Test Contest 1 Problem C

Find all real solutions to the system of equations: $$\left\{ \begin{array}{l} xy = z - x - y,\\ xz = y - x - z,\\ yz = x - y - z. \end{array} \right.$$

Replace sorry in the template below with your solution. See Answer Bank for acceptible answer declarations. Mathlib version used by the checker is v4.31.0.

import Mathlib.Data.Real.Basic

def answer : Set (ℝ × ℝ × ℝ) := sorry

theorem solution (x y z : ℝ) : 
    x * y = z - x - y ∧
    x * z = y - x - z ∧
    y * z = x - y - z ↔
    (x, y, z) ∈ answer := sorry

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