$1$) Cho $x,y,z>0$ thỏa mãn $x^{2}+y^{2}+z^{2}=6$. Tìm min $P=\frac{xy+yz+zx}{4}+\frac{6}{x+y+z-2}-\frac{16z}{z^{2}+12}$
$2$) Cho $a,b,c>0$ thỏa mãn $(a+b)(b+c)(c+a)=8$. CMR: $\frac{a}{b(c+1)(b+c)^{2}}+\frac{b}{c(a+1)(c+a)^{2}}+\frac{c}{a(b+1)(a+b)^{2}} \geq \frac{3}{8}$
$3$) Cho $x,y,z \geq 0$ thỏa mãn $x^{2}+y^{2}+z^{2}=6$. Tìm max $P=\sqrt{x+xy}+\sqrt{(5y+2)(z+3)}$
$4$) Cho $x,y,z>0$. CMR: $\frac{2x}{2x+y+z}+\frac{2y}{2y+z+x}+\frac{2z}{2z+x+y} \geq 1+\frac{4xyz}{(x+y)(y+z)(z+x)}$
$5$) Cho $x,y \geq 0$ thỏa mãn $x^{2}+y^{2}=2$. Tìm max $T=\frac{y}{x+y^{2}}+\frac{x}{y+x^{2}}+\frac{1-xy}{x+y+2}$
$6$) Cho $a,b,c>0$. CMR: $\sqrt{2}(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}) \geq a+b+c+\sqrt{3(a^{2}+b^{2}+c^{2})}$
$7$) Cho $a,b,c>0$. CMR: $(\sqrt{\frac{a}{b+a}}+\sqrt{\frac{c}{b+c}})(\frac{1}{\sqrt{b+a}}+\frac{1}{\sqrt{b+c}}) \leq \frac{2}{\sqrt{b}}$
