Question

Tìm các số nguyên x,y,z thỏa mãn: x/y+y/z+z/x=y/x+z/y+x/z=x+y+z=3

Answer 1

Ta có: 

\(\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)\left(\frac{y}{x}+\frac{z}{y}+\frac{x}{z}\right)=3+\left(\frac{xz}{y^2}+\frac{y^2}{xz}\right)+\left(\frac{x^2}{yz}+\frac{yz}{x^2}\right)+\left(\frac{z^2}{xy}+\frac{xy}{z^2}\right)\)

\(\ge3+2\sqrt{\frac{xy^2z}{y^2xz}}+2\sqrt{\frac{x^2yz}{yzx^2}}+2\sqrt{\frac{z^2xy}{xyz^2}}=3+2+2+2=9\)

Dấu \(=\)xảy ra khi \(x=y=z\). 

Suy ra giả thiết xảy ra khi \(x=y=z\)suy ra \(x=y=z=1\).