2023 AIME II Problem 4 - Math Gold Medalist

Lor

2023 AIME II

Problem 4

Let $x,y,$ and $z$ be real numbers satisfying the system of equations $\begin{align*} xy + 4z &= 60 \\ yz + 4x &= 60 \\ zx + 4y &= 60. \end{align*}$ Let $S$ be the set of possible values of $x.$ Find the sum of the squares of the elements of $S.$

Related Ideas

Replacement

Similar Equations

Factorization

Hint

Sol1.(Replacement)

Find z in terms of x and y then find x in terms of y

Sol2.(Similar Equations)

Subtract equations from each other