Identitate conditionata

[mihai miculita]

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)

[Ciprian.]

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)

$\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ac+c+1}=\frac{1}{b+1+bc}+\frac{b}{bc+b+1}+\frac{bc}{1+b+bc}=\frac{1+b+bc}{1+bc+b}=1$
Nu e cam simpla pentru un test de selectie? .

[Mr. Ady]

Vezi aici http://forum.gil.ro/viewtopic.php?f=16&t=689

[math111]

Se face mai usor cu substitutiile $a=\dfrac{u}{v},b=\dfrac{v}{w},c=\dfrac{w}{u}$.