Question

Bài 1: Cho - 1 \(\le\) x; y; z \(\le\)2 và x + y + z = 0. CMR x² + y² + z² \(\le\) 6

Bài 2: CMR: Nếu ( x - y )² + ( y - z )² + ( z - x )² = ( y + z - 2x )² + ( z + x - 2y )² + ( x + y - 2z )² thì x = y = z

Answer 1

2. Phân tích vế trái ta được:

\(2.\left[x^2+y^2+z^2-\left(xy+yz+zx\right)\right]\)

Phân tích vế phải ta được:

\(6.\left[x^2+y^2+z^2-\left(xy+yz+zx\right)\right]\)

Vì \(VT=VP\) nên \(VP-VT=0.\)

\(\Rightarrow4.\left[x^2+y^2+z^2-\left(xy+yz+zx\right)\right]=0\)

\(\Rightarrow2.\left\{2.\left[x^2+y^2+z^2-\left(xy+yz+zx\right)\right]\right\}=0\)

\(\Rightarrow2.\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]=0\)

\(\Rightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\)

\(\Rightarrow\left[{}\begin{matrix}\left(x-y\right)^2=0\\\left(y-z\right)^2=0\\\left(z-x\right)^2=0\end{matrix}\right.\)

\(\Rightarrow x=y=z\) ( đpcm )