a. Tứ giác CEHD có \(\widehat{HEC}=\widehat{HDC}=90^o\Rightarrow\) nó là tứ giác nội tiếp.

b. Tứ giác BFEC có \(\widehat{BEC}=\widehat{BFC}=90^o\Rightarrow\)nó là tứ giác nội tiếp. Vậy 4 điểm B, C, E, F cùng thuộc một đường tròn.

c. Ta thấy \(\Delta HAE\sim\Delta CAD\left(g-g\right)\Rightarrow\frac{AH}{AC}=\frac{AE}{AD}\Rightarrow AE.AC=AH.AD\)

Ta thấy \(\Delta CBE\sim\Delta CAD\left(g-g\right)\Rightarrow\frac{BC}{AC}=\frac{BE}{AD}\Rightarrow AD.BC=BE.AC\)

d. Ta thấy ngay \(\widehat{PCB}=\widehat{BAM}\) (Cùng phụ với góc ABC)

Mà \(\widehat{BAM}=\widehat{BCM}\) (Góc nội tiếp cùng chắn cung BM)

Vậy nên \(\widehat{PCB}=\widehat{BCM}\) hay CM là phân giác góc \(\widehat{PCB}\)

Lại có \(CM⊥HD\) nên HCM là tam giác cân. Vậy CB là trung trực của HM hay H, M đối xứng nhau qua BC.

e. Ta thấy BFHD là tứ giác nội tiếp nên \(\widehat{FDH}=\widehat{FBH}\) (Góc nội tiếp cùng chẵn cung FH)

 DHEC cùng là tứ giác nội tiếp nên \(\widehat{HDE}=\widehat{HCE}\) (Góc nội tiếp cùng chẵn cung HE)

Mà \(\widehat{FBH}=\widehat{HCE}\) ( Cùng phụ với góc \(\widehat{BAC}\) )

nên \(\widehat{FDH}=\widehat{HDE}\) hay DH là phân giác góc FDE.

Tương tự FH, EH cũng là phân giác góc DFE và DEF.

Vậy tâm đường tròn nội tiếp tam giác DEF chính là H.

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

hình 2

_ I see a man is holding a child and a woman is going to work.

_I can see that in the future, men will take care of children, cook rice, ... and women will be the backbone of the family to make money to feed everyone in the house. But today, in complete contrast, men are the breadwinner of the family, and women stay home to take care of their babies, and cook housework.

_I think this transition is not very appropriate for men, which are healthier than women, so they will do heavy work. Women are weaker than men, so they will do light work such as taking care of children , cooking rice , doing housework

