Áp dụng bất đẳng thức Cauchy, ta có: \(\sqrt{\frac{b+c}{a}}=\sqrt{\left(\frac{b+c}{a}\right).1}\le\frac{\frac{b+c}{a}+1}{2}=\frac{a+b+c}{2a}\)\(\Rightarrow\sqrt{\frac{a}{b+c}}\ge\frac{2a}{a+b+c}\)

Tương tự:\(\sqrt{\frac{b}{c+a}}\ge\frac{2b}{a+b+c}\); \(\sqrt{\frac{c}{a+b}}\ge\frac{2c}{a+b+c}\)

Cộng theo vế ba bất đẳng thức trên, ta được: \(\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\ge\frac{2\left(a+b+c\right)}{a+b+c}=2\)

Đẳng thức xảy ra khi \(b+c=a,c+a=b,a+b=c\Rightarrow a+b+c=0\)

Nhưng do a, b, c > 0 (gt) nên a + b + c > 0

Vậy đẳng thức không xảy ra

a/

\(BC^2=5^2=25\)

\(AB^2+AC^2=4^2+3^2=16+9=25\)

\(\Rightarrow BC^2=AB^2+AC^2=25\) => tg ABC vuông tại A

b/

Theo t/c đường phân giác 

\(\frac{BD}{CD}=\frac{AB}{AC}=\frac{4}{3}\) (Trong tam giác, đường phân giác của một góc chia cạnh đối diện thành hai đoạn tỉ lệ với hai cạnh kề của hai đoạn ấy)

\(\Rightarrow BD=\frac{5}{4+3}x4=\frac{20}{7}\Rightarrow CD=5-\frac{20}{7}=\frac{15}{7}\)

là bustling

Nhộn nhịp = bustling

PRESERVING THE PAST

Life has changed a lot over the past 50 years, and there are many good pastimes which seem to be dying out. Work in groups and search for a past tradition or pastime which you highly appreciate give reasons why you it work out a plan to help preserve it. Then make a poster presenting your ideas and share it with your class

Life has changed a lot over the past 50 years and there are many good pastimes and traditions such as folk music, folk games, paintings… But the tradition that I appreciate is writing letters because of some reasons: In the past, writing letters was such an important means of communication to contact people with people. As far as I know in the past Ho Chi Minh president used to use letters to contact with his comrade to make a plan for upcoming resistance. Otherwise, I feel very surprised and interested when I open a letter with some tiny skers. Writing to friends and family was a great to keep in touch. It also enabled me to prace my literacy skills and communicate effectively. Writing letters can be seen as a gift because someone has taken his/her time to write, think and express love. But now, this pastime seems to be dying out. Technology is growing, so people communicate by email more than writing a letter. In conclusion, you should write a letter to your friends or family in your free time to preserve them for future generations, I am sure that it will be very interesting.

PRESERVING THE PAST

Life has changed a lot over the past 50 years, and there are many good pastimes which seem to be dying out. Work in groups and search for a past tradition or pastime which you highly appreciate give reasons why you it work out a plan to help preserve it. Then make a poster presenting your ideas and share it with your class

many other traditions, the spirit of patriotism is a feature of our country's long-standing culture, it is expressed from the past to the present and goes into every action and thought of each person. Since ancient times, the spirit of patriotism is most clearly disclosed in the mirror heroes Hai Ba Trung, Ba Trieu, Tran Hung Dao and so on ... But it is in times of war. Also now - peacetime - the modernization period with machines and tools being increasingly useful and pracal. The society is more and more advanced, everything has changed a lot, only the patriotism of each individual is still not blurred. In the life of thoughts, jobs to help develop the country's economy are partly patrio.