Question

Tìm x, y, z thoả mãn điều kiện: \(\left\{{}\begin{matrix}x+y=\sqrt{4z-1}\\y+z=\sqrt{4x-1}\\z+x=\sqrt{4y-1}\end{matrix}\right.\)

Answer 1

ĐK:\(x,y,z\ge \frac{1}{2}\)

Cộng theo vế 3 BĐT trên ta có:

\(2x+2y+2z-\sqrt{4x-1}-\sqrt{4y-1}-\sqrt{4z-1}=0\)

\(\Leftrightarrow\left(4x-1-2\sqrt{4x-1}+1\right)+\left(4y-1-2\sqrt{4y-1}+1\right)+\left(4z-1-2\sqrt{4z-1}+1\right)=0\)

\(\Leftrightarrow\left(\sqrt{4x-1}-1\right)^2+\left(\sqrt{4y-1}-1\right)^2+\left(\sqrt{4z-1}-1\right)^2=0\)

Dễ thấy: \(VT\ge0\forall x,y,z\)

\("="\Leftrightarrow\left\{{}\begin{matrix}\sqrt{4x-1}=1\\\sqrt{4y-1}=1\\\sqrt{4z-1}=1\end{matrix}\right.\)\(\Leftrightarrow x=y=z=\dfrac{1}{2}\)