a}+\mathrm{b}+\mathrm{d}}+\frac{\mathrm{b}}{\mathrm{a}+\mathrm{b}+\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{b}+\mathrm{c}+\mathrm{d}}+\frac{\mathrm{d}}{\mathrm{a}+\mathrm{c}+\mathrm{d}}$$$$\mathrm{when}\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\mathrm{vary}\mathrm{over}\mathrm{positive}$$$$\mathrm{reals}$$

Commented by Frix last updated on 10/Nov/23

$$\mathrm{No}\mathrm{problem}!{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{also}\mathrm{trying}...$$$$\mathrm{I}\mathrm{think}1⩽f\mathrm{is}\mathrm{true}\mathrm{but}f<2?$$

Commented by witcher3 last updated on 10/Nov/23

$$\mathrm{i}\mathrm{didnt}\mathrm{want}\mathrm{delate}\mathrm{Y}\mathrm{re}\mathrm{coment}$$

Commented by witcher3 last updated on 10/Nov/23

$$\mathrm{sorry}\mathrm{frix}\mathrm{my}\mathrm{solution}\mathrm{is}\mathrm{false}$$

Commented by witcher3 last updated on 12/Nov/23

$$\mathrm{f}<2\mathrm{i}?\mathrm{think}\mathrm{also}$$

Commented by witcher3 last updated on 12/Nov/23

$$\mathrm{if}\mathrm{a}+\mathrm{b}+\mathrm{c}>\mathrm{d}\mathrm{or}\mathrm{evry}\mathrm{sum}\mathrm{of}\mathrm{three}⩾\mathrm{forthe}$$$$\mathrm{we}\mathrm{get}\mathrm{f}⩽\frac{4}{3}$$$$$$

Commented by Frix last updated on 12/Nov/23

$$f(a,b,c,d)=\frac{a}{a+b+0c+d}+\frac{b}{a+b+c+0d}+\frac{c}{0a+b+c+d}+\frac{d}{a+0b+c+d}$$$$f(x,x,y,y)=\frac{2(x^2+4xy+y^2)}{(x+2y)(2x+y)}\Rightarrow 1⩽f⩽\frac{4}{3}$$$$f(x,y,x,y)=\frac{4(x^2+xy+y^2)}{(x+2y)(2x+y)}\Rightarrow \frac{4}{3}⩽f⩽2$$$$\mathrm{This}\mathrm{is}\mathrm{still}\mathrm{no}\mathrm{proof}...$$

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