Question

Cho a,b,c \(\in\) R, biết \(a+b+c=n\) và \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{n}\) CMR: Trong 3 số a,b,c có ít nhất 1 số bằng n

Answer 1

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

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{a+b+c}=0\)

\(\Leftrightarrow\frac{a+b}{ab}+\frac{a+b+c-c}{c\left(a+b+c\right)}=0\)

\(\Leftrightarrow\left(a+b\right)\left(\frac{1}{ab}+\frac{1}{ac+bc+c^2}\right)=0\)

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

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

\(\Leftrightarrow\left[{}\begin{matrix}a+b=0\Rightarrow c=n-a-b=n\\b+c=0\Rightarrow a=n\\a+c=0\Rightarrow b=n\end{matrix}\right.\)