1. It was a terrible film and I was bored from start to finish

2. I'm very excited because I'm going to Florida tomorrow

3. Are you surprised or have you expected her news?

4. Lisa is reading a very interesting book at the moment

5. I've had a very tiring day at work today and I want to go to the bed

6. Most people were surprised that she won the prize

7. I'm boring. Let's go out for a drink somewhere

8. Visit our exciting new shop!

9. His speech was very long and very boring

10. Jerry was totally convinced by Alice's explanation

\(\frac{1}{2}\left(4x-6\right)-1=2\left(\frac{1}{2}x+3\right)\)

\(\Leftrightarrow2x-3-1=x+6\)

\(\Leftrightarrow2x-4=x+6\)

\(\Leftrightarrow x=10\)

vậy......

Đặt \(\hept{\begin{cases}b+c=x\\a+c=y\\a+b=z\end{cases}}\)với x,y,z dương và \(a=\frac{y+z-x}{2};b=\frac{x+z-y}{2};c=\frac{x+y-z}{2}\)

Ta có \(\frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c}=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{y+z-x}{2x}+\frac{x+z-y}{2y}+\frac{x+y-z}{2z}\)

\(=\frac{1}{2}\left(\frac{y}{x}+\frac{x}{y}\right)+\frac{1}{2}\left(\frac{z}{x}+\frac{x}{z}\right)+\frac{1}{2}\left(\frac{z}{y}+\frac{y}{z}\right)-\frac{3}{2}\ge1+1+1-\frac{3}{2}=\frac{3}{2}\)

Dấu "=" xảy ra khi và chỉ khi x=y=z

Với x=y=z thì a=b=c => tam giác ABC đều

Cách khác :

Chu vi tam giác bằng 1 suy ra \(a+b+c=1\Rightarrow\hept{\begin{cases}1-a=b+c\\1-b=c+a\\1-c=a+b\end{cases}}\)

Nên đẳng thức viết lại thành: \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\)\(=\frac{3}{2}\)

Ta sẽ chứng minh \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}\)

Thật vậy, áp dụng bất đẳng thức Cauchy-Schwarz dạng Engel: 

\(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{a^2}{ab+ca}+\frac{b^2}{bc+ab}+\frac{c^2}{ac+bc}\ge\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\)

\(\ge\frac{3\left(ab+bc+ca\right)}{2\left(ab+bc+ca\right)}=\frac{3}{2}\)

Dấu "=" xảy ra khi \(a=b=c\)

Vậy tam giác ABC đều.

số mét đường làm ngày thứ 3 là : 30*3=90m

90-29,6-[29,6+1,8]=29,3

\(\left(\frac{1}{x+2\sqrt{x}}-\frac{1}{\sqrt{x}+2}\right)\div\frac{1-\sqrt{x}}{x+4\sqrt{x}+4}\)

ĐKXĐ : \(\hept{\begin{cases}x>0\\x\ne1\end{cases}}\)

\(=\left(\frac{1}{\sqrt{x}\left(\sqrt{x}+2\right)}-\frac{1}{\sqrt{x}+2}\right)\div\frac{1-\sqrt{x}}{\left(\sqrt{x}+2\right)^2}\)

\(=\left(\frac{1}{\sqrt{x}\left(\sqrt{x}+2\right)}-\frac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+2\right)}\right)\div\frac{1-\sqrt{x}}{\left(\sqrt{x}+2\right)^2}\)

\(=\frac{1-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+2\right)}\times\frac{\left(\sqrt{x}+2\right)^2}{1-\sqrt{x}}\)

\(=\frac{\sqrt{x}+2}{\sqrt{x}}\)

=> đpcm

Vài cái sai đề

với lại đề kiểu đó ai đọc đc

Do \(a,b,c>\frac{25}{4}\)(gt) nên suy ra \(2\sqrt{a}-5>0,2\sqrt{b}-5>0,2\sqrt{c}-5>0\)

Áp dụng bđt cô - si cho 2 số không âm, ta được:

\(\frac{a}{2\sqrt{b}-5}+2\sqrt{b}-5\ge2\sqrt{a}\)

