cho a,b,c >0 ; \(a+b+c=a^2+b^2+c^2=2\)

Tinnsh \(A=a\sqrt{\dfrac{\left(1+b^2\right)\left(1+c^2\right)}{1+a^2}}+b\sqrt{\dfrac{\left(1+a^2\right)\left(1+c^2\right)}{1+b^2}}+c\sqrt{\dfrac{\left(1+a^2\right)\left(1+b^2\right)}{1+c^2}}\)

\(\left\{{}\begin{matrix}xy-3x-2y=3\\x^2+y^2-x-3y=38\end{matrix}\right.\)

1. cho \(a=\dfrac{1-\sqrt{2}}{2}\). Tính A = \(\sqrt{16a^8-51a}\)

2. Cho 100 số tự nhiên a1,a2,...,a100 thoả mãn \(\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_2}}+...+\dfrac{1}{\sqrt{a_{100}}}=19\)

Chứng minh rằng trong 100 số tự nhiên đó tồn tại 2 số bằng nhau

1) \(2\sqrt{x+3}=x-1+4\sqrt{2x-1}\)

2) \(\sqrt[4]{x-1}+\sqrt[4]{5-x}=2\)

3) \(\sqrt[3]{1-2x}+\sqrt{x+3}=1\)

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

1) \(\dfrac{x-3x^2}{2}+\sqrt{2x^4-x^3+7x^2-3x+3}=2\)

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

3) \(x+y+z+\dfrac{3}{x-1}+\dfrac{3}{y-1}+\dfrac{3}{z-1}=2\left(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2}\right)\) với x ,y ,z > 1

4) \(\sqrt[3]{x+6}+x^2=7-\sqrt{x-1}\)

5) \(x^4-2x^3+x-\sqrt{2\left(x^2-x\right)}=0\)

\(1,x=\sqrt{3-x}.\sqrt{4-x}+\sqrt{4-x}.\sqrt{5-x}+\sqrt{5-x}.\sqrt{3-x}\)

2) \(\dfrac{x^2-4}{x}+\dfrac{y^2-4}{y}+8=4\left(\sqrt{x-1}+\sqrt{y-1}\right)\)

Giải phương trình:
1, \(3x^2+6x-3=\sqrt{\dfrac{x+7}{3}}\) (2 cách khác nhau )

2, \(\left(\sqrt{3x+1}-\sqrt{x-2}\right)\left(\sqrt{3x^2+7x+2}+4\right)=4x-2\)

3, \(\sqrt{-3x-1}+\sqrt{9x^2+9x+3}=-9x^2-6x\)

4, \(\sqrt{x^2+x-6}+3\sqrt{x-1}=\sqrt{5x^2-1}\)

5, \(\left(\sqrt{x+4}+2\right)\left(x+2\sqrt{x-5}+1\right)=6x\)

6, \(\sqrt{5-x^4}-\sqrt[3]{3x^2-2}=1\)

7, \(3x^2+11+\sqrt{x-2}+\sqrt{2x+3}=14x\)

8, \(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-7}}}}=7\)

9, \(\sqrt{2x^2-1}+3x\sqrt{x^2-1}=3x^3+2x^2-9x-7\) ( với \(x>0\) )

sai roi

chứng minh \(A=\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+\dfrac{1}{\sqrt{5}+\sqrt{6}}+...+\dfrac{1}{\sqrt{2023}+\sqrt{2024}}>22\)

1. Biết a, b, c đôi 1 khác nhau . Chứng miinh rằng :
\(\dfrac{b-c}{\left(a-b\right)\left(a-c\right)}+\dfrac{c-a}{\left(b-c\right)\left(b-a\right)}+\dfrac{a-b}{\left(c-a\right)\left(c-b\right)}=\dfrac{2}{a-b}+\dfrac{2}{b-c}+\dfrac{2}{c-a}\).

2. Cho x,y,z đôi một khác nhau thoả mãn \(\dfrac{xy+1}{y}+\dfrac{yz+1}{z}+\dfrac{zx+1}{x}\). Chứng minh rằng : \(\left|xyz\right|=1\)