Demonstrati ca: $abc=1\Rightarrow \frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}=1;\,a,b,c>0.$
(Test Selectie IMO, Malaysia-2011)
Identitate conditionata
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- Mesaje: 1493
- Membru din: Mar Oct 26, 2010 9:21 pm
- Localitate: ORADEA
Re: Identitate conditionata
$\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}=\frac{1}{b+1+bc}+\frac{b}{bc+b+1}+\frac{bc}{1+b+bc}=\frac{1+b+bc}{1+bc+b}=1$mihai miculita scrie:Demonstrati ca: $abc=1\Rightarrow \frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}=1;\,a,b,c>0.$
(Test Selectie IMO, Malaysia-2011)
Nu e cam simpla pentru un test de selectie? .
Re: Identitate conditionata
Catană Adrian,
Elev la CNIV, Targoviste, clasa a X-a
Elev la CNIV, Targoviste, clasa a X-a
Re: Identiate conditionala
Se face mai usor cu substitutiile $a=\dfrac{u}{v},b=\dfrac{v}{w},c=\dfrac{w}{u}$.