Question

Biết x = by + cz

y = ax + cz

z = ax + by

CMR \(\frac{1}{1+a}\)+ \(\frac{1}{1+b}\)+\(\frac{1}{1+c}\)= 2

Answer 1

\(\hept{\begin{cases}x=by+cz\left(1\right)\\y=ax+cz\left(2\right)\\z=ax+by\left(3\right)\end{cases}}\)

Cộng theo vế 3 đẳng thức trên:

\(x+y+z=2ax+2by+2cz=2\left(ax+by\right)+2cz=2z+2cz=2z\left(c+1\right)\)

\(=>\frac{1}{c+1}=\frac{2z}{x+y+z}\left(4\right)\)

Tương tự,ta có \(\frac{1}{a+1}=\frac{2x}{x+y+z}\left(5\right);\frac{1}{b+1}=\frac{2y}{x+y+z}\left(6\right)\)

cộng theo vế (4),(5),(6) ta đc:

\(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}=\frac{2x}{x+y+z}+\frac{2y}{x+y+z}+\frac{2z}{x+y+z}=\frac{2\left(x+y+z\right)}{x+y+z}=2\) (đpcm)

Answer 2

http://olm.vn/hoi-dap/question/580063.html   (Câu hỏi của Anh Cao Ngọc)