Question

[Nguyễn Hồng Hạnh]

Chứng minh rằng : nếu a,c,b là 3 số thỏa mãn hệ thức 

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}\) thì 2 trong 3 số đó phải là đối nhau

Answer 1

[ST]

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}\Leftrightarrow\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b+c}-\frac{1}{c}\)

\(\Leftrightarrow\frac{a+b}{ab}=\frac{c-a-b-c}{c\left(a+b+c\right)}\Leftrightarrow\frac{a+b}{ab}=\frac{-\left(a+b\right)}{ac+bc+c^2}\)

\(\Leftrightarrow\left(a+b\right)\left(ac+bc+c^2\right)=-\left(a+b\right)ab\)

\(\Leftrightarrow\left(a+b\right)\left(ac+bc+c^2\right)+\left(a+b\right)ab=0\)

\(\Leftrightarrow\left(a+b\right)\left(ac+bc+c^2+ab\right)=0\)

\(\Leftrightarrow\left(a+b\right)\left[a\left(b+c\right)+c\left(b+c\right)\right]=0\)

\(\Leftrightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)=0\)

<=> a +  b = 0 hoặc b + c = 0 hoặc c + a = 0

<=> a = -b hoặc b = -c hoặc c = -a

Vậy...

Answer 2

[Trần Thùy Dương]

Ta có : 

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

\(\Rightarrow\frac{bc+ca+ab}{abc}=\frac{1}{a+b+c}\)

\(\Rightarrow\left(bc+ca+ab\right)\left(a+b+c\right)=abc\)

\(\Rightarrow\left(bc+ac+ab\right)\left(a+b+c\right)-abc=0\)

\(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)=0\)

\(\Rightarrow\hept{\begin{cases}a+b=0\\b+c=0\\c+a=0\end{cases}}\)