Ta có: \(x^2+y^2=\left(x^2+4y^2\right)-3y^2\)

                           \(\ge4xy-3y^2\)

                            \(\ge4xy-3y.\frac{x}{2}\)

                              \(=\frac{5}{2}xy\)

Khi đó \(A=\frac{x^2+y^2}{2017xy}\ge\frac{\frac{5xy}{2}}{2017xy}=\frac{5}{4034}\)

Dấu "=" xảy ra <=> x = 2y

Bài này chắc dùng phương pháp hạ bậc + chọn điểm rơi. :v

                         Lời giải:

Dự đoán dấu "=" xảy ra tại a = b = 1

Ta có: \(1+a^2\ge2a;1+b^2\ge2b\) (cô si)

Suy ra \(\frac{1}{1+a^2}+\frac{1}{1+b^2}\le\frac{1}{2a}+\frac{1}{2b}\) (1)

Áp dụng BĐT Am-Gm (Cô si),ta có: \(ab\le\frac{a^2+b^2}{2}\)

Lại có: \(\frac{2}{1+ab}\ge\frac{2}{1+\frac{a^2+b^2}{2}}\ge\frac{2}{1+\frac{2}{2}}=1\) (2)

Ta sẽ c/m: \(\frac{1}{2a}+\frac{1}{2b}\le1\Leftrightarrow\frac{1}{a}+\frac{1}{b}\le2\)

Chứng minh tiếp đi:v,bí r:v

: ở đâu có nhãn xanh thế tth?

Từ \(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge2\)

\(\Rightarrow\frac{1}{1+x}\ge\left(1-\frac{1}{1+y}\right)+\left(1-\frac{1}{1+z}\right)\)          

                    \(=\frac{y}{1+y}+\frac{z}{1+z}\ge2\sqrt{\frac{yz}{\left(1+y\right)\left(1+z\right)}}\)

C/m tương tự cũng có \(\frac{1}{1+y}\ge2\sqrt{\frac{xz}{\left(1+x\right)\left(1+z\right)}}\)

                                    \(\frac{1}{1+z}\ge2\sqrt{\frac{xy}{\left(1+x\right)\left(1+y\right)}}\)

Nhân 3 vế của các bất đẳng thức trên lại ta được

\(\frac{1}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\ge\frac{8xyz}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\)

\(\Rightarrow1\ge8xyz\)

\(\Leftrightarrow xyz\le\frac{1}{8}\)

Dấu "='' khi \(x=y=z=\frac{1}{2}\)

Vậy .......

Đây là môn toán mà!

Dấu BĐT ngược 1 chút \(\frac{1}{1+a^2}+\frac{1}{1+b^2}\ge\frac{2}{1+ab}\)

Xét hiệu 2 vế của BĐT

\(\frac{1}{1+a^2}+\frac{1}{1+b^2}-\frac{2}{1+ab}=\frac{1}{1+a^2}-\frac{1}{1+ab}+\frac{1}{1+b^2}-\frac{1}{1+ab}\)

\(=\frac{\left(a-b\right)^2\left(ab-1\right)}{\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)}\ge0\)

=> \(\frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}\ge\frac{2}{1+ab}\)

Dấu "=" xảy ra <=> a=b=1

a/  ĐKXĐ: \(x\ne3;-3;2\)

\(\frac{x+2}{x+3}-\frac{5}{x^2+x-6}+\frac{1}{3-x}=\frac{x+2}{x+3}-\frac{5}{x^2+3x-2x-6}+\frac{-1}{x-3}=\frac{x+2}{x+3}-\frac{5}{\left(x^2-2x\right)+\left(3-6x\right)}\)

\(+\frac{-1}{x-3}=\frac{x+2}{x+3}-\frac{5}{\left(x-2\right)\left(x+3\right)}+\frac{-1}{x-3}\)

Đến đây bạn tự quy đồng nhé! Chúc sớm giải được. Cố lên!!!!!

6c

7c

8b

9a

10b

11a

12a

13c

14a

15a

16b

17b

18a

19b

20a

6. The front yard is_______ to play soccer in.

A. small too                B. so small                  C. too small                D. very small

7. My brother feels_______ again after his operation.

A. strongly                  B. strength                 C. strong                     D. enough strong

8. Their family had a few_______ rest after a long trip.

A. hours                      B. hour’s                     C. hours’                     D. hour

9. Her son was absent from school_______ of his illness last Tuesday.

A. because                  B. instead                   C. though                    D. but

10. Susan is the_______ of the three girls.

            A. prettier                   B. prettiest                 C. pretty                      D. prettily

11. _______ the beginning of the day called morning or evening?

            A. Is                             B. Are                          C. Does                       D.Can

12. Nam_______ familiar with getting up early in winter two years ago.

            A. was                         B. were                       C. is                             D. has been

13.The school yard was_______ with students.

A. crowd                     B. crowds                   C. crowded                 D. crowding

14. After a month, Hoa got used to_______ in her new school.

A. study                      B. studying                 C. studied                   D. is studying

15. My father always drives_______. He is a_______ driver.

            A. carefully/ careful  B. carefully/carefullyC. carefully/ care      D. careful/carefully

16. Unless you work hard, you_______ fail the exams.

A. won’t fail                B. will fail                    C. don’t fail                 D. didn’t fail

17. Mr Minh used to_______ alot when he was young.

            A. smoke                     B. smoking                 C. smoked                   D.be smoking

18. She told me_______ up too late.

A. don’t stay               B. not to stay              C. not stay                  D. to not stay

19. Why are you turning on the TV? I_______ the news.
            A. will watch               B. am going to watch            C. watch                      D. watched

20. Hung is_______ to ride motorbike to school.

A. not old enough      B. old not enough      C. enough old not      D. enough not old

30 phút = 1/2 giờ

Gọi độ dài quãng đường AB là x(km) (x > 0)

Ta có: \(\frac{x}{30}-\frac{x}{40}=\frac{1}{2}\Leftrightarrow\frac{x}{120}=\frac{1}{2}\Leftrightarrow x=60\) (thỏa mãn)

Vậy độ dài quãng đường AB là 60 km.