Question

Cho các số thực dương a,b,c thỏa mãn \(\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a}=1\). CMR: \(a^2\left(1+c\right)+b^2\left(1+a\right)+c^2\left(1+b\right)=1+abc\)

Answer 1

Quy đồng giả thiết lên:

\(\Leftrightarrow a\left(1+a\right)\left(1+c\right)+b\left(1+b\right)\left(1+a\right)+c\left(1+c\right)\left(1+b\right)=\left(1+a\right)\left(1+b\right)\left(1+c\right)\)

\(\Leftrightarrow a+b+c+ab+bc+ca+a^2\left(1+c\right)+b^2\left(1+a\right)+c^2\left(1+b\right)=1+a+b+c+ab+bc+ca+abc\)

\(\Leftrightarrow a^2\left(1+c\right)+b^2\left(1+a\right)+c^2\left(1+b\right)=1+abc\)