\(\frac{b}{2\sqrt{c}-5}+2\sqrt{c}-5\ge2\sqrt{b}\)

\(\frac{c}{2\sqrt{a}-5}+2\sqrt{a}-5\ge2\sqrt{c}\)

Cộng từng vế của các bđt trên, ta được:

\(\text{ Σ}_{cyc}\frac{a}{2\sqrt{b}-5}+\text{ Σ}_{cyc}\left(2\sqrt{b}\right)-15\ge\text{ Σ}_{cyc}\left(2\sqrt{a}\right)\)

Suy ra \(\text{​​}\text{​​}\text{Σ}_{cyc}\frac{a}{2\sqrt{b}-5}\ge15\)

hay \(Q\ge15\)

(Dấu "="\(\Leftrightarrow a=b=c=25\))

a) Để hàm xác định thì \(\hept{\begin{cases}x\ge0\\\sqrt{x}-1\ne0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)

b) Ta có: \(f\left(x\right)=\frac{\sqrt{x}+1}{\sqrt{x}-1}\)

\(\Rightarrow f\left(4-2\sqrt{3}\right)=\frac{\sqrt{4-2\sqrt{3}}+1}{\sqrt{4-2\sqrt{3}}-1}=\frac{\sqrt{\left(\sqrt{3}-1\right)^2}+1}{\sqrt{\left(\sqrt{3}-1\right)^2}-1}=\frac{\sqrt{3}}{\sqrt{3}-2}\)

và \(f\left(a^2\right)=\frac{\sqrt{a^2}+1}{\sqrt{a^2}-1}=\frac{\left|a\right|+1}{\left|a\right|-1}\)(với \(a\ne\pm1\))

* Nếu \(a\ge0;a\ne1\)thì \(f\left(a^2\right)=\frac{a+1}{a-1}\)

* Nếu \(a< 0;a\ne-1\)thì \(f\left(a^2\right)=\frac{a-1}{a+1}\)

c) \(f\left(x\right)=\frac{\sqrt{x}+1}{\sqrt{x}-1}=\frac{\sqrt{x}-1+2}{\sqrt{x}-1}=1+\frac{2}{\sqrt{x}-1}\)

Để f(x) nguyên thì \(\frac{2}{\sqrt{x}-1}\)nguyên hay \(2⋮\sqrt{x}-1\Rightarrow\sqrt{x}-1\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\)

Mà \(\sqrt{x}-1\ge-1\)nên ta xét ba trường hợp:

+) \(\sqrt{x}-1=-1\Rightarrow x=0\left(tmđk\right)\)

+) \(\sqrt{x}-1=1\Rightarrow x=4\left(tmđk\right)\)

+) \(\sqrt{x}-1=2\Rightarrow x=9\left(tmđk\right)\)

Vậy \(x\in\left\{0;4;9\right\}\)thì f(x) có giá trị nguyên 

d) \(f\left(x\right)=\frac{\sqrt{x}+1}{\sqrt{x}-1}\); \(f\left(2x\right)=\frac{\sqrt{2x}+1}{\sqrt{2x}-1}\)

f(x) = f(2x) khi \(\frac{\sqrt{x}+1}{\sqrt{x}-1}=\frac{\sqrt{2x}+1}{\sqrt{2x}-1}\Leftrightarrow\left(\sqrt{x}+1\right)\left(\sqrt{2x}-1\right)=\left(\sqrt{x}-1\right)\left(\sqrt{2x}+1\right)\)\(\Leftrightarrow\sqrt{2}x+\sqrt{2x}-\sqrt{x}-1=\sqrt{2}x-\sqrt{2x}+\sqrt{x}-1\)\(\Leftrightarrow\sqrt{2x}-\sqrt{x}=-\sqrt{2x}+\sqrt{x}\Leftrightarrow2\sqrt{2x}=2\sqrt{x}\Leftrightarrow\sqrt{2x}=\sqrt{x}\Leftrightarrow x=0\)(tmđk)

Vậy x = 0 thì f(x) = f(2x)