Întrebare Ex clasa a7-a

Fie a,b,c,d numere reale pozitive astfel incat a*b*c*d=1. Calculati:

E= (7 + a) / (1 + a + a*b + a*b*c) + (7 + b) / (1 + b + b*c + b*c*d) + (7 + c) / (1 + c + c*d + c*d*a) + (7 + d) / (1 + d + d*a + d*a*b).

Buna seara:

$abc=\frac{1}{d}\Rightarrow \frac{a+7}{1+a+ab+abc}=\frac{a+7}{1+a+ab+\frac{1}{d}}=\frac{ad+7d}{1+d+da+dab}$.

In mod similar

$bc=\frac{1}{ad}\Rightarrow \frac{b+7}{1+b+bc+bcd}=\frac{abd+7ad}{1+d+da+dab}$ si

$c=\frac{1}{abd}\Rightarrow \frac{c+7}{1+c+cd+cda}=\frac{1+7abd}{1+d+da+dab}$. Atunci

$E=\frac{ad+7d+abd+7ad+1+7abd+7+d}{1+d+da+dab}=\frac{8\cdot\left ( 1+d+da+dab \right )}{1+d+da+dab}=8$.