A two equations system in four variables
Source: German Bundeswettbewerb Mathematik 2022, Round 2, Problem 1
September 14, 2022
algebraAM-GM
Problem Statement
Find all quadrupels of positive real numbers that satisfy the following two equations:
\begin{align*}
ab + cd &= 8,\\
abcd &= 8 + a + b + c + d.
\end{align*}