perfectly symmetrical equations
Source: 2023 AIME II/4
February 16, 2023
AMCAIME
Problem Statement
Let , , and be real numbers satisfying the system of equations
\begin{align*}
xy+4z&=60\\
yz+4x&=60\\
zx+4y&=60.
\end{align*}
Let be the set of possible values of . Find the sum of the squares of the elements of .