giải giúp mấy bài sau nha mn
thanks nhiều

1. Tìm nghiệm nguyên của pt:

a) \(\left(x^2+y\right)\left(x+y^2\right)=\left(x-y\right)^3\)

b) \(12x^2+6xy+3y^2=28\left(x+y\right)\)

2. Cho x,y,z>0 và \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=4\)

C/m: \(\dfrac{1}{2x+y+z}+\dfrac{1}{x+2y+z}+\dfrac{1}{x+y+2z}=< 1\)

3. Cho a,b,c>0 và abc=1

C/m: \(\dfrac{1}{a^3\left(b+c\right)}+\dfrac{1}{b^3\left(c+a\right)}+\dfrac{1}{c^3\left(a+b\right)}>=\dfrac{3}{2}\)

4. Cho x,y>0 và x + y >= 2

Tìm GTNN của biểu thức \(A=4\left(x+y\right)+\dfrac{1}{x+1}+\dfrac{1}{y+1}+1\)

Đề: Cho \(\left\{{}\begin{matrix}x,y,z>0\\x+y\le z\end{matrix}\right.\) tìm Min của \(\left(x^2+y^2+z^2\right)\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\right)\) Làm thế này không biết đúng ko

Ta có :A= \(\left(x^2+y^2+z^2\right)\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\right)=3+\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}+\dfrac{z^2}{x^2}+\dfrac{x^2}{z^2}+\dfrac{z^2}{y^2}+\dfrac{y^2}{z^2}\)

=> A \(=3+\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+\left(\dfrac{x^2}{z^2}+\dfrac{z^2}{16x^2}\right)+\left(\dfrac{y^2}{z^2}+\dfrac{z^2}{16y^2}\right)+\dfrac{15}{16}\left(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\right)\)

Áp dụng BĐT Cauchy ta có

\(A\ge3+2+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{15}{16}\left(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\right)=6+\dfrac{15}{16}\left(\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}\right)\)

Do \(x+y\le z\Rightarrow\dfrac{x}{z}+\dfrac{y}{z}\le1\) ; Đặt \(u=\dfrac{x}{z}\); \(v=\dfrac{y}{z}\)

\(\Rightarrow\dfrac{z^2}{x^2}+\dfrac{z^2}{y^2}=\dfrac{1}{u^2}+\dfrac{1}{v^2}\ge\dfrac{2}{uv}\ge\dfrac{2}{\dfrac{\left(u+v\right)^2}{4}}\ge\dfrac{2}{\dfrac{1}{4}}=8\)

\(\Rightarrow A\ge6+\dfrac{15}{16}.8=\dfrac{27}{2}\) Vậy minA = \(\dfrac{27}{2}\) khi \(x=y=\dfrac{z}{2}\)

chứng minh rằng :

a, x+2y+\(\dfrac{25}{x}\)+\(\dfrac{27}{y^2}\)\(\ge\) 19 ( \(\forall\)x,y \(\)> 0 )

b, \(x+\dfrac{1}{\left(x-y\right)y}\ge3\) ( \(\forall\)x>y>0 )

c,\(\dfrac{x}{2}+\dfrac{16}{x-2}\ge13\left(\forall x>2\right)\)

d, \(a+\dfrac{1}{a^2}\ge\dfrac{9}{4}\left(\forall x\ge2\right)\)

e, a+\(\dfrac{1}{a\left(a-b\right)^2}\ge2\sqrt{2}\) ( \(\forall x>y\ge0\))

f, \(\dfrac{2a^3+1}{4b\left(a-b\right)}\ge3[\forall a\ge\dfrac{1}{2};\dfrac{a}{b}>1]\)

g, x+\(\dfrac{4}{\left(x-y\right)\left(y+1\right)^2}\ge3\left(\forall x>y\ge0\right)\)

h, \(2a^4+\dfrac{1}{1+a^2}\ge3a^2-1\)