1. a,b,c>0 và a+b+c=2017

\(CM:\Sigma\dfrac{2017a-a^2}{bc}\ge\sqrt{2}\left(\Sigma\sqrt{\dfrac{2017-a}{a}}\right)\)

2. cho x,y,z tm: \(x^2+y^2+z^2=3\)

\(CM:8\left(2-x\right)\left(2-y\right)\left(2-z\right)\ge\left(x+yz\right)\left(y+zx\right)\left(z+xy\right)\)

3. a,b,c>0 và \(a^2+b^2+c^2\ge6\)

\(CM:\Sigma\dfrac{1}{1+ab}\ge\dfrac{3}{2}\)

Giải phương trình nghiệm nguyên:

\(\left(2x-y-2\right)^2=7\left(x-2y-y^2-1\right)\)

Giải hệ:

\(\left\{{}\begin{matrix}\sqrt{x+y}-\sqrt{x-y}=2\\\sqrt{x^2+y^2}+\sqrt{x^2-y^2}=4\end{matrix}\right.\)

+) Tìm min 

\(E=\dfrac{1+\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}}{xy+yz+zx}\)

+) Tìm max và min

\(F=\dfrac{a-b}{c}+\dfrac{b-c}{a}+\dfrac{c-a}{b}\)

Trong đó a,b,c>0 và \(min\left\{a,b,c\right\}\ge\dfrac{1}{4}max\left\{a,b,c\right\}\)

Cho x,y,z>0 

\(CM:\sqrt{\dfrac{x}{z+3x}}+\sqrt{\dfrac{y}{x+3y}}+\sqrt{\dfrac{z}{y+3z}}\le\dfrac{3}{2}\)

Cho 0≤x,y,z≤1

Tìm max \(D=\sqrt{\dfrac{x}{1+yz}}+\sqrt{\dfrac{y}{1+zx}}+\dfrac{z}{2+2xy}\)

+) Cho a,b,c>0 tm: abc=1 

\(CMR:a^3+b^3+c^3+\dfrac{ab}{a^2+b^2}+\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}\ge\dfrac{9}{2}\)

Cho a,b,c>0 và ab+bc+ca=8

Tìm min \(A=3\left(a^2+b^2+c^2\right)+\dfrac{27\left(a+b\right)\left(b+c\right)\left(c+a\right)}{\left(a+b+c\right)^2}\)