Now, we live in a big city. The advantages of the city often bring us comfort and its disadvantages always cause us much trouble and unhappiness. We try to address two aspects of the problem. The advantages of the city are many and obvious. However, if we look at the downside, we can realize that city life has more disadvantages. Firstly, the exhaust fumes of cars and machines have seriously polluted the air in the city, because of the smoke from the factory chimneys, because of the dirt and the spinning garbage. Rivers and lakes contaminated with toxic chemicals are irresponsibly discarded. Secondly, moving in the city is sometimes difficult due to traffic jams, degraded roads, clogged drains due to garbage that always causes flooding in the rainy season. Third, the noisy and hurried life in the city, the fierce competition to find work in the overcrowded city often discourages city people and makes them old on the shoulders of young people. Fourth, the bohemian lifestyle with a culture of debauchery, injecting, drug trafficking and prostitution has poisoned the lives of city dwellers. Fifth, city citizens are always haunted by sudden death due to traffic accidents or racing. Worse yet, somewhere in the city, they dare not go out at night for fear of being tortured and violated. We all expect the state to take drastic measures to correct the shortcomings so that city people can enjoy a perfect life in a modern civilized city.

\(P=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{3\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{6\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)+3\left(\sqrt{x}-1\right)-\left(6\sqrt{x}-4\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

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

\(=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)

b.

\(P=-1\Rightarrow\dfrac{\sqrt{x}-1}{\sqrt{x}+1}=-1\)

\(\Rightarrow\sqrt{x}-1=-\sqrt{x}-1\)

\(\Rightarrow2\sqrt{x}=0\Rightarrow\sqrt{x}=0\)

\(\Rightarrow x=0\)

c.

\(P=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}=\dfrac{\sqrt{x}+1-2}{\sqrt{x}+1}=1-\dfrac{2}{\sqrt{x}+1}\)

\(P\in Z\Rightarrow\dfrac{2}{\sqrt{x}+1}\in Z\Rightarrow\sqrt{x}+1=Ư\left(2\right)\)

Mà \(\sqrt{x}+1\ge1\)

\(\Rightarrow\left[{}\begin{matrix}\sqrt{x}+1=1\\\sqrt{x}+1=2\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\sqrt{x}=0\\\sqrt{x}=1\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x=1\left(loại\right)\\\end{matrix}\right.\)

Phương trình hoành độ giao điểm : 

x² = (m - 1)x - 1 

<=> x² - (m - 1)x + 1 = 0

Có nghiệm khi (m - 1)² - 4 \(\ge0\)

<=> \(\left[{}\begin{matrix}m\ge3\\m\le-1\end{matrix}\right.\)

Hệ thức Viere : \(\left\{{}\begin{matrix}x_1+x_2=m-1\\x_1x_2=1\end{matrix}\right.\) 

Thay A(x₁ ; y₁) ; B(x₂ ; y₂) vào (P) được 

y₁ = x₁² ; y₂ = x₂²

Khi đó ta được x₁⁶ - x₂⁶ = 18(x₁³ - x₂³) 

 <=> (x₁³ - x₂³)(x₁³ + x₂³ - 18) = 0 

<=> x₁ = x₂ hoặc (x₁ + x₂)³ - 3x₁x₂(x₁ + x₂) = 18 

Khi x₁ = x₂ => x₁ = x₂ = \(\pm1\)

(*) x₁ = x₂ = 1  <=> 2 = m - 1 <=> m = 3 (tm) 

(*) x₁ = x₂ = -1 <=> -2 = m - 1 <=> m = -1 (tm) 

Khi (x₁ + x₂)³ - 3x₁x₂(x₁ + x₂) = 18 

<=> (m - 1)³ - 3(m - 1) = 18

<=> (m - 1)³ - 27 - 3(m - 1) + 9 = 0

<=> (m - 4)[(m - 1)² + 3(m - 1) + 6] = 0

<=> (m - 4)(m² + m + 4) = 0

<=> m = 4 (vì m² + m + 4 > 0) (tm)

Vậy m \(\in\) {4 ; -1 ; 3} 

a, gọi giao điểm của hai đt là A(x,y) 
vậy tọa độ A là nghiệm của
     2x + 1 = 3x -2
=> x = 3
thay x = 3 vào y = 2x + 1  ta có y = 3.2 + 1 = 7
vậy giao hai đường thẳng là A( 3;7)
b, ba đường thẳng cùng đi qua 1 điểm khi và chỉ khi đt y = (2m+1)x + m - 3 đi qua C(3; 7)
vậy tọa độ điểm C thỏa mãn pt :  y = ( 2m + 1) x + m - 3
ta có : (2m +1).3 + m - 3 = 7
           6m + 3 + m - 3 = 7
                         7m = 7 
                           m = 1
vậy với m = 1 thì đt y = ( 2m +1)x + m - 3 đồng quy với d và d'

Lời giải:

Đặt $\frac{x}{y}+\frac{y}{x}=a$

BĐT cần chứng minh tương đương với:
$(\frac{x}{y}+\frac{y}{x})^2+2\geq 3(\frac{x}{y}+\frac{y}{x})$

$\Leftrightarrow a^2+2\geq 3a$

$\Leftrightarrow a^2-3a+2\geq 0$

$\Leftrightarrow (a-1)(a-2)\geq 0$

$\Leftrightarrow (\frac{x}{y}+\frac{y}{x}-1)(\frac{x}{y}+\frac{y}{x}-2)\geq 0$

$\Leftrightarrow \frac{(x^2-xy+y^2)(x-y)^2}{x^2y^2}\geq 0$

Điều này luôn đúng do:
$(x-y)^2\geq 0$

$x^2y^2>0$

$x^2-xy+y^2=(x-\frac{y}{2})^2+\frac{3}{4}y^2>0$ với mọi $x,y\neq 0$

Do đó ta có đpcm.