ta có 

\(4\left(x^2+xy+y^2\right)\ge3\left(x+y\right)^2\Leftrightarrow\left(x-y\right)^2\ge0\) vì thế \(\sqrt{x^2+xy+y^2}\ge\frac{\sqrt{3}}{2}\left(x+y\right)\)

hoàn toàn tương tự ta sẽ có 

\(P\ge\frac{\sqrt{3}}{2}\left(x+y\right)+\frac{\sqrt{3}}{2}\left(y+z\right)+\frac{\sqrt{3}}{2}\left(x+z\right)\)

hay

\(P\ge\sqrt{3}\left(x+y+z\right)=3\sqrt{3}\)

dấu bằng xảy ra khi x=y=z=1

 Ta có: \(P=\sqrt{x^2+xy+y^2}+\sqrt{y^2+yz+z^2}+\sqrt{z^2+zx+x^2}\)\(=\sqrt{\frac{3}{4}\left(x+y\right)^2+\frac{1}{4}\left(x-y\right)^2}+\sqrt{\frac{3}{4}\left(y+z\right)^2+\frac{1}{4}\left(y-z\right)^2}+\sqrt{\frac{3}{4}\left(z+x\right)^2+\frac{1}{4}\left(x-y\right)^2}\)\(\ge\sqrt{\frac{3}{4}\left(x+y\right)^2}+\sqrt{\frac{3}{4}\left(y+z\right)^2}+\sqrt{\frac{3}{4}\left(z+x\right)^2}=\sqrt{3}\left(x+y+z\right)=3\sqrt{3}\)

Đẳng thức xảy ra khi x = y = z = 1

I. Choose the best option A,B,C or D to complete the sentence

1 I wish I ______ people with the same interests

A.meet B.can meet C. could meet D.will meet

2. My mother says she used to _____ a diary when she was young.

A.keep B.use C.write D.preserve

3 The ______ between generations seems to be bigger and bigger than ever

A.blank B.space C.gap D.distance

4. Our teacher's is that we set up a project on preservation of our traditional games

A.offer B.suggestion C,scheme D.order

5 I found the letter in my bag. -I _____ to post it

A.forgot B.would forget C. was forgetting D. forgotten 

6. Does he tell you how he is getting _____ his new colleagues?

A.on with B.on of C.away with D.out of

7 Unfortunately the new system didn't _____our expectations

A. catch up with B. bring about C.come across D.come up to

8. The doctors say it'll take him a long to get _____ the shock

A.past B.above C.through D.over

9. I wish you ____ stop interrupting me whenever I speak.

A.will B.would C.could D.might

10. The policeman is happy ______ the kids safe and sound now.

A.finding B.be found C.to find D.have found

11. _______ she is young, she is very independent

A.Although B.because C.in order that D. in spite of

12. _______ he had finished his studies, he went to America

A.while B.before C.after D.until

Áp dụng giả thiết và bất đẳng thức AM - GM, ta có: \(VT=\frac{x}{\sqrt{x^2+xy+yz+zx}}+\frac{y}{\sqrt{y^2+xy+yz+zx}}+\frac{z}{\sqrt{z^2+xy+yz+zx}}\)\(=\frac{x}{\sqrt{\left(x+y\right)\left(x+z\right)}}+\frac{y}{\sqrt{\left(y+x\right)\left(y+z\right)}}+\frac{z}{\sqrt{\left(z+x\right)\left(z+y\right)}}\)\(=\sqrt{\frac{x}{x+y}.\frac{x}{x+z}}+\sqrt{\frac{y}{x+y}.\frac{y}{y+z}}+\sqrt{\frac{z}{z+x}.\frac{z}{y+z}}\)\(\le\frac{\frac{x}{x+y}+\frac{x}{x+z}+\frac{y}{x+y}+\frac{y}{y+z}+\frac{z}{z+x}+\frac{z}{y+z}}{2}=\frac{3}{2}\)

Đẳng thức xảy ra khi \(x=y=z=\frac{1}{\sqrt{3}}\)