Help with Holder's Inequality I was studying Holder's inequality and I came across the second problem used at the 2001 IMO because it involved Holder. The question I want to ask is what is the reasoning behind the first line? Where did all those terms come from and how is Holder even involved in it? I can't really understand the solution so can someone please explain this step clearly and transparently? I would really appreciate any help with understanding this. Also please don't downvote this question, I just want help.
Thanks
The link to the solution

Holder's inequality for length-3 real vectors $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ states that for all $p$ and $q$ for which $\frac{1}{p}+\frac{1}{q}=1$,
$$\lvert x_1y_1+x_2y_2+x_3y_3\rvert\le(x_1^p+x_2^p+x_3^p)^{\frac{1}{p}}(y_1^q+y_2^q+y_3^q)^{\frac{1}{q}}.$$
Here, we use $p=\frac{3}{2}$, $q=3$, $x_1=\left(\frac{a}{\sqrt{a^2+8bc}}\right)^{\frac{2}{3}}$, $y_1=\left(a(a^2+8bc)\right)^{\frac{1}{3}}$, $x_2$/$y_2$/$x_3$/$y_3$ defined analogously. Once the Holder application is set up, the solution cubed everything. The idea is to set up a Holder-type inequality that bounds the objective term without using any radicals.
