Question

Cho a,b,c > 0 thỏa mãn

\(a+b+c=\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)

Cm a=b=c

Answer 1

\(\Leftrightarrow2a+2b+2c=2\sqrt{ab}+2\sqrt{bc}+2\sqrt{ca}\)

\(\Leftrightarrow\left(a-2\sqrt{ab}+b\right)+\left(b-2\sqrt{bc}+c\right)+\left(c-2\sqrt{ca}+a\right)=0\)

\(\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2+\left(\sqrt{b}-\sqrt{c}\right)^2+\left(\sqrt{c}-\sqrt{a}\right)^2=0\)

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