Question

Stiind ca a,b,c∈(0,∞) si $b^{2} +bc-ac=c ^{2}$ , respectiv a+b=2c , calculati $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$

Answer 1

$Inmultim~prima~relatie~cu~4: \\ \\ 4b^2+4bc-4ac=4c^2. \\ \\ Substituim~2c~cu~a+b: \\ \\ 4b^2+2ab+2b^2-2a^2-2ab=a^2+2ab+b^2 \Leftrightarrow \\ \\ \Leftrightarrow 5b^2-2ab-3a^2=0 \Leftrightarrow 2b(b-a)+3(b^2-a^2)=0 \Leftrightarrow \\ \\ \Leftrightarrow (b-a)(3a+5b)=0.$

$Distingem~doua~cazuri: \\ \\ i)~3a+5b=0,~imposibil,~deoarece~a,b\ \textgreater \ 0. \\ \\ ii))~a-b=0 \Rightarrow a=b,~deci~a+b=2c \Leftrightarrow 2a=2b=2c \Leftrightarrow a=b=c. \\ \\ E=1+1=1=3.$