1. Tam Coc - Bich Dong, which belongs to Trang An Scenic Landscape Complex, is well known for its poe and inspirational natural scenery.

2. It is the second nicest cavern in Viet Nam, after Huong rich Cavern.

3. Bich Dong Pagoda was built in Le Dynasty in Bich Dong Cavern which/whose name means “Green Pearl Cavern”.

4. Tam Coc Cavern consists of Ca cave, Hai cave and Ba cave, all of which offer both beauty and mystery to tourists.

5. Visiting Tam Coc Cavern, tourists will feel  getting lost in such a hidden fairy site.

6. In brief, it is an ideal ecological spot for lovers of nature.

,mkjh

1. Progress in science is being made day after day. → People are making progress in science day after day.

2. A French architect designed the palace.

3. They have paved the road in front of my house.

4. You can see that they haven't washed the dishes.

5. They will not increase our salaries this year.

6. Didn't they build that theatre two years ago?

7. They may discuss the problem again.

8. They have offered my brother a well – paid job.

9. People reported that the war started again in South American.

10. He told me that his football team had played well last season.

2. A French architect designed the palace.

3. They have paved the road in front of my house.

4. You can see that they haven't washed the dishes.

5. They will not increase our salaries this year.

6. Didn't they build that theatre two years ago?

7. They may discuss the problem again.

8. They have offered my brother a well – paid job.

9. People reported that the war started again in South American.

10. He told me that his football team had played well last season.

2. The man is believed to have been killed by terrorists.

3. The company is thought to be planning a new advertising campaign.

4. The President was reported to have suffered a heart attack.

5. The man is alleged to have been driving at 110 miles an hour.

6. The expedition is known to have reached the South Pole in May.

7. There is said to be a secret tunnel between them.

8. She is considered to have been the best singer that Australia has ever produced.

9. The weather is expected to be good tomorrow.

10. The Prime Minister and his wife are believed to have separated.

2. The man is believed to have been killed by terrorists.

3. The company is thought to be planning a new advertising campaign.

4. The President was reported to have suffered a heart attack.

5. The man is alleged to have been driving at 110 miles an hour.

6. The expedition is known to have reached the South Pole in May.

7. There is said to be a secret tunnel between them.

8. She is considered to have been the best singer that Australia has ever produced.

9. The weather is expected to be good tomorrow.

10. The Prime Minister and his wife are believed to have separated.

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a, xét từ giác AMNC có 
$={90}^{\circ }$ (Ac là tiếp tuyến của (O) , $\stackrel{}{}$

${}^{}={90}^{\circ }$ (MN vuông góc với CD) => $\stackrel{}{}$\(\widehat{CAM}+\widehat{CNM}\)=180

=> AMNC nội tiếp

Xét tứ giác BMND có $\stackrel{ˆ}{}$=90 ( BD là tiếp tuyến của (O) , \(\widehat{CND}\)=90 ( MN vuông góc với CD)

=> \(\widehat{MND}+\widehat{NAC}\)=180

=> Tứ giác BDMN nội tiếp

b, Ta có \(\widehat{CMN}=\widehat{NAC}\) (cùng chắn CN)

=> ${\mathrm{\backslash (\backslash widehat\{CMN\}\backslash )}}^{}$=$\frac{1}{2}$ cung AN(1)

Ta cũng có\(\widehat{NMD}+\widehat{NMD}\) (cùng chắn cung ND)

\(\widehat{NMD}\)=$\frac{1}{2}$ cung NB(2)

Từ (1) và (2) => \(\widehat{CMD}+\widehat{NMD}\)= $\frac{1}{2}$ (cung AN + cung NB) 

=> \(\widehat{CMD}\)= $\frac{1}{2}$ cung AB = $\frac{180}{2}$=90

=> tam giác CMD vuông tại M

Vì NMBD nội tiếp => \(\widehat{NDM}+\widehat{NBM}\) ( góc nội tiếp cùng chắn cung AM) 

Mà \(\widehat{MCD}+\widehat{NBM}\)=90

=> \(\widehat{MCD}+\widehat{NBM}\)=90 (1)

Mặt khác \(\widehat{NAB}+\widehat{NBA}\)=90 (2)

Từ (1) và (2) => \(\widehat{MCD}=\widehat{NAB}\)

Xét tam giác ANB và CMD ta cs

\(\widehat{ANB}=\widehat{CMD}\) (=90)

\(\widehat{MCD}=\widehat{NAD}\)

=> 2 tam giác này bằng nhau

Add: Tr Ph Thảo (hpthaoo)

Áp dụng bất đẳng thức Bunhiacopxki ta có \(\left(a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}\right)^2\le2\left(a+b+c\right)\left(a^2+b^2+c^2\right)\)\(=abc\left(a+b+c\right)\left(a^2+b^2+c^2\right)\)

Theo một bất đẳng thức quen thuộc ta có \(abc\left(a+b+c\right)\le\frac{1}{3}\left(ab+bc+ca\right)^2\)

Từ đó ta được \(abc\left(a+b+c\right)\left(a^2+b^2+c^2\right)\le\frac{\left(a^2+b^2+c^2\right)\left(ab+bc+ca\right)^2}{3}\)\(\le\frac{\left(a^2+b^2+c^2+ab+bc+ca+ab+bc+ca\right)^3}{3^4}=\frac{\left(a+b+c\right)^6}{3^4}\)

Do đó ta có \(\left(a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}\right)^2\le\frac{\left(a+b+c\right)^6}{3^4}\)hay \(a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}\le\frac{\left(a+b+c\right)^3}{3^2}\)(*)

Dễ dàng chứng minh được \(a^3+b^3+c^3\ge\frac{\left(a+b+c\right)^3}{9}\)(**)

Từ (*) và (**) suy ra \(a^3+b^3+c^3\ge a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}\)

Vậy bất đẳng thức được chứng minh

Đẳng thức xảy ra khi \(a=b=c=\sqrt[3]{2}\)

Xét hiệu : \(a^3+b^3-ab\left(a+b\right)=\left(a-b\right)^2\left(a+b\right)\ge0,\forall a,b>0\)

\(\Rightarrow a^3+b^3\ge ab\left(a+b\right)\)

Áp dụng BĐT AM-GM :
\(a^3+b^3+2c^3\ge ab\left(a+b\right)+2c^3\ge2\sqrt{ab\left(a+b\right).2c^3}=2\sqrt{4c^2\left(a+b\right)}\)

\(=4c\sqrt{a+b}\)

Hoàn toàn tương tự

\(a^3+2b^3+c^3\ge4b\sqrt{a+c};2a^3+b^3+c^3\ge4a\sqrt{b+c}\)

Cộng thao vế bất đẳng thức vừa thu được

\(\Rightarrow a^3+b^3+c^3\ge a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}\left(đpcm\right)\)

Dấu " = " xảy ra khi \(a=b=c=\sqrt[3]{2}\)

Chúc bạn học tốt !!!