Find the real solution(s) to the system $\sqrt{4a-b^2}=\sqrt{b+2}+\sqrt{4a^2+b}$.

Find the real solution(s) to the system

$\sqrt{4a-b^2}=\sqrt{b+2}+\sqrt{4a^2+b}$.

My solution:

Prove that [MATH]\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+b^3+abc}\le \frac{1}{abc}[/MATH] for all positive real $a,\,b$ and $c$.

Prove that [MATH]\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+b^3+abc}\le \frac{1}{abc}[/MATH] for all positive real $a,\,b$ and $c$.

My solution:

Let $a,\,b,\,c,\,x,\,y$ and $z$ be strictly positive real numbers, prove that [MATH](a+x)(b+y)(c+z)+4\left(\frac{1}{ax}+\frac{1}{by}+\frac{1}{cz}\right)\ge 20[/MATH].

Let $a,\,b,\,c,\,x,\,y$ and $z$ be strictly positive real numbers, prove that

[MATH](a+x)(b+y)(c+z)+4\left(\frac{1}{ax}+\frac{1}{by}+\frac{1}{cz}\right)\ge 20[/MATH].

Given that [MATH]\frac{\sin 3x}{\sin x}=\frac{6}{5}[/MATH], what is the ratio of [MATH]\frac{\sin 5x}{\sin x}[/MATH]?

Given that [MATH]\frac{\sin 3x}{\sin x}=\frac{6}{5}[/MATH], what is the ratio of [MATH]\frac{\sin 5x}{\sin x}[/MATH]?

My solution: