If you want to shuffle multiple boxes around the page, use tikz.
\documentclass[12pt,a4paper]{article}
%\usepackage[utf8]{inputenc}% not with my editor
\usepackage{graphicx} % to be able to embed graphical content
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{color}
\definecolor{pink}{HTML}{FDDBC7}
\definecolor{red}{HTML}{B2182B}
\definecolor{blue}{HTML}{4393C3}
\newlength{\tempwidth}
\begin{document}
\thispagestyle{empty}
\begin{tikzpicture}[overlay,remember picture,font=\huge]
\node[below,minimum width={\paperwidth},minimum height={0.5\textwidth},
fill=red,text=white] at (current page.north) {An awesome report};
\node[minimum width={\paperwidth},minimum height={0.5\textwidth},
fill=red,text=white] at (current page.center) {An awesome person};
\node[above right,minimum width={0.5\paperwidth},minimum height={0.25\paperwidth},
draw=blue,line width=5mm,fill=red,text=white] at (current page.south west) {\today};
\node[above left,minimum width={0.5\paperwidth},minimum height={0.25\paperwidth},
draw=blue,line width=5mm,fill=red,text=white] at (current page.south east)
{\includegraphics[height=4cm]{example-image}};
\end{tikzpicture}
\newpage
The rest of the document
\end{document}
Load theorems library (I do it with class option most) and use ams or any other maths options.
ams set upper and lower parts to mathematical mode with \displaystyle. You don't need equation environments is these boxes. And you don't need valign.
\documentclass{beamer}
\usepackage[most]{tcolorbox}
\newtcolorbox{equationframe}{
math
}
\begin{document}
\begin{frame}
\begin{itemize}
\item[]
\begin{equationframe}
p(x,y)=\begin{cases}
\min\{q(x,y),\frac{\pi({y})q(y,{x})}{\pi(x)}\}, \quad x\neq y\\
1-\int_{x\neq y} p(x,y)
\end{cases}
\end{equationframe}
\begin{equationframe}
\pi(x)p(x,y)=\min\{\pi(x)q(x,y),\pi(y)q(y,x)\}=\pi(y)p(y,x)%
\end{equationframe}
\end{itemize}
\end{frame}
\end{document}
Update: ams equation*, ams align*:
Instead of math, tcolorbox also offers some other mathematical boxes: ams equation, ams align, ams gather, ...
\documentclass{beamer}
\usepackage[most]{tcolorbox}
\begin{document}
\begin{frame}
\begin{tcolorbox}[ams equation*]
p(x,y)=\begin{cases}
\min\{q(x,y),\frac{\pi({y})q(y,{x})}{\pi(x)}\}, \quad x\neq y\\
1-\int_{x\neq y} p(x,y)
\end{cases}
\end{tcolorbox}
\begin{tcolorbox}[ams nodisplayskip, ams align*]
\pi(x)p(x,y) & =\min\{\pi(x)q(x,y),\pi(y)q(y,x)\}\\
& =\pi(y)p(y,x)%
\end{tcolorbox}
\end{frame}
\end{document}
Best Answer
It seems that using the lowerpart as a subtitle is the only solution. The subtitle is a new colorbox inside the colorbox. In the tcolorbox documentation at page 22 is written that "upperpart is mandatory" so it cannot be suppressed. I came up with a similar solution: