Chứng minh các bất đẳng thức sau(nếu không nói gì thêm thì các biến dương)

1. $$\frac{{{a}^{2}}}{{{\left( b+2c \right)}^{2}}\left( a+b \right)}+\frac{{{b}^{2}}}{{{(c+2a)}^{2}}(b+c)}+\frac{{{c}^{2}}}{{{(a+2b)}^{2}}(c+a)}\ge \frac{1}{2}a+b+c=1$$.

2. $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b}}\ge \frac{\sqrt{2}}{2}\left( a+b+c \right){{a}^{2}}+{{b}^{2}}+{{c}^{2}}\le 3$$.

3. $$\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\ge \sqrt{a}+\sqrt{b}+\sqrt{c}+3abc=1$$

4. $$\frac{b\sqrt{a}}{4b\sqrt{c}-c\sqrt{a}}+\frac{c\sqrt{b}}{4c\sqrt{a}-a\sqrt{b}}+\frac{a\sqrt{c}}{4a\sqrt{b}-b\sqrt{c}}\ge 1a,b,c\in \left( 1;2 \right)$$

5. $$\frac{{{a}^{2}}+b}{b+c}+\frac{{{b}^{2}}+c}{c+a}+\frac{{{c}^{2}}+a}{a+b}\ge 2a+b+c=1$$

6. $$\frac{a+b}{b+c}.\frac{a}{2a+b+c}+\frac{b+c}{c+a}.\frac{b}{2b+c+a}+\frac{c+a}{a+b}.\frac{c}{2c+a+b}\ge \frac{3}{4}$$

7. $$\frac{{{a}^{3}}}{{{b}^{2}}-bc+{{c}^{2}}}+\frac{{{b}^{3}}}{{{c}^{2}}-ca+{{a}^{2}}}+\frac{{{c}^{3}}}{{{c}^{2}}-ca+{{a}^{2}}}\ge \frac{3(ab+bc+ca)}{a+b+c}$$

8. $$\frac{ab}{a+b+2c}+\frac{bc}{b+c+2a}+\frac{ca}{c+a+2b}\le \frac{a+b+c}{4}$$

9.

10. $$\frac{1+{{x}^{2}}}{1+y+{{z}^{2}}}+\frac{1+{{y}^{2}}}{1+z+{{x}^{2}}}+\frac{1+{{z}^{2}}}{1+x+{{y}^{2}}}\le 2x,y,z>-1$$.