hình 2

2. The problems faced by the company, which I'll look at in detail in a moment, are being resolved.

3. She was greatly influenced by her father, who/ whom she adored.

4. Parents whose children are between four and six are being asked to take part in the survey.

5. He isn't looking forward to the time when he will have to leave.

6. The Roman coins, which a local farmer came across in a field, are now on display in the National Museum.

7. He pointed to the stairs which/that led down to the cellar.

8. These drugs, which are used to treat stomach ulcers, have been withdrawn from sale.

9. The singer, who was recovering from flu, had to cancel her concert.

10. We went to the Riverside Restaurant where I once had lunch with Henry.

11. My aunt, whose first job was filling shelves in a supermarket, is now a manager of a department store.

12. John Graham‟s latest film, which is set in the north of Australia, is his first for more than five years.

13. The newspaper is owned by the Mears group, whose chairperson is Sir James Hex.

14. The Master's course, which I took in 1990, is no longer taught.

15. The minister talked about the plans for tax reform (that/which) he will reveal next month.

2. The problems faced by the company, which I'll look at in detail in a moment, are being resolved.

3. She was greatly influenced by her father, who/ whom she adored.

4. Parents whose children are between four and six are being asked to take part in the survey.

5. He isn't looking forward to the time when he will have to leave.

6. The Roman coins, which a local farmer came across in a field, are now on display in the National Museum.

7. He pointed to the stairs which/that led down to the cellar.

8. These drugs, which are used to treat stomach ulcers, have been withdrawn from sale.

9. The singer, who was recovering from flu, had to cancel her concert.

10. We went to the Riverside Restaurant where I once had lunch with Henry.

11. My aunt, whose first job was filling shelves in a supermarket, is now a manager of a department store.

12. John Graham‟s latest film, which is set in the north of Australia, is his first for more than five years.

13. The newspaper is owned by the Mears group, whose chairperson is Sir James Hex.

14. The Master's course, which I took in 1990, is no longer taught.

15. The minister talked about the plans for tax reform that he will reveal next month.

1. Ann is very friendly. She lives next door.

Ann who lives next door is very friendly

2. The man is a famous actor. You met him at the party last night.

The man whom you met at the party last night is a famous actor

3. There are some words. They are very difficult to translate.

There are some words which are very difficult to translate

4. The sun provides us with heat and light. It is one of millions of stars in the universe.

The sun, which is one of millions of stars in the universe, provides us with heat and light

5. Students will be punished. Their homework is late.

 Students who homework is late will be punished

6. I was looking for a book this morning. I've found it now.

I've found the book I was looking for this morning

7. Is that the car? You want to buy it.

  Is that the car which you want to buy ?

8. Sandra works in advertising. You were talking to her.

Sandra, who you were talking to, works in advertising.

9. Lake Prespa is a lonely beautiful lake. It‟s on the north Greek border.

Lake Prespa, which is a lonely beautiful lake, is on the north Greek border

10. The little girl ate sweets the whole way. She sat next to me on the coach.

The little girl who sat next to me on the coach ate candy the whole way

1. Ann, who lives next door, is very friendly.

2. The man who/that you met at the party last night is a famous actor.

3. There are some words that/which are very difficult to translate.

4. The sun, which is one of millions of stars in the universe, provides us with heat and light.

5. Students whose homework is late will be punished.

6. I've found the book that/which I was looking for in the morning.

7. Is that the car that/which you want to buy?

8. Sandra, who you were talking to, works in advertising.

9. Lake Prespa, which is on the north Greek border, is a lonely beautiful lake.

10. The little girl who/that sat next to me on the coach ate sweets the whole way.

Tìm min:

Theo BĐT AM-GM thì: $P=a^2+b^2+c^2\ge ab+bc+ac$ hay $P\ge 9$

Vậy $P_{min}=9$. Giá trị này đạt tại $a=b=c=\sqrt{3}$

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Tìm max:

$P=a^2+b^2+c^2=(a+b+c{)}^2-2(ab+bc+ac)=(a+b+c{)}^2-18$

Vì $a,b,c\ge 1$ nên:

$(a-1)(b-1)\ge 0\iff ab+1\ge a+b$

Hoàn toàn tương tự: $bc+1\ge b+c;ac+1\ge a+c$

Cộng lại: $2(a+b+c)\le ab+bc+ac+3=12$

$\Rightarrow a+b+c\le 6$

$\Rightarrow P=(a+b+c{)}^2-18\le 6^2-18=18$

Vậy $P_{max}=18$. Giá trị này đạt tại $(a,b,c)=(1,1,4)$ và hoán vị

Vì 2n + 1 là số chính phương . Mà 2n + 1 là số lẻ

=> 2n + 1 = 1(mod8)

=> n chia hết cho 4

=> n + 1 là số lẻ

=> n + 1 = 1(mod8)

=> n chia hết cho 8

Mặt khác :

3n + 2 = 2(mod3)

=> (n + 1) + (2n + 1) = 2(mod3)

Mà n + 1 và 2n + 1 là các số chính phương lẻ

=> (n + 1) = (2n + 1) = 1(mod3)

=. n chia hết cho 3

Mà (3;8) = 1

Vậy n chia hết cho 24

Vì 2n + 1 là số chính phương . Mà 2n + 1 là số lẻ

=> 2n + 1 = 1(mod8)

=> n chia hết cho 4

=> n + 1 là số lẻ

=> n + 1 = 1(mod8)

=> n chia hết cho 8

Mặt khác :

3n + 2 = 2(mod3)

=> (n + 1) + (2n + 1) = 2(mod3)

Mà n + 1 và 2n + 1 là các số chính phương lẻ

=> (n + 1) = (2n + 1) = 1(mod3)

=. n chia hết cho 3

Mà (3;8) = 1

Vậy n chia hết cho 24

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