Problem

Source: German Bundeswettbewerb Mathematik 2022, Round 2, Problem 1

Tags: algebra, AM-GM



Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations: \begin{align*} ab + cd &= 8,\\ abcd &= 8 + a + b + c + d. \end{align*}