Općinsko natjecanje 2006 SŠ1 3
Dodao/la:
arhiva2. travnja 2012. Ako su
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
i
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
realni brojevi za koje je
![a+b\neq0](/media/m/3/5/1/35114236218fe03142d987bfda00bd08.png)
,
![b+c\neq0](/media/m/4/3/1/4314b5e08740220959e59f4e7e66fb27.png)
i
![a+c\neq0](/media/m/e/6/6/e6654c640e70e89400e0ac55d440e398.png)
, dokaži da izraz:
![\left(1+\dfrac{c}{a+b}\right) \left( 1+\dfrac{a}{b+c}\right)
\left( 1+\dfrac{b}{a+c}\right) -\dfrac{a^{3}+b^{3}+c^{3}}
{\left(a+b\right) \left(b+c\right) \left(a+c\right)}](/media/m/d/1/e/d1e2c2fa02922c252346cdd17bb4ea3c.png)
ne ovisi o vrijednostima brojeva
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
i
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
.
%V0
Ako su $a$, $b$ i $c$ realni brojevi za koje je $a+b\neq0$, $b+c\neq0$ i $a+c\neq0$, dokaži da izraz: $$
\left(1+\dfrac{c}{a+b}\right) \left( 1+\dfrac{a}{b+c}\right)
\left( 1+\dfrac{b}{a+c}\right) -\dfrac{a^{3}+b^{3}+c^{3}}
{\left(a+b\right) \left(b+c\right) \left(a+c\right)}
$$ ne ovisi o vrijednostima brojeva $a$, $b$ i $c$.
Izvor: Općinsko natjecanje iz matematike 2006