Question

Nếu a+b+c=\(\sqrt{ab}\)+\(\sqrt{bc}\)+\(\sqrt{ca}\)( a,b,c là ba số thực dương ) thì ?

Answer 1

\(\Leftrightarrow2a+2b+2c-2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca}=0\)

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

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